Parallel Subgradient-Like Extragradient Approaches for Variational Inequality and Fixed Point Problems with Bregman Relatively Asymptotical Nonexpansivity

1. Introduction

Let $\varnothing \ne C\subset H$ and $P_C$ be the metric projection from H onto C with C being convex and closed in a real Hilbert space H. Suppose that the $\langle ·,·\rangle$ and $\parallel ·\parallel$ are the inner product and induced norm in H, respectively. Given a nonlinear operator $S:C\to C$. We denote by $\mathrm{Fix}\left(S\right)$ the fixed-point set of S. Also, the $\mathbf{R},\rightharpoonup$ and → are used to stand for the real-number set, the weak convergence and the strong convergence, respectively. A self-mapping S on C is known as being of asymptotical nonexpansivity if ∃ (nonnegative real sequence) $\left\{{\theta }_n\right\}$ s.t.

$$({\theta }_n+1)\parallel v-u\parallel \ge \parallel S^{n}v-S^{n}u\parallel \forall v,u\in C,n\ge 1,$$

(1.1)

with ${\theta }_n\to 0$. In particular, in case ${\theta }_n=0\forall n\ge 1$, S reduces to a nonexpansive mapping. Let $A:H\to H$ be a mapping. Recall that so-called variational inequality problem (VIP) is to find $v\in C$ such that

$$\langle Av,x-v\rangle \ge 0\forall x\in C,$$

(1.2)

Here $\mathrm{VI}(C,A)$ denotes the set of solutions of the VIP. In 1976, under weaker assumptions, Korpelevich [24] put forward the extragradient rule for approximating an element of $\mathrm{VI}(C,A)$, i.e., for any starting $t_0\in C$, $\left\{t_n\right\}$ is the sequence generated by

$\left\{\begin{array}{cc}& s_n=P_C(t_n-ϵA{t}_n),\hfill \\ & t_{n+1}=P_C(t_n-ϵA{s}_n)\forall n\ge 0,\hfill \end{array}\right.$

with $ϵ\in (0,\frac{1}{L})$. If $\mathrm{VI}(C,A)\ne \varnothing$, then $\left\{t_n\right\}$ converges weakly to an element in $\mathrm{VI}(C,A)$. To the best of our understanding, the Korpelevich extragradient rule is one of the most effective approaches for solving the VIP till now. The literature on the VIP is vast and the Korpelevich extragradient rule has acquired the extensive attention paid by numerous scholars, who ameliorated it in various ways; see e.g., [1-6, 8-9, 13-16, 19, 21-23, 25-28, 31, 34].

Recently, Thong and Hieu [21] first put forth the inertial subgradient extragradient rule, i.e., for any starting $t_0,t_1\in H$, $\left\{t_n\right\}$ is the sequence generated by

$\left\{\begin{array}{cc}& y_n=t_n+{\alpha }_n(t_n-t_{n-1}),\hfill \\ & s_n=P_C(y_n-\ell A{y}_n),\hfill \\ & C_n=\{v\in H:\langle y_n-\ell A{y}_n-s_n,v-s_n\rangle \le 0\},\hfill \\ & t_{n+1}=P_{C_n}(y_n-\ell A{s}_n)\forall n\ge 1,\hfill \end{array}\right.$

with constant $\ell \in (0,\frac{1}{L})$. Under suitable assumptions, they proved the weak convergence of $\left\{t_n\right\}$ to an element of $\mathrm{VI}(C,A)$. Subsequently, Thong and Hieu [15] introduced two inertial subgradient extragradient algorithms with linear-search process for solving the VIP with Lipschitz continuous monotone mapping A and the fixed-point problem (FPP) of a quasi-nonexpansive mapping S with demiclosedness property in H.

Algorithm 1.1 (see [15, Algorithm 1]). Initialization: Given $\gamma >0,l\in (0,1),\mu \in (0,1)$. Choose any initial $t_0,t_1\in H$.

Iterations: Compute $t_{n+1}$ below:

Step 1. Put $s_n=t_n+{\alpha }_n(t_n-t_{n-1})$ and calculate $y_n=P_C(s_n-{\ell }_{n}A{s}_n)$, wherein ${\ell }_n$ is picked as the largest $\ell \in \{\gamma ,\gamma l,\gamma l^2,...\}$ s.t. $\ell \parallel A{s}_n-A{y}_n\parallel \le \mu \parallel s_n-y_n\parallel$.

Step 2. Calculate $u_n=P_{C_n}(s_n-{\ell }_{n}A{y}_n)$, where $C_n:=\{u\in H:\langle s_n-{\ell }_{n}A{s}_n-y_n,u-y_n\rangle \le 0\}$.

Step 3. Calculate $t_{n+1}=(1-{\beta }_n)s_n+{\beta }_{n}S{u}_n$. When $s_n=u_n=t_{n+1}$, one has $s_n\in \mathrm{Fix}\left(T\right)\cap \mathrm{VI}(C,A)$. Put $n:=n+1$ and return to Step 1.

Algorithm 1.2 (see [15, Algorithm 2]). Initialization: Given $\gamma >0,l\in (0,1),\mu \in (0,1)$. Choose any initial $t_0,t_1\in H$.

Iterative steps: Compute $t_{n+1}$ below:

Step 1. Putt $s_n=t_n+{\alpha }_n(t_n-t_{n-1})$ and calculate $y_n=P_C(s_n-{\ell }_{n}A{s}_n)$, wherein ${\ell }_n$ is picked as the largest $\ell \in \{\gamma ,\gamma l,\gamma l^2,...\}$ s.t. $\ell \parallel A{s}_n-A{y}_n\parallel \le \mu \parallel s_n-y_n\parallel$.

Step 2. Calculate $u_n=P_{C_n}(s_n-{\ell }_{n}A{y}_n)$, where $C_n:=\{u\in H:\langle s_n-{\ell }_{n}A{s}_n-y_n,u-y_n\rangle \le 0\}$.

Step 3. Calculate $t_{n+1}=(1-{\beta }_n)t_n+{\beta }_{n}S{u}_n$. When $s_n=u_n=t_n=t_{n+1}$, one has $t_n\in \mathrm{Fix}\left(T\right)\cap \mathrm{VI}(C,A)$. Put $n:=n+1$ and return to Step 1.

With the help of suitable assumptions, it was proved in [15] that the sequences generated by the suggested algorithms converge weakly to a point in $\mathrm{VI}(C,A)\cap \mathrm{Fix}\left(S\right)$. In addition, combining the subgradient extragradient method and the Halpern’s iteration rule, Kraikaew and Saejung [22] proposed the Halpern subgradient extragradient rule for solving the VIP, and showed that the sequence generated by the proposed rule converges strongly to a point in $\mathrm{VI}(C,A)$. Recently, Reich et al. [27] put forward 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 that the sequences generated by two algorithms converge weakly and strongly to a point in $\mathrm{VI}(C,A)$, respectively.

On the other hand, let $\varnothing \ne C\subset E$ where C is convex and closed in a uniformly smooth and p-uniformly convex Banach space E for $p,q>1$ satisfying $\frac{1}{p}+\frac{1}{q}=1$. Let $J_E^p$ be the duality mapping of E, and let $E^{*}$ be the dual of E with the duality $J_{E^{*}}^q$. Suppose that the norm and the duality pairing between E and $E^{*}$ are denoted by $\parallel ·\parallel$ and $\langle ·,·\rangle$, 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 ${\Pi }_C$ be Bregman’s projection w.r.t. $f_p$ from E onto C, and presume that $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ s.t. ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, ${lim}_{n\to \infty }{\alpha }_n=0$ and $0<{lim\; inf}_{n\to \infty }{\beta }_n\le {lim\; sup}_{n\to \infty }{\beta }_n<1$. Assume that $A:E\to E^{*}$ is uniformly continuous and pseudomonotone operator and S is Bregman relatively nonexpansive self-mapping on C. Very recently, inspired by the research works in [27], Eskandani et al. [31] proposed the hybrid projection approach with linesearch process for approximating a point in $\mathrm{VI}(C,A)\cap \mathrm{Fix}\left(S\right)$.

Algorithm 1.3 (see [31]). Initialization: Given $l\in (0,1),\nu >0,\lambda \in (0,\frac{1}{\nu })$ and choose $u,t_1\in C$ randomly.

Iterations: Compute $t_{n+1}(n\ge 1)$ below:

Step 1. Calculate $y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{t}_n-\lambda A{t}_n)\right)$ and $r_{\lambda }\left(t_n\right):=t_n-y_n$. If $r_{\lambda }\left(t_n\right)=0$ and $S{t}_n=t_n$, then stop; $t_n\in \Omega =\mathrm{VI}(C,A)\cap \mathrm{Fix}\left(S\right)$. If this case does not occur, then,

Step 2. Calculate $s_n=t_n-{ϵ}_n{r}_{\lambda }\left(t_n\right)$, with both ${ϵ}_n:=l^{k_n}$ and $k_n$ being the smallest $k\ge 0$ s.t. $\langle A{t}_n-A(t_n-l^k{r}_{\lambda }\left(t_n\right)),r_{\lambda }\left(t_n\right)\rangle \le \frac{\nu }{2}{D}_{f_p}(t_n,y_n)$.

Step 3. Calculate $u_n=J_{E^{*}}^q({\beta }_n{J}_E^p{t}_n+(1-{\beta }_n)J_E^p\left(S{\Pi }_{C_n}{t}_n\right))$ and $t_{n+1}={\Pi }_C\left(J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{u}_n)\right)$, with $C_n:=\{y\in C:0\ge h_n\left(y\right)\}$ and $h_n\left(y\right)=\langle A{s}_n,y-t_n\rangle +\frac{{ϵ}_n}{2\lambda }{D}_{f_p}(t_n,y_n)$.

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

With the help of suitable conditions, it was proven in [31] that $\left\{t_n\right\}$ converges strongly to ${\Pi }_{\Omega }u$.

This article designs two parallel subgradient-like extragradient algorithms with inertial effect for resolving a pair of variational inequality and fixed point problems (VIFPPs) in uniformly smooth and p-uniformly convex Banach space E. Here two variational inequality problems (VIPs) involve two uniformly continuous pseudomonotone operators and two fixed point problems implicate two uniformly continuous Bregman relatively asymptotically nonexpansive mappings. Moreover, each algorithm consists of two parts which are of symmetric structure mutually. With the help of appropriate registrations, it is proven that the sequences generated by the suggested algorithms converge weakly and strongly to a solution of this pair of VIFPPs, respectively. Lastly, an illustrative instance is furnished to check the implementability and applicability of the proposed approaches.

The structure of the article is described as follows: Section 2 releases certain terminologies and preliminary results for later applications. Section 3 is focused on discussing the convergence of the suggested algorithms. In Section 4, the major outcomes are utilized to deal with the CFPP and VIPs in an illustrative instance. Our results improve and develop the revelent ones obtained previously in [15, 27, 31].

2. Preliminaries

Let ($E,\parallel ·\parallel$) be a real Banach space, whose dual is denoted by $E^{*}$. We use the $y_n\to y$ and $y_n\rightharpoonup y$ to represent the strong and weak convergence of $\left\{y_n\right\}$ to $y\in E$, respectively. Moreover, the set of weak cluster points of $\left\{y_n\right\}$ is denoted by ${\omega }_w\left(y_n\right)$, i.e., ${\omega }_w\left(y_n\right)=\{y^{†}\in E:y_{n_k}\rightharpoonup y^{†}\mathrm{for}\mathrm{some}\left\{y_{n_k}\right\}\subset \left\{y_n\right\}\}$. Let $U=\{y\in E:\parallel y\parallel =1\}$ 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 $y,x\in U$ with $y\ne x$, one has $\parallel y+x\parallel /2<1$. E is referred to as being uniformly convex if $\forall \varsigma \in (0,2]$, $\exists \delta >0$ s.t. $\forall y,x\in U$ with $\parallel y-x\parallel \ge \varsigma$, one has $\parallel y+x\parallel /2\le 1-\delta$. 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 $\delta \left(\varsigma \right)=inf\{1-\parallel y+x\parallel /2:y,x\in U\mathrm{with}\parallel y-x\parallel \ge \varsigma \}$. It is also known that E is uniformly convex if and only if $\delta \left(\varsigma \right)>0\forall \varsigma \in (0,2]$. Moreover, E is referred to as being p-uniformly convex if $\exists c>0$ s.t. $\delta \left(\varsigma \right)\ge c{\varsigma }^p$ for $0\le \varsigma \le 2$.

The nonnegative function ${\rho }_E(·)$ on $[0,\infty )$ is called the modulus of smoothness of E if ${\rho }_E\left(\varsigma \right):=sup\left\{\right(\parallel y+\varsigma x\parallel +\parallel y-\varsigma x\parallel )/2-1:y,x\in U\}\forall \varsigma \in [0,\infty )$. E is said to be uniformly smooth if ${lim}_{\varsigma \to 0}{\rho }_E\left(\varsigma \right)/\varsigma =0$, and q-uniformly smooth if $\exists C_q>0$ s.t. ${\rho }_E\left(\varsigma \right)\le C_q{\varsigma }^q\forall \varsigma >0$. Recall that E is of p-uniform convexity iff $E^{*}$ is of q-uniform smoothness; see e.g., [32] for more details. Putting $B(0,r)=\{y\in E:\parallel y\parallel \le r\}$ for each $r>0$, we say that $f:E\to \mathbf{R}$ is uniformly convex on bounded sets (see [31]) if ${\rho }_r\left(\varsigma \right)>0\forall r,\varsigma >0$, where ${\rho }_r\left(\varsigma \right):[0,\infty )\to [0,\infty ]$ is specified below

$\begin{array}{cc}{\rho }_r\left(\varsigma \right)\hfill & =inf\left\{\right[ϵf\left(y\right)+(1-ϵ)f\left(x\right)-f(ϵy+(1-ϵ\left)x\right)]/ϵ(1-ϵ):\hfill \\ & ϵ\in (0,1)\mathrm{and}y,x\in B(0,r)\mathrm{with}\parallel y-x\parallel =\varsigma \}\forall \varsigma \ge 0.\hfill \end{array}$

The ${\rho }_r$ is known as the gauge function of f with uniform convexity. It is clear that the gauge ${\rho }_r$ is nondecreasing.

Let $f:E\to \mathbf{R}$ be a convex function. If the limit ${lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f\left(y\right)}{\varsigma }$ exists for each $x\in E$, then f is referred to as being Gâteaux differentiable at y. In this case, the gradient $\nabla f\left(y\right)$ of f at y is of linearity, and is formulated as $\langle \nabla f\left(y\right),x\rangle :={lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f\left(y\right)}{\varsigma }\forall x\in E$. The f is referred to as being of Gâteaux differentiablility if it is of Gâteaux differentiablility at any $y\in E$. In case ${lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f\left(y\right)}{\varsigma }$ is achieved uniformly for any $x\in U$, we say that f is of Fréchet differentiablility at y. Besides, f is referred to as being of uniform Fréchet differentiablility on a subset $K\subset E$ if ${lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f\left(y\right)}{\varsigma }$ is achieved uniformly for $(y,x)\in K\times U$. When the norm of E is of Gâteaux differentiablility, E is said to be of smoothness.

Let $\frac{1}{p}+\frac{1}{q}=1$ for $p,q>1$. The $J_E^p:E\to E^{*}$ is specified below

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

It is known that E is of smoothness iff $J_E^p$ is of single value from E into $E^{*}$. Also, E is of reflexivity iff $J_E^p$ is of surjectivity, and E is strictly convex iff $J_E^p$ is of one-to-one property. So it follows that, when the smooth Banach space E is of both strict convexity and reflexivity, $J_E^p$ is the bijection and in this case, $J_{E^{*}}^q={\left(J_E^p\right)}^{-1}$. Also, recall that E is of uniform smoothness iff the function $f_p\left(y\right)={\parallel y\parallel }^p/p$ is of uniform Fréchet differentiablility on bounded sets iff $J_E^p$ is of uniform continuity on bounded sets. Moreover, E is of uniform convexity iff the function $f_p$ is of uniform convexity (see [32]).

Let the function $f:E\to \mathbf{R}$ be of both Gâteaux differentiablility and convexity. Bregman’s distance w.r.t. f is specified below

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

It is worth mentioning that the $D_f(·,·)$ is not a metric in the common sense of the terminology. Evidently, $D_f(t,t)=0$ but $D_f(t,s)=0$ can not lead to $t=s$. Generally, $D_f$ is not of symmetry and fulfill no triangle inequality. However, $D_f$ fulfills the three point equequality

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

See [20] for many details.

It is remarkable that the $J_E^p$ on the smooth E is Gâteaux’s derivative of $f_p$. Thus, Bregman’s distance w.r.t. $f_p$ is specified below

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

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

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

(2.1)

where $\tau >0$ is some fixed number (see [12]). Via (2.1) it can be easily seen that for each $\left\{y_n\right\}\subset E$ of boundedness, the relation is valid:

$$y_n\to y\iff D_{f_p}(y_n,y)\mathrm{converges}\mathrm{to}0\mathrm{as}n\to \infty .$$

Let $\varnothing \ne C\subset E$ with C being convex and closed in a strictly convex, smooth and reflexive Banach space E. Bregman’s projection is formulated as minimizers of Bregman’s distance. Bregman’s projection of $y\in E$ onto C w.r.t. $f_p$ indicates a unique point ${\Pi }_{C}y\in C$ s.t. $D_{f_p}({\Pi }_{C}y,y)={min}_{x\in C}{D}_{f_p}(x,y)$. In the case of Hilbert space, Bregman’s projection w.r.t. $f_2$ reduces to the metric projection. Using [30, Theorem 2.1] and [18, Corollary 4.4], in a uniformly convex Banach space, the characterization of Bregman’s projection is formulated by:

$$\langle J_E^p\left(y\right)-J_E^p\left({\Pi }_{C}y\right),x-{\Pi }_{C}y\rangle \le 0\forall x\in C.$$

(2.2)

Meantime, (2.2) is equivalent to the descent property

$$D_{f_p}(x,{\Pi }_{C}y)+D_{f_p}({\Pi }_{C}y,y)\le D_{f_p}(x,y)\forall x\in C.$$

(2.3)

When $p=2$, $J_E^p$ reduces to the normalized duality mapping and is written as J. The $\varphi :E^2\to \mathbf{R}$ is formulated below

$$\varphi (t,s)={\parallel t\parallel }^2-2\langle Js,t\rangle +{\parallel s\parallel }^2\forall t,s\in E,$$

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

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

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

(2.4)

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

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

(2.5)

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

$$D_{f_p}(z,J_{E^{*}}^q\left(\sum _{j=1}^n{\varsigma }_j{J}_E^p\left(y_j\right)\right)\le \sum _{j=1}^n{\varsigma }_j{D}_{f_p}(z,y_j)\forall z\in E,{\left\{y_j\right\}}_{j=1}^n\subset E,{\left\{{\varsigma }_j\right\}}_{j=1}^n\subset [0,1]\mathrm{with}\sum _{j=1}^n{\varsigma }_j=1.$$

(2.6)

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

Assume that S is a self-mapping on C. Let the $\mathrm{Fix}\left(S\right)$ indicate the set of fixed points of S, that is, $\mathrm{Fix}\left(S\right)=\{y\in C:y=Sy\}$. A point $y^{†}\in C$ is referred to as an asymptotic fixed point of S if $\exists \left\{y_n\right\}\subset C$ s.t. $y_n\rightharpoonup y^{†}$ and $(I-S)y_n\to 0$. Let the $\widehat{\mathrm{Fix}}\left(S\right)$ denote the asymptotic fixed point set of S. The terminology of asymptotic fixed points was invented in [11]. A self-mapping S on C is known as being the one of Bregman’s relatively asymptotical nonexpansivity w.r.t. $f_p$ if $\mathrm{Fix}\left(S\right)=\widehat{\mathrm{Fix}}\left(S\right)\ne \varnothing$, and $\exists \left\{{\theta }_n\right\}\subset [0,\infty )$ with both ${\theta }_n\to 0(n\to \infty )$ and

$$({\theta }_n+1)D_{f_p}(y,x)\ge D_{f_p}(y,S^{n}x)\forall y\in \mathrm{Fix}\left(S\right),x\in C,n\ge 1.$$

In particular, if ${\theta }_n=0\forall n\ge 1$, then S reduces to the one of Bregman’s relatively nonexpansivity w.r.t. $f_p$, that is, $\mathrm{Fix}\left(S\right)=\widehat{\mathrm{Fix}}\left(S\right)\ne \varnothing$ and $D_{f_p}(y,Sx)\le D_{f_p}(y,x)\forall y\in \mathrm{Fix}\left(S\right),x\in C$. In addition, a mapping $A:C\to E^{*}$ is known as being

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

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

(iii)ℓ-Lipschitz continuous or ℓ-Lipschitzian if $\exists \ell >0$ s.t. $\parallel At-Ay\parallel \le \ell \parallel t-y\parallel \forall t,y\in C$;

(iv) of weakly sequential continuity if $\forall \left\{t_n\right\}\subset C$,the relation holds: $t_n\rightharpoonup t\Rightarrow A{t}_n\rightharpoonup At$.

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

$$f\left(\sum _{k=1}^n{\alpha }_k{t}_k\right)\le \sum _{k=1}^n{\alpha }_{k}f\left(t_k\right)-{\alpha }_i{\alpha }_j{\rho }_r(\parallel t_i-t_j\parallel ),$$

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

Proof. It is easy to show the conclusion.

Lemma 2.3 ([28]). Let $E_i$ be a Banach space for $i=1,2$ and suppose that $A:E_1\to E_2$ is of uniform continuity on any bounded subset of $E_1$ and $D\subset E_1$ is of boundedness. Then $A\left(D\right)\subset E_2$ is of boundedness.

Lemma 2.4 ([10]). Assume $\varnothing \ne C\subset E$ with C being convex and closed, and let $A:C\to E^{*}$ be of both pseudomonotonicity and continuity. Given $y^{†}\in C$. Then $\langle A{y}^{†},y-y^{†}\rangle \ge 0\forall y\in C\iff \langle Ay,y-y^{†}\rangle \ge 0\forall y\in C$.

Lemma 2.5. Suppose that E is a smooth and p-uniformly convex Banach space for $p\ge 2$, where $J_E^p$ is of weakly sequential continuity. Assume $\left\{s_n\right\}\subset E$ and $\varnothing \ne \Omega \subset E$. If ${\omega }_w\left(s_n\right)\subset \Omega$, and $\left\{D_{f_p}(z,s_n)\right\}$ converges for each $z\in \Omega$. Then one has the weak convergence of $\left\{s_n\right\}$ to an element of $\Omega$.

Proof. First, one has $\tau \parallel z-s_n{\parallel }^p\le D_{f_p}(z,s_n)\forall z\in \Omega$ by (2.1). Thus we obtain that $\left\{s_n\right\}$ is of boundedness. Since E is reflexive, we get ${\omega }_w\left(s_n\right)\ne \varnothing$. Also, one claims that $\left\{s_n\right\}$ converges weakly to an element of $\Omega$. Indeed, let $\overline{s},\widehat{s}\in {\omega }_w\left(s_n\right)$ with $\overline{s}\ne \widehat{s}$. Then, $\exists \left\{s_{n_k}\right\}\subset \left\{s_n\right\}$ and $\exists \left\{s_{m_k}\right\}\subset \left\{s_n\right\}$ s.t. $s_{n_k}\rightharpoonup \overline{s}$ and $s_{m_k}\rightharpoonup \widehat{s}$. Since $J_E^p$ is weakly sequentially continuous, we obtain both $J_E^p\left(s_{n_k}\right)\rightharpoonup J_E^p\overline{s}$ and $J_E^p\left(s_{m_k}\right)\rightharpoonup J_E^p\widehat{s}$. Note that $D_{f_p}(\overline{s},\widehat{s})+D_{f_p}(\widehat{s},s_n)=D_{f_p}(\overline{s},s_n)-\langle J_E^p\widehat{s}-J_E^p{s}_n,\overline{s}-\widehat{s}\rangle$. So, utilizing the convergence of the sequences $\left\{D_{f_p}(\overline{s},s_n)\right\}$ and $\left\{D_{f_p}(\widehat{s},s_n)\right\}$, we conclude that

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

which hence yields $\langle J_E^p\overline{s}-J_E^p\widehat{s},\overline{s}-\widehat{s}\rangle =0$. From (2.1) we get $0<\tau \parallel \overline{s}-\widehat{s}{\parallel }^p\le D_{f_p}(\overline{s},\widehat{s})\le \langle J_E^p\overline{s}-J_E^p\widehat{s},\overline{s}-\widehat{s}\rangle =0$. This arrives at a contradiction. Thereupon, this means that $\left\{s_n\right\}$ converges weakly to an element of $\Omega$.

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 ${\mathbf{R}}^m$.

Lemma 2.6. Assume $\varnothing \ne C\subset E$ with C being convex and closed. Suppose that $K:=\{y\in C:h(y)\le 0\}$ where $h:E\to \mathbf{R}$ is defined 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 ([17]). Let $\left\{{\Gamma }_n\right\}$ be a sequence of real numbers that does not decrease at infinity in the sense that, $\exists \left\{{\Gamma }_{n_k}\right\}\subset \left\{{\Gamma }_n\right\}$ s.t. ${\Gamma }_{n_k}<{\Gamma }_{n_k+1}$ for all k. Assume that ${\left\{\phi \left(n\right)\right\}}_{n\ge n_0}$ is defined as $\phi \left(n\right)=max\{k\le n:{\Gamma }_k<{\Gamma }_{k+1}\}$, with integer $n_0\ge 1$ satisfying $\{k\le n_0:{\Gamma }_k<{\Gamma }_{k+1}\}\ne \varnothing$. Then the following hold:

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

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

Lemma 2.8 ([7]). Let $\left\{{\sigma }_n\right\}$ be a sequence in $[0,\infty )$ satisfying ${\sigma }_{n+1}\le (1-{\mu }_n){\sigma }_n+{\mu }_n{c}_n\forall n\ge 1$, with $\left\{{\mu }_n\right\}$ and $\left\{c_n\right\}$ being real sequences satisfying the conditions: (i) $\left\{{\mu }_n\right\}\subset [0,1]$ and ${\sum }_{n=1}^{\infty }{\mu }_n=\infty$; and (ii) ${lim\; sup}_{n\to \infty }{c}_n\le 0$ or ${\sum }_{n=1}^{\infty }\left|{\mu }_n{c}_n\right|<\infty$. Then ${\sigma }_n\to 0$ as $n\to \infty$.

Lemma 2.9 ([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 $\varnothing \ne C\subset E$ with C being convex and closed in uniformly smooth and p-uniformly convex Banach space E for $p\ge 2$. We are now in a position to present and analyze our iterative algorithms for approximating a common solution of a pair of VIFPPs, where each algorithm consists of two parts which are of symmetric structure mutually. Assume always that the following conditions hold:

(C1) $S_1,S_2:C\to C$ are the mappings of both uniform continuity and Bregman’s relatively asymptotical nonexpansivity with sequences ${\left\{{\theta }_n\right\}}_{n=1}^{\infty }$ and ${\left\{{\overline{\theta }}_n\right\}}_{n=1}^{\infty }$, respectively.

(C2) For $i=1,2$, $A_i:E\to E^{*}$ is of both uniform continuity and pseudomonotonicity on C, s.t. $\parallel A_i{y}^{†}\parallel \le {lim\; inf}_{n\to \infty }\parallel A_i{y}_n\parallel \forall \left\{y_n\right\}\subset C$ with $y_n\rightharpoonup y^{†}$.

(C3) $\Omega ={\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\cap \mathrm{Fix}\left(S_i\right)\ne \varnothing$.

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 $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ and $\left\{{\ell }_n\right\}\subset (0,\infty )$ s.t. $0<{lim\; inf}_{n\to \infty }{\alpha }_n(1-{\alpha }_n)$, $0<{lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)$ and ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$. Moreover, assume ${\sum }_{n=1}^{\infty }{\theta }_n<\infty$, and 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

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

Iterations: Compute $x_{n+1}$ below:

Step 1. Put $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$ and calculate $s_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, $y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{s}_n-{\lambda }_1{A}_1{s}_n)\right)$, $e_{{\lambda }_1}\left(s_n\right):=s_n-y_n$ and $t_n=s_n-{\tau }_n{e}_{{\lambda }_1}\left(s_n\right)$, with ${\tau }_n:=l_1^{k_n}$ and $k_n$ being the smallest $k\ge 0$ s.t.

$$\frac{{\mu }_1}{2}{D}_{f_p}(s_n,y_n)\ge \langle A_1{s}_n-A_1(s_n-l_1^k{e}_{{\lambda }_1}\left(s_n\right)),s_n-y_n\rangle .$$

(3.1)

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

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

(3.2)

Step 3. Calculate ${\overline{y}}_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n)\right)$, $e_{{\lambda }_2}\left(w_n\right):=w_n-{\overline{y}}_n$ and ${\overline{t}}_n=w_n-{\overline{\tau }}_n{e}_{{\lambda }_2}\left(w_n\right)$, with ${\overline{\tau }}_n:=l_2^{j_n}$ and $j_n$ being the smallest $j\ge 0$ s.t.

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

(3.3)

Step 4. Calculate $v_n=J_{E^{*}}^q({\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p\left(S_2^n{w}_n\right))$ and $x_{n+1}={\Pi }_{{\overline{C}}_n\cap Q_n}\left(w_n\right)$, with $Q_n:=\{y\in C:(1+{\overline{\theta }}_n)D_{f_p}(y,w_n)\ge D_{f_p}(y,v_n)\}$, ${\overline{C}}_n:=\{y\in C:{\overline{h}}_n\left(y\right)\le 0\}$ and

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

(3.4)

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 following hold: $\frac{1}{{\lambda }_1}{D}_{f_p}(s_n,y_n)\le \langle A_1{s}_n,e_{{\lambda }_1}\left(s_n\right)\rangle$ and $\frac{1}{{\lambda }_2}{D}_{f_p}(w_n,{\overline{y}}_n)\le \langle A_2{w}_n,e_{{\lambda }_2}\left(w_n\right)\rangle$.

Proof. Note that the former inequality is analogous to the latter. So it suffices to show that the latter holds. Indeed, using the definition of ${\overline{y}}_n$ and properties of ${\Pi }_C$, one has

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

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

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

which completes the proof.

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

Proof. Observe that the rule (3.1) is analogous to the one (3.3). So it suffices to show that the latter is valid. Using the uniform continuity of $A_2$ on C, from $l_2\in (0,1)$ one gets ${lim}_{j\to \infty }\langle A_2{w}_n-A_2(w_n-l_2^j{e}_{{\lambda }_2}\left(w_n\right)),e_{{\lambda }_2}\left(w_n\right)\rangle =0$. In the case of $e_{{\lambda }_2}\left(w_n\right)=0$, it is explicit that $j_n=0$. In the case of $e_{{\lambda }_2}\left(w_n\right)\ne 0$, we obtain that $\exists j_n\ge 0$ s.t. (3.3) holds.

It is not hard to check that $Q_n$ and ${\overline{C}}_n$ are convex and closed for all n. Let us show that $Q_n\cap {\overline{C}}_n\supset \Omega$. Choose a $z\in \Omega$ arbitrarily. Since $S_2$ is Bregman’s relatively asymptotically nonexpansive mapping, by Lemma 2.2 one gets

$$\begin{array}{cc}& D_{f_p}(z,v_n)\le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)D_{f_p}(z,S_2^n{w}_n)-{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_2^n{w}_n\parallel \hfill \\ & \le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)(1+{\overline{\theta }}_n)D_{f_p}(z,w_n)-{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_2^n{w}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(z,w_n)-{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_2^n{w}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(z,w_n),\hfill \end{array}$$

which hence leads to $z\in Q_n$. Meanwhile, from Lemma 2.4, we get $\langle A_2{\overline{t}}_n,{\overline{t}}_n-z\rangle \ge 0$. Thus,

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

(3.5)

So it follows from (3.3) that

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

By Lemma 3.1 we have

$$(\frac{1}{{\lambda }_2}-\frac{{\mu }_2}{2})D_{f_p}(w_n,{\overline{y}}_n)\le \langle A_2{w}_n,e_{{\lambda }_2}\left(w_n\right)\rangle -\frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\overline{y}}_n)\le \langle A_2{\overline{t}}_n,e_{{\lambda }_2}\left(w_n\right)\rangle ,$$

which together with (3.5), attains

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

Therefore, $\Omega \subset {\overline{C}}_n\cap Q_n$. As a result, the sequence $\left\{x_n\right\}$ is well defined.

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

Proof. Note that the former inclusion is analogous to the latter. So it suffices to show that the latter is valid. Indeed, taking a $z\in {\omega }_w\left(w_n\right)$ arbitrarily, we know that $\exists \left\{w_{n_k}\right\}\subset \left\{w_n\right\}$, s.t. $w_{n_k}-{\overline{y}}_{n_k}\to 0$ and $w_{n_k}\rightharpoonup z$. So, we have ${\overline{y}}_{n_k}\rightharpoonup z$. Noticing the convexity and closedness of C, according to ${\overline{y}}_{n_k}\rightharpoonup z$ and $\left\{{\overline{y}}_n\right\}\subset C$, one gets $z\in C$. In what follows, ones consider two aspects. If $A_{2}z=0$, then $z\in \mathrm{VI}(C,A_2)$ (due to $\langle A_{2}z,x-z\rangle \ge 0$ for all $x\in C$). If $A_{2}z\ne 0$, by the condition on $A_2$, one gets $0<\parallel A_{2}z\parallel \le {lim\; inf}_{k\to \infty }\parallel A_2{w}_{n_k}\parallel$. So, we might assume that $\parallel A_2{w}_{n_k}\parallel \ne 0\forall k\ge 1$. From (2.2), we get

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

and hence

$$\frac{1}{{\lambda }_2}\langle J_E^p{w}_{n_k}-J_E^p{\overline{y}}_{n_k},y-{\overline{y}}_{n_k}\rangle +\langle A_2{w}_{n_k},{\overline{y}}_{n_k}-w_{n_k}\rangle \le \langle A_2{w}_{n_k},y-w_{n_k}\rangle \forall y\in C.$$

(3.6)

Since $A_2$ is uniformly continuous, using Lemma 2.3 we deduce that $\left\{A_2{w}_{n_k}\right\}$ is of boundedness. Observe that $\left\{{\overline{y}}_{n_k}\right\}$ is also of boundedness. So, using the uniform continuity of $J_E^p$ on any bounded subset of E, from (3.6) we have

$$\underset{k\to \infty }{lim\; inf}\langle A_2{w}_{n_k},y-w_{n_k}\rangle \ge 0\forall y\in C.$$

(3.7)

To prove that z lies in $\mathrm{VI}(C,A_2)$, one picks $\left\{{\kappa }_k\right\}$ in $(0,1)$ s.t. ${\kappa }_k\downarrow 0$. For any k, we choose the smallest $m_k>0$ s.t. for all $j\ge m_k$,

$$\langle A_2{w}_{n_j},y-w_{n_j}\rangle +{\kappa }_k\ge 0.$$

(3.8)

Because $\left\{{\kappa }_k\right\}$ is decreasing, we get the increasing property of $\left\{m_k\right\}$. For the sake of simplicity, $\left\{A_2{w}_{n_{m_k}}\right\}$ is still written as $\left\{A_2{w}_{m_k}\right\}$. It is known that $A_2{w}_{m_k}\ne 0$ for all k (due to $\left\{A_2{w}_{m_k}\right\}\subset \left\{A_2{w}_{n_k}\right\}$). Then, putting ${\overline{g}}_{m_k}=\frac{A_2{w}_{m_k}}{\parallel A_2{w}_{m_k}{\parallel }^{\frac{q}{q-1}}}$, one gets $\langle A_2{w}_{m_k},J_{E^{*}}^q{\overline{g}}_{m_k}\rangle =1\forall k\ge 1$. Indeed, it is evident that $\langle A_2{w}_{m_k},J_{E^{*}}^q{\overline{g}}_{m_k}\rangle =\langle A_2{w}_{m_k},{\left(\frac{1}{\parallel A_2{w}_{m_k}{\parallel }^{\frac{q}{q-1}}}\right)}^{q-1}{J}_{E^{*}}^q{A}_2{w}_{m_k}\rangle ={\left(\frac{1}{\parallel A_2{w}_{m_k}{\parallel }^{\frac{q}{q-1}}}\right)}^{q-1}{\parallel A_2{w}_{m_k}\parallel }^q=1\forall k\ge 1$. So, by (3.8) one has $\langle A_2{w}_{m_k},y+{\kappa }_k{J}_{E^{*}}^q{\overline{g}}_{m_k}-w_{m_k}\rangle \ge 0\forall k\ge 1$. Again from the pseudomonotonicity of $A_2$ one has

$$\langle A_2(y+{\kappa }_k{J}_{E^{*}}^q{\overline{g}}_{m_k}),y+{\kappa }_k{J}_{E^{*}}^q{\overline{g}}_{m_k}-w_{m_k}\rangle \ge 0\forall y\in C.$$

(3.9)

Let us show that ${lim}_{k\to \infty }{\kappa }_k{J}_{E^{*}}^q{\overline{g}}_{m_k}=0$. Indeed, noticing ${\kappa }_k\downarrow 0$ and $\left\{w_{m_k}\right\}\subset \left\{w_{n_k}\right\}$, we obtain that

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

Hence one gets ${\kappa }_k{J}_{E^{*}}^q{\overline{g}}_{m_k}\to 0$ as $k\to \infty$. Thus, taking the limit as $k\to \infty$ in (3.9), from (C3) one has $\langle A_{2}y,y-z\rangle \ge 0$ for all $y\in C$. In terms of Lemma 2.4 we conclude that z lies in $\mathrm{VI}(C,A_2)$.

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

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

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

Proof. Note that the claim (i) is analogous to the one (ii). So it suffices to show that the second is valid. To verify the second, we discuss two cases. In case ${lim\; inf}_{n\to \infty }{\overline{\tau }}_n>0$, one may presume that $\exists \overline{\tau }>0$ satisfying ${\overline{\tau }}_n\ge \overline{\tau }>0$ for all n, which immediately leads to

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

(3.10)

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

In case ${lim\; inf}_{n\to \infty }{\overline{\tau }}_n=0$, we assume that ${lim\; sup}_{n\to \infty }{D}_{f_p}(w_n,{\overline{y}}_n)=\widehat{a}>0$. This ensures that $\exists \left\{m_j\right\}\subset \left\{n\right\}$ satisfying

$$\underset{j\to \infty }{lim}{\overline{\tau }}_{m_j}=0\mathrm{and}\underset{j\to \infty }{lim}{D}_{f_p}(w_{m_j},{\overline{y}}_{m_j})=\widehat{a}>0.$$

We define $\widehat{t_{m_j}}=\frac{1}{l_2}{\overline{\tau }}_{m_j}{\overline{y}}_{m_j}+(1-\frac{1}{l_2}{\overline{\tau }}_{m_j})w_{m_j}\forall j\ge 1$. Noticing ${lim}_{j\to \infty }{\overline{\tau }}_{m_j}{D}_{f_p}(w_{m_j},{\overline{y}}_{m_j})=0$, From (2.1) we get ${lim}_{j\to \infty }{\overline{\tau }}_{m_j}{\parallel w_{m_j}-{\overline{y}}_{m_j}\parallel }^p=0$ and hence

$$\underset{j\to \infty }{lim}\parallel \widehat{t_{m_j}}-w_{m_j}{\parallel }^p=\underset{j\to \infty }{lim}\frac{{\overline{\tau }}_{m_j}^{p-1}}{l_2^p}·{\overline{\tau }}_{m_j}{\parallel w_{m_j}-{\overline{y}}_{m_j}\parallel }^p=0.$$

(3.11)

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

$$\underset{j\to \infty }{lim}\parallel A_2{w}_{m_j}-A_2\widehat{t_{m_j}}\parallel =0.$$

(3.12)

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

$$\langle A_2{w}_{m_j}-A_2\widehat{t_{m_j}},w_{m_j}-{\overline{y}}_{m_j}\rangle >\frac{{\mu }_2}{2}{D}_{f_p}(w_{m_j},{\overline{y}}_{m_j}).$$

(3.13)

Now, taking the limit as $j\to \infty$, from (3.12) we have ${lim}_{j\to \infty }{D}_{f_p}(w_{m_j},{\overline{y}}_{m_j})=0$. This, however, yields a contradiction. As a result, $D_{f_p}(w_n,{\overline{y}}_n)\to 0$ as $n\to \infty$.

In what follows, we show the first main result.

Theorem 3.1. Suppose that E is uniformly smooth and p-uniformly convex, where $J_E^p$ is of weakly sequential continuity. If under Algorithm 3.1, $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{w}_n-S_2^n{w}_n\to 0$, then $x_n\rightharpoonup z\in \Omega \iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Proof. Note that that the necessity is valid. So we need to only show the statement of sufficiency. Presume ${sup}_{n\ge 0}\parallel x_n\parallel <\infty$. Choose a $z\in \Omega$ arbitrarily. Clearly, $x_{n-1}\ne x_n\iff J_E^p{S}_1^n{x}_n\ne J_E^p(S_1^n{x}_n-x_{n-1}+x_n)$. Using the definition of ${ϵ}_n$, we get ${ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \le {\ell }_n\forall n\ge 1$. From (2.1), (2.6) and the three point identity of $D_{f_p}$ we get

$\begin{array}{cc}& D_{f_p}(z,g_n)\le (1-{ϵ}_n)D_{f_p}(z,S_1^n{x}_n)+{ϵ}_n{D}_{f_p}(z,S_1^n{x}_n+x_n-x_{n-1})\hfill \\ & =D_{f_p}(z,S_1^n{x}_n)+{ϵ}_n[D_{f_p}(z,S_1^n{x}_n+x_n-x_{n-1})-D_{f_p}(z,S_1^n{x}_n)]\hfill \\ & =D_{f_p}(z,S_1^n{x}_n)+{ϵ}_n[D_{f_p}(S_1^n{x}_n,S_1^n{x}_n+x_n-x_{n-1})\hfill \\ & +\langle J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1}),z-S_1^n{x}_n\rangle ]\hfill \\ & \le (1+{\theta }_n)D_{f_p}(z,x_n)+{ϵ}_n\langle J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1}),z+x_{n-1}-x_n-S_1^n{x}_n\rangle \hfill \\ & \le (1+{\theta }_n)D_{f_p}(z,x_n)+{ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \parallel z+x_{n-1}-x_n-S_1^n{x}_n\parallel \hfill \\ & \le (1+{\theta }_n)D_{f_p}(z,x_n)+{\ell }_{n}M,\hfill \end{array}$

where ${sup}_{n\ge 1}\parallel z+x_{n-1}-x_n-S_1^n{x}_n\parallel \le M$ for some $M>0$. By Lemma 2.2 we get

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

Noticing $w_n={\Pi }_{C_n}{s}_n$, by (2.1) and (2.3) we get

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

Because $x_{n+1}={\Pi }_{{\overline{C}}_n\cap Q_n}{w}_n$, by (2.1) and (2.3) we have

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

Combining these inequalities and (3.13), leads to

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

(3.14)

which hence leads to

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

Since ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$ and ${\sum }_{n=1}^{\infty }{\theta }_n<\infty$, 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 $\left\{g_n\right\},\left\{s_n\right\},\left\{v_n\right\},\left\{w_n\right\},\left\{y_n\right\},\left\{{\overline{y}}_n\right\}$, $\left\{t_n\right\},\left\{{\overline{t}}_n\right\},\left\{S_1^n{x}_n\right\}$ and $\left\{S_2^n{w}_n\right\}$ are also bounded. From (3.14) we obtain

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

which immediately yields

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

Since ${lim}_{n\to \infty }{\ell }_n=0$, ${lim}_{n\to \infty }{\theta }_n=0$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exists, it follows that ${lim}_{n\to \infty }{D}_{f_p}(w_n,s_n)=0$, ${lim}_{n\to \infty }{D}_{f_p}(x_{n+1},w_n)=0$, 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 $s_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, it is readily known that ${lim}_{n\to \infty }\parallel J_E^p{s}_n-J_E^p{x}_n\parallel =0$. Noticing $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$, we obtain from ${lim}_{n\to \infty }{\ell }_n=0$ and the definition of ${ϵ}_n$ that

$$\parallel J_E^p{g}_n-J_E^p{S}_1^n{x}_n\parallel ={ϵ}_n\parallel J_E^p(S_1^n{x}_n+x_n-x_{n-1})-J_E^p{S}_1^n{x}_n\parallel \le {\ell }_n\to 0(n\to \infty ).$$

Hence, using (2.1) and uniform continuity of $J_{E^{*}}^q$ on bounded subsets of $E^{*}$, we conclude that ${lim}_{n\to \infty }\parallel g_n-S_1^n{x}_n\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_n-s_n\parallel =\underset{n\to \infty }{lim}\parallel x_{n+1}-w_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-S_1^n{x}_n\parallel =\underset{n\to \infty }{lim}\parallel s_n-x_n\parallel =0.$$

(3.15)

Since $\left\{x_n\right\}$ is of boundedness and E is of reflexivity, we obtain that ${\omega }_w\left(x_n\right)$ is nonempty. Next, let us show that $\Omega \supset {\omega }_w\left(x_n\right)$. Choose a z in ${\omega }_w\left(x_n\right)$ arbitrarily. It is known that $\exists \left\{x_{n_k}\right\}\subset \left\{x_n\right\}$ satisfying $x_{n_k}\rightharpoonup z$. By (3.15) one gets $w_{n_k}\rightharpoonup z$. Since $\left\{A_1{t}_n\right\}$ is of boundedness, one knows that $\exists L_1>0$ satisfying $\parallel A_1{t}_n\parallel \le L_1$. So it follows that for all $u,v\in C_n$,

$$|h_n\left(u\right)-h_n\left(v\right)|=|\langle A_1{t}_n,u-v\rangle |\le \parallel A_1{t}_n\parallel \parallel u-v\parallel \le L_1\parallel u-v\parallel ,$$

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

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

(3.16)

Since $x_{n+1}$ lies in $Q_n$, by (3.14) one has

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

Since ${lim}_{n\to \infty }{\theta }_n=0$, ${lim}_{n\to \infty }{\overline{\theta }}_n=0$, ${lim}_{n\to \infty }{\ell }_n=0$ and ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exists, we have $D_{f_p}(x_{n+1},v_n)\to 0$ and thus $x_{n+1}-v_n\to 0$. By (3.15) we get

$$\underset{n\to \infty }{lim}\parallel w_n-v_n\parallel =0.$$

(3.17)

Furthermore, by Lemma 2.2, we have

$$\begin{array}{cc}& D_{f_p}(z,v_n)=V_{f_p}(z,{\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p{S}_2^n{w}_n)\hfill \\ & \le \frac{1}{p}{\parallel z\parallel }^p-{\alpha }_n\langle J_E^p{w}_n,z\rangle -(1-{\alpha }_n)\langle J_E^p{S}_2^n{w}_n,z\rangle +\frac{{\alpha }_n}{q}{\parallel J_E^p{w}_n\parallel }^q\hfill \\ & +\frac{(1-{\alpha }_n)}{q}\parallel J_E^p{S}_2^n{w}_n{\parallel }^q-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel \hfill \\ & =\frac{1}{p}{\parallel z\parallel }^p-{\alpha }_n\langle J_E^p{w}_n,z\rangle -(1-{\alpha }_n)\langle J_E^p{S}_2^n{w}_n,z\rangle +\frac{{\alpha }_n}{q}{\parallel w_n\parallel }^p\hfill \\ & +\frac{(1-{\alpha }_n)}{q}\parallel S_2^n{w}_n{\parallel }^p-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel \hfill \\ & ={\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)D_{f_p}(z,S_2^n{w}_n)-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(z,w_n)-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel .\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}& {\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel \le (1+{\overline{\theta }}_n)D_{f_p}(z,w_n)-D_{f_p}(z,v_n)\hfill \\ & \le D_{f_p}(z,w_n)-D_{f_p}(z,v_n)+D_{f_p}(w_n,v_n)+{\overline{\theta }}_n{D}_{f_p}(z,w_n)\hfill \\ & =\langle J_E^p{v}_n-J_E^p{w}_n,z-w_n\rangle +{\overline{\theta }}_n{D}_{f_p}(z,w_n).\hfill \end{array}$$

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 ${lim\; inf}_{n\to \infty }{\alpha }_n(1-{\alpha }_n)>0$, we get ${lim}_{n\to \infty }{\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel =0$ and hence ${lim}_{n\to \infty }\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel =0$. Since $J_{E^{*}}^q$ is uniformly continuous on any bounded subset of $E^{*}$, we deduce that

$$\underset{n\to \infty }{lim}\parallel w_n-S_2^n{w}_n\parallel =0.$$

(3.18)

Now let us show $z\in {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. Since $\left\{A_2{\overline{t}}_n\right\}$ is of boundedness, it follows that $\exists L_2>0$ satisfying $\parallel A_2{\overline{t}}_n\parallel \le L_2$. Thus we obtain that for all $u,v\in {\overline{C}}_n$,

$$|{\overline{h}}_n\left(u\right)-{\overline{h}}_n\left(v\right)|=|\langle A_2{\overline{t}}_n,u-v\rangle |\le \parallel A_2{\overline{t}}_n\parallel \parallel u-v\parallel \le L_2\parallel u-v\parallel ,$$

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

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

(3.19)

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

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

(3.20)

Thus,

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

According to Lemma 3.4, we have

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

In addition, from (3.15) and $x_{n_k}\rightharpoonup z$ we infer that $s_{n_k}\rightharpoonup z$ and $w_{n_k}\rightharpoonup z$. By Lemma 3.3 we obtain that $z\in {\omega }_w\left(s_n\right)\subset \mathrm{VI}(C,A_1)$ 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 $z\in {\bigcap }_{i=1}^2\mathrm{Fix}\left(S_i\right)$. Indeed, by (3.15) we immediately get

$$\parallel x_{n+1}-x_n\parallel \le \parallel x_{n+1}-w_n\parallel +\parallel w_n-s_n\parallel +\parallel s_n-x_n\parallel \to 0(n\to \infty ).$$

(3.21)

We first claim that ${lim}_{n\to \infty }\parallel x_n-S_1{x}_n\parallel =0$ and ${lim}_{n\to \infty }\parallel w_n-S_2{w}_n\parallel =0$. Actually, using (3.15), (3.18) and uniform continuity of $S_i$ on C for $i=1,2$, we obtain that $S_1{x}_n-S_1^{n+1}{x}_n\to 0$ and $S_2{w}_n-S_2^{n+1}{w}_n\to 0$. Thus, from $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{w}_n-S_2^n{w}_n\to 0$ (due to the assumptions) we deduce that

$$\parallel x_n-S_1{x}_n\parallel \le \parallel x_n-S_1^n{x}_n\parallel +\parallel S_1^n{x}_n-S_1^{n+1}{x}_n\parallel +\parallel S_1^{n+1}{x}_n-S_1{x}_n\parallel \to 0(n\to \infty )$$

and

$$\parallel w_n-S_2{w}_n\parallel \le \parallel w_n-S_2^n{w}_n\parallel +\parallel S_2^n{w}_n-S_2^{n+1}{w}_n\parallel +\parallel S_2^{n+1}{w}_n-S_2{w}_n\parallel \to 0(n\to \infty ).$$

These together with $x_{n_k}\rightharpoonup z$ and $w_{n_k}\rightharpoonup z$, ensure that $z\in {\bigcap }_{i=1}^2\widehat{\mathrm{Fix}}\left(S_i\right)={\bigcap }_{i=1}^2\mathrm{Fix}\left(S_i\right)$. Therefore, $z\in \Omega$. This means that ${\omega }_w\left(x_n\right)\subset \Omega$. As a result, by Lemma 2.5 one gets the desired conclusion.

In what follows, we prove the second main outcome for finding a solution of a pair of VIFPPs for two operators of both uniform continuity and pseudomonotonicity, and two mappings of both uniform continuity and Bregman’s relatively asymptotical nonexpansivity in E.

Algorithm 3.2. Initialization: Given $x_0,x_1\in C$ arbitrarily and let $ϵ>0,{\mu }_{\iota }>0,l_{\iota }\in (0,1)$ and ${\lambda }_{\iota }\in (0,\frac{1}{{\mu }_{\iota }})$ for $\iota =1,2$. Choose $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\subset (0,1)$ and $\left\{{\ell }_n\right\}\subset (0,\infty )$ s.t. ${lim}_{n\to \infty }{\ell }_n=0,{\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, ${lim}_{n\to \infty }{\alpha }_n=0$, ${lim}_{n\to \infty }\frac{{\theta }_n+{\overline{\theta }}_n}{{\alpha }_n}=0$, $0<{lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)$ and $0<{lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n)$. 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

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

Iterations: Compute $x_{n+1}$ below:

Step 1. Put $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$, and calculate $s_n=J_{E^{*}}^q({\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)$, $y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{s}_n-{\lambda }_1{A}_1{s}_n)\right)$, $e_{{\lambda }_1}\left(s_n\right):=s_n-y_n$ and $t_n=s_n-{\tau }_n{e}_{{\lambda }_1}\left(s_n\right)$, where ${\tau }_n:=l_1^{k_n}$ and $k_n$ is the smallest $k\ge 0$ s.t.

$\frac{{\mu }_1}{2}{D}_{f_p}(s_n,y_n)\ge \langle A_1{s}_n-A_1(s_n-l_1^k{e}_{{\lambda }_1}\left(s_n\right)),s_n-y_n\rangle$.

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

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

Step 3. Calculate ${\overline{y}}_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n)\right)$, $e_{{\lambda }_2}\left(w_n\right):=w_n-{\overline{y}}_n$ and ${\overline{t}}_n=w_n-{\overline{\tau }}_n{e}_{{\lambda }_2}\left(w_n\right)$, where ${\overline{\tau }}_n:=l_2^{j_n}$ and $j_n$ is the smallest $j\ge 0$ s.t.

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

Step 4. Set $z_n={\Pi }_{{\overline{C}}_n}\left(w_n\right)$, and calculate $v_n=J_{E^{*}}^q({\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p\left(S_2^n{z}_n\right))$ and $x_{n+1}={\Pi }_C(J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$, where ${\overline{C}}_n:=\{y\in C:{\overline{h}}_n\left(y\right)\le 0\}$ and

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

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

Theorem 3.2. Suppose that the conditions (C1)-(C3) hold. If under Algorithm 3.2, $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{z}_n-S_2^n{z}_n\to 0$, then $x_n\to {\Pi }_{\Omega }u\iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Proof. It is explicit that the necessity of Theorem 3.2 holds. Hence we need to only prove the statement of sufficiency. Assume that ${sup}_{n\ge 0}\parallel x_n\parallel <\infty$. In what follows, we divide our proof into four claims.

Claim 1. One shows that

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

for some $M>0$. In fact, put $\widehat{u}={\Pi }_{\Omega }u$. Noticing $w_n={\Pi }_{C_n}{s}_n$ and $z_n={\Pi }_{{\overline{C}}_n}{w}_n$, we obtain from (2.1) and (2.3) that

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

and

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

By the similar reasonings to those in the proof of the above theorem, we obtain

$\begin{array}{cc}{D}_{f_p}(\widehat{u},g_n)\hfill & \le D_{f_p}(\widehat{u},S_1^n{x}_n)+{ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel \le (1+{\theta }_n)D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M,\hfill \end{array}$

where ${sup}_{n\ge 1}\parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel \le M$ 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 $\left\{{\beta }_n\right\}\subset (0,1)$ we obtain

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

which immediately arrives at the desired claim. In addition, it is easily known that $\left\{s_n\right\},\left\{v_n\right\}$, $\left\{w_n\right\},\left\{y_n\right\},\left\{{\overline{y}}_n\right\},\left\{z_n\right\},\left\{t_n\right\},\left\{{\overline{t}}_n\right\}$ and $\left\{S_2^n{z}_n\right\}$ are of boundedness.

Claim 2. One shows that

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

Indeed, set $b={sup}_{n\ge 1}\{\parallel x_n{\parallel }^{p-1},\parallel g_n{\parallel }^{p-1},\parallel z_n{\parallel }^{p-1},\parallel S_2^n{z}_n{\parallel }^{p-1}\}$. By Lemma 2.2 we get

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

and

$$\begin{array}{cc}& D_{f_p}(\widehat{u},v_n)=V_{f_p}(\widehat{u},{\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p{S}_2^n{z}_n)\hfill \\ & \le {\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)D_{f_p}(\widehat{u},S_2^n{z}_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ & \le {\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)(1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},z_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},z_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel .\hfill \end{array}$$

(3.23)

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

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le V_{f_p}(\widehat{u},{\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)\hfill \\ & \le V_{f_p}(\widehat{u},{\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n-{\alpha }_n(J_E^{p}u-J_E^p\widehat{u}))+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel +{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

(3.24)

Furthermore, from (3.23) one has

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},v_n)\hfill & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},z_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)[D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)]-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)[D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)].\hfill \end{array}$$

This together with (3.24), arrives at

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

which immediately yields

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

(3.25)

Claim 3. One shows that

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

Indeed, by the analogous reasonings to those of (3.20), one gets

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

(3.26)

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

(3.27)

Claim 4. One shows that ${lim}_{n\to \infty }\parallel x_n-\widehat{u}\parallel =0$. Indeed, since E is reflexive and $\left\{x_n\right\}$ is bounded, one has ${\omega }_w\left(x_n\right)\ne \varnothing$. Choose a z in ${\omega }_w\left(x_n\right)$ arbitrarily. It is known that $\exists \left\{x_{n_k}\right\}\subset \left\{x_n\right\}$ satisfying $x_{n_k}\rightharpoonup z$. One writes ${\Gamma }_n=D_{f_p}(\widehat{u},x_n)$ for all n. In what follows, let us prove $\left\{{\Gamma }_n\right\}\to 0(n\to \infty )$ in the two possible aspects below.

Aspect 1. Assume that $\exists n_0\ge 1$ s.t. ${\left\{{\Gamma }_n\right\}}_{n=n_0}^{\infty }$ is non-increasing. It is known that ${\Gamma }_n\to d<+\infty$ and hence ${\Gamma }_n-{\Gamma }_{n+1}\to 0$. From (3.25) and (3.22) we get

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

which hence yields

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

Since ${\ell }_n\to 0,{\overline{\theta }}_n\to 0,{\theta }_n\to 0,{\alpha }_n\to 0,{lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n)>0,{\Gamma }_n\to d$ and $\left\{{\zeta }_n\right\}$ is of boundedness, one obtains ${lim}_{n\to \infty }{D}_{f_p}(w_n,s_n)=0$, ${lim}_{n\to \infty }{D}_{f_p}(z_n,w_n)=0$, 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 ${lim}_{n\to \infty }\parallel J_E^p{s}_n-J_E^p{x}_n\parallel =0$. Noticing $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$, we infer from ${lim}_{n\to \infty }{\ell }_n=0$ and the definition of ${ϵ}_n$ that

$$\parallel J_E^p{g}_n-J_E^p{S}_1^n{x}_n\parallel ={ϵ}_n\parallel J_E^p(S_1^n{x}_n+x_n-x_{n-1})-J_E^p{S}_1^n{x}_n\parallel \le {\ell }_n\to 0(n\to \infty ).$$

Hence, using (2.1) and uniform continuity of $J_{E^{*}}^q$ on any bounded subset of $E^{*}$, we conclude that ${lim}_{n\to \infty }\parallel g_n-S_1^n{x}_n\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_n-s_n\parallel =\underset{n\to \infty }{lim}\parallel z_n-w_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-S_1^n{x}_n\parallel =\underset{n\to \infty }{lim}\parallel s_n-x_n\parallel =0.$$

(3.28)

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

$$\begin{array}{cc}& (1-{\alpha }_n){\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)-D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1+{\overline{\theta }}_n)(1+{\theta }_n)D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+(1+{\overline{\theta }}_n){\ell }_{n}M+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

By the similar reasonings, we deduce that ${lim}_{n\to \infty }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel =0$, which hence leads to ${lim}_{n\to \infty }\parallel J_E^p{v}_n-J_E^p{z}_n\parallel =0$ (due to $v_n=J_{E^{*}}^q({\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p{S}_2^n{z}_n)$). Using uniform continuity of $J_{E^{*}}^q$ on bounded subsets of $E^{*}$, we get

$$\underset{n\to \infty }{lim}\parallel z_n-S_2^n{z}_n\parallel =\underset{n\to \infty }{lim}\parallel v_n-z_n\parallel =0.$$

(3.29)

This together with (3.28) implies that

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

It is clear that

$$\underset{n\to \infty }{lim}\parallel z_n-x_n\parallel =0.$$

(3.30)

Let us show that $z\in {\bigcap }_{i=1}^2\mathrm{Fix}\left(S_i\right)$. 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

$$\underset{n\to \infty }{lim}\parallel {\zeta }_n-x_n\parallel =0.$$

(3.31)

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

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

which hence yields

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

So it follows that $D_{f_p}(x_{n+1},{\zeta }_n)\to 0$ and hence $\parallel x_{n+1}-{\zeta }_n\parallel \to 0$. Thus, from (3.31) we get

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

(3.32)

We now claim that ${lim}_{n\to \infty }\parallel x_n-S_1{x}_n\parallel =0$ and ${lim}_{n\to \infty }\parallel w_n-S_2{w}_n\parallel =0$. Indeed, using (3.28), (3.29) and uniform continuity of $S_i$ on C for $i=1,2$, we obtain that $S_1{x}_n-S_1^{n+1}{x}_n\to 0$ and $S_2{z}_n-S_2^{n+1}{z}_n\to 0$. Thus, from $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{z}_n-S_2^n{z}_n\to 0$ (due to the assumptions) we deduce that

$$\parallel x_n-S_1{x}_n\parallel \le \parallel x_n-S_1^n{x}_n\parallel +\parallel S_1^n{x}_n-S_1^{n+1}{x}_n\parallel +\parallel S_1^{n+1}{x}_n-S_1{x}_n\parallel \to 0(n\to \infty )$$

and

$$\parallel z_n-S_2{z}_n\parallel \le \parallel z_n-S_2^n{z}_n\parallel +\parallel S_2^n{z}_n-S_2^{n+1}{z}_n\parallel +\parallel S_2^{n+1}{z}_n-S_2{z}_n\parallel \to 0(n\to \infty ).$$

These together with $x_{n_k}\rightharpoonup z$ and $z_{n_k}\rightharpoonup z$ (due to (3.30)), ensure that $z\in {\bigcap }_{i=1}^2\widehat{\mathrm{Fix}}\left(S_i\right)={\bigcap }_{i=1}^2\mathrm{Fix}\left(S_i\right)$.

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

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

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

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

(3.33)

By Lemma 3.4, we obtain

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

(3.34)

Applying (3.34) and Lemma 3.3, one gets $z\in {\bigcap }_{j=1}^2\mathrm{VI}(C,A_j)$. Thus one has ${\omega }_w\left(x_n\right)\subset {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. Consequently, $\Omega \supset {\omega }_w\left(x_n\right)$. Finally, let us prove ${lim\; sup}_{n\to \infty }\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \le 0$. One can pick $\left\{x_{n_j}\right\}\subset \left\{x_n\right\}$ s.t.

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

Because E is reflexive and $\left\{x_n\right\}$ is bounded, we might assume $x_{n_j}\rightharpoonup \overline{z}$. Using (2.2) and $\overline{z}\in \Omega$ we infer that

$$\underset{n\to \infty }{lim\; sup}\langle J_E^{p}u-J_E^p\widehat{u},x_n-\widehat{u}\rangle =\underset{j\to \infty }{lim}\langle J_E^{p}u-J_E^p\widehat{u},x_{n_j}-\widehat{u}\rangle =\langle J_E^{p}u-J_E^p\widehat{u},\overline{z}-\widehat{u}\rangle \le 0,$$

(3.35)

which along with (3.31), arrives at

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

From (3.24) and (3.22), we get

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le (1-{\alpha }_n)(1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)(1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},s_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},s_n)+{\overline{\theta }}_n{D}_{f_p}(\widehat{u},s_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)[(1+{\theta }_n)D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel ]+{\overline{\theta }}_n{D}_{f_p}(\widehat{u},s_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel +{\theta }_n{D}_{f_p}(\widehat{u},x_n)+{\overline{\theta }}_n{D}_{f_p}(\widehat{u},s_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & =(1-{\alpha }_n)D_{f_p}(\widehat{u},x_n)+{\alpha }_n\{\frac{{ϵ}_n}{{\alpha }_n}\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel +\frac{{\theta }_n}{{\alpha }_n}{D}_{f_p}(\widehat{u},x_n)+\frac{{\overline{\theta }}_n}{{\alpha }_n}{D}_{f_p}(\widehat{u},s_n)+\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \}.\hfill \end{array}$$

(3.36)

Using uniform continuity of $J_E^p$ on any bounded subset of E, from (3.32) and the boundedness of $\left\{x_n\right\}$ we get

$$\underset{n\to \infty }{lim}\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel =0.$$

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

$$\begin{array}{cc}& {\displaystyle \underset{n\to \infty }{lim\; sup}}\{\frac{{ϵ}_n}{{\alpha }_n}\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel \hfill \\ & +\frac{{\theta }_n}{{\alpha }_n}{D}_{f_p}(\widehat{u},x_n)+\frac{{\overline{\theta }}_n}{{\alpha }_n}{D}_{f_p}(\widehat{u},s_n)+\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \}\le 0.\hfill \end{array}$$

Thanks to $\left\{{\alpha }_n\right\}\subset (0,1)$ with ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, utilizing Lemma 2.8 to (3.36) one gets $D_{f_p}(\widehat{u},x_n)\to 0$ and hence $\parallel x_n-\widehat{u}\parallel \to 0$.

Aspect 2. Assume that $\exists \left\{{\Gamma }_{n_k}\right\}\subset \left\{{\Gamma }_n\right\}$ satisfying ${\Gamma }_{n_k}<{\Gamma }_{n_k+1}$ for all k, with $\mathcal{N}$ being the natural-number set. Let $\phi :\mathcal{N}\to \mathcal{N}$ be formulated below

$$\phi \left(n\right):=max\{j\le n:{\Gamma }_j<{\Gamma }_{j+1}\}.$$

Using Lemma 2.7, one has

$$max\{{\Gamma }_{\phi \left(n\right)},{\Gamma }_n\}\le {\Gamma }_{\phi \left(n\right)+1}.$$

(3.37)

From (3.25) and (3.22) it follows that

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

Noticing $g_{\phi \left(n\right)}=J_{E^{*}}^q((1-{ϵ}_{\phi \left(n\right)})J_E^p{S}_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}+{ϵ}_{\phi \left(n\right)}{J}_E^p(S_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}+x_{\phi \left(n\right)}-x_{\phi \left(n\right)-1}))$ and $s_{\phi \left(n\right)}=J_{E^{*}}^q({\gamma }_{\phi \left(n\right)}{J}_E^p{x}_{\phi \left(n\right)}+(1-{\gamma }_{\phi \left(n\right)})J_E^p{g}_{\phi \left(n\right)}))$, we obtain that ${lim}_{n\to \infty }\parallel g_{\phi \left(n\right)}-S_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_{\phi \left(n\right)}-s_{\phi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel z_{\phi \left(n\right)}-w_{\phi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel x_{\phi \left(n\right)}-S_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel s_{\phi \left(n\right)}-x_{\phi \left(n\right)}\parallel =0.$$

(3.38)

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

$$\begin{array}{cc}& (1-{\alpha }_{\phi \left(n\right)}){\beta }_{\phi \left(n\right)}(1-{\beta }_{\phi \left(n\right)}){\rho }_b^{*}\parallel J_E^p{z}_{\phi \left(n\right)}-J_E^p{S}_2^{\phi \left(n\right)}{z}_{\phi \left(n\right)}\parallel \hfill \\ & \le (1+{\overline{\theta }}_{\phi \left(n\right)})(1+{\theta }_{\phi \left(n\right)}){\Gamma }_{\phi \left(n\right)}-{\Gamma }_{\phi \left(n\right)+1}+(1+{\overline{\theta }}_{\phi \left(n\right)}){\ell }_{\phi \left(n\right)}M+{\alpha }_{\phi \left(n\right)}\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\phi \left(n\right)}-\widehat{u}\rangle .\hfill \end{array}$$

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

$$\underset{n\to \infty }{lim}\parallel z_{\phi \left(n\right)}-S_2^{\phi \left(n\right)}{z}_{\phi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel v_{\phi \left(n\right)}-z_{\phi \left(n\right)}\parallel =0.$$

This together with (3.38) implies that

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

(3.39)

Noticing ${\zeta }_{\phi \left(n\right)}=J_{E^{*}}^q({\alpha }_{\phi \left(n\right)}{J}_E^{p}u+(1-{\alpha }_{\phi \left(n\right)})J_E^p{v}_{\phi \left(n\right)})$, by (3.39) one gets

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

(3.40)

Applying the same reasonings as in Case 1, one has that ${lim}_{n\to \infty }\parallel x_{\phi \left(n\right)}-x_{\phi \left(n\right)+1}\parallel =0$,

$$\underset{n\to \infty }{lim}\parallel s_{\phi \left(n\right)}-y_{\phi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel w_{\phi \left(n\right)}-{\overline{y}}_{\phi \left(n\right)}\parallel =0,$$

(3.41)

and

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

(3.42)

Using (3.36), we get

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{\phi \left(n\right)+1})\hfill \\ & \le (1-{\alpha }_{\phi \left(n\right)})D_{f_p}(\widehat{u},x_{\phi \left(n\right)})+{\alpha }_{\phi \left(n\right)}\{\frac{{ϵ}_{\phi \left(n\right)}}{{\alpha }_{\phi \left(n\right)}}\parallel J_E^p{S}_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}-J_E^p(S_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}+x_{\phi \left(n\right)}-x_{\phi \left(n\right)-1})\parallel \hfill \\ & \times \parallel \widehat{u}+x_{\phi \left(n\right)-1}-x_{\phi \left(n\right)}-S_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}\parallel +\frac{{\theta }_{\phi \left(n\right)}}{{\alpha }_{\phi \left(n\right)}}{D}_{f_p}(\widehat{u},x_{\phi \left(n\right)})+\frac{{\overline{\theta }}_{\phi \left(n\right)}}{{\alpha }_{\phi \left(n\right)}}{D}_{f_p}(\widehat{u},s_{\phi \left(n\right)})\hfill \\ & +\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\phi \left(n\right)}-\widehat{u}\rangle \},\hfill \end{array}$$

(3.43)

which together with (3.37), hence yields

$$\begin{array}{cc}& {\Gamma }_{\phi \left(n\right)}\hfill \\ & \le \frac{{ϵ}_{\phi \left(n\right)}}{{\alpha }_{\phi \left(n\right)}}\parallel J_E^p{S}_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}-J_E^p(S_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}+x_{\phi \left(n\right)}-x_{\phi \left(n\right)-1})\parallel \parallel \widehat{u}+x_{\phi \left(n\right)-1}-x_{\phi \left(n\right)}-S_1^{\phi \left(n\right)}{x}_{\phi \left(n\right)}\parallel \hfill \\ & +\frac{{\theta }_{\phi \left(n\right)}}{{\alpha }_{\phi \left(n\right)}}{D}_{f_p}(\widehat{u},x_{\phi \left(n\right)})+\frac{{\overline{\theta }}_{\phi \left(n\right)}}{{\alpha }_{\phi \left(n\right)}}{D}_{f_p}(\widehat{u},s_{\phi \left(n\right)})+\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\phi \left(n\right)}-\widehat{u}\rangle .\hfill \end{array}$$

As a result, from (3.42) we deduce that

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

(3.44)

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

$${\Gamma }_{\phi \left(n\right)+1}\to 0(n\to \infty ).$$

(3.45)

Again using (3.37), one gets ${\Gamma }_n\to 0$. Therefore, $x_n-\widehat{u}\to 0$. 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.

Under Algorithm 3.1, setting $A_2=0$ one obtains the algorithm below for approximating a point in $\Omega =\mathrm{VI}(C,A_1)\cap \left({\bigcap }_{i=1}^2\mathrm{Fix}\left(S_i\right)\right)$.

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 $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ and $\left\{{\ell }_n\right\}\subset (0,\infty )$ s.t. ${lim\; inf}_{n\to \infty }{\alpha }_n(1-{\alpha }_n)>0$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$. Moreover, assume ${\sum }_{n=1}^{\infty }{\theta }_n<\infty$, and 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

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

Iterations: Compute $x_{n+1}$ below:

Step 1. Put $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$, and calculate $s_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, $y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{s}_n-{\lambda }_1{A}_1{s}_n)\right)$, $e_{{\lambda }_1}\left(s_n\right):=s_n-y_n$ and $t_n=s_n-{\tau }_n{e}_{{\lambda }_1}\left(s_n\right)$, with ${\tau }_n:=l_1^{k_n}$ and $k_n$ being the smallest $k\ge 0$ s.t.

$$\frac{{\mu }_1}{2}{D}_{f_p}(s_n,y_n)\ge \langle A_1{s}_n-A_1(s_n-l_1^k{e}_{{\lambda }_1}\left(s_n\right)),s_n-y_n\rangle .$$

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

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

Step 3. Calculate $v_n=J_{E^{*}}^q({\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p\left(S_2^n{w}_n\right))$ and $x_{n+1}={\Pi }_{Q_n}\left(w_n\right)$, with $Q_n:=\{y\in C:D_{f_p}(y,v_n)\le ({\overline{\theta }}_n+1)D_{f_p}(y,w_n)\}$.

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

Corollary 3.1. Let the terms (C1)-(C2) with $A_2=0$, be valid, and assume $\Omega =\mathrm{VI}(C,A_1)\cap \left({\bigcap }_{i=1}^2\mathrm{Fix}\left(S_i\right)\right)\ne \varnothing$. If under Algorithm 3.3, $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{w}_n-S_2^n{w}_n\to 0$, then $x_n\rightharpoonup z\in \Omega \iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Next, put $S_2=I$ the identity mapping of E. Then we get $\Omega =\mathrm{Fix}\left(S_1\right)\cap \left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)$. In this case, Algorithm 3.2 can be rewritten as the iterative scheme below for settling a pair of VIPs and the FPP of $S_1$. By Theorem 3.2 one derives the strong convergence outcome below.

Corollary 3.2. Suppose that the condition (C2) 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

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

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

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

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

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

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

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

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

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

Then, $x_n\to {\Pi }_{\Omega }u\iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$ provided $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$.

4. Implementability and Applicability

In this section, we provide an illustrative example to demonstrate the applicability and implementability of our suggested approaches. For $i=1,2$, we take $ϵ=\frac{1}{3}$, ${\mu }_i=1$ and $l_i={\lambda }_i=\frac{1}{3}$. First, we present an instance involving the mappings $A_1,A_2:E\to E^{*}$ of both uniform continuity and pseudomonotonicity, and the mappings $S_1,S_2:C\to C$ of both uniform continuity and Bregman’s relatively asymptotical nonexpansivity satisfying $\Omega \ne \varnothing$. Put $C=[-2,2]$ and $E=H=\mathbf{R}$ with the inner product and induced norm being written as $\langle a,b\rangle =ab$ and $\parallel ·\parallel =|·|$, respectively. The starting $x_0,x_1$ are randomly chosen in C. For $i=1,2$, let $A_i:H\to H$ be defined as $A_{1}y:=\frac{1}{1+|siny|}-\frac{1}{1+\left|y\right|}$ and $A_{2}y:=y+siny$ for all $y\in H$. Next, let us prove that $A_1$ is the mapping of both Lipschitz continuity and pseudomonotonicity. In fact, for each $v,w\in H$ one has

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

Thus, $A_1$ is of Lipschitz continuity. Also, one shows that $A_1$ is of pseudomonotonicity. For any $v,w\in H$, it is easily known that

$\begin{array}{cc}& \langle A_{1}v,w-v\rangle =(\frac{1}{1+|sinv|}-\frac{1}{1+\left|v\right|})(w-v)\ge 0\hfill \\ & \Rightarrow \langle A_{1}w,w-v\rangle =(\frac{1}{1+|sinw|}-\frac{1}{1+\left|w\right|})(w-v)\ge 0.\hfill \end{array}$

It is easy to see that $A_2$ is of both Lipschitz continuity and monotonicity. Indeed, we deduce that $\parallel A_{2}v-A_{2}y\parallel \le \parallel v-y\parallel +\parallel sinv-siny\parallel \le 2\parallel v-y\parallel$ and

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

Now, let $S_1:C\to C$ and $S_2:C\to C$ be defined as $S_{1}y=S_{2}y:=Sy=\frac{4}{5}siny$. It is clear that $\mathrm{Fix}\left(S_i\right)=\mathrm{Fix}\left(S\right)=\left\{0\right\}$ for $i=1,2$.

Also, $S:C\to C$ is the mapping of Bregman’s relatively asymptotical nonexpansivity with ${\theta }_n={\left(\frac{4}{5}\right)}^n$, and $\forall \left\{{\varrho }_n\right\}\subset C$ we get $\parallel S^{n+1}{\varrho }_n-S^n{\varrho }_n\parallel \to 0$. In fact, note that

$$\parallel S^{n}v-S^n{w\parallel }^2\le {\left(\frac{4}{5}\right)}^2\parallel S^{n-1}v-S^{n-1}{w\parallel }^2\le \cdots \le {\left(\frac{4}{5}\right)}^{2n}{\parallel v-w\parallel }^2\le (1+{\theta }_n){\parallel v-w\parallel }^2,$$

$$\parallel S^{n+1}{\varrho }_n-S^n{\varrho }_n\parallel \le {\left(\frac{4}{5}\right)}^{n-1}\parallel S^2{\varrho }_n-S{\varrho }_n\parallel ={\left(\frac{4}{5}\right)}^{n-1}\parallel \frac{4}{5}sin\left(S{\varrho }_n\right)-\frac{4}{5}sin{\varrho }_n\parallel \le 2{\left(\frac{4}{5}\right)}^n\to 0(n\to \infty ),$$

and

$$\underset{n\to \infty }{lim}\frac{{\theta }_n}{1/2(n+1)}=\underset{n\to \infty }{lim}\frac{{(4/5)}^n}{1/2(n+1)}=0.$$

Consequently,

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

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

$$\underset{n\to \infty }{lim}{\beta }_n(1-{\beta }_n)=\underset{n\to \infty }{lim}\frac{n+2}{2(n+1)}(1-\frac{n+2}{2(n+1)})=\frac{1}{2}(1-\frac{1}{2})=\frac{1}{4}>0$$

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

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

Algorithm 3.1 is rewritten as follows:

$$\left\{\begin{array}{cc}& g_n=S^n{x}_n+{ϵ}_n(x_n-x_{n-1}),\hfill \\ & s_n=\frac{n+2}{2(n+1)}{x}_n+\frac{n}{2(n+1)}{g}_n,\hfill \\ & y_n=P_C(s_n-\frac{1}{3}{A}_1{s}_n),\hfill \\ & t_n=(1-{\tau }_n)s_n+{\tau }_n{y}_n,\hfill \\ & w_n=P_{C_n}{s}_n,\hfill \\ & {\overline{y}}_n=P_C(w_n-\frac{1}{3}{A}_2{w}_n),\hfill \\ & {\overline{t}}_n=(1-{\overline{\tau }}_n)w_n+{\overline{\tau }}_n{\overline{y}}_n,\hfill \\ & v_n=\frac{n}{2(n+1)}{S}^n{w}_n+\frac{n+2}{2(n+1)}{w}_n,\hfill \\ & Q_n=\{y\in C:\parallel v_n{-y\parallel }^2\le (1+{\left(\frac{4}{5}\right)}^n)\parallel w_n-y{\parallel }^2\},\hfill \\ & x_{n+1}=P_{{\overline{C}}_n\cap Q_n}{w}_n,\hfill \end{array}\right.$$

(4.1)

with the sets $C_n,{\overline{C}}_n$ and the step-sizes ${\tau }_n,{\overline{\tau }}_n$ being picked as in Algorithm 3.1. By Theorem 3.1, one obtain $x_n\to 0\in \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \mathrm{Fix}\left(S\right))$.

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

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

Algorithm 3.2 is rewritten as follows:

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

(4.2)

with the sets $C_n,{\overline{C}}_n$ and the step-sizes ${\tau }_n,{\overline{\tau }}_n$ being picked as in Algorithm 3.2. By Theorem 3.2, we deduce that $x_n\to 0\in \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \mathrm{Fix}\left(S\right)$.

5. Conclusions

This article designs iterative algorithms for resolving a pair of VIFPPs in uniformly smooth and p-uniformly convex Banach spaces. With the help of parallel subgradient-like extragradient methods with both inertial effect and linesearch process, we fabricate two algorithms for approximating a common solution of the two pseudomonotone VIPs and the CFPP of two mappings of Bregman’s relatively asymptotical nonexpansivity. We are focused on discussing the weak and strong convergence of the proposed algorithms by using standard terms and novel manoeuvres. Besides, an illustrative example is furnished to bear up the applicability and implementability of our proposed approaches. Finally, it is worth mentioning that part of our future research is aimed at achieving the weak and strong convergence results for the modifications of our proposed approaches with Nesterov double inertial extrapolation steps (see [34]) and adaptive stepsizes.