Modified inertial-type subgradient extragradient methods for variational inequalities and fixed points of finite Bregman relatively nonexpansive and demicontractive mappings

1. Introduction

Suppose that ($H,\langle ·,·\rangle$) is a real Hilbert space with the induced norm $\parallel ·\parallel$. Let $P_C$ be the metric projection from H onto a nonempty, convex and closed $C\subset H$. Given a nonlinear operator $S:C\to C$. We denote by $\mathrm{Fix}\left(S\right)$ the fixed-point set of S. Also, the $\mathbf{R},\to$ and ⇀ are used to represent the set of all real numbers, the strong convergence and the weak convergence, respectively. A mapping $S:C\to C$ is said to be strictly pseudocontractive (see [1]) if $\exists \xi \in [0,1)$ s.t. ${\parallel Sx-Sy\parallel }^2\le {\parallel x-y\parallel }^2+\xi {\parallel (I-S)x-(I-S)y\parallel }^2\forall x,y\in C$. In particular, in case $\xi =0$, S reduces to a nonexpansive mapping. Moreover, S is said to be demicontractive if $\mathrm{Fix}\left(S\right)\ne \varnothing$ and $\exists \xi \in [0,1)$ s.t. ${\parallel Sx-y\parallel }^2\le {\parallel x-y\parallel }^2+\xi {\parallel x-Sx\parallel }^2\forall x\in C,y\in \mathrm{Fix}\left(S\right)$. In particular, in case $\xi =0$, S reduces to a quasi-nonexpansive mapping.

Let $A:H\to H$ be a mapping. Consider the classical variational inequality problem (VIP) of finding $u\in C$ s.t. $\langle Au,v-u\rangle \ge 0\forall v\in C$. The solution set of the VIP is denoted by $\mathrm{VI}(C,A)$. In 1976, to seek an element of $\mathrm{VI}(C,A)$ under weaker conditions, Korpelevich [24] put forward the extragradient approach below, i.e., for any initial $x_0\in C$, the sequence $\left\{x_n\right\}$ is generated by

$\left\{\begin{array}{cc}& y_n=P_C(x_n-\tau A{x}_n),\hfill \\ & x_{n+1}=P_C(x_n-\tau A{y}_n)\forall n\ge 0,\hfill \end{array}\right.$

with $\tau \in (0,\frac{1}{L})$. If $\mathrm{VI}(C,A)\ne \varnothing$, then the sequence $\left\{x_n\right\}$ converges weakly to an element in $\mathrm{VI}(C,A)$. To the best of our knowledge, the Korpelevich extragradient approach is one of the most effective methods for solving the VIP at present. The literature on the VIP is vast and the Korpelevich extragradient approach has attained wide attention paid by many scholars, who ameliorated it in various forms; see e.g., [1-6, 8-9, 13-16, 19, 21-23, 25-28, 31, 34].

Furthermore, in 2018, Thong and Hieu [21] first put forward the inertial subgradient extragradient method, that is, for any initial $x_0,x_1\in H$, the sequence $\left\{x_n\right\}$ is generated by

$\left\{\begin{array}{cc}& u_n=x_n+{\alpha }_n(x_n-x_{n-1}),\hfill \\ & y_n=P_C(u_n-\ell A{u}_n),\hfill \\ & C_n=\{v\in H:\langle u_n-\ell A{u}_n-y_n,v-y_n\rangle \le 0\},\hfill \\ & x_{n+1}=P_{C_n}(u_n-\ell A{y}_n)\forall n\ge 1,\hfill \end{array}\right.$ with constant $\ell \in (0,\frac{1}{L})$. Under suitable conditions, they proved the weak convergence of $\left\{x_n\right\}$ to an element of $\mathrm{VI}(C,A)$. Subsequently, Ceng et al. [14] introduced a modified inertial subgradient extragradient method for solving the pseudomonotone VIP and common fixed point problem (CFPP) of finite nonexpansive mappings. Let $S_i:H\to H$ be nonexpansive for $i=1,...,N$, $A:H\to H$ be L-Lipschitz continuous pseudomonotone on H, and sequentially weakly continuous on C, s.t. $\Omega ={\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\cap \mathrm{VI}(C,A)\ne \varnothing$. Let $f:H\to H$ be a contraction with constant $\delta \in [0,1)$ and $F:H\to H$ be $\eta$-strongly monotone and $\kappa$-Lipschitzian s.t. $\delta <\tau :=1-\sqrt{1-\rho (2\eta -\rho {\kappa }^2)}$ for $\rho \in (0,2\eta /{\kappa }^2)$. Presume that $\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\tau }_n\right\}$ are positive sequences s.t. ${\beta }_n+{\gamma }_n<1,{\sum }_{n=1}^{\infty }{\beta }_n=\infty ,{lim}_{n\to \infty }{\beta }_n=0,{lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n)>0$ and ${\tau }_n=o\left({\beta }_n\right)$. Moreover, one writes $S_n:=S_{n\mathrm{mod}N}$ for integer $n\ge 1$ with the mod function taking values in the set $\{1,...,N\}$, i.e., if $n=jN+m$ for some integers $j\ge 0$ and $0\le m<N$, then $S_n=S_N$ if $m=0$ and $S_n=S_m$ if $0<m<N$.

Algorithm 1.1 (see [14, Algorithm 3.1]). Initialization: Given ${\lambda }_1>0,\alpha >0,\mu \in (0,1)$. Let $x_0,x_1\in H$ be arbitrary.

Iterative steps: Calculate $x_{n+1}$ as follows:

Step 1. Given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${\alpha }_n$ s.t. $0\le {\alpha }_n\le \overline{{\alpha }_n}$, where

$\overline{{\alpha }_n}=\left\{\begin{array}{cc}& min\{\alpha ,\frac{{\tau }_n}{\parallel x_n-x_{n-1}\parallel }\}\mathrm{if}{x}_n\ne x_{n-1},\hfill \\ & \alpha \mathrm{otherwise}.\hfill \end{array}\right.$

Step 2. Compute $w_n=S_n{x}_n+{\alpha }_n(S_n{x}_n-S_n{x}_{n-1})$ and $y_n=P_C(w_n-{\lambda }_{n}A{w}_n)$.

Step 3. Construct the half-space $C_n:=\{z\in H:\langle w_n-{\lambda }_{n}A{w}_n-y_n,z-y_n\rangle \le 0\}$, and compute $z_n=P_{C_n}(w_n-{\lambda }_{n}A{y}_n)$.

Step 4. Calculate $x_{n+1}={\beta }_{n}f\left(x_n\right)+{\gamma }_n{x}_n+((1-{\gamma }_n)I-{\beta }_n\rho F)z_n$, and update

${\lambda }_{n+1}=\left\{\begin{array}{cc}& min\{\mu \frac{\parallel w_n-y_n{\parallel }^2+{\parallel z_n-y_n\parallel }^2}{2\langle A{w}_n-A{y}_n,z_n-y_n\rangle },{\lambda }_n\}\langle A{w}_n-A{y}_n,z_n-y_n\rangle >0,\hfill \\ & {\lambda }_n\mathrm{otherwise}.\hfill \end{array}\right.$

Set $n:=n+1$ and go to Step 1.

Under appropriate conditions, they proved the strong convergence of $\left\{x_n\right\}$ to an element of $\Omega ={\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\cap \mathrm{VI}(C,A)$. In addition, combining the subgradient extragradient method and the Halpern’s iteration method, Kraikaew and Saejung [22] proposed the Halpern subgradient extragradient rule for solving the VIP in 2014. They proved the strong convergence of the proposed method to an element in $\mathrm{VI}(C,A)$. Recently, Reich et al. [27] introduced two gradient-projection algorithms for solving the VIP for uniformly continuous pseudomonotone mapping. In particular, they used a novel Armijo-type line search to acquire a hyperplane which strictly separates the current iterate from the solutions of the VIP under consideration. They proved the weak and strong convergence of two algorithms to a solution of the VIP for uniformly continuous pseudomonotone mapping, respectively.

On the other hand, let $p,q\in (1,\infty )$ with $\frac{1}{p}+\frac{1}{q}=1$, let E be a p-uniformly convex and uniformly smooth Banach space and C be a convex, closed and nonempty set in E. We denote by $E^{*}$ the dual space of E. The norm and the duality pairing between E and $E^{*}$ are denoted by $\parallel ·\parallel$ and $\langle ·,·\rangle$, respectively. Let $J_E^p$ and $J_{E^{*}}^q$ be the duality mappings of E and $E^{*}$, respectively. Let $f_p\left(u\right)={\parallel u\parallel }^p/p\forall u\in E$, $D_{f_p}$ be the Bregman distance with respect to (w.r.t) $f_p$ and ${\mathsf{\Pi }}_C$ be the Bregman projection of E onto C w.r.t. $f_p$, and presume that $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ s.t. ${lim}_{n\to \infty }{\alpha }_n=0$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$. Assume that $A:E\to E^{*}$ is uniformly continuous and pseudo-monotone mapping and $S:C\to C$ is Bregman relatively nonexpansive mapping. Very recently, inspired by the research outcomes in [27], Eskandani et al. [31] invented the hybrid projection method with line-search process for seeking a solution of the VIP with the FPP constraint of S.

Algorithm 1.2 (see [31]). Initialization: Let $\nu >0,l\in (0,1),\lambda \in (0,\frac{1}{\nu })$, and put $u,u_1\in C$ arbitrarily.

Iterative steps: Given the current iterate $u_n$, calculate $u_{n+1}$ below:

Step 1. Compute $v_n={\mathsf{\Pi }}_C\left(J_{E^{*}}^q(J_E^p{u}_n-\lambda A{u}_n)\right)$ and $r_{\lambda }\left(u_n\right):=u_n-v_n$. If $r_{\lambda }\left(u_n\right)=0$ and $S{u}_n=u_n$, then stop; $u_n\in \Omega =\mathrm{Fix}\left(S\right)\cap \mathrm{VI}(C,A)$. Otherwise,

Step 2. Compute $t_n=u_n-{\tau }_n{r}_{\lambda }\left(u_n\right)$, where ${\tau }_n:=l^{k_n}$ and $k_n$ is the smallest nonnegative integer k satisfying $\langle A{u}_n-A(u_n-l^k{r}_{\lambda }\left(u_n\right)),r_{\lambda }\left(u_n\right)\rangle \le \frac{\nu }{2}{D}_{f_p}(u_n,v_n)$.

Step 3. Compute $w_n=J_{E^{*}}^q({\beta }_n{J}_E^p{u}_n+(1-{\beta }_n)J_E^p\left(S{\mathsf{\Pi }}_{C_n}{u}_n\right))$ and $u_{n+1}={\mathsf{\Pi }}_C\left(J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{w}_n)\right)$, where $C_n:=\{v\in C:{\hslash }_n\left(v\right)\le 0\}$ and ${\hslash }_n\left(v\right)=\langle A{t}_n,v-u_n\rangle +\frac{{\tau }_n}{2\lambda }{D}_{f_p}(u_n,v_n)$.

Again set $n:=n+1$ and return to Step 1.

Under suitable conditions, they proved the strong convergence of Algorithm 1.2 to ${\mathsf{\Pi }}_{\Omega }u$. Inspired by the above research outcomes, we design two inertial-type subgradient extragradient algorithms with line-search process for solving the pseudomonotone variational inequality problems (VIPs) and common fixed-point problem (CFPP) of finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping in p-uniformly convex and uniformly smooth Banach spaces. Under mild conditions, we prove weak and strong convergence of the suggested algorithms to a common solution of the VIPs and CFPP, respectively. Additionally, an illustrated example is furnished to back up the feasibility and implementability of our proposed approaches.

The structure of this paper is built below: In Sect. 2, we release some concepts and basic results for further use. In Sect. 3, we discuss the convergence analysis of the suggested algorithms. In Sect. 4, our main results are employed to solve the VIPs and CFPP in an illustrated example. Our algorithms are more advantageous and more flexible than the above Algorithms 1.1-1.2 because they involve solving the VIPs for uniformly continuous pseudomonotone mappings and the CFPP of finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping. Our results improve and extend the corresponding results announced by some others, e.g., Ceng et al. [14], Eskandani et al. [31] and Reich et al. [27].

2. Preliminaries

Let ($E,\parallel ·\parallel$) be a real Banach space, whose dual is denoted by $E^{*}$. We use the $u_n\to u$ and $u_n\rightharpoonup u$ to indicate the strong and weak convergence of $\left\{u_n\right\}$ to $u\in E$, respectively. Moreover, the set of weak cluster points of $\left\{u_n\right\}$ is denoted by ${\omega }_w\left(u_n\right)$, i.e., ${\omega }_w\left(u_n\right)=\{u^{†}\in E:u_{n_k}\rightharpoonup u^{†}\mathrm{for}\mathrm{some}\left\{u_{n_k}\right\}\subset \left\{u_n\right\}\}$. Let $U=\{u\in E:\parallel u\parallel =1\}$ and $1<q\le 2\le p$ with $\frac{1}{p}+\frac{1}{q}=1$. A Banach space E is referred to as being strictly convex if for each $u,v\in U$ with $u\ne v$, one has $\parallel u+v\parallel /2<1$. E is referred to as being uniformly convex if $\forall ϵ\in (0,2]$, $\exists \delta >0$ s.t. $\forall u,v\in U$ with $\parallel u-v\parallel \ge ϵ$, one has $\parallel u+v\parallel /2\le 1-\delta$. It is known that a uniformly convex Banach space is reflexive and strictly convex. The modulus of convexity of E is the function $\delta :[0,2]\to [0,1]$ defined by $\delta \left(ϵ\right)=inf\{1-\parallel u+v\parallel /2:u,v\in U\mathrm{with}\parallel u-v\parallel \ge ϵ\}$. It is also known that E is uniformly convex if and only if $\delta \left(ϵ\right)>0\forall ϵ\in (0,2]$. Moreover, E is referred to as being p-uniformly convex if $\exists c>0$ s.t. $\delta \left(ϵ\right)\ge c{ϵ}^p\forall ϵ\in [0,2]$.

The modulus of smoothness ${\rho }_E:[0,\infty )\to [0,\infty )$ is defined as ${\rho }_E\left(\tau \right)=sup\left\{\right(\parallel u+\tau v\parallel +\parallel u-\tau v\parallel )/2-1:u,v\in U\}$. E is said to be uniformly smooth if ${lim}_{\tau \to 0}{\rho }_E\left(\tau \right)/\tau =0$, and q-uniformly smooth if $\exists C_q>0$ s.t. ${\rho }_E\left(\tau \right)\le C_q{\tau }^q\forall \tau >0$. It is known that E is p-uniformly convex if and only if $E^{*}$ is q-uniformly smooth. For example, see [32] for more details. Putting $B(0,r)=\{u\in E:\parallel u\parallel \le r\}$ for each $r>0$, we say that $f:E\to \mathbf{R}$ is uniformly convex on bounded sets (see [31]) if ${\rho }_r\left(t\right)>0\forall r,t>0$, where ${\rho }_r\left(t\right):[0,\infty )\to [0,\infty ]$ is specified below

$\begin{array}{cc}{\rho }_r\left(t\right)\hfill & =inf\left\{\right[\alpha f\left(u\right)+(1-\alpha )f\left(v\right)-f(\alpha u+(1-\alpha \left)v\right)]/\alpha (1-\alpha ):\hfill \\ & \alpha \in (0,1)\mathrm{and}u,v\in B(0,r)\mathrm{with}\parallel u-v\parallel =t\},\hfill \end{array}$ for all $t\ge 0$. The function ${\rho }_r$ is called the gauge of uniform convexity of f. It is known that ${\rho }_r$ is a nondecreasing function.

Let $f:E\to \mathbf{R}$ be a convex function. If the limit ${lim}_{t\to 0^{+}}\frac{f(u+tv)-f\left(u\right)}{t}$ exists for each $v\in E$, then f is referred to as being Gâteaux differentiable at u. In this case, the gradient of f at u is the linear function $\nabla f\left(u\right)$, which is defined by $\langle \nabla f\left(u\right),v\rangle :={lim}_{t\to 0^{+}}\frac{f(u+tv)-f\left(u\right)}{t}$ for each $v\in E$. The function f is referred to as being Gâteaux differentiable if it is Gâteaux differentiable at each $u\in E$. Whenever the limit ${lim}_{t\to 0^{+}}\frac{f(u+tv)-f\left(u\right)}{t}$ is attained uniformly for any $v\in U$, we say that f is Fréchet differentiable at u. Besides, f is referred to as being uniformly Fréchet differentiable on a subset $K\subset E$ if the limit ${lim}_{t\to 0^{+}}\frac{f(u+tv)-f\left(u\right)}{t}$ is attained uniformly for $(u,v)\in K\times U$. A Banach space E is called smooth if its norm is Gâteaux differentiable.

Let $p,q\in (1,\infty )$ with $\frac{1}{p}+\frac{1}{q}=1$. The duality mapping $J_E^p:E\to E^{*}$ is specified below

$$J_E^p\left(u\right)=\{\psi \in E^{*}:\langle \psi ,u\rangle ={\parallel u\parallel }^p\mathrm{and}\parallel \psi \parallel ={\parallel u\parallel }^{p-1}\}\forall u\in E.$$

It is known that E is smooth if and only if $J_E^p$ is single-valued mapping of E into $E^{*}$. Also, E is reflexive if and only if $J_E^p$ is surjective, and E is strictly convex if and only if $J_E^p$ is one-to-one. So it follows that, if E is smooth, strictly convex and reflexive Banach space, then $J_E^p$ is a single-valued bijection and in this case, $J_E^p={\left(J_{E^{*}}^q\right)}^{-1}$ where $J_{E^{*}}^q$ is the duality mapping of $E^{*}$. Besides, it is known that E is uniformly smooth if and only if the function $f_p\left(u\right)={\parallel u\parallel }^p/p$ is uniformly Fréchet differentiable on bounded sets if and only if $J_E^p$ is single-valued and uniformly continuous on bounded sets. It is also known that E is uniformly convex if and only if the function $f_p$ is uniformly convex (see [32]).

Let $f:E\to \mathbf{R}$ be a Gâteaux differentiable convex function. The Bregman distance w.r.t. f is specified below

$$D_f(u,v):=f\left(u\right)-f\left(v\right)-\langle \nabla f\left(v\right),u-v\rangle \forall u,v\in E.$$

It is worth mentioning that the Bregman distance is not a metric in the usual sense of the term. Clearly, $D_f(u,u)=0$ but $D_f(u,v)=0$ can not yield $u=v$. Generally, $D_f$ is not symmetric and does not satisfy the triangle inequality. However, $D_f$ satisfies the three point identity

$$D_f(u,v)+D_f(v,w)=D_f(u,w)-\langle \nabla f\left(v\right)-\nabla f\left(w\right),u-v\rangle .$$

See [20] for more details on Bregman functions and distances.

It is noteworthy that the duality mapping $J_E^p$ on the smooth Banach space E is the Gâteaux derivative of the function $f_p$. Then the Bregman distance w.r.t. $f_p$ is specified below

$$\begin{array}{cc}{D}_{f_p}(u,v)\hfill & ={\parallel u\parallel }^p/p-{\parallel v\parallel }^p/p-\langle J_E^p\left(v\right),u-v\rangle \hfill \\ & ={\parallel u\parallel }^p/p+{\parallel v\parallel }^p/q-\langle J_E^p\left(v\right),u\rangle \hfill \\ & ={(\parallel v\parallel }^p-{\parallel u\parallel }^p)/q-\langle J_E^p\left(v\right)-J_E^p\left(u\right),u\rangle .\hfill \end{array}$$

In the smooth and p-uniformly convex Banach space E with $2\le p<\infty$, there holds the following relationship between the metric and Bregman distance:

$${\tau \parallel u-v\parallel }^p\le D_{f_p}(u,v)\le \langle J_E^p\left(u\right)-J_E^p\left(v\right),u-v\rangle ,\left(2.1\right)$$

where $\tau >0$ is some fixed number (see [12]). From (2.1) it is readily known that for any bounded sequence $\left\{u_n\right\}\subset E$, the following holds:

$$u_n\to u\iff D_{f_p}(u,u_n)\to 0(n\to \infty ).$$

Let C be a nonempty closed convex subset of reflexive, smooth and strictly convex Banach space E. Bregman projections are defined as minimizers of Bregman distances. The Bregman projection of $u\in E$ onto C w.r.t. $f_p$ is the unique element ${\mathsf{\Pi }}_{C}u\in C$ s.t. $D_{f_p}({\mathsf{\Pi }}_{C}u,u)={min}_{v\in C}{D}_{f_p}(v,u)$. In Hilbert spaces the Bregman projection w.r.t. $f_2$ reduces to the metric projection. Using [18, Corollary 4.4] and [30, Theorem 2.1], in uniformly convex Banach spaces Bregman projections can be characterized by the following inequality:

$$\langle J_E^p\left(u\right)-J_E^p\left({\mathsf{\Pi }}_{C}u\right),v-{\mathsf{\Pi }}_{C}u\rangle \le 0\forall v\in C.\left(2.2\right)$$

Moreover, this inequality is equivalent to the descent property

$$D_{f_p}(v,{\mathsf{\Pi }}_{C}u)+D_{f_p}({\mathsf{\Pi }}_{C}u,u)\le D_{f_p}(v,u)\forall v\in C.\left(2.3\right)$$

In case $p=2$, the duality mapping $J_E^p$ reduces to the normalized duality mapping and is denoted by J. The function $\varphi :E^2\to \mathbf{R}$ is formulated below

$$\varphi (u,v)={\parallel u\parallel }^2-2\langle Jv,u\rangle +{\parallel v\parallel }^2\forall u,v\in E,$$

and ${\mathsf{\Pi }}_C\left(u\right)={\mathrm{argmin}}_{v\in C}\varphi (v,u)\forall u\in E$.

In terms of [31], the function $V_{f_p}:E\times E^{*}\to [0,\infty )$ associated with $f_p$ is specified below

$$V_{f_p}(u,u^{*})={\parallel u\parallel }^p/p-\langle u^{*},u\rangle +{\parallel u^{*}\parallel }^q/q\forall (u,u^{*})\in E\times E^{*}.\left(2.4\right)$$

So, $V_{f_p}(u,u^{*})=D_{f_p}(u,J_{E^{*}}^q\left(u^{*}\right))\forall (u,u^{*})\in E\times E^{*}$. Moreover, by the subdifferential inequality, we obtain

$$V_{f_p}(u,u^{*})+\langle v^{*},J_{E^{*}}^q\left(u^{*}\right)-u\rangle \le V_{f_p}(u,u^{*}+v^{*})\forall u\in E,u^{*},v^{*}\in E^{*}.\left(2.5\right)$$

In addition, $V_{f_p}$ is convex in the second variable. Thus one has

$$D_{f_p}(z,J_{E^{*}}^q\left(\sum _{i=1}^n{t}_i{J}_E^p\left(u_i\right)\right))\le \sum _{i=1}^n{t}_i{D}_{f_p}(z,u_i)\forall z\in E,{\left\{u_i\right\}}_{i=1}^n\subset E,{\left\{t_i\right\}}_{i=1}^n\subset [0,1]\mathrm{with}\sum _{i=1}^n{t}_i=1.\left(2.6\right)$$

Lemma 2.1 (see [30]). Let E be a uniformly convex Banach space and $\left\{u_n\right\},\left\{v_n\right\}$ be two sequences in E such that the first one is bounded. If ${lim}_{n\to \infty }{D}_{f_p}(v_n,u_n)=0$, then ${lim}_{n\to \infty }\parallel v_n-u_n\parallel =0$.

Let $S:C\to C$ be a mapping. We denote by $\mathrm{Fix}\left(S\right)$ the set of fixed points of S, that is, $\mathrm{Fix}\left(S\right)=\{u\in C:u=Su\}$. A point $u\in C$ is referred to as an asymptotic fixed point of S if $\exists \left\{u_n\right\}\subset C$ s.t. $u_n\rightharpoonup u$ and $u_n-S{u}_n\to 0$. We denote by $\widehat{\mathrm{Fix}}\left(S\right)$ the set of asymptotic fixed points of S. The notion of asymptotic fixed points was invented in Reich [11]. A mapping $S:C\to C$ is known as being Bregman relatively $\xi$-demicontractive w.r.t. $f_p$ if $\mathrm{Fix}\left(S\right)=\widehat{\mathrm{Fix}}\left(S\right)\ne \varnothing$, and $\exists \xi \in [0,1)$ s.t. for each bounded $\left\{v_n\right\}\subset C$ satisfying ${sup}_{n\ge 1}\parallel S{v}_n\parallel <\infty$, the following holds:

$$D_{f_p}(u,S{v}_n)\le D_{f_p}(u,v_n)+\xi {\rho }_b^{*}\parallel J_E^p{v}_n-J_E^{p}S{v}_n\parallel \forall u\in \mathrm{Fix}\left(S\right),$$

with $b={sup}_{n\ge 1}\{\parallel v_n{\parallel }^{p-1},\parallel S{v}_n{\parallel }^{p-1}\}<\infty$. In particular, putting $b_x={max\{\parallel x\parallel }^{p-1}{,\parallel Sx\parallel }^{p-1}\}$ for each $x\in C$, one has

$$D_{f_p}(u,Sx)\le D_{f_p}(u,x)+\xi {\rho }_{b_x}^{*}\parallel J_E^{p}x-J_E^{p}Sx\parallel \forall u\in \mathrm{Fix}\left(S\right).$$

In addition, if $\xi =0$, then S reduces to a Bregman relatively nonexpansive mapping w.r.t. $f_p$, that is, S is said to be Bregman relatively nonexpansive w.r.t. $f_p$ if $\mathrm{Fix}\left(S\right)=\widehat{\mathrm{Fix}}\left(S\right)\ne \varnothing$ and $D_{f_p}(u,Sv)\le D_{f_p}(u,v)\forall v\in C,u\in \mathrm{Fix}\left(S\right)$.

Definition 2.1. Let C be a nonempty closed convex subset of E. A mapping $A:C\to E^{*}$ is referred to as being

(i) monotone on C if $\langle Au-Av,u-v\rangle \ge 0\forall u,v\in C$;

(ii) pseudo-monotone if $\langle Au,v-u\rangle \ge 0\Rightarrow \langle Av,v-u\rangle \ge 0\forall u,v\in C$;

(iii)L-Lipschitz continuous or L-Lipschitzian if $\exists L>0$ s.t. $\parallel Au-Av\parallel \le L\parallel u-v\parallel \forall u,v\in C$;

(iv) weakly sequentially continuous if for each $\left\{x_n\right\}\subset C$, the relation holds: $x_n\rightharpoonup x\Rightarrow A{x}_n\rightharpoonup Ax$.

Lemma 2.2 (see [31]). Let $r>0$ be a constant and suppose that $f:E\to \mathbf{R}$ is a uniformly convex function on bounded subsets of a Banach space E. Then

$$f\left(\sum _{k=1}^n{\alpha }_k{x}_k\right)\le \sum _{k=1}^n{\alpha }_{k}f\left(x_k\right)-{\alpha }_i{\alpha }_j{\rho }_r(\parallel x_i-x_j\parallel ),$$

$\forall i,j\in \{1,2,...,n\},{\left\{x_k\right\}}_{k=1}^n\subset B(0,r)$ and ${\left\{{\alpha }_k\right\}}_{k=1}^n\subset (0,1)$ with ${\sum }_{k=1}^n{\alpha }_k=1$, where ${\rho }_r$ is the gauge of uniform convexity of f.

Proof. It is easy to show the conclusion.

Lemma 2.3 (see [28]). Let $E_1$ and $E_2$ be two Banach spaces. Suppose that $A:E_1\to E_2$ is uniformly continuous on bounded subsets of $E_1$ and D is a bounded subset of $E_1$. Then $A\left(D\right)$ is bounded.

Lemma 2.4 (see [10]). Let $\varnothing \ne C\subset E$ with C being closed and convex in a Banach space E and suppose $A:C\to E^{*}$ is pseudo-monotone and continuous. Then $x^{†}\in C$ is a solution to the VIP $\langle A{x}^{†},y-x^{†}\rangle \ge 0\forall y\in C$, if and only if $\langle Ay,y-x^{†}\rangle \ge 0\forall y\in C$.

Lemma 2.5. Let $2\le p<\infty$ and suppose that E is a smooth and p-uniformly convex Banach space with the weakly sequentially continuous duality mapping $J_E^p$. Let $\left\{q_n\right\}\subset E$ and $\varnothing \ne \Omega \subset E$. If $\left\{D_{f_p}(x,q_n)\right\}$ converges for each $x\in \Omega$, and ${\omega }_w\left(q_n\right)\subset \Omega$. Then $\left\{q_n\right\}$ converges weakly to a point in $\Omega$.

Proof. 

Using (2.1) we get $\tau \parallel x-q_n{\parallel }^p\le D_{f_p}(x,q_n)\forall x\in \Omega$. This ensures that $\left\{q_n\right\}$ is bounded. Hence, from the reflexivity of E we have ${\omega }_w\left(q_n\right)\ne \varnothing$. Also, let us show the weak convergence of $\left\{q_n\right\}$ to a point in $\Omega$. Indeed, let $\overline{q},\widehat{q}\in {\omega }_w\left(q_n\right)$ with $\overline{q}\ne \widehat{q}$. Then, $\exists \left\{q_{n_k}\right\}\subset \left\{q_n\right\}$ and $\exists \left\{q_{m_k}\right\}\subset \left\{q_n\right\}$ s.t. $q_{n_k}\rightharpoonup \overline{q}$ and $q_{m_k}\rightharpoonup \widehat{q}$. By the weakly sequential continuity of $J_E^p$ one deduces that $J_E^p\left(q_{n_k}\right)\rightharpoonup J_E^p\overline{q}$ and $J_E^p\left(q_{m_k}\right)\rightharpoonup J_E^p\widehat{q}$. Note that $D_{f_p}(\overline{q},\widehat{q})+D_{f_p}(\widehat{q},q_n)=D_{f_p}(\overline{q},q_n)-\langle J_E^p\widehat{q}-J_E^p{q}_n,\overline{q}-\widehat{q}\rangle$. So, exploiting the convergence of the sequences $\left\{D_{f_p}(\overline{q},q_n)\right\}$ and $\left\{D_{f_p}(\widehat{q},q_n)\right\}$, we deduce that

$$\begin{array}{cc}& -\langle J_E^p\widehat{q}-J_E^p\overline{q},\overline{q}-\widehat{q}\rangle ={\displaystyle \underset{k\to \infty }{lim}}[-\langle J_E^p\widehat{q}-J_E^p{q}_{n_k},\overline{q}-\widehat{q}\rangle ]\hfill \\ & ={\displaystyle \underset{n\to \infty }{lim}}[D_{f_p}(\overline{q},\widehat{q})+D_{f_p}(\widehat{q},q_n)-D_{f_p}(\overline{q},q_n)]\hfill \\ & ={\displaystyle \underset{k\to \infty }{lim}}[-\langle J_E^p\widehat{q}-J_E^p{q}_{m_k},\overline{q}-\widehat{q}\rangle ]=-\langle J_E^p\widehat{q}-J_E^p\widehat{q},\overline{q}-\widehat{q}\rangle =0,\hfill \end{array}$$

which hence yields $\langle J_E^p\overline{q}-J_E^p\widehat{q},\overline{q}-\widehat{q}\rangle =0$. From (2.1) we get $0<\tau \parallel \overline{q}-\widehat{q}{\parallel }^p\le D_{f_p}(\overline{q},\widehat{q})\le \langle J_E^p\overline{q}-J_E^p\widehat{q},\overline{q}-\widehat{q}\rangle =0$. This arrives at a contradiction. Consequently, the sequence $\left\{q_n\right\}$ converges weakly to a point in $\Omega$. □

The lemma below was put forth in ${\mathbf{R}}^m$ by [29]. It is easy to verify that the proof of the lemma in Banach spaces is actually the same as in ${\mathbf{R}}^m$. Here, we present the lemma but omit the proof in Banach spaces.

Lemma 2.6. Let $\varnothing \ne C\subset E$ with C being closed and convex in a Banach space E. Suppose that $K:=\{x\in C:h(x)\le 0\}$ where h is a real-valued function on E. If $K\ne \varnothing$ and h is Lipschitz continuous on C with modulus $\theta >0$, then $\theta \mathrm{dist}(x,K)\ge max\left\{h\right(x),0\}\forall x\in C$, where $\mathrm{dist}(x,K)$ stands for the distance of x to K.

Lemma 2.7 (see [17]). Let $\left\{{\mathsf{\Gamma }}_n\right\}$ be a sequence of real numbers that does not decrease at infinity in the sense that, $\exists \left\{{\mathsf{\Gamma }}_{n_k}\right\}\subset \left\{{\mathsf{\Gamma }}_n\right\}$ s.t. ${\mathsf{\Gamma }}_{n_k}<{\mathsf{\Gamma }}_{n_k+1}\forall k\ge 1$. Let the sequence ${\left\{\psi \left(n\right)\right\}}_{n\ge n_0}$ of integers be defined as $\psi \left(n\right)=max\{k\le n:{\mathsf{\Gamma }}_k<{\mathsf{\Gamma }}_{k+1}\}$, with integer $n_0\ge 1$ satisfying $\{k\le n_0:{\mathsf{\Gamma }}_k<{\mathsf{\Gamma }}_{k+1}\}\ne \varnothing$. Then the following hold:

(i) $\psi \left(n_0\right)\le \psi (n_0+1)\le \cdots$ and $\psi \left(n\right)\to \infty$;

(ii) ${\mathsf{\Gamma }}_{\psi \left(n\right)}\le {\mathsf{\Gamma }}_{\psi \left(n\right)+1}$ and ${\mathsf{\Gamma }}_n\le {\mathsf{\Gamma }}_{\psi \left(n\right)+1}\forall n\ge n_0$.

Lemma 2.8 (see [7]). Let $\left\{a_n\right\}$ be a sequence in $[0,\infty )$ satisfying $a_{n+1}\le (1-{\mu }_n)a_n+{\mu }_n{\nu }_n\forall n\ge 1$, where $\left\{{\mu }_n\right\}$ and $\left\{{\nu }_n\right\}$ both are real sequences such that (i) $\left\{{\mu }_n\right\}\subset [0,1]$ and ${\sum }_{n=1}^{\infty }{\mu }_n=\infty$, and (ii) ${lim\; sup}_{n\to \infty }{\nu }_n\le 0$ or ${\sum }_{n=1}^{\infty }\left|{\mu }_n{\nu }_n\right|<\infty$. Then ${lim}_{n\to \infty }{a}_n=0$.

Lemma 2.9 (see [33]). Let $\left\{a_n\right\},\left\{b_n\right\}$ and $\left\{{\delta }_n\right\}$ be sequences of nonnegative real numbers satisfying the inequality $a_{n+1}\le (1+{\delta }_n)a_n+b_n\forall n\ge 1$. If ${\sum }_{n=1}^{\infty }{\delta }_n<\infty$ and ${\sum }_{n=1}^{\infty }{b}_n<\infty$, then ${lim}_{n\to \infty }{a}_n$ exists.

3. Main Results

In this section, let E be a p-uniformly convex and uniformly smooth Banach space with $2\le p<\infty$. Let $\varnothing \ne C\subset E$ with C being closed and convex in E. We are now in a position to state and analyze our iterative algorithms for settling the VIPs for uniformly continuous pseudomonotone mappings and the CFPP of finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping in E. Assume always that the conditions hold below:

(C1) For $i=1,...,N$, $S_i:C\to C$ is a uniformly continuous and Bregman relatively nonexpansive mapping and $S_0:C\to C$ is a uniformly continuous and Bregman relatively $\xi$-demicontractive mapping.

(C2) ${\left\{S_n\right\}}_{n=1}^{\infty }$ is defined as $S_n:=S_{n\mathrm{mod}N}$ for integer $n\ge 1$ with the mod function taking values in the set $\{1,...,N\}$, i.e., if $n=jN+m$ for some integers $j\ge 0$ and $0\le m<N$, then $S_n=S_N$ if $m=0$ and $S_n=S_m$ if $0<m<N$.

(C3) For $i=1,2$, $A_i:E\to E^{*}$ is pseudomonotone and uniformly continuous on C, s.t. $\parallel A_i{x}^{†}\parallel \le {lim\; inf}_{n\to \infty }\parallel A_i{x}_n\parallel \forall \left\{x_n\right\}\subset C$ with $x_n\rightharpoonup x^{†}$.

(C4) $\Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)\ne \varnothing$.

Algorithm 3.1. Initialization: Given $x_0,x_1\in C$ arbitrarily and let $ϵ>0,{\mu }_i>0,{\lambda }_i\in (0,\frac{1}{{\mu }_i}),l_i\in (0,1)$ for $i=1,2$. Choose $\left\{{\ell }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ and $\left\{{\alpha }_n\right\}\subset (\xi ,1)$ s.t. ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${lim\; inf}_{n\to \infty }({\alpha }_n-\xi )(1-{\alpha }_n)>0$. Moreover, given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel }\}\mathrm{if}{S}_n{x}_n\ne S_n{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$

Iterative steps: Calculate $x_{n+1}$ as follows:

Step 1. Calculate $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$ and calculate $u_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, $y_n={\mathsf{\Pi }}_C\left(J_{E^{*}}^q(J_E^p{u}_n-{\lambda }_1{A}_1{u}_n)\right)$, $r_{{\lambda }_1}\left(u_n\right):=u_n-y_n$ and $s_n=u_n-{\tau }_n{r}_{{\lambda }_1}\left(u_n\right)$, where ${\tau }_n:=l_1^{k_n}$ and $k_n$ is the smallest nonnegative integer k satisfying

$$\langle A_1{u}_n-A_1(u_n-l_1^k{r}_{{\lambda }_1}\left(u_n\right)),u_n-y_n\rangle \le \frac{{\mu }_1}{2}{D}_{f_p}(u_n,y_n).\left(3.1\right)$$

Step 2. Calculate $w_n={\mathsf{\Pi }}_{C_n}\left(u_n\right)$, with $C_n:=\{x\in C:h_n\left(x\right)\le 0\}$ and

$$h_n\left(x\right)=\langle A_1{s}_n,x-u_n\rangle +\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(u_n,y_n).\left(3.2\right)$$

Step 3. Calculate ${\tilde{y}}_n={\mathsf{\Pi }}_C\left(J_{E^{*}}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n)\right)$, $r_{{\lambda }_2}\left(w_n\right):=w_n-{\tilde{y}}_n$ and $t_n=w_n-{\tilde{\tau }}_n{r}_{{\lambda }_2}\left(w_n\right)$, where ${\tilde{\tau }}_n:=l_2^{j_n}$ and $j_n$ is the smallest nonnegative integer j satisfying

$$\langle A_2{w}_n-A_2(w_n-l_2^j{r}_{{\lambda }_2}\left(w_n\right)),w_n-{\tilde{y}}_n\rangle \le \frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n).\left(3.3\right)$$

Step 4. Calculate $v_n=J_{E^{*}}^q({\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p\left(S_0{w}_n\right))$ and $x_{n+1}={\mathsf{\Pi }}_{{\tilde{C}}_n\cap Q_n}\left(w_n\right)$, where $Q_n:=\{x\in C:D_{f_p}(x,v_n)\le D_{f_p}(x,w_n)\}$, ${\tilde{C}}_n:=\{x\in C:{\tilde{h}}_n\left(x\right)\le 0\}$ and

$${\tilde{h}}_n\left(x\right)=\langle A_2{t}_n,x-w_n\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n).\left(3.4\right)$$

Again set $n:=n+1$ and go to Step 1.

The following lemmas are used in the proofs of our main results in the sequel.

Lemma 3.1. Suppose that $\left\{x_n\right\}$ is the sequence constructed in Algorithm 3.1. Then the relations hold: $\frac{1}{{\lambda }_1}{D}_{f_p}(u_n,y_n)\le \langle A_1{u}_n,r_{{\lambda }_1}\left(u_n\right)\rangle$ and $\frac{1}{{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\le \langle A_2{w}_n,r_{{\lambda }_2}\left(w_n\right)\rangle$.

Proof. 

Observe that the last two relations are similar. Then it suffices to show that the latter relation holds. In fact, using the definition of ${\tilde{y}}_n$ and properties of ${\mathsf{\Pi }}_C$, one has

$$\langle J_E^p{w}_n-{\lambda }_2{A}_2{w}_n-J_E^p{\tilde{y}}_n,y-{\tilde{y}}_n\rangle \le 0\forall y\in C.$$

Setting $y=w_n$ in the last inequality, from (2.1) we get

$$D_{f_p}(w_n,{\tilde{y}}_n)\le \langle J_E^p{w}_n-J_E^p{\tilde{y}}_n,w_n-{\tilde{y}}_n\rangle \le {\lambda }_2\langle A_2{w}_n,w_n-{\tilde{y}}_n\rangle .$$

This completes the proof. □

Lemma 3.2. The Armijo-type search rules (3.1), (3.3) and the sequence $\left\{x_n\right\}$ constructed in Algorithm 3.1 are well defined.

Proof. 

Observe that the rules (3.1) and (3.3) are similar. Then it suffices to show that the latter rule (3.3) is valid. Using the uniform continuity of $A_2$ on C, from $l_2\in (0,1)$ one gets ${lim}_{j\to \infty }\langle A_2{w}_n-A_2(w_n-l_2^j{r}_{{\lambda }_2}\left(w_n\right)),r_{{\lambda }_2}\left(w_n\right)\rangle =0$. In case $r_{{\lambda }_2}\left(w_n\right)=0$, it is evident that $j_n=0$. In case $r_{{\lambda }_2}\left(w_n\right)\ne 0$, we know that $\exists j_n\ge 0$ s.t. (3.3) holds.

It is easy to check that for each $n\ge 1,{\tilde{C}}_n$ and $Q_n$ are convex and closed. We assert that $\Omega \subset {\tilde{C}}_n\cap Q_n$. Let $z\in \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)$. Using Lemma 2.2 and the Bregman relative $\xi$-demicontractivity of $S_0$, from $\left\{{\alpha }_n\right\}\subset (\xi ,1)$ we get

$$\begin{array}{cc}& D_{f_p}(z,v_n)\le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)D_{f_p}(z,S_0{w}_n)\hfill \\ & -{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_0{w}_n\parallel \hfill \\ & \le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)[D_{f_p}(z,w_n)+\xi {\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_0{w}_n\parallel ]\hfill \\ & -{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_0{w}_n\parallel \hfill \\ & =D_{f_p}(z,w_n)-({\alpha }_n-\xi )(1-{\alpha }_n){\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_0{w}_n\parallel \hfill \\ & \le D_{f_p}(z,w_n),\hfill \end{array}$$

which hence leads to $z\in Q_n$. Moreover, by Lemma 2.4, we get $\langle A_2{t}_n,t_n-z\rangle \ge 0$. Therefore,

$$\begin{array}{cc}{\tilde{h}}_n\left(z\right)\hfill & =\langle A_2{t}_n,z-w_n\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\hfill \\ & =-\langle A_2{t}_n,w_n-t_n\rangle -\langle A_2{t}_n,t_n-z\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\hfill \\ & \le -{\tilde{\tau }}_n\langle A_2{t}_n,r_{{\lambda }_2}\left(w_n\right)\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n).\hfill \end{array}\left(3.5\right)$$

So it follows from (3.3) that

$$\langle A_2{w}_n-A_2{t}_n,r_{{\lambda }_2}\left(w_n\right)\rangle \le \frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n).$$

Using Lemma 3.1 we have

$$\begin{array}{cc}\langle A_2{t}_n,r_{{\lambda }_2}\left(w_n\right)\rangle \hfill & \ge \langle A_2{w}_n,r_{{\lambda }_2}\left(w_n\right)\rangle -\frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n)\hfill \\ & \ge (\frac{1}{{\lambda }_2}-\frac{{\mu }_2}{2})D_{f_p}(w_n,{\tilde{y}}_n).\hfill \end{array}$$

This together with (3.5), arrives at

$${\tilde{h}}_n\left(z\right)\le -\frac{{\tilde{\tau }}_n}{2}(\frac{1}{{\lambda }_2}-{\mu }_2)D_{f_p}(w_n,{\tilde{y}}_n)\le 0.$$

Consequently, $\Omega \subset {\tilde{C}}_n\cap Q_n$. So, the sequence $\left\{x_n\right\}$ is well defined. □

Lemma 3.3. Suppose that $\left\{y_n\right\}$ and $\left\{{\tilde{y}}_n\right\}$ are the sequences generated by Algorithm 3.1. If ${lim}_{n\to \infty }\parallel u_n-y_n\parallel =0$ and ${lim}_{n\to \infty }\parallel w_n-{\tilde{y}}_n\parallel =0$, then ${\omega }_w\left(u_n\right)\subset \mathrm{VI}(C,A_1)$ and ${\omega }_w\left(w_n\right)\subset \mathrm{VI}(C,A_2)$.

Proof. 

Observe that the last two inclusions are similar. Then it suffices to show that the latter inclusion is valid. In fact, let $z\in {\omega }_w\left(w_n\right)$. Then, $\exists \left\{w_{n_k}\right\}\subset \left\{w_n\right\}$, s.t. $w_{n_k}\rightharpoonup z$ and ${lim}_{n\to \infty }\parallel w_{n_k}-{\tilde{y}}_{n_k}\parallel =0$. Hence, it is known that ${\tilde{y}}_{n_k}\rightharpoonup z$. Since C is of both convexity and closedness, from $\left\{{\tilde{y}}_n\right\}\subset C$ and ${\tilde{y}}_{n_k}\rightharpoonup z$ we get $z\in C$. Next, we consider two cases. If $A_{2}z=0$, then $z\in \mathrm{VI}(C,A_2)$ because $\langle A_{2}z,y-z\rangle \ge 0\forall y\in C$. If $A_{2}z\ne 0$, using the assumption on $A_2$, instead of the weakly sequential continuity of $A_2$, we get $0<\parallel A_{2}z\parallel \le {lim\; inf}_{k\to \infty }\parallel A_2{w}_{n_k}\parallel$. So, we could assume that $\parallel A_2{w}_{n_k}\parallel \ne 0\forall k\ge 1$. From (2.2), we get

$$\langle J_E^p{w}_{n_k}-{\lambda }_2{A}_2{w}_{n_k}-J_E^p{\tilde{y}}_{n_k},x-{\tilde{y}}_{n_k}\rangle \le 0\forall x\in C,$$

and hence

$$\frac{1}{{\lambda }_2}\langle J_E^p{w}_{n_k}-J_E^p{\tilde{y}}_{n_k},x-{\tilde{y}}_{n_k}\rangle +\langle A_2{w}_{n_k},{\tilde{y}}_{n_k}-w_{n_k}\rangle \le \langle A_2{w}_{n_k},x-w_{n_k}\rangle \forall x\in C.\left(3.6\right)$$

According to the uniform continuity of $A_2$, one knows that $\left\{A_2{w}_{n_k}\right\}$ is bounded by Lemma 2.3. Note that $\left\{{\tilde{y}}_{n_k}\right\}$ is bounded as well. So, using the uniform continuity of $J_E^p$ on bounded subsets of E, from (3.6) we have

$$\underset{k\to \infty }{lim\; inf}\langle A_2{w}_{n_k},x-w_{n_k}\rangle \ge 0\forall x\in C.\left(3.7\right)$$

To prove that $z\in \mathrm{VI}(C,A_2)$, we now pick a sequence $\left\{{\tilde{\epsilon }}_k\right\}\subset (0,1)$ satisfying ${\tilde{\epsilon }}_k\downarrow 0$ as $k\to \infty$. For each $k\ge 1$, we denote by $l_k$ the smallest positive integer such that

$$\langle A_2{w}_{n_j},y-w_{n_j}\rangle +{\tilde{\epsilon }}_k\ge 0\forall j\ge l_k.\left(3.8\right)$$

Because $\left\{{\tilde{\epsilon }}_k\right\}$ is decreasing, it is easily known that $\left\{l_k\right\}$ is increasing. For convenience, we still denote $\left\{A_2{w}_{n_{l_k}}\right\}$ by $\left\{A_2{w}_{l_k}\right\}$. Note that $A_2{w}_{l_k}\ne 0\forall k\ge 1$ (due to $\left\{A_2{w}_{l_k}\right\}\subset \left\{A_2{w}_{n_k}\right\}$). Then, putting ${\tilde{g}}_{l_k}=\frac{A_2{w}_{l_k}}{\parallel A_2{w}_{l_k}{\parallel }^{\frac{q}{q-1}}}$, one gets $\langle A_2{w}_{l_k},J_{E^{*}}^q{\tilde{g}}_{l_k}\rangle =1\forall k\ge 1$. Indeed, it is evident that $\langle A_2{w}_{l_k},J_{E^{*}}^q{\tilde{g}}_{l_k}\rangle =\langle A_2{w}_{l_k},{\left(\frac{1}{\parallel A_2{w}_{l_k}{\parallel }^{\frac{q}{q-1}}}\right)}^{q-1}{J}_{E^{*}}^q{A}_2{w}_{l_k}\rangle ={\left(\frac{1}{\parallel A_2{w}_{l_k}{\parallel }^{\frac{q}{q-1}}}\right)}^{q-1}{\parallel A_2{w}_{l_k}\parallel }^q=1\forall k\ge 1$. So, by (3.8) one has $\langle A_2{w}_{l_k},y+{\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}-w_{l_k}\rangle \ge 0\forall k\ge 1$. Again from the pseudomonotonicity of $A_2$ one has

$$\langle A_2(y+{\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}),y+{\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}-w_{l_k}\rangle \ge 0\forall y\in C.\left(3.9\right)$$

We assert that ${lim}_{k\to \infty }{\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}=0$. Indeed, since $\left\{w_{l_k}\right\}\subset \left\{w_{n_k}\right\}$ and ${\tilde{\epsilon }}_k\downarrow 0$ as $k\to \infty$, it follows that

$$0\le \underset{k\to \infty }{lim\; sup}\parallel {\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}\parallel =\underset{k\to \infty }{lim\; sup}\frac{{\tilde{\epsilon }}_k}{\parallel A_2{w}_{l_k}\parallel }\le \frac{{lim\; sup}_{k\to \infty }{\tilde{\epsilon }}_k}{{lim\; inf}_{k\to \infty }\parallel A_2{w}_{n_k}\parallel }=0.$$

Hence one gets ${\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}\to 0$ as $k\to \infty$. Thus, taking the limit as $k\to \infty$ in (3.9), by condition (C2) we have $\langle A_{2}y,y-z\rangle \ge 0\forall y\in C$. By Lemma 2.4 one obtains $z\in \mathrm{VI}(C,A_2)$. □

Lemma 3.4. Suppose that $\left\{y_n\right\}$ and $\left\{{\tilde{y}}_n\right\}$ are the sequences generated by Algorithm 3.1. Then the following hold:

(i) ${lim}_{n\to \infty }{\tau }_n{D}_{f_p}(u_n,y_n)=0\Rightarrow {lim}_{n\to \infty }{D}_{f_p}(u_n,y_n)=0$;

(ii) ${lim}_{n\to \infty }{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)=0\Rightarrow {lim}_{n\to \infty }{D}_{f_p}(w_n,{\tilde{y}}_n)=0$.

Proof. 

Observe that the assertions (i) and (ii) are similar. Then it suffices to show that assertion (ii) is valid. To verify assertion (ii), we consider two cases. In case ${lim\; inf}_{n\to \infty }{\tilde{\tau }}_n>0$, we might presume that $\exists \tilde{\tau }>0$ s.t. ${\tilde{\tau }}_n\ge \tilde{\tau }>0\forall n\ge 1$, which hence arrives at

$$D_{f_p}(w_n,{\tilde{y}}_n)=\frac{1}{{\tilde{\tau }}_n}{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)\le \frac{1}{\tilde{\tau }}·{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n).\left(3.10\right)$$

This together with ${lim}_{n\to \infty }{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)=0$, leads to ${lim}_{n\to \infty }{D}_{f_p}(w_n,{\tilde{y}}_n)=0$.

In case ${lim\; inf}_{n\to \infty }{\tilde{\tau }}_n=0$, we presume that ${lim\; sup}_{n\to \infty }{D}_{f_p}(w_n,{\tilde{y}}_n)=a_2>0$. Then we know that $\exists \left\{n_k\right\}\subset \left\{n\right\}$ s.t.

$$\underset{k\to \infty }{lim}{\tilde{\tau }}_{n_k}=0\mathrm{and}\underset{k\to \infty }{lim}{D}_{f_p}(w_{n_k},{\tilde{y}}_{n_k})=a_2>0.$$

We define $\widehat{t_{n_k}}=\frac{1}{l_2}{\tilde{\tau }}_{n_k}{\tilde{y}}_{n_k}+(1-\frac{1}{l_2}{\tilde{\tau }}_{n_k})w_{n_k}$ for each $k\ge 1$. Applying (2.1) and noticing ${lim}_{k\to \infty }{\tilde{\tau }}_{n_k}{D}_{f_p}(w_{n_k},{\tilde{y}}_{n_k})=0$, we have ${lim}_{k\to \infty }{\tilde{\tau }}_{n_k}{\parallel w_{n_k}-{\tilde{y}}_{n_k}\parallel }^p=0$ and hence

$$\underset{k\to \infty }{lim}\parallel \widehat{t_{n_k}}-w_{n_k}{\parallel }^p=\underset{k\to \infty }{lim}\frac{{\tilde{\tau }}_{n_k}^{p-1}}{l_2^p}·{\tilde{\tau }}_{n_k}{\parallel w_{n_k}-{\tilde{y}}_{n_k}\parallel }^p=0.\left(3.11\right)$$

Because $A_2$ is uniformly continuous on bounded subsets of C, we obtain

$$\underset{k\to \infty }{lim}\parallel A_2{w}_{n_k}-A_2\widehat{t_{n_k}}\parallel =0.\left(3.12\right)$$

From the step size rule (3.3) and the definition of $\widehat{t_{n_k}}$, it follows that

$$\langle A_2{w}_{n_k}-A_2\widehat{t_{n_k}},w_{n_k}-{\tilde{y}}_{n_k}\rangle >\frac{{\mu }_2}{2}{D}_{f_p}(w_{n_k},{\tilde{y}}_{n_k}).\left(3.13\right)$$

Now, taking the limit as $k\to \infty$, from (3.12) we have ${lim}_{k\to \infty }{D}_{f_p}(w_{n_k},{\tilde{y}}_{n_k})=0$. This, however, reaches a contradiction. So it follows that ${lim}_{n\to \infty }{D}_{f_p}(w_n,{\tilde{y}}_n)=0$. □

Now, we are ready to prove the weak convergence theorem.

Theorem 3.1. Suppose that E is a p-uniformly convex and uniformly smooth Banach space with the weakly sequentially continuous duality mapping $J_E^p$. If $\left\{x_n\right\}$ is the sequence generated by Algorithm 3.1, then $x_n\rightharpoonup z\in \Omega \iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Proof. 

It is clear that the necessity of Theorem 3.1 is valid. Next it suffices to show that the sufficiency is valid. Assume that ${sup}_{n\ge 0}\parallel x_n\parallel <\infty$. Let $z\in \Omega$. It is clear that $S_n{x}_n\ne S_n{x}_{n-1}\iff J_E^p{S}_n{x}_n\ne J_E^p(2S_n{x}_n-S_n{x}_{n-1})$. Using the definition of ${ϵ}_n$, we get ${ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \le {\ell }_n\forall n\ge 1$. From (2.1), (2.6) and the three point identity of $D_{f_p}$ we get

$\begin{array}{cc}& D_{f_p}(z,g_n)\le (1-{ϵ}_n)D_{f_p}(z,S_n{x}_n)+{ϵ}_n{D}_{f_p}(z,2S_n{x}_n-S_n{x}_{n-1})\hfill \\ & =D_{f_p}(z,S_n{x}_n)+{ϵ}_n[D_{f_p}(z,2S_n{x}_n-S_n{x}_{n-1})-D_{f_p}(z,S_n{x}_n)]\hfill \\ & =D_{f_p}(z,S_n{x}_n)+{ϵ}_n[D_{f_p}(S_n{x}_n,2S_n{x}_n-S_n{x}_{n-1})\hfill \\ & +\langle J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1}),z-S_n{x}_n\rangle ]\hfill \\ & \le D_{f_p}(z,x_n)+{ϵ}_n\langle J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1}),z+S_n{x}_{n-1}-2S_n{x}_n\rangle \hfill \\ & \le D_{f_p}(z,x_n)+{ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \parallel z+S_n{x}_{n-1}-2S_n{x}_n\parallel \hfill \\ & \le D_{f_p}(z,x_n)+{\ell }_{n}M,\hfill \end{array}$

where ${sup}_{n\ge 1}\parallel z+S_n{x}_{n-1}-2S_n{x}_n\parallel \le M$ for some $M>0$. Using Lemma 2.2, we get

$$\begin{array}{cc}& D_{f_p}(z,u_n)=V_{f_p}(z,{\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)\hfill \\ & \le \frac{1}{p}{\parallel z\parallel }^p-{\beta }_n\langle J_E^p{x}_n,z\rangle -(1-{\beta }_n)\langle J_E^p{g}_n,z\rangle +\frac{{\beta }_n}{q}{\parallel J_E^p{x}_n\parallel }^q\hfill \\ & +\frac{(1-{\beta }_n)}{q}\parallel J_E^p{g}_n{\parallel }^q-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & =\frac{1}{p}{\parallel z\parallel }^p-{\beta }_n\langle J_E^p{x}_n,z\rangle -(1-{\beta }_n)\langle J_E^p{g}_n,z\rangle +\frac{{\beta }_n}{q}{\parallel x_n\parallel }^p\hfill \\ & +\frac{(1-{\beta }_n)}{q}\parallel g_n{\parallel }^p-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & ={\beta }_n{D}_{f_p}(z,x_n)+(1-{\beta }_n)D_{f_p}(z,g_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le {\beta }_n{D}_{f_p}(z,x_n)+(1-{\beta }_n)[D_{f_p}(z,x_n)+{\ell }_{n}M]-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le D_{f_p}(z,x_n)+{\ell }_{n}M-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel .\hfill \end{array}$$

Since $w_n={\mathsf{\Pi }}_{C_n}{u}_n$, by (2.1) and (2.3) we get

$$\begin{array}{cc}{D}_{f_p}(z,w_n)\hfill & \le D_{f_p}(z,u_n)-D_{f_p}(w_n,u_n)\hfill \\ & =D_{f_p}(z,u_n)-D_{f_p}({\mathsf{\Pi }}_{C_n}{u}_n,u_n)\hfill \\ & \le D_{f_p}(z,u_n)-\tau {\parallel {\mathsf{\Pi }}_{C_n}{u}_n-u_n\parallel }^p\hfill \\ & \le D_{f_p}(z,u_n)-\tau {\parallel P_{C_n}{u}_n-u_n\parallel }^p\hfill \\ & =D_{f_p}(z,u_n)-\tau {\left[\mathrm{dist}(C_n,u_n)\right]}^p.\hfill \end{array}$$

Because $x_{n+1}={\mathsf{\Pi }}_{{\tilde{C}}_n\cap Q_n}{w}_n$, from (2.1) and (2.3) we get

$$\begin{array}{cc}{D}_{f_p}(z,x_{n+1})\hfill & \le D_{f_p}(z,w_n)-D_{f_p}(x_{n+1},w_n)\hfill \\ & =D_{f_p}(z,w_n)-D_{f_p}({\mathsf{\Pi }}_{{\tilde{C}}_n\cap Q_n}{w}_n,w_n)\hfill \\ & \le D_{f_p}(z,w_n)-D_{f_p}({\mathsf{\Pi }}_{{\tilde{C}}_n}{w}_n,w_n)\hfill \\ & \le D_{f_p}(z,w_n)-\tau {\parallel {\mathsf{\Pi }}_{{\tilde{C}}_n}{w}_n-w_n\parallel }^p\hfill \\ & \le D_{f_p}(z,w_n)-\tau {\parallel P_{{\tilde{C}}_n}{w}_n-w_n\parallel }^p\hfill \\ & =D_{f_p}(z,w_n)-\tau {\left[\mathrm{dist}({\tilde{C}}_n,w_n)\right]}^p.\hfill \end{array}$$

Combining (3.13) and the last two inequalities, we obtain

$$\begin{array}{cc}& D_{f_p}(z,x_{n+1})\le D_{f_p}(z,w_n)-D_{f_p}(x_{n+1},w_n)\hfill \\ & \le D_{f_p}(z,u_n)-D_{f_p}(w_n,u_n)-D_{f_p}(x_{n+1},w_n)\hfill \\ & \le D_{f_p}(z,u_n)-\tau {\left[\mathrm{dist}(C_n,u_n)\right]}^p-\tau {\left[\mathrm{dist}({\tilde{C}}_n,w_n)\right]}^p\hfill \\ & \le D_{f_p}(z,x_n)+{\ell }_{n}M-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & -\tau {\left[\mathrm{dist}(C_n,u_n)\right]}^p-\tau {\left[\mathrm{dist}({\tilde{C}}_n,w_n)\right]}^p,\hfill \end{array}\left(3.14\right)$$

which hence arrives at

$$D_{f_p}(z,x_{n+1})\le D_{f_p}(z,x_n)+{\ell }_{n}M.$$

Since ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$, by Lemma 2.9 we deduce that ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exists. In addition, by the boundedness of $\left\{x_n\right\}$, we conclude that $\left\{g_n\right\},\left\{u_n\right\},\left\{v_n\right\},\left\{w_n\right\},\left\{y_n\right\},\left\{{\tilde{y}}_n\right\}$, $\left\{s_n\right\},\left\{t_n\right\},\left\{S_n{x}_n\right\}$ and $\left\{S_0{w}_n\right\}$ are also bounded. Using (3.14) we obtain

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(x_{n+1},w_n)\le D_{f_p}(z,u_n)-D_{f_p}(z,x_{n+1})\hfill \\ & \le D_{f_p}(z,x_n)+{\ell }_{n}M-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel -D_{f_p}(z,x_{n+1}),\hfill \end{array}$$

which immediately yields

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(x_{n+1},w_n)+{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le D_{f_p}(z,x_n)-D_{f_p}(z,x_{n+1})+{\ell }_{n}M.\hfill \end{array}$$

Since ${lim}_{n\to \infty }{\ell }_n=0$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exists, it follows that ${lim}_{n\to \infty }{D}_{f_p}(w_n,u_n)=0$, ${lim}_{n\to \infty }{D}_{f_p}(x_{n+1},w_n)=0$, and ${lim}_{n\to \infty }{\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$, which hence yields ${lim}_{n\to \infty }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$. From $u_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, it can be readily seen that ${lim}_{n\to \infty }\parallel J_E^p{u}_n-J_E^p{x}_n\parallel =0$. Noticing $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$, we obtain from ${lim}_{n\to \infty }{\ell }_n=0$ and the definition of ${ϵ}_n$ that

$$\parallel J_E^p{g}_n-J_E^p{S}_n{x}_n\parallel ={ϵ}_n\parallel J_E^p(2S_n{x}_n-S_n{x}_{n-1})-J_E^p{S}_n{x}_n\parallel \le {\ell }_n\to 0(n\to \infty ).$$

Hence, using (2.1) and uniform continuity of $J_E^p$ on bounded subsets of E, we conclude that ${lim}_{n\to \infty }\parallel g_n-S_n{x}_n\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_n-u_n\parallel =\underset{n\to \infty }{lim}\parallel x_{n+1}-w_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-S_n{x}_n\parallel =\underset{n\to \infty }{lim}\parallel u_n-x_n\parallel =0.\left(3.15\right)$$

Since $\left\{x_n\right\}$ is bounded and E is reflexive, then we know that ${\omega }_w\left(x_n\right)\ne \varnothing$. In what follows, we claim that ${\omega }_w\left(x_n\right)\subset \Omega$. Let $z\in {\omega }_w\left(x_n\right)$. Then, $\exists \left\{x_{n_k}\right\}\subset \left\{x_n\right\}$ s.t. $x_{n_k}\rightharpoonup z$. From (3.15) one gets $w_{n_k}\rightharpoonup z$. Since $\left\{A_1{s}_n\right\}$ is bounded, we know that $\exists L_1>0$ s.t. $\parallel A_1{s}_n\parallel \le L_1$. This ensures that for each $x,y\in C_n$,

$$|h_n\left(x\right)-h_n\left(y\right)|=|\langle A_1{s}_n,x-y\rangle |\le \parallel A_1{s}_n\parallel \parallel x-y\parallel \le L_1\parallel x-y\parallel ,$$

which implies that $h_n\left(x\right)$ is $L_1$-Lipschitz continuous on $C_n$. Using Lemma 2.6, we get

$$\mathrm{dist}(C_n,u_n)\ge \frac{1}{L_1}{h}_n\left(u_n\right)=\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n).\left(3.16\right)$$

Noticing $x_{n+1}\in Q_n$, from the definition of $Q_n$ and (3.14), we have

$$\begin{array}{cc}{D}_{f_p}(x_{n+1},v_n)\hfill & \le D_{f_p}(x_{n+1},w_n)\hfill \\ & \le D_{f_p}(z,w_n)-D_{f_p}(z,x_{n+1})\hfill \\ & \le D_{f_p}(z,u_n)-D_{f_p}(z,x_{n+1})\hfill \\ & \le D_{f_p}(z,x_n)-D_{f_p}(z,x_{n+1})+{\ell }_{n}M.\hfill \end{array}$$

Since ${lim}_{n\to \infty }{\ell }_n=0$ and ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exists, we have ${lim}_{n\to \infty }{D}_{f_p}(x_{n+1},v_n)$$=0$ and hence ${lim}_{n\to \infty }\parallel x_{n+1}-v_n\parallel =0$. This together with (3.15), arrives at

$$\underset{n\to \infty }{lim}\parallel w_n-v_n\parallel =0.\left(3.17\right)$$

On the other hand, using Lemma 2.2, we get

$$\begin{array}{cc}& D_{f_p}(z,v_n)=V_{f_p}(z,{\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p{S}_0{w}_n)\hfill \\ & \le \frac{1}{p}{\parallel z\parallel }^p-{\alpha }_n\langle J_E^p{w}_n,z\rangle -(1-{\alpha }_n)\langle J_E^p{S}_0{w}_n,z\rangle +\frac{{\alpha }_n}{q}{\parallel J_E^p{w}_n\parallel }^q\hfill \\ & +\frac{(1-{\alpha }_n)}{q}\parallel J_E^p{S}_0{w}_n{\parallel }^q-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill \\ & =\frac{1}{p}{\parallel z\parallel }^p-{\alpha }_n\langle J_E^p{w}_n,z\rangle -(1-{\alpha }_n)\langle J_E^p{S}_0{w}_n,z\rangle +\frac{{\alpha }_n}{q}{\parallel w_n\parallel }^p\hfill \\ & +\frac{(1-{\alpha }_n)}{q}\parallel S_0{w}_n{\parallel }^p-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill \\ & ={\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)D_{f_p}(z,S_0{w}_n)-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill \\ & \le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)[D_{f_p}(z,w_n)+\xi {\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel ]\hfill \\ & -{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill \\ & =D_{f_p}(z,w_n)-({\alpha }_n-\xi )(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel .\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}({\alpha }_n-\xi )(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill & \le D_{f_p}(z,w_n)-D_{f_p}(z,v_n)\hfill \\ & \le D_{f_p}(z,w_n)-D_{f_p}(z,v_n)+D_{f_p}(w_n,v_n)\hfill \\ & =\langle J_E^p{v}_n-J_E^p{w}_n,z-w_n\rangle .\hfill \end{array}$$

Taking the limit in the last inequality as $n\to \infty$, and using uniform continuity of $J_E^p$ on bounded subsets of E, (3.17) and ${lim\; inf}_{n\to \infty }({\alpha }_n-\xi )(1-{\alpha }_n)>0$, we get ${lim}_{n\to \infty }{\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel =0$ and hence ${lim}_{n\to \infty }\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel =0$. This together with uniform continuity of $J_{E^{*}}^q$ on bounded subsets of $E^{*}$ implies that

$$\underset{n\to \infty }{lim}\parallel w_n-S_0{w}_n\parallel =0.\left(3.18\right)$$

Now let us show that $z\in {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. Since $\left\{A_2{t}_n\right\}$ is bounded, we know that $\exists L_2>0$ s.t. $\parallel A_2{t}_n\parallel \le L_2$. This ensures that for each $x,y\in {\tilde{C}}_n$,

$$|{\tilde{h}}_n\left(x\right)-{\tilde{h}}_n\left(y\right)|=|\langle A_2{t}_n,x-y\rangle |\le \parallel A_2{t}_n\parallel \parallel x-y\parallel \le L_2\parallel x-y\parallel ,$$

which guarantees that ${\tilde{h}}_n\left(x\right)$ is $L_2$-Lipschitz continuous on ${\tilde{C}}_n$. By Lemma 2.6, we get

$$\mathrm{dist}({\tilde{C}}_n,w_n)\ge \frac{1}{L_2}{\tilde{h}}_n\left(w_n\right)=\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n).\left(3.19\right)$$

Combining (3.14), (3.16) and (3.19), we obtain

$$\begin{array}{cc}& D_{f_p}(z,x_n)-D_{f_p}(z,x_{n+1})+{\ell }_{n}M\hfill \\ & \ge D_{f_p}(z,u_n)-D_{f_p}(z,x_{n+1})\hfill \\ & \ge \tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p.\hfill \end{array}\left(3.20\right)$$

Thus,

$$\underset{n\to \infty }{lim}{\tau }_n{D}_{f_p}(u_n,y_n)=\underset{n\to \infty }{lim}{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)=0.$$

By Lemma 3.4, we get

$$\underset{n\to \infty }{lim}\parallel u_n-y_n\parallel =\underset{n\to \infty }{lim}\parallel w_n-{\tilde{y}}_n\parallel =0.$$

Besides, combining (3.15) and $x_{n_k}\rightharpoonup z$ guarantees that $u_{n_k}\rightharpoonup z$ and $w_{n_k}\rightharpoonup z$. By Lemma 3.3 we deduce that $z\in {\omega }_w\left(u_n\right)\subset \mathrm{VI}(C,A_1)$ and $z\in {\omega }_w\left(w_n\right)\subset \mathrm{VI}(C,A_2)$. Consequently,

$$z\in \bigcap _{i=1}^2\mathrm{VI}(C,A_i).$$

Next we claim that $z\in {\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)$. Indeed, by (3.15) we immediately get

$$\parallel x_{n+1}-x_n\parallel \le \parallel x_{n+1}-w_n\parallel +\parallel w_n-u_n\parallel +\parallel u_n-x_n\parallel \to 0(n\to \infty ).\left(3.21\right)$$

We first claim that ${lim}_{n\to \infty }\parallel x_n-S_r{x}_n\parallel =0$ for $r=1,...,N$. Actually, by the definition of $S_n$, we obtain that $S_n\in \{S_1,...,S_N\}\forall n\ge 1$, which hence leads to $S_{n+i}\in \{S_1,...,S_N\}\forall n\ge 1,i=1,...,N$. Note that for $i=1,...,N$,

$$\begin{array}{cc}\parallel x_n-S_{n+i}{x}_n\parallel \hfill & \le \parallel x_n-x_{n+i}\parallel +\parallel x_{n+i}-S_{n+i}{x}_{n+i}\parallel +\parallel S_{n+i}{x}_{n+i}-S_{n+i}{x}_n\parallel \hfill \\ & \le \parallel x_n-x_{n+i}\parallel +\parallel x_{n+i}-S_{n+i}{x}_{n+i}\parallel +{\displaystyle \sum _{j=1}^N}\parallel S_j{x}_{n+i}-S_j{x}_n\parallel .\hfill \end{array}$$

Utilizing the uniform continuity of each $S_j$ on C, we deduce from (3.15) and (3.21) that $x_{n+i}-S_{n+i}{x}_{n+i}\to 0$ and $S_j{x}_{n+i}-S_j{x}_n\to 0$ for $i,j=1,...,N$. Thus, we get ${lim}_{n\to \infty }\parallel x_n-S_{n+i}{x}_n\parallel =0$ for $i=1,...,N$. This immediately implies that

$$\underset{n\to \infty }{lim}\parallel x_n-S_r{x}_n\parallel =0\mathrm{for}r=1,...,N.$$

So it follows from $x_{n_k}\rightharpoonup z$ that $z\in \widehat{\mathrm{Fix}}\left(S_r\right)=\mathrm{Fix}\left(S_r\right)$ for $r=1,...,N$. Therefore, $z\in {\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)$. In addition, from (3.15) and $x_{n_k}\rightharpoonup z$, one has that $w_{n_k}\rightharpoonup z$. Thus, using (3.18) we get $z\in \widehat{\mathrm{Fix}}\left(S_0\right)=\mathrm{Fix}\left(S_0\right)$. Consequently, $z\in {\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)$, and hence $z\in \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)$. This means that ${\omega }_w\left(x_n\right)\subset \Omega$. As a result, applying Lemma 2.5 we conclude that $x_n\rightharpoonup z$. □

Next, we prove a strong convergence theorem for approximating a common solution of the VIPs for uniformly continuous pseudomonotone mappings and the CFPP of finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping in E.

Algorithm 3.2. Initialization: Given $x_0,x_1\in C$ arbitrarily and let $ϵ>0,{\mu }_i>0,l_i\in (0,1)$ and ${\lambda }_i\in (0,\frac{1}{{\mu }_i})$ for $i=1,2$. Choose $\left\{{\ell }_n\right\},\left\{{\gamma }_n\right\},\left\{{\alpha }_n\right\}\subset (0,1)$ and $\left\{{\beta }_n\right\}\subset (\xi ,1)$ s.t. ${lim}_{n\to \infty }{\ell }_n=0,{\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, ${lim}_{n\to \infty }{\alpha }_n=0$, ${lim\; inf}_{n\to \infty }({\beta }_n-\xi )(1-{\beta }_n)>0$ and ${lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n)>0$. Moreover, given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where ${sup}_{n\ge 1}\frac{{ϵ}_n}{{\alpha }_n}<\infty$ and

$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel }\}\mathrm{if}{S}_n{x}_n\ne S_n{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$

Iterative steps: Calculate $x_{n+1}$ as follows:

Step 1. Set $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$, and calculate $u_n=J_{E^{*}}^q({\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)$, $y_n={\mathsf{\Pi }}_C\left(J_{E^{*}}^q(J_E^p{u}_n-{\lambda }_1{A}_1{u}_n)\right)$, $r_{{\lambda }_1}\left(u_n\right):=u_n-y_n$ and $s_n=u_n-{\tau }_n{r}_{{\lambda }_1}\left(u_n\right)$, where ${\tau }_n:=l_1^{k_n}$ and $k_n$ is the smallest nonnegative integer k satisfying

$\langle A_1{u}_n-A_1(u_n-l_1^k{r}_{{\lambda }_1}\left(u_n\right)),u_n-y_n\rangle \le \frac{{\mu }_1}{2}{D}_{f_p}(u_n,y_n)$.

Step 2. Calculate $w_n={\mathsf{\Pi }}_{C_n}\left(u_n\right)$, with $C_n:=\{x\in C:h_n\left(x\right)\le 0\}$ and

$h_n\left(x\right)=\langle A_1{s}_n,x-u_n\rangle +\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(u_n,y_n)$.

Step 3. Calculate ${\tilde{y}}_n={\mathsf{\Pi }}_C\left(J_{E^{*}}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n)\right)$, $r_{{\lambda }_2}\left(w_n\right):=w_n-{\tilde{y}}_n$ and $t_n=w_n-{\tilde{\tau }}_n{r}_{{\lambda }_2}\left(w_n\right)$, where ${\tilde{\tau }}_n:=l_2^{j_n}$ and $j_n$ is the smallest nonnegative integer j satisfying

$\langle A_2{w}_n-A_2(w_n-l_2^j{r}_{{\lambda }_2}\left(w_n\right)),w_n-{\tilde{y}}_n\rangle \le \frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n)$.

Step 4. Set $z_n={\mathsf{\Pi }}_{{\tilde{C}}_n}\left(w_n\right)$, and calculate $v_n=J_{E^{*}}^q({\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p\left(S_0{z}_n\right))$ and $x_{n+1}={\mathsf{\Pi }}_C(J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$, where ${\tilde{C}}_n:=\{x\in C:{\tilde{h}}_n\left(x\right)\le 0\}$ and

${\tilde{h}}_n\left(x\right)=\langle A_2{t}_n,x-w_n\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n)$.

Again set $n:=n+1$ and go to Step 1.

Theorem 3.2. Suppose that the conditions (C1)-(C3) hold. If $\left\{x_n\right\}$ is the sequence generated by Algorithm 3.2, then $x_n\to {\mathsf{\Pi }}_{\Omega }u\iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Proof. 

It is clear that the necessity of Theorem 3.2 is valid. Next it suffices to show that the sufficiency is valid. Assume that ${sup}_{n\ge 0}\parallel x_n\parallel <\infty$. In what follows, we divide our proof into four claims.

Claim 1. We show that

$$\begin{array}{cc}& (1-{\alpha }_n){\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(z,x_n)-D_{f_p}(\widehat{u},x_{n+1})+{\ell }_{n}M,\hfill \end{array}$$

for some $M>0$. Indeed, put $\widehat{u}={\mathsf{\Pi }}_{\Omega }u$. Noticing $w_n={\mathsf{\Pi }}_{C_n}{u}_n$ and $z_n={\mathsf{\Pi }}_{{\tilde{C}}_n}{w}_n$, we deduce from (2.1) and (2.3) that

$\begin{array}{cc}{D}_{f_p}(\widehat{u},w_n)\hfill & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(w_n,u_n)\hfill \\ & \le D_{f_p}(\widehat{u},u_n)-\tau {\left[\mathrm{dist}(C_n,u_n)\right]}^p,\hfill \end{array}$

and

$\begin{array}{cc}{D}_{f_p}(\widehat{u},z_n)\hfill & \le D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)\hfill \\ & \le D_{f_p}(\widehat{u},w_n)-\tau {\left[\mathrm{dist}({\tilde{C}}_n,w_n)\right]}^p.\hfill \end{array}$

Using the same inferences as in the proof of Theorem 3.1, we know that

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},g_n)\hfill & \le D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel \le D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M,\hfill \end{array}$$

where ${sup}_{n\ge 1}\parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel \le M$ for some $M>0$. This ensures that $\left\{g_n\right\}$ is bounded.

Using (2.6) and the last two inequalities, from $\left\{{\gamma }_n\right\}\subset (0,1)$ and $\left\{{\beta }_n\right\}\subset (\xi ,1)$ we obtain

$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[{\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)D_{f_p}(\widehat{u},S_0{z}_n)\hfill \\ & -{\beta }_n(1-{\beta }_n){\rho }_{b_{z_n}}^{*}\parallel z_n-S_0{z}_n\parallel ]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)\{{\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)[D_{f_p}(\widehat{u},z_n)+\xi {\rho }_{b_{z_n}}^{*}\parallel z_n-S_0{z}_n\parallel ]\hfill \\ & -{\beta }_n(1-{\beta }_n){\rho }_{b_{z_n}}^{*}\parallel z_n-S_0{z}_n\parallel \}\hfill \\ & ={\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[D_{f_p}(\widehat{u},z_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_{b_{z_n}}^{*}\parallel z_n-S_0{z}_n\parallel ]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[D_{f_p}(\widehat{u},u_n)-D_{f_p}(w_n,u_n)-D_{f_p}(z_n,w_n)]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)D_{f_p}(\widehat{u},u_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[{\gamma }_n{D}_{f_p}(\widehat{u},x_n)+(1-{\gamma }_n)D_{f_p}(\widehat{u},g_n)\hfill \\ & -{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[D_{f_p}(z,x_n)+{\ell }_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(z,x_n)+{\ell }_{n}M-(1-{\alpha }_n){\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ,\hfill \end{array}$

which immediately arrives at the desired claim. In addition, it is easily known that $\left\{u_n\right\},\left\{v_n\right\}$, $\left\{w_n\right\},\left\{y_n\right\},\left\{{\tilde{y}}_n\right\},\left\{z_n\right\},\left\{s_n\right\},\left\{t_n\right\}$ and $\left\{S_0{z}_n\right\}$ are also bounded.

Claim 2. We show that

$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(z_n,w_n)\hfill \\ & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$

Indeed, set $b={sup}_{n\ge 1}\{\parallel x_n{\parallel }^{p-1},\parallel g_n{\parallel }^{p-1},\parallel z_n{\parallel }^{p-1},\parallel S_0{z}_n{\parallel }^{p-1}\}$. By Lemma 2.2 we get

$$\begin{array}{cc}& D_{f_p}(\widehat{u},u_n)=V_{f_p}(\widehat{u},{\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)\hfill \\ & \le \frac{1}{p}\parallel \widehat{u}{\parallel }^p-{\gamma }_n\langle J_E^p{x}_n,\widehat{u}\rangle -(1-{\gamma }_n)\langle J_E^p{g}_n,\widehat{u}\rangle +\frac{{\gamma }_n}{q}{\parallel J_E^p{x}_n\parallel }^q\hfill \\ & +\frac{(1-{\gamma }_n)}{q}\parallel J_E^p{g}_n{\parallel }^q-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & =\frac{1}{p}\parallel \widehat{u}{\parallel }^p-{\gamma }_n\langle J_E^p{x}_n,\widehat{u}\rangle -(1-{\gamma }_n)\langle J_E^p{g}_n,\widehat{u}\rangle +\frac{{\gamma }_n}{q}{\parallel x_n\parallel }^p\hfill \\ & +\frac{(1-{\gamma }_n)}{q}\parallel g_n{\parallel }^p-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & ={\gamma }_n{D}_{f_p}(\widehat{u},x_n)+(1-{\gamma }_n)D_{f_p}(\widehat{u},g_n)-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le {\gamma }_n{D}_{f_p}(\widehat{u},x_n)+(1-{\gamma }_n)[D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M]-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ,\hfill \end{array}\left(3.22\right)$$

and

$$\begin{array}{cc}& D_{f_p}(\widehat{u},v_n)=V_{f_p}(\widehat{u},{\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p{S}_0{z}_n)\hfill \\ & \le \frac{1}{p}\parallel \widehat{u}{\parallel }^p-{\beta }_n\langle J_E^p{z}_n,\widehat{u}\rangle -(1-{\beta }_n)\langle J_E^p{S}_0{z}_n,\widehat{u}\rangle +\frac{{\beta }_n}{q}{\parallel J_E^p{z}_n\parallel }^q\hfill \\ & +\frac{(1-{\beta }_n)}{q}\parallel J_E^p{S}_0{z}_n{\parallel }^q-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & =\frac{1}{p}\parallel \widehat{u}{\parallel }^p-{\beta }_n\langle J_E^p{z}_n,\widehat{u}\rangle -(1-{\beta }_n)\langle J_E^p{S}_0{z}_n,\widehat{u}\rangle +\frac{{\beta }_n}{q}{\parallel z_n\parallel }^p\hfill \\ & +\frac{(1-{\beta }_n)}{q}\parallel S_0{z}_n{\parallel }^p-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & ={\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)D_{f_p}(\widehat{u},S_0{z}_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & \le {\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)[D_{f_p}(\widehat{u},z_n)+\xi {\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel ]\hfill \\ & -{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & =D_{f_p}(\widehat{u},z_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & \le D_{f_p}(\widehat{u},w_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel .\hfill \end{array}\left(3.23\right)$$

Set ${\zeta }_n=J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$. From (2.5), we have

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le D_{f_p}(\widehat{u},J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n))=V_{f_p}(\widehat{u},{\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)\hfill \\ & \le V_{f_p}(\widehat{u},{\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n-{\alpha }_n(J_E^{p}u-J_E^p\widehat{u}))+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},\widehat{u})+(1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & =(1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)[D_{f_p}(\widehat{u},w_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel ]+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}\left(3.24\right)$$

On the other hand, from (3.23) we have

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},v_n)\hfill & \le D_{f_p}(\widehat{u},z_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & \le D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & \le D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n).\hfill \end{array}$$

Substituting the above inequality into (3.24), we get

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le (1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(w_n,u_n)-D_{f_p}(z_n,w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

This immediately arrives at

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(z_n,w_n)\hfill \\ & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}\left(3.25\right)$$

Claim 3. We show that

$$\begin{array}{cc}& (1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+{\ell }_{n}M.\hfill \end{array}$$

Indeed, using the similar inferences to these of (3.20) in the proof of Theorem 3.1, we get

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},z_n)\hfill & \le D_{f_p}(\widehat{u},w_n)-\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\hfill \\ & \le D_{f_p}(\widehat{u},u_n)-\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p-\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p.\hfill \end{array}\left(3.26\right)$$

Applying (3.26), (3.23) and (3.22), we have

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le D_{f_p}(\widehat{u},J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n))\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[D_{f_p}(\widehat{u},z_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel z_n-S_0{z}_n\parallel ]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)\{D_{f_p}(\widehat{u},u_n)-\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p-\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},u_n)-(1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & -(1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M\hfill \\ & -(1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}.\hfill \end{array}\left(3.27\right)$$

Claim 4. We show that $x_n\to \widehat{u}$ as $n\to \infty$. Indeed, since E is reflexive and $\left\{x_n\right\}$ is bounded, we know that ${\omega }_w\left(x_n\right)\ne \varnothing$. Let $z\in {\omega }_w\left(x_n\right)$. Then, $\exists \left\{x_{n_k}\right\}\subset \left\{x_n\right\}$ s.t. $x_{n_k}\rightharpoonup z$. For each $n\ge 1$, we write ${\mathsf{\Gamma }}_n=D_{f_p}(\widehat{u},x_n)$. In what follows, we show the convergence of $\left\{{\mathsf{\Gamma }}_n\right\}$ to zero in the following two possible cases.

Case 1. Suppose that ∃ (integer) $n_0\ge 1$ such that ${\left\{{\mathsf{\Gamma }}_n\right\}}_{n=n_0}^{\infty }$ is nonincreasing. Then ${lim}_{n\to \infty }{\mathsf{\Gamma }}_n=d<+\infty$ and ${lim}_{n\to \infty }({\mathsf{\Gamma }}_n-{\mathsf{\Gamma }}_{n+1})=0$. From (3.25) and (3.22) we get

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(z_n,w_n)\hfill \\ & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel -D_{f_p}(\widehat{u},x_{n+1})\hfill \\ & +{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle ,\hfill \end{array}$$

which hence yields

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(z_n,w_n)+{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+{\ell }_{n}M+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & ={\mathsf{\Gamma }}_n-{\mathsf{\Gamma }}_{n+1}+{\ell }_{n}M+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

Since ${lim}_{n\to \infty }{\ell }_n={lim}_{n\to \infty }{\alpha }_n=0,{lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n)>0,{lim}_{n\to \infty }({\mathsf{\Gamma }}_n-{\mathsf{\Gamma }}_{n+1})=0$ and the sequence $\left\{{\zeta }_n\right\}$ is bounded, we obtain that ${lim}_{n\to \infty }{D}_{f_p}(w_n,u_n)=0$, ${lim}_{n\to \infty }$$D_{f_p}(z_n,w_n)=0$, and ${lim}_{n\to \infty }{\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$, which hence yields ${lim}_{n\to \infty }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$. From $u_n=J_{E^{*}}^q({\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)$, it is easily known that ${lim}_{n\to \infty }\parallel J_E^p{u}_n$ $-J_E^p{x}_n\parallel =0$. Noticing $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$, we deduce from ${lim}_{n\to \infty }{\ell }_n=0$ and the definition of ${ϵ}_n$ that

$$\parallel J_E^p{g}_n-J_E^p{S}_n{x}_n\parallel ={ϵ}_n\parallel J_E^p(2S_n{x}_n-S_n{x}_{n-1})-J_E^p{S}_n{x}_n\parallel \le {\ell }_n\to 0(n\to \infty ).$$

Hence, using (2.1) and uniform continuity of $J_E^p$ on bounded subsets of E, we conclude that ${lim}_{n\to \infty }\parallel g_n-x_n\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_n-u_n\parallel =\underset{n\to \infty }{lim}\parallel z_n-w_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-S_n{x}_n\parallel =\underset{n\to \infty }{lim}\parallel u_n-x_n\parallel =0.\left(3.28\right)$$

Furthermore, from (3.24) and (3.22) we have

$$\begin{array}{cc}& (1-{\alpha }_n)({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},w_n)-D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+{\ell }_{n}M+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

By the similar inferences, we infer that ${lim}_{n\to \infty }\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel =0$, which hence leads to ${lim}_{n\to \infty }\parallel J_E^p{v}_n-J_E^p{z}_n\parallel =0$ (due to $v_n=J_{E^{*}}^q({\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p{S}_0{z}_n)$). Using uniform continuity of $J_{E^{*}}^q$ on bounded subsets of $E^{*}$, we get

$$\underset{n\to \infty }{lim}\parallel z_n-S_0{z}_n\parallel =\underset{n\to \infty }{lim}\parallel v_n-z_n\parallel =0.\left(3.29\right)$$

This together with (3.28) implies that

$$\parallel v_n-x_n\parallel \le \parallel v_n-z_n\parallel +\parallel z_n-w_n\parallel +\parallel w_n-u_n\parallel +\parallel u_n-x_n\parallel \to 0(n\to \infty ).$$

It is clear that

$$\underset{n\to \infty }{lim}\parallel z_n-x_n\parallel =0.\left(3.30\right)$$

Let us show that $z\in {\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)$. Indeed, since ${\zeta }_n=J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$, it can be readily seen that

$$\underset{n\to \infty }{lim}\parallel {\zeta }_n-x_n\parallel =0.\left(3.31\right)$$

In addition, using (2.3), (3.22) and (3.23), we have

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le D_{f_p}(\widehat{u},{\zeta }_n)-D_{f_p}(x_{n+1},{\zeta }_n)\hfill \\ & =D_{f_p}(\widehat{u},J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)-D_{f_p}(x_{n+1},{\zeta }_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)-D_{f_p}(x_{n+1},{\zeta }_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},w_n)-D_{f_p}(x_{n+1},{\zeta }_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},u_n)-D_{f_p}(x_{n+1},{\zeta }_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M-D_{f_p}(x_{n+1},{\zeta }_n),\hfill \end{array}$$

which hence arrives at

$$\begin{array}{cc}{D}_{f_p}(x_{n+1},{\zeta }_n)\hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+{\ell }_{n}M\hfill \\ & ={\alpha }_n{D}_{f_p}(\widehat{u},u)+{\mathsf{\Gamma }}_n-{\mathsf{\Gamma }}_{n+1}+{\ell }_{n}M.\hfill \end{array}$$

So it follows that ${lim}_{n\to \infty }{D}_{f_p}(x_{n+1},{\zeta }_n)=0$ and hence ${lim}_{n\to \infty }\parallel x_{n+1}-{\zeta }_n\parallel =0$. This together with (3.31), leads to

$$\parallel x_{n+1}-x_n\parallel \le \parallel x_{n+1}-{\zeta }_n\parallel +\parallel {\zeta }_n-x_n\parallel \to 0(n\to \infty ).\left(3.32\right)$$

Observe that for $i=1,...,N$,

$$\begin{array}{cc}\parallel x_n-S_{n+i}{x}_n\parallel \hfill & \le \parallel x_n-x_{n+i}\parallel +\parallel x_{n+i}-S_{n+i}{x}_{n+i}\parallel +\parallel S_{n+i}{x}_{n+i}-S_{n+i}{x}_n\parallel \hfill \\ & \le \parallel x_n-x_{n+i}\parallel +\parallel x_{n+i}-S_{n+i}{x}_{n+i}\parallel +{\displaystyle \sum _{j=1}^N}\parallel S_j{x}_{n+i}-S_j{x}_n\parallel .\hfill \end{array}$$

Exploiting the uniform continuity of each $S_j$ on C, we deduce from (3.28) and (3.32) that $x_{n+i}-S_{n+i}{x}_{n+i}\to 0$ and $S_j{x}_{n+i}-S_j{x}_n\to 0$ for $i,j=1,...,N$. Thus, we get ${lim}_{n\to \infty }\parallel x_n-S_{n+i}{x}_n\parallel =0$ for $i=1,...,N$. This immediately implies that ${lim}_{n\to \infty }\parallel x_n-S_r{x}_n\parallel =0$ for $r=1,...,N$. So it follows from $x_{n_k}\rightharpoonup z$ that $z\in \widehat{\mathrm{Fix}}\left(S_r\right)=\mathrm{Fix}\left(S_r\right)$ for $r=1,...,N$. Therefore, $z\in {\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)$. In addition, from (3.30) and $x_{n_k}\rightharpoonup z$, one has that $z_{n_k}\rightharpoonup z$. Thus, using (3.29) we get $z\in \widehat{\mathrm{Fix}}\left(S_0\right)=\mathrm{Fix}\left(S_0\right)$. Consequently, $z\in {\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)$,

In what follows, we show that $z\in {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. From (3.27), we have

$$\begin{array}{cc}& (1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+{\ell }_{n}M\hfill \\ & ={\alpha }_n{D}_{f_p}(\widehat{u},u)+{\mathsf{\Gamma }}_n-{\mathsf{\Gamma }}_{n+1}+{\ell }_{n}M.\hfill \end{array}$$

So it follows that ${lim}_{n\to \infty }\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)={lim}_{n\to \infty }\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)=0$, and hence

$$\underset{n\to \infty }{lim}{\tau }_n{D}_{f_p}(u_n,y_n)=\underset{n\to \infty }{lim}{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)=0.\left(3.33\right)$$

Using Lemma 3.4, we infer that

$$\underset{n\to \infty }{lim}\parallel u_n-y_n\parallel =\underset{n\to \infty }{lim}\parallel w_n-{\tilde{y}}_n\parallel =0.\left(3.34\right)$$

Applying Lemma 3.3 and (3.34), we obtain that $z\in {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. Hence we get ${\omega }_w\left(x_n\right)\subset {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. Consequently, ${\omega }_w\left(x_n\right)\subset \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)$. Lastly, we show that ${lim\; sup}_{n\to \infty }\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \le 0$. We can pick a subsequence $\left\{x_{n_j}\right\}$ of $\left\{x_n\right\}$ such that

$$\underset{n\to \infty }{lim\; sup}\langle J_E^{p}u-J_E^p\widehat{u},x_n-\widehat{u}\rangle =\underset{j\to \infty }{lim}\langle J_E^{p}u-J_E^p\widehat{u},x_{n_j}-\widehat{u}\rangle .$$

Because E is reflexive and $\left\{x_n\right\}$ is bounded, we may assume, without loss of generality, that $x_{n_j}\rightharpoonup \tilde{z}$. So it follows from (2.2) and $\tilde{z}\in \Omega$ that

$$\underset{n\to \infty }{lim\; sup}\langle J_E^{p}u-J_E^p\widehat{u},x_n-\widehat{u}\rangle =\underset{j\to \infty }{lim}\langle J_E^{p}u-J_E^p\widehat{u},x_{n_j}-\widehat{u}\rangle =\langle J_E^{p}u-J_E^p\widehat{u},\tilde{z}-\widehat{u}\rangle \le 0.\left(3.35\right)$$

This together with (3.31) ensures that

$$\underset{n\to \infty }{lim\; sup}\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \le 0.$$

From (3.24) and (3.22), we get

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le (1-{\alpha }_n)D_{f_p}(\widehat{u},w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},u_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)[D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel ]+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel +{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & =(1-{\alpha }_n)D_{f_p}(\widehat{u},x_n)+{\alpha }_n\{\frac{{ϵ}_n}{{\alpha }_n}\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel +\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \}.\hfill \end{array}\left(3.36\right)$$

Using uniform continuity of each $S_i(1\le i\le N)$ on C, and uniform continuity of $J_E^p$ on bounded subsets of E, from (3.32) and the boundedness of $\left\{x_n\right\}$ we get

$$\underset{n\to \infty }{lim}\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel =0.$$

Noticing ${sup}_{n\ge 1}\frac{{ϵ}_n}{{\alpha }_n}<\infty$ and ${lim\; sup}_{n\to \infty }\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \le 0$, we infer that

$$\begin{array}{cc}& {\displaystyle \underset{n\to \infty }{lim\; sup}}\{\frac{{ϵ}_n}{{\alpha }_n}\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel +\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \}\le 0.\hfill \end{array}$$

Since $\left\{{\alpha }_n\right\}\subset (0,1)$ and ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, applying Lemma 2.8 to (3.36) we obtain that ${lim}_{n\to \infty }{D}_{f_p}(\widehat{u},x_n)=0$ and hence ${lim}_{n\to \infty }\parallel \widehat{u}-x_n\parallel =0$.

Case 2. Suppose that $\exists \left\{{\mathsf{\Gamma }}_{n_k}\right\}\subset \left\{{\mathsf{\Gamma }}_n\right\}$ s.t. ${\mathsf{\Gamma }}_{n_k}<{\mathsf{\Gamma }}_{n_k+1}\forall k\in \mathcal{N}$, where $\mathcal{N}$ is the set of all positive integers. Define the mapping $\psi :\mathcal{N}\to \mathcal{N}$ by

$$\psi \left(n\right):=max\{k\le n:{\mathsf{\Gamma }}_k<{\mathsf{\Gamma }}_{k+1}\}.$$

By Lemma 2.7, we get

$${\mathsf{\Gamma }}_{\psi \left(n\right)}\le {\mathsf{\Gamma }}_{\psi \left(n\right)+1}\mathrm{and}{\mathsf{\Gamma }}_n\le {\mathsf{\Gamma }}_{\psi \left(n\right)+1}.\left(3.37\right)$$

From (3.25) and (3.22) it follows that

$$\begin{array}{cc}& D_{f_p}(w_{\psi \left(n\right)},u_{\psi \left(n\right)})+D_{f_p}(z_{\psi \left(n\right)},w_{\psi \left(n\right)})+{\gamma }_{\psi \left(n\right)}(1-{\gamma }_{\psi \left(n\right)}){\rho }_b^{*}\parallel J_E^p{x}_{\psi \left(n\right)}-J_E^p{g}_{\psi \left(n\right)}\parallel \hfill \\ & \le {\mathsf{\Gamma }}_{\psi \left(n\right)}-{\mathsf{\Gamma }}_{\psi \left(n\right)+1}+{\ell }_{\psi \left(n\right)}M+{\alpha }_{\psi \left(n\right)}\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle .\hfill \end{array}$$

Noticing $g_{\psi \left(n\right)}=J_{E^{*}}^q((1-{ϵ}_{\psi \left(n\right)})J_E^p{S}_{\psi \left(n\right)}{x}_{\psi \left(n\right)}+{ϵ}_{\psi \left(n\right)}{J}_E^p(2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1}))$ and $u_{\psi \left(n\right)}=J_{E^{*}}^q({\gamma }_{\psi \left(n\right)}{J}_E^p{x}_{\psi \left(n\right)}+(1-{\gamma }_{\psi \left(n\right)})J_E^p{g}_{\psi \left(n\right)}))$, we obtain that ${lim}_{n\to \infty }\parallel g_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_{\psi \left(n\right)}-u_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel z_{\psi \left(n\right)}-w_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel x_{\psi \left(n\right)}-S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel u_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =0.\left(3.38\right)$$

Also, from (3.24) and (3.22) we have

$$\begin{array}{cc}& (1-{\alpha }_{\psi \left(n\right)})({\beta }_{\psi \left(n\right)}-\xi )(1-{\beta }_{\psi \left(n\right)}){\rho }_b^{*}\parallel J_E^p{z}_{\psi \left(n\right)}-J_E^p{S}_0{z}_{\psi \left(n\right)}\parallel \hfill \\ & \le {\mathsf{\Gamma }}_{\psi \left(n\right)}-{\mathsf{\Gamma }}_{\psi \left(n\right)+1}+{\ell }_{\psi \left(n\right)}M+{\alpha }_{\psi \left(n\right)}\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle .\hfill \end{array}$$

Noticing $v_{\psi \left(n\right)}=J_{E^{*}}^q({\beta }_{\psi \left(n\right)}{J}_E^p{z}_{\psi \left(n\right)}+(1-{\beta }_{\psi \left(n\right)})J_E^p{S}_0{z}_{\psi \left(n\right)})$ and using the similar inferences to those in Case 1, we get

$$\underset{n\to \infty }{lim}\parallel z_{\psi \left(n\right)}-S_0{z}_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel v_{\psi \left(n\right)}-z_{\psi \left(n\right)}\parallel =0.$$

This together with (3.38) implies that

$$\underset{n\to \infty }{lim}\parallel v_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel z_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =0.\left(3.39\right)$$

Noticing ${\zeta }_{\psi \left(n\right)}=J_{E^{*}}^q({\alpha }_{\psi \left(n\right)}{J}_E^{p}u+(1-{\alpha }_{\psi \left(n\right)})J_E^p{v}_{\psi \left(n\right)})$, from (3.39) we get

$$\underset{n\to \infty }{lim}\parallel {\zeta }_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =0.\left(3.40\right)$$

Using the similar inferences to those in Case 1, we conclude that ${lim}_{n\to \infty }\parallel x_{\psi \left(n\right)+1}-x_{\psi \left(n\right)}\parallel =0$,

$$\underset{n\to \infty }{lim}\parallel u_{\psi \left(n\right)}-y_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel w_{\psi \left(n\right)}-{\tilde{y}}_{\psi \left(n\right)}\parallel =0,\left(3.41\right)$$

and

$$\underset{n\to \infty }{lim\; sup}\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle \le 0.\left(3.42\right)$$

Using (3.36), we get

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{\psi \left(n\right)+1})\le (1-{\alpha }_{\psi \left(n\right)})D_{f_p}(\widehat{u},x_{\psi \left(n\right)})\hfill \\ & +{\alpha }_{\psi \left(n\right)}\{\frac{{ϵ}_{\psi \left(n\right)}}{{\alpha }_{\psi \left(n\right)}}\parallel J_E^p{S}_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-J_E^p(2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1}-2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}\parallel +\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle \},\hfill \end{array}\left(3.43\right)$$

which together with (3.37), hence yields

$$\begin{array}{cc}& {\mathsf{\Gamma }}_{\psi \left(n\right)}\le \frac{{ϵ}_{\psi \left(n\right)}}{{\alpha }_{\psi \left(n\right)}}\parallel J_E^p{S}_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-J_E^p(2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1}-2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}\parallel +\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle .\hfill \end{array}$$

As a result, from (3.42) we deduce that

$$\underset{n\to \infty }{lim}{\mathsf{\Gamma }}_{\psi \left(n\right)}=0.\left(3.44\right)$$

From (3.42), (3.43) and (3.44), one has that

$$\underset{n\to \infty }{lim}{\mathsf{\Gamma }}_{\psi \left(n\right)+1}=0.\left(3.45\right)$$

Again from (3.37), we have ${lim}_{n\to \infty }{D}_{f_p}(\widehat{u},x_n)={lim}_{n\to \infty }{\mathsf{\Gamma }}_n=0$. Hence ${lim}_{n\to \infty }\parallel x_n-\widehat{u}\parallel =0$. This completes the proof. □

Remark 3.1. It can be easily seen from the proof of Theorem 3.2 that if the assumption that ${lim}_{n\to \infty }\frac{{\ell }_n}{{\alpha }_n}=0$, is used in place of the one that ${lim}_{n\to \infty }{\ell }_n=0$ and ${sup}_{n\ge 1}\frac{{ϵ}_n}{{\alpha }_n}<\infty$, then Theorem 3.2 is still valid.

Setting $A_2=0$ in Algorithm 3.1, we immediately obtain the following algorithm for finding an element of $\Omega =\mathrm{VI}(C,A_1)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)$.

Algorithm 3.3. Initialization: Given $x_0,x_1\in C$ arbitrarily and let $ϵ>0,{\mu }_1>0,{\lambda }_1\in (0,\frac{1}{{\mu }_1}),l_1\in (0,1)$. Choose $\left\{{\ell }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ and $\left\{{\alpha }_n\right\}\subset (\xi ,1)$ s.t. ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${lim\; inf}_{n\to \infty }({\alpha }_n-\xi )(1-{\alpha }_n)>0$. Moreover, given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel }\}\mathrm{if}{S}_n{x}_n\ne S_n{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$

Iterative steps: Calculate $x_{n+1}$ as follows:

Step 1. Set $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$, and calculate $u_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, $y_n={\mathsf{\Pi }}_C\left(J_{E^{*}}^q(J_E^p{u}_n-{\lambda }_1{A}_1{u}_n)\right)$, $r_{{\lambda }_1}\left(u_n\right):=u_n-y_n$ and $s_n=u_n-{\tau }_n{r}_{{\lambda }_1}\left(u_n\right)$, where ${\tau }_n:=l_1^{k_n}$ and $k_n$ is the smallest nonnegative integer k satisfying

$$\langle A_1{u}_n-A_1(u_n-l_1^k{r}_{{\lambda }_1}\left(u_n\right)),u_n-y_n\rangle \le \frac{{\mu }_1}{2}{D}_{f_p}(u_n,y_n).$$

Step 2. Calculate $w_n={\mathsf{\Pi }}_{C_n}\left(u_n\right)$, with $C_n:=\{x\in C:h_n\left(x\right)\le 0\}$ and

$$h_n\left(x\right)=\langle A_1{s}_n,x-u_n\rangle +\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(u_n,y_n).$$

Step 3. Calculate $v_n=J_{E^{*}}^q({\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p\left(S_0{w}_n\right))$ and $x_{n+1}={\mathsf{\Pi }}_{Q_n}\left(w_n\right)$, where $Q_n:=\{x\in C:D_{f_p}(x,v_n)\le D_{f_p}(x,w_n)\}$.

Again set $n:=n+1$ and go to Step 1.

Corollary 3.1. Suppose that the conditions (C1)-(C3) with $A_2=0$, hold, and $\Omega =\mathrm{VI}(C,A_1)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)\ne \varnothing$. Let $\left\{x_n\right\}$ be the sequence constructed in Algorithm 3.3. Then $x_n\rightharpoonup z\in \Omega \iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Next, let $S_1:E\to C$ be a Bregman relatively nonexpansive mapping and $S_i=S=I$ the identity mapping of E for $i=2,...,N$. Then we get $\Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)=\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \mathrm{Fix}\left(S_1\right)$. In this case, Algorithm 3.2 reduces to the following iterative scheme for solving a pair of VIPs and the FPP of $S_1$. By Theorem 3.2 we obtain the following strong convergence result.

Corollary 3.2. Suppose that the condition (C3) holds, and let $\Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \mathrm{Fix}\left(S_1\right)\ne \varnothing$. For initial $x_0,x_1\in C$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel J_E^p{S}_1{x}_n-J_E^p(2S_1{x}_n-S_1{x}_{n-1})\parallel }\}\mathrm{if}{S}_1{x}_n\ne S_1{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$

Suppose that $\left\{x_n\right\}$ is the sequence constructed by

$$\left\{\begin{array}{cc}& g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_1{x}_n+{ϵ}_n{J}_E^p(2S_1{x}_n-S_1{x}_{n-1})),\hfill \\ & u_n=J_{E^{*}}^q({\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n),\hfill \\ & y_n={\mathsf{\Pi }}_C\left(J_{E^{*}}^q(J_E^p{u}_n-{\lambda }_1{A}_1{u}_n)\right),\hfill \\ & s_n=(1-{\tau }_n)u_n+{\tau }_n{y}_n,\hfill \\ & w_n={\mathsf{\Pi }}_{C_n}{u}_n,\hfill \\ & {\tilde{y}}_n={\mathsf{\Pi }}_C\left(J_{E^{*}}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n)\right),\hfill \\ & t_n=(1-{\tilde{\tau }}_n)w_n+{\tilde{\tau }}_n{\tilde{y}}_n,\hfill \\ & z_n={\mathsf{\Pi }}_{{\tilde{C}}_n}{w}_n,\hfill \\ & x_{n+1}={\mathsf{\Pi }}_C(J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{z}_n)\forall n\ge 1,\hfill \end{array}\right.$$

where ${\tau }_n:=l_1^{k_n},{\tilde{\tau }}_n:=l_2^{j_n}$ and $k_n,j_n$ are the smallest nonnegative integers k and j satisfying, respectively,

$$\langle A_1{u}_n-A_1(u_n-l_1^k(u_n-y_n)),u_n-y_n\rangle \le \frac{{\mu }_1}{2}{D}_{f_p}(u_n,y_n),$$

$$\langle A_2{w}_n-A_2(w_n-l_2^j(w_n-{\tilde{y}}_n)),w_n-{\tilde{y}}_n\rangle \le \frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n),$$

and the sets $C_n,{\tilde{C}}_n$, are constructed below

(i) $C_n:=\{x\in C:h_n\left(x\right)\le 0\}$ and $h_n\left(x\right)=\langle A_1{s}_n,x-u_n\rangle +\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(u_n,y_n)$;

(ii) ${\tilde{C}}_n:=\{x\in C:{\tilde{h}}_n\left(x\right)\le 0\}$ and ${\tilde{h}}_n\left(x\right)=\langle A_2{t}_n,x-w_n\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n)$.

Then, $x_n\to {\mathsf{\Pi }}_{\Omega }u\iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

4. Examples

In this section, we provide an illustrated example to demonstrate the feasibility and implementability of our proposed approaches. Put $ϵ=\frac{1}{3}$, ${\mu }_i=1$ and $l_i={\lambda }_i=\frac{1}{3}$ for $i=1,2$. We first provide an example of uniformly continuous and pseudomonotone mappings $A_i:E\to E^{*},i=1,2$, Bregman relatively nonexpansive mapping $S_1:C\to C$ and Bregman relatively demicontractive mapping $S_0:C\to C$ with $\Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^1\mathrm{Fix}\left(S_i\right)\right)\ne \varnothing$. Let $C=[-2,2]$ and $E=H=\mathbf{R}$ with the inner product $\langle a,b\rangle =ab$ and induced norm $\parallel ·\parallel =|·|$. The initial points $x_0,x_1$ are randomly chosen in C. For $i=1,2$, let $A_i:H\to H$ be defined as $A_{1}x:=\frac{1}{1+|sinx|}-\frac{1}{1+\left|x\right|}$ and $A_{2}x:=x+sinx$ for all $x\in H$. Now, we first show that $A_1$ is Lipschitz continuous and pseudomonotone. Indeed, for all $x,y\in H$ we have

$$\begin{array}{cc}\parallel A_{1}x-A_{1}y\parallel \hfill & =|\frac{1}{1+\parallel sinx\parallel }-\frac{1}{1+\parallel x\parallel }-\frac{1}{1+\parallel siny\parallel }+\frac{1}{1+\parallel y\parallel }|\hfill \\ & \le |\frac{\parallel y\parallel -\parallel x\parallel }{(1+\parallel x\parallel )(1+\parallel y\parallel )}|+|\frac{\parallel siny\parallel -\parallel sinx\parallel }{(1+\parallel sinx\parallel )(1+\parallel siny\parallel )}|\hfill \\ & \le \parallel x-y\parallel +\parallel sinx-siny\parallel \le 2\parallel x-y\parallel .\hfill \end{array}$$

This implies that $A_1$ is Lipschitz continuous. Also, we show that $A_1$ is pseudomonotone. For each $x,y\in H$, it is easy to see that

$$\begin{array}{cc}& \langle A_{1}x,y-x\rangle =(\frac{1}{1+|sinx|}-\frac{1}{1+\left|x\right|})(y-x)\ge 0\hfill \\ & \Rightarrow \langle A_{1}y,y-x\rangle =(\frac{1}{1+|siny|}-\frac{1}{1+\left|y\right|})(y-x)\ge 0.\hfill \end{array}$$

It is readily known that $A_2$ is Lipschitz continuous and monotone. Indeed, we deduce that $\parallel A_{2}x-A_{2}y\parallel \le \parallel x-y\parallel +\parallel sinx-siny\parallel \le 2\parallel x-y\parallel$ and

$$\langle A_{2}x-A_{2}y,x-y\rangle ={\parallel x-y\parallel }^2+\langle sinx-siny,x-y\rangle \ge {\parallel x-y\parallel }^2-{\parallel x-y\parallel }^2=0.$$

Now, let $S_1:C\to C$ and $S_0:C\to C$ be defined as $S_{1}x=sinx$ and $S_{0}x=\frac{1}{5}x+\frac{3}{5}sinx$. It is easy to verify that $\mathrm{Fix}\left(S_1\right)=\mathrm{Fix}\left(S_0\right)=\left\{0\right\}$ and $S_1:C\to C$ is Bregman relatively nonexpansive. Also, $S_0:C\to C$ is Bregman relatively $\xi$-demicontractive with $\xi =\frac{1}{5}$. Indeed, note that

$$\parallel S_{0}x-S_0{y\parallel }^2=\parallel \frac{1}{5}(x-y)+\frac{3}{5}{(sinx-siny)\parallel }^2\le {\parallel x-y\parallel }^2+\frac{2}{5}{\parallel (I-S_0)x-(I-S_0)y\parallel }^2.$$

Consequently,

$$\Omega =\left(\bigcap _{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left(\bigcap _{i=0}^1\mathrm{Fix}\left(S_i\right)\right)=\left\{0\right\}\ne \varnothing .$$

In addition, putting ${\beta }_n=\frac{n+2}{2(n+1)}\forall n\ge 1$, we obtain

$$\underset{n\to \infty }{lim}({\beta }_n-\xi )(1-{\beta }_n)=\underset{n\to \infty }{lim}(\frac{n+2}{2(n+1)}-\frac{1}{5})(1-\frac{n+2}{2(n+1)})=(\frac{1}{2}-\frac{1}{5})(1-\frac{1}{2})=\frac{3}{20}>0$$

In this case, the conditions (C1)-(C3) are satisfied.

Example 4.1. Let ${\ell }_n=\frac{1}{2{(n+1)}^2}$ and ${\alpha }_n={\beta }_n=\frac{n+2}{2(n+1)}\forall n\ge 1$. Given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel S_1{x}_n-S_1{x}_{n-1}\parallel }\}\mathrm{if}{S}_1{x}_n\ne S_1{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$

Algorithm 3.1 is rewritten as follows:

$$\left\{\begin{array}{cc}& g_n=S_1{x}_n+{ϵ}_n(S_1{x}_n-S_1{x}_{n-1}),\hfill \\ & u_n=\frac{n+2}{2(n+1)}{x}_n+\frac{n}{2(n+1)}{g}_n,\hfill \\ & y_n=P_C(u_n-\frac{1}{3}{A}_1{u}_n),\hfill \\ & s_n=(1-{\tau }_n)u_n+{\tau }_n{y}_n,\hfill \\ & w_n=P_{C_n}{u}_n,\hfill \\ & {\tilde{y}}_n=P_C(w_n-\frac{1}{3}{A}_2{w}_n),\hfill \\ & t_n=(1-{\tilde{\tau }}_n)w_n+{\tilde{\tau }}_n{\tilde{y}}_n,\hfill \\ & v_n=\frac{n+2}{2(n+1)}{w}_n+\frac{n}{2(n+1)}{S}_0{w}_n,\hfill \\ & Q_n=\{x\in C:\parallel x-v_n\parallel \le \parallel x-w_n\parallel \},\hfill \\ & x_{n+1}=P_{{\tilde{C}}_n\cap Q_n}{w}_n\forall n\ge 1,\hfill \end{array}\right.\left(4.1\right)$$

where for each $n\ge 1$, the sets $C_n,{\tilde{C}}_n$ and the step-sizes ${\tau }_n,{\tilde{\tau }}_n$ are chosen as in Algorithm 3.1. Then, by Theorem 3.1, we deduce that $\left\{x_n\right\}$ converges to $0\in \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^1\mathrm{Fix}\left(S_i\right)\right)$.

Example 4.2. Let ${\ell }_n=\frac{1}{2{(n+1)}^2},{\alpha }_n=\frac{1}{2(n+1)}$ and ${\beta }_n={\gamma }_n=\frac{n+2}{2(n+1)}\forall n\ge 1$. Given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel S_1{x}_n-S_1{x}_{n-1}\parallel }\}\mathrm{if}{S}_1{x}_n\ne S_1{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$

Algorithm 3.2 is rewritten as follows:

$$\left\{\begin{array}{cc}& g_n=S_1{x}_n+{ϵ}_n(S_1{x}_n-S_1{x}_{n-1}),\hfill \\ & u_n=\frac{n+2}{2(n+1)}{x}_n+\frac{n}{2(n+1)}{g}_n,\hfill \\ & y_n=P_C(u_n-\frac{1}{3}{A}_1{u}_n),\hfill \\ & s_n=(1-{\tau }_n)u_n+{\tau }_n{y}_n,\hfill \\ & w_n=P_{C_n}{u}_n,\hfill \\ & {\tilde{y}}_n=P_C(w_n-\frac{1}{3}{A}_2{w}_n),\hfill \\ & t_n=(1-{\tilde{\tau }}_n)w_n+{\tilde{\tau }}_n{\tilde{y}}_n,\hfill \\ & z_n=P_{{\tilde{C}}_n}{w}_n,\hfill \\ & v_n=\frac{n+2}{2(n+1)}{z}_n+\frac{n}{2(n+1)}{S}_0{z}_n,\hfill \\ & x_{n+1}=P_C(\frac{1}{2(n+1)}u+\frac{2n+1}{2(n+1)}{v}_n)\forall n\ge 1,\hfill \end{array}\right.\left(4.2\right)$$

where for each $n\ge 1$, the sets $C_n,{\tilde{C}}_n$ and the step-sizes ${\tau }_n,{\tilde{\tau }}_n$ are chosen as in Algorithm 3.2. Then, by Theorem 3.2, we deduce that $\left\{x_n\right\}$ converges to $0\in \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^1\mathrm{Fix}\left(S_i\right)\right)$.

5. Conclusions

Let $1<q\le 2\le p<\infty$ with $\frac{1}{p}+\frac{1}{q}=1$ and let E be a p-uniformly convex and uniformly smooth Banach space. Then its dual space $E^{*}$ is q-uniformly smooth Banach space with $1<q\le 2$. Utilizing the geometric properties of E and $E^{*}$, we design two inertial-type subgradient extragradient algorithms with line-search process for solving the pseudomonotone variational inequality problems (VIPs) and common fixed-point problem (CFPP), where the geometric properties involve the properties of the generalized duality mappings $J_E^p,J_{E^{*}}^q$ and Bregman projection operator ${\mathsf{\Pi }}_C$. Here the CFPP indicates the common fixed-point problem of finite Bregman relatively nonexpansive mapping and a Bregman relatively demicontractive mapping in E. Under the properties of the generalized duality mappings $J_E^p,J_{E^{*}}^q$ and Bregman projection operator ${\mathsf{\Pi }}_C$, we prove weak and strong convergence of the suggested algorithms to a common solution of the VIPs and CFPP, respectively. Additionally, an illustrated example is furnished to demonstrate the feasibility and implementability of our proposed approaches. In the end, it is noteworthy that part of our future research is aimed at attaining the weak and strong convergence results for the modifications of our proposed approaches with Nesterov double inertial-type extrapolation steps (see [34]) and adaptive stepsizes.