On Generalized Fractional Operator and Related Fractional Integral Equations in Orlicz Spaces

1. Introduction

Fractional calculus is an important tool for describing memory and hereditary properties related to various processes and materials. It is an important scientific field due to its applications in economics, biology, physics, viscoelasticity, engineering, fluid dynamics, electrical circuits, earthquakes, electrochemistry, and traffic models (cf. [2,3,4,5,6]).

The "g-fractional operator", which is the fractional integral of a Lebesgue function y concerning another function g was introduced in ([5], Sect.18.2 and [2], Sect.2.5.). It combines and unifies the Hadamard, Riemann-Liouville, and Erdlyi–Kober fractional operators into one form (see [7,8,9,10,11,12]), and is better able to present memory properties connected to various types of materials and processes.

Quadratic integral equations using the g-fractional operator are more appropriate in the kinetic theory of gases [13], radiative transfer [16], neutron transport [14] etc. The primary objective of the work is to prove and describe the vital properties of the g-fractional type operator, encompassing boundedness, continuity, and monotonicity in Orlicz spaces $L_{\psi }$. We will apply these properties to illustrate and analyze the monotonic solutions of the equation

$$y\left(v\right)=f\left(v\right)+{\displaystyle \frac{h_1(v,y\left(v\right))}{\Gamma \left(\alpha \right)}}·{\int }_0^v{\displaystyle \frac{h_2(s,y\left(s\right))}{{\left(g\left(v\right)-g\left(s\right)\right)}^{1-\alpha }}}{g}^{\prime }\left(s\right)ds,v\in [0,d],$$

(1)

where $0<\alpha <1$ in the mentioned spaces.

This is inspired by statistical physics, physics models (cf. [17,18]), and various applications of partial differential equations or integral equations in Orlicz spaces $L_{\psi }$ [31,32].

Our method covers and generalizes different types of fractional integrals that have been examined separately and encourages us to recall some of them. The author in [22] presented some basic properties of the Riemann-Liouville type fractional integral operator and explored equation solutions

$$y\left(v\right)=f\left(v\right)+G\left(y\right)\left(v\right){\int }_0^v{\displaystyle \frac{{(v-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}f(s,y\left(s\right))ds,0<\alpha <1,v\in [0,d]$$

in Orlicz spaces $L_{\psi }$.

The author in [23] demonstrated and studied fundamental features of the Hadamard type fractional operator within $L_{\psi }$-spaces and utilized them to solve the equation:

$$y\left(v\right)=G_2\left(y\right)\left(v\right)+{\displaystyle \frac{G_1\left(y\right)\left(v\right)}{\Gamma \left(\alpha \right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{s}}\right)}^{\alpha -1}{G}_2\left(y\right)\left(s\right)ds,v\in [1,e],0<\alpha <1.$$

The authors in [30] showed the basic characteristics of the Erdlyi–Kober fractional operators in Lebesgue and Orlicz spaces and used them to analyze the problem.

$$y\left(v\right)=f\left(v\right)+f_1(v,y\left(v\right))+f_2\left(v,{\displaystyle \frac{\beta h_1(v,y\left(v\right))}{\Gamma \left(\alpha \right)}}·{\int }_0^v{\displaystyle \frac{t^{\beta -1}{h}_2(s,y\left(s\right))}{{(v^{\beta }-s^{\beta })}^{1-\alpha }}}ds\right),v\in [0,d],$$

where $0<\alpha <1$ & $\beta >0$ in the indicated spaces.

Furthermore, the noncompactness measure ($\mathcal{MNC}$) and Darboe’s fixed point hypothesis ($\mathcal{FPT}$) were used to study different types of quadratic integral equations in Orlicz spaces $L_{\psi }$ under various sets of assumptions (cf. [19,20,21]).

This article aims to illustrate and demonstrate some essential aspects of the g-fractional operator, including boundedness, action, continuity, and monotonicity within $L_{\psi }$-spaces. We utilize such properties through the measure of noncompactness ($\mathcal{MNC}$) and fixed-point theorem ($\mathcal{FPT}$) to investigate the existence and uniqueness of the solution for a quadratic integral equation (1) within the given spaces.

2. Preliminaries

Let $\mathbb{R}=(-\infty ,\infty )$ and $\mathbb{I}=[0,d]\subset {\mathbb{R}}^{+}=[0,\infty )$. Denote the Young function (YF) by $P:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+},$ where

$$P\left(\theta \right)={\int }_0^{\theta }g\left(s\right)ds,\mathrm{for}\theta \ge 0$$

and $g:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is neither identically zero nor infinite and an increasing and left-continuous function on ${\mathbb{R}}^{+}$. The pair $(P,Q)$ is said to be a complementary pair of YF if $Q\left(y\right)={sup}_{x\ge 0}(yx-P\left(y\right))$.

The function P is known as N-function when it is a finite-valued and verifies ${lim}_{\theta \to \infty }{\displaystyle \frac{P\left(\theta \right)}{\theta }}=\infty$, ${lim}_{\theta \to 0}{\displaystyle \frac{P\left(\theta \right)}{\theta }}=0$, and $P\left(\theta \right)>0$ if $\theta >0$ ($P\left(\theta \right)=0\iff \theta =0$).

The Orlicz space $L_P=L_P\left(\mathbb{I}\right)$ is the space of all measurable functions $y:\mathbb{I}\to \mathbb{R}$ with the norm

$${\parallel y\parallel }_P=\underset{\lambda >0}{inf}\left\{{\int }_{\mathbb{I}}P\left({\displaystyle \frac{y\left(s\right)}{\lambda }}\right)ds\le 1\right\}<\infty .$$

It is important to recall that for any YF P, we have $P(\theta +s)\le P\left(\theta \right)+P\left(s\right)$ and $P\left(\rho \theta \right)\le \rho P\left(\theta \right)$, where $\theta ,s\in \mathbb{R}$, and $\rho \in [0,1]$.

Assume that $E_P\left(\mathbb{I}\right)$ is the set of all bounded functions in $L_P\left(\mathbb{I}\right)$ contain absolutely continuous norms.

Moreover, we get $L_P=E_P$ if P satisfies the ${\Delta }_2$ condition, i.e.:

$$\left({\Delta }_2\right)\exists \omega ,{\theta }_0\ge 0\mathrm{such} \mathrm{that}P\left(2\theta \right)\le \omega P\left(\theta \right),\theta \ge {\theta }_0.$$

It is noting that, the classical Lebesgue spaces $L_p\left(\mathbb{I}\right)$ shall be considered as a particular case of Orlicz spaces $L_{P_p}\left(\mathbb{I}\right)$ with corresponding N-function $P_p=s^p,p>1$ satisfies the above ${\Delta }_2$ condition.

Lemma 1.

[17] Assume that, the function $h(v,y):\mathbb{I}\times \mathbb{R}\to \mathbb{R}$ verifies Carathéodory conditions (i.e., it is continuous in the variable y for almost all $v\in \mathbb{I}$ and measurable in v for any $y\in \mathbb{R}$). The superposition operator $F_h=h(v,y):E_{{\psi }_1}\to L_P=E_P$ is bounded and continuous if

$$\left|h(v,y)\right|\le a\left(v\right)+b{P}^{-1}\left({\psi }_1\left(y\right)\right),y\in \mathbb{R},v\in \mathbb{I},$$

where $b\ge 0$, $a\in L_P$ and the N-function $P\left(y\right)$ verifies the ${\Delta }_2$ condition.

Lemma 2.

[24] Assume that ψ, ${\psi }_1$, and ${\psi }_2$ are arbitrary different N-functions. The following given conditions are equivalent:

(1)

For every functions $y_1\in L_{{\psi }_1}$ and $y_2\in L_{{\psi }_2}$, $y_1·y_2\in L_{\psi }$.

(2)

$\exists k>0$ s.t. for all measurable $y_1,y_2$ on $\mathbb{I}$, we have $\parallel y_1{y}_2{\parallel }_{\psi }\le k\parallel y_1{\parallel }_{{\psi }_1}{\parallel y_1\parallel }_{{\psi }_2}$.

(3)

$\exists C>0$, $s_0\ge 0$ s.t. for all $v,s\ge s_0,\psi \left({\displaystyle \frac{sv}{C}}\right)\le {\psi }_1\left(s\right)+{\psi }_2\left(v\right)$.

(4)

${lim\; sup}_{v\to \infty }{\displaystyle \frac{{\psi }_1^{-1}\left(v\right){\psi }_2^{-1}\left(v\right)}{\psi \left(v\right)}}<\infty$.

The set $S=S\left(\mathbb{I}\right)$ is the set of Lebesgue measurable functions "means" on $\left(\mathbb{I}\right)$ connected with the metric

$$d(y,x)=\underset{\rho >0}{inf}[ϵ+meas\{s:|y\left(s\right)-x\left(s\right)|\ge \rho \left\}\right]$$

is a complete space. Additionally, the convergence in measure on $\mathbb{I}$ is equivalent to the convergence regarding d (cf. [25]). The compactness in S is called "compactness in measure".

Lemma 3

( [19]). Assume that $Y\subset L_{\psi }\left(\mathbb{I}\right)$ is a bounded set, and ∃ a family ${\left({\Omega }_r\right)}_{0\le r\le d}\subset \mathbb{I}$ s.t. meas ${\Omega }_r=r$ for every $r\in [0,d]$, and for every $y\in Y$,

$$y\left(s_1\right)\ge y\left(s_2\right),(s_1\in {\Omega }_r,t_2\notin {\Omega }_r).$$

Then, Y is compact in measure in $L_{\psi }\left(\mathbb{I}\right)$.

Definition 1

([26]). The Hausdorff measure of noncompactness ($\mathcal{MNC}$) ${\beta }_H\left(X\right)$ for a bounded set $\varnothing \ne Y\subset L_{\psi }$ is known as

$${\beta }_H\left(Y\right)=inf\{r>0:\exists X\subset L_{\psi }s.t.Y\subset X+B_r\},$$

where $B_r=\{y\in L_{\psi }{\left(\mathbb{I}\right):\parallel y\parallel }_{\psi }\le r\}$ is the ball centered at the origin with radius r.

Denote a measure of equi-integrability c of $Y\in L_{\psi }\left(\mathbb{I}\right)$ by :

$$c\left(Y\right)=\underset{ϵ\to 0}{lim}\underset{measD\le ϵ}{sup}\underset{y\in Y}{sup}{\parallel y·{\chi }_D\parallel }_{\psi },$$

where $ϵ>0$ and ${\chi }_A$ points to the characteristic function $A\subset \mathbb{I}$ (see [25] or [27]).

Lemma 4

([19,27]). Assume that $\varnothing \ne Y\subset L_{\psi }$ is a bounded set and compact in measure. Then, we have:

$${\beta }_H\left(Y\right)=c\left(Y\right).$$

Theorem 1

([26]). Assume that $\varnothing \ne \Omega \subset L_{\psi }$ is a convex, bounded, and closed set and $T:\Omega \to \Omega$ is continuous mapping and a contraction regarding to ${\beta }_H$, i.e.:

$${\beta }_H\left(T\left(Y\right)\right)\le k{\beta }_H\left(Y\right),0\le k<1$$

for any $\varnothing \ne Y\subset \Omega$. Then, the map T has at least one fixed point in Ω.

3. Generalized Fractional Operators

We provide and prove some properties and concepts of the generalized fractional (or g-fractional) integral operator In $L_{\psi }\left(\mathbb{I}\right)$.

Definition 2

([2,5]). The g-fractional, or generalized fractional, integral of a well-defined function y of order α concerning a different function $g\left(s\right)$ is given by

$$J_g^{\alpha }y\left(v\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^v{\displaystyle \frac{y\left(s\right)}{{\left(g\left(v\right)-g\left(s\right)\right)}^{1-\alpha }}}{g}^{\prime }\left(s\right)ds,\alpha >0,$$

(2)

where g is a positive-increasing function on $(0,\infty ]$ and has a continuous derivative on $(0,\infty )$.

Remark 1.

• If $g\left(v\right)=v$, the operator $J_g^{\alpha }$ (2) becomes the Riemann–Liouville fractional operator, which has been analyzed in [2,3,22]:

  $$J_{\nu }^{\alpha }y\left(v\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^v{\displaystyle \frac{y\left(s\right)}{{(v-s)}^{1-\alpha }}}ds.$$
• If $g\left(v\right)=log\left(v\right)$, the operator $J_g^{\alpha }$ (2) becomes the Hadamard fractional operator, which has been analyzed in [2,3,23]:

  $$J_{log\left(v\right)}^{\alpha }y\left(v\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{s}}\right)}^{\alpha -1}{\displaystyle \frac{y\left(s\right)}{s}}ds.$$
• If $g\left(v\right)=v^{\beta },\beta >0$, the operator $J_g^{\alpha }$ (2) becomes the Erdlyi–Kober’s operator, which has been analyzed in [2,3,30]:

  $$J_{v^{\beta }}^{\alpha }y\left(v\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^v{\displaystyle \frac{\beta s^{\beta -1}y\left(s\right)}{{(v^{\beta }-s^{\beta })}^{1-\alpha }}}ds.$$
• If $g\left(v\right)=v^2$, the operator $J_g^{\alpha }$ (2) becomes the fractional integral operator of the form (Sneddon [28]):

  $$J_{v^2}^{\alpha }y\left(v\right)={\displaystyle \frac{2}{\Gamma \left(\alpha \right)}}{\int }_0^v{\displaystyle \frac{y\left(s\right)}{{(v^2-s^2)}^{1-\alpha }}}sds.$$

Now, we examine the monotonicity of the operator $J_g^{\alpha }$ (2).

Lemma 5.

The operator $J_g^{\alpha },\alpha >0$ with $g\left(0\right)=0$ maps nonnegative and a.e. non-decreasing functions into functions having similar properties.

Proof. 

Let $v_1,v_2\in \mathbb{I},v_1\le v_2$, and y be a.e. nondecreasing-nonnegative function, then by putting $g\left(s\right)=g\left(v_1\right)·u$, we have

$$\begin{array}{ccc}\hfill J_g^{\alpha }y\left(v_1\right)& =& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{v_1}{\displaystyle \frac{y\left(s\right)}{{\left(g\left(v_1\right)-g\left(s\right)\right)}^{1-\alpha }}}{g}^{\prime }\left(s\right)ds\hfill \\ & =& {\displaystyle \frac{g^{\alpha }\left(v_1\right)}{\Gamma \left(\alpha \right)}}{\int }_0^1{\displaystyle \frac{y\left(g^{-1}\left(g\left(v_1\right)·u\right)\right)}{{\left(1-u\right)}^{1-\alpha }}}du.\hfill \end{array}$$

Since the function g is an increasing and positive function on $(0,\infty ]$, then its inverse $g^{-1}$ exists and has also the same properties (cf. [29]), then we get

$$J_g^{\alpha }y\left(v_1\right)\le {\displaystyle \frac{g^{\alpha }\left(v_2\right)}{\Gamma \left(\alpha \right)}}{\int }_0^1{\displaystyle \frac{y\left(g^{-1}\left(g\left(v_2\right)·u\right)\right)}{{\left(1-u\right)}^{1-\alpha }}}du=J_g^{\alpha }y\left(v_2\right).$$

Therefore, $0\le J_g^{\alpha }y\left(v_1\right)\le J_g^{\alpha }y\left(v_2\right)$ for $v_1\le v_2$. □

Proposition 1

([15]). Let P be a (YF) Young function, then for $v\in {\mathbb{R}}^{+}$ and any $0<\alpha <1$, the set

$$\mathbb{P}\left(s\right)=\left\{ϵ>0:{\displaystyle \frac{1}{\parallel g^{\prime }{\parallel }_P}}{\int }_0^{g\left(v\right){\sigma }^{{\displaystyle \frac{1}{1-\alpha }}}}P\left(u^{\alpha -1}\right)du\le {\sigma }^{{\displaystyle \frac{1}{1-\alpha }}}\right\}\ne \varnothing ,\sigma ={\displaystyle \frac{ϵ}{\parallel g^{\prime }{\parallel }_P}},$$

is increasing and continuous functions with $\mathbb{P}\left(0\right)=0$, where the function g is defined in Definition 2.

Lemma 6.

Let $(P,Q)$ be a complementary pair of N-functions and ψ be an N-function, where P verifies ${\int }_0^{g\left(v\right)}P\left(u^{\alpha -1}\right)du<\infty ,\alpha \in (0,1).$ Then the operator $J_g^{\alpha }:L_Q\left(\mathbb{I}\right)\to L_{\psi }\left(\mathbb{I}\right)$ is continuous, where

$$k\left(v\right)={\displaystyle \frac{{\sigma }^{\frac{1}{\alpha -1}}}{\parallel g^{\prime }{\parallel }_P}}{\int }_0^{g\left(v\right){\sigma }^{{\displaystyle \frac{1}{1-\alpha }}}}P\left(u^{\alpha -1}\right)du\in E_{\psi }\left(\mathbb{I}\right),ϵ>0,\sigma ={\displaystyle \frac{ϵ}{\parallel g^{\prime }{\parallel }_P}}$$

for a.e. $v\in \mathbb{I}$.

Proof. 

Assume that

$$K(v,s)=\left\{\begin{array}{c}{\displaystyle \frac{{\left(g\left(v\right)-g\left(s\right)\right)}^{\alpha -1}{g}^{\prime }\left(s\right)}{\Gamma \left(\alpha \right)}}\mathrm{if}s\in [0,v],v>0,\\ 0\mathrm{otherwise}.\end{array}\right.$$

For $y\in L_Q\left(\mathbb{I}\right)$ and by utilizing Hölder inequality, we get:

$$\begin{array}{ccc}\hfill |J_g^{\alpha }y\left(v\right)|& =& \left|{\int }_0^{\infty }K(v,s)y\left(s\right)ds\right|\hfill \\ & \le & {2\parallel K(v,·)\parallel }_P{\parallel y\parallel }_Q\hfill \\ & \le & {\displaystyle \frac{2}{\Gamma \left(\alpha \right)}}\underset{ϵ>0}{inf}\left\{{\int }_{\mathbb{I}}P\left({\displaystyle \frac{{\left(g\left(v\right)-g\left(s\right)\right)}^{\alpha -1}{g}^{\prime }\left(s\right)}{ϵ}}\right)ds\le 1\right\}{\parallel y\parallel }_Q\hfill \\ & =& {\displaystyle \frac{2}{\Gamma \left(\alpha \right)}}\underset{ϵ>0}{inf}\left\{{\int }_{\mathbb{I}}P\left({\displaystyle \frac{{\left(g\left(v\right)-g\left(s\right)\right)}^{\alpha -1}{g}^{\prime }\left(s\right)\parallel g^{\prime }{\parallel }_P}{ϵ·\parallel g^{\prime }{\parallel }_P}}\right)ds\le 1\right\}{\parallel y\parallel }_Q\hfill \\ & \le & {\displaystyle \frac{2}{\Gamma \left(\alpha \right)}}\underset{ϵ>0}{inf}\left\{{\int }_{\mathbb{I}}P\left({\displaystyle \frac{{\left(g\left(v\right)-g\left(s\right)\right)}^{\alpha -1}\parallel g^{\prime }{\parallel }_P}{ϵ}}\right){\displaystyle \frac{g^{\prime }\left(s\right)}{\parallel g^{\prime }{\parallel }_P}}ds\le 1\right\}{\parallel y\parallel }_Q.\hfill \end{array}$$

Put $u=\left(g\left(v\right)-g\left(s\right)\right){\sigma }^{{\displaystyle \frac{1}{1-\alpha }}}$, where $\sigma ={\displaystyle \frac{ϵ}{\parallel g^{\prime }{\parallel }_P}}$, we have

$$\begin{array}{ccc}\hfill \parallel J_g^{\alpha }{y\parallel }_{\psi }& \le & {\displaystyle \frac{2}{\Gamma \left(\alpha \right)}}{∥\underset{ϵ>0}{inf}\left\{{\displaystyle \frac{1}{\parallel g^{\prime }{\parallel }_P}}{\int }_0^{g\left(v\right){\sigma }^{{\displaystyle \frac{1}{1-\alpha }}}}P\left(u^{\alpha -1}\right)du\le {\sigma }^{{\displaystyle \frac{1}{1-\alpha }}}\right\}∥}_{\psi }{\parallel y\parallel }_Q\hfill \\ & \le & {\displaystyle \frac{2}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{\psi }{\parallel y\parallel }_Q,\hfill \end{array}$$

where $k\in E_{\psi }\left(\mathbb{I}\right)$. Then, by recalling Proposition 1 and [17], we have $J_g^{\alpha }:L_Q\left(\mathbb{I}\right)\to L_{\psi }\left(\mathbb{I}\right)$ and is continuous. □

4. Main Results

Equation (1) can take the form:

$$y=B\left(y\right)=f+U\left(y\right),$$

where

$$U\left(y\right)=F_{h_1}\left(y\right)·A\left(y\right),A\left(y\right)\left(v\right)=J_g^{\alpha }{F}_{h_2}\left(y\right)\left(v\right),$$

such that $J_g^{\alpha }$ is defined in Definition 2 and $F_{h_i},i=1,2$ are known as the superposition operators.

Next, we shall discuss our existence theorem in $L_{\psi }$-spaces in the most interesting case, where the generating N-functions verifying the ${\Delta }_2$-condition (cf. [1,20,21,30]). These allow us to utilize some general conditions for the studied functions.

Theorem 2.

Assume that ${\psi }_1,{\psi }_2,and\psi$ are N-functions and $(P,Q)$ is a complementary pair of N-functions, such that $Q,\psi ,{\psi }_1$ satisfy the ${\Delta }_2$ condition and ${\int }_0^{g\left(v\right)}P\left(u^{\alpha -1}\right)du<\infty ,\alpha \in (0,1)$ and that:

(G1)

$\exists k_1>0$ s.t. for $y_1\in L_{{\psi }_1}\left(\mathbb{I}\right)$ and $y_2\in L_{{\psi }_2}\left(\mathbb{I}\right)$ we get $\parallel y_1{y}_2{\parallel }_{\psi }\le k_1\parallel y_1{\parallel }_{{\psi }_1}{\parallel y_2\parallel }_{{\psi }_2}$.

(C1)

$f\in E_{\psi }\left(\mathbb{I}\right)$ is a.e. nondecreasing on $\mathbb{I}$.

(C2)

$h_i:\mathbb{I}\times \mathbb{R}\to \mathbb{R},i=1,2$ satisfy Carathéodory conditions and $(s,y)\to h_i(s,y)$ are nondecreasing.

(C3)

$\exists d_i\ge 0,i=1,2$ and functions $b_1\in E_{{\psi }_1}\left(\mathbb{I}\right)$, and $b_2\in E_Q\left(\mathbb{I}\right)$, s.t.

$$|h_1(s,y)|\le b_1\left(s\right)+d_1{\psi }_1^{-1}\left(\psi \left(y\right)\right),\left|h_2(s,y)\right|\le b_2\left(s\right)+d_2{Q}^{-1}\left(\psi \left(y\right)\right).$$

(3)

(C4)

Assume that for a.e. $v\in \mathbb{I}$, $\exists ϵ>0$, s.t.

$$k\left(v\right)={\displaystyle \frac{{\sigma }^{\frac{1}{\alpha -1}}}{\parallel g^{\prime }{\parallel }_P}}{\int }_0^{{\sigma }^{{\displaystyle \frac{1}{1-\alpha }}}g\left(v\right)}P\left(u^{\alpha -1}\right)du\in E_{\psi }\left(\mathbb{I}\right),\sigma ={\displaystyle \frac{ϵ}{\parallel g^{\prime }{\parallel }_P}}.$$

(C5)

Assume that, $r>0$ on $I_0=[0,d_0]\subset \mathbb{I}$ verifying

$${\int }_{I_0}\psi \left(\left|f\left(v\right)\right|+{\displaystyle \frac{2k_1}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_1{\parallel }_{{\psi }_1}+d_{1}r\right)\left(\parallel b_2{\parallel }_Q+d_{2}r\right)\right)dv\le r$$

and

$$\left[{\displaystyle \frac{2d_1{k}_1}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_2{\parallel }_Q+d_2·r\right)\right]<1.$$

Then, there is a.e. nondecreasing solution $y\in E_{\psi }\left(I_0\right)$ of (1) on $I_0\subset \mathbb{I}$.

Proof.I. Lemma 1 and assumptions (C2), (C3), imply that $F_{h_1}:E_{\psi }\left(\mathbb{I}\right)){\to }_{{\psi }_1}\left(\mathbb{I}\right),F_{h_2}:E_{\psi }\left(\mathbb{I}\right)){\to }_Q\left(\mathbb{I}\right)$ and are continuous. Lemma 6) gives us that $A=J_g^{\alpha }{F}_{h_2}:E_{\psi }\left(\mathbb{I}\right)\to E_{{\psi }_2}\left(\mathbb{I}\right)$ is continuous. Assumption (G1) implies that $U:E_{\psi }\left(\mathbb{I}\right)\to E_{\psi }\left(\mathbb{I}\right)$, and by assumption $\left(C1\right)$, $B:E_{\psi }\left(\mathbb{I}\right)\to E_{\psi }\left(\mathbb{I}\right)$ is continuous.

II. Next, we should inspect and show that the operator B is bounded in $E_{\psi }\left(\mathbb{I}\right)$.

Let $\Omega$ point to the closure of the set $\{y\in E_{\psi }\left(I_0\right):{\int }_0^{d_0}\psi \left(\right|y\left(s\right)\left|\right)ds\le r-1\}$. Clearly, $\Omega$ is not a ball in $E_{\psi }\left(I_0\right)$, but $\Omega \subset B_r\left(E_{\psi }\left(I_0\right)\right)$ (cf. [17] p. 222) and the set $\overline{\Omega }$ is a bounded, convex, and closed subset of $E_{\psi }\left(I_0\right)$.

By using Theorem 10.5 with constant k = 1 [17], then for arbitrary $y\in \Omega$ and $v\in I_0$, we get:

$$\parallel {\psi }_1^{-1}\left(\psi \left(\right|y\left|\right)\right){\parallel }_{{\psi }_1}\le {\parallel y\parallel }_{\psi }=1+{\int }_{I_0}\psi \left(y\left(s\right)\right)dsand\parallel Q^{-1}\left(\psi \left(y\right)\right){\parallel }_Q\le {\parallel y\parallel }_{\psi }=1+{\int }_{I_0}\psi \left(y\left(s\right)\right)ds.$$

(4)

Therefore, by using Lemma 6 and our assumptions, we get

$$\begin{array}{cc}\hfill \left|B\right(y\left)\right(v\left)\right|& \le \left|f\right(v\left)\right|+\left|U\right(y\left)\right(v\left)\right|\hfill \\ \hfill & \le \left|f\left(v\right)\right|+k_1\parallel F_{h_1}{\left(y\right)\parallel }_{{\psi }_1}{\parallel A\left(y\right)\parallel }_{{\psi }_2}\hfill \\ \hfill & \le \left|f\left(v\right)\right|+k_1\parallel b_1+d_1{\psi }_1^{-1}\left(\psi \left(\right|y\left|\right)\right){\parallel }_{{\psi }_1}·{\parallel J_g^{\alpha }{F}_{h_2}\left(y\right)\parallel }_{{\psi }_2}\hfill \\ \hfill & \le \left|f\left(v\right)\right|+k_1\left(\parallel b_1{\parallel }_{{\psi }_1}+d_1\parallel {\psi }_1^{-1}\left(\psi \left(\right|y\left|\right)\right){\parallel }_{{\psi }_1}\right){\displaystyle \frac{2}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_2{\parallel }_Q+d_2\parallel Q^{-1}\left(\psi \left(\right|y\left|\right)\right){\parallel }_Q\right)\hfill \\ \hfill & \le \left|f\left(v\right)\right|+{\displaystyle \frac{2k_1}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_1{\parallel }_{{\psi }_1}+d_1+d_1{\int }_{I_0}\psi \left(y\left(s\right)\right)ds\right)\left(\parallel b_2{\parallel }_Q+d_2+d_2{\int }_{I_0}\psi \left(y\left(s\right)\right)ds\right)\hfill \\ \hfill & \le \left|f\left(v\right)\right|+{\displaystyle \frac{2k_1}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_1{\parallel }_{{\psi }_1}+d_1+d_1(r-1)\right)\left(\parallel b_2{\parallel }_Q+d_2+d_2(r-1)\right).\hfill \end{array}$$

Recalling assumption (C5), we get

$$\begin{array}{cc}\hfill {\int }_{I_0}\psi \left(B\left(y\right)\left(v\right)\right)dv& \le {\int }_{I_0}\psi \left[\left|f\left(v\right)\right|+{\displaystyle \frac{2k_1}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_1{\parallel }_{{\psi }_1}+d_{1}r\right)\left(\parallel b_2{\parallel }_Q+d_{2}r\right)\right]dv\le r,\hfill \end{array}$$

then $B(\Omega )\subset \Omega$, and $B\left(\overline{\Omega }\right)\subset \overline{B(\Omega )}\subset \overline{\Omega }=\Omega$. Then, the operator $B:\Omega \to \Omega$ is continuous on $\Omega \subset B_r\left(E_{\psi }\left(I_0\right)\right)$.

III. Let ${\Omega }_r\subset \Omega$ contain all a.e. monotonic (nondecreasing) functions on $I_0$. The set $\varnothing \ne {\Omega }_r$ is bounded, closed, compact in measure, and convex in $L_{\psi }\left(I_0\right)$ (cf. [20]).

IV. The monotonicity of the functions is preserved via the operator B.

Take $y\in {\Omega }_r$, then y is a.e. nondecreasing on $I_0$ and, consequently, the operators $F_{h_i},i=1,2$ are also a.e. nondecreasing on $I_0$. By Lemma 5), A is a.e. nondecreasing on $I_0$, then $U=F_{h_1}A$ is also a.e. nondecreasing on $I_0$. Using (C1), we get $B:{\Omega }_r\to {\Omega }_r$ is continuous.

V. Now, we show that B satisfies contraction condition w.r. to ${\beta }_H$.

Suppose there is a set $D\subset I_0$, with meas $D\le \epsilon ,\epsilon >0$. Therefore, for $y\in Y$ and $\varnothing \ne Y\subset Q_r$, we have:

$$\begin{array}{cc}\hfill \parallel B\left(y\right)·{\chi }_D{\parallel }_{\psi }& \le \parallel f·{\chi }_D{\parallel }_{\psi }+{\parallel F_{h_1}\left(y\right)A\left(y\right)·{\chi }_D\parallel }_{\psi }\hfill \\ \hfill & \le \parallel f·{\chi }_D{\parallel }_{\psi }+k_1\parallel F_{h_1}\left(y\right)·{\chi }_D{\parallel }_{{\psi }_1}{\parallel A\left(y\right)·{\chi }_D\parallel }_{{\psi }_2}\hfill \\ \hfill & \le \parallel f·{\chi }_D{\parallel }_{\psi }+k_1\parallel \left(b_1+d_1{\psi }_1^{-1}\left(\psi \left(\right|y\left|\right)\right)\right)·{\chi }_D{\parallel }_{{\psi }_1}·\parallel J_g^{\alpha }{F}_{h_2}{\parallel }_{{\psi }_2}\hfill \\ \hfill & \le \parallel f·{\chi }_D{\parallel }_{\psi }+k_1\left(\parallel b_1·{\chi }_D{\parallel }_{{\psi }_1}+d_1\parallel {\psi }_1^{-1}\left(\psi \left(\right|y\left|\right)\right)·{\chi }_D{\parallel }_{{\psi }_1}\right)\hfill \\ \hfill & \times {\displaystyle \frac{2}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_2{\parallel }_Q+d_2\parallel Q^{-1}\left(\psi \left(\right|y\left|\right)\right){\parallel }_Q\right)\hfill \\ \hfill & \le \parallel f·{\chi }_D{\parallel }_{\psi }+{\displaystyle \frac{2k_1}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_1·{\chi }_D{\parallel }_{{\psi }_1}+d_1{\parallel y·{\chi }_D\parallel }_{\psi }\right)\left(\parallel b_2{\parallel }_Q+d_2·r\right).\hfill \end{array}$$

Since $f\in E_{\psi }$, $b_1\in E_{{\psi }_1}$, then we have

$$\underset{\epsilon \to 0}{lim}\{\underset{measD\le \epsilon }{sup}[\underset{y\in Y}{sup}\{\parallel f·{\chi }_D{\parallel }_{\psi }\left\}\right]\}=0$$

and

$$\underset{\epsilon \to 0}{lim}\{\underset{measD\le \epsilon }{sup}[\underset{y\in Y}{sup}\{\parallel b_1·{\chi }_D{\parallel }_{{\psi }_1}\left\}\right]\}=0.$$

By using the formula of of $c\left(Y\right)$, we get

$$c\left(B\left(Y\right)\right)\le \left({\displaystyle \frac{2d_1{k}_1}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_2{\parallel }_Q+d_2·r\right)\right)c\left(Y\right).$$

Based on the previously established properties, we may apply Lemma 4 to get

$${\beta }_H\left(B\left(Y\right)\right)\le \left({\displaystyle \frac{2d_1{k}_1}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left(\parallel b_2{\parallel }_Q+d_2·r\right)\right){\beta }_H\left(Y\right).$$

The above inequality with $\left({\displaystyle \frac{2d_1{k}_1}{\Gamma \left(\alpha \right)}}{\parallel k\parallel }_{{\psi }_2}\left[\left(\right]\parallel b_2{\parallel }_Q+d_2·r\right)\right]<1$ allows us to apply Theorem 1. That ends the proof. □

4.1. Uniqueness of the Solution

Now, we may prove and discuss the uniqueness of the solutions of Eq. (1).

Theorem 3.

Assume that assumptions of Theorem 2 are verified, but replace the inequalities (3) with:

(C6)

$|h_i(v,0)|\le b_i\left(v\right),b_1\in E_{{\psi }_1}\left(\mathbb{I}\right),b_2\in E_Q\left(\mathbb{I}\right)$ and

$$|h_1(v,y)-h_1(v,z)|\le d_1{\psi }_1^{-1}\left(\psi \left(|y-z|\right)\right),|h_2(v,y)-h_2(v,z)|\le d_2{Q}^{-1}\left(\psi \left(|y-z|\right)\right),y,z\in {\Omega }_r,$$

where $d_i\ge 0,$ and ${\Omega }_r$ is as in Theorem 2 for $i=1,2.$

(C7)

Assume that

$$\begin{array}{ccc}& & {\displaystyle \frac{2k_1{\parallel k\parallel }_{{\psi }_2}}{\Gamma \left(\alpha \right)}}\left(d_2\left(\parallel b_1{\parallel }_{{\psi }_1}+d_1·r\right)+d_1\left(\parallel b_2{\parallel }_{{\psi }_2}+d_2·r\right)\right)<1,\hfill \end{array}$$

where r is given in assumption (C5), then (1) has a unique solution $y\in E_{\psi }$ in ${\Omega }_r$.

Proof. 

Using assumption (C6), we get

$$\begin{array}{ccc}\hfill \left|\right|h_1(v,y)|-|h_1(v,0)\left|\right|& \le & |h_1(v,x)-h_1(v,0)|\le d_1{\psi }_1^{-1}\left(\psi \left(y\right)\right)\hfill \\ \hfill \Rightarrow |h_1(v,y)|& \le & |h_1(v,0)|+d_1{\psi }_1^{-1}\left(\psi \left(y\right)\right)\le b_1\left(v\right)+d_1{\psi }_1^{-1}\left(\psi \left(y\right)\right).\hfill \end{array}$$

Similarly, $|h_2(v,y)|\le b_2\left(v\right)+d_2{Q}^{-1}\left(\psi \left(y\right)\right)$. Thus, Theorem 2 implies that, there exists a.e. nondecreasing solution $y\in E_{\psi }$ of (1) in ${\Omega }_r$.

Next, Let $y,z\in {\Omega }_r$ be two distinct solutions of equation (1), then by using the inequalities (4) and assumption (C6), we obtain

$$\begin{array}{ccc}\hfill {\parallel y-z\parallel }_{\psi }& \le & \parallel F_{h_1}\left(y\right)A\left(y\right)-F_{h_1}{\left(z\right)A\left(z\right)\parallel }_{\psi }\hfill \\ & \le & \parallel F_{h_1}\left(y\right)A\left(y\right)-F_{h_1}{\left(y\right)A\left(z\right)\parallel }_{\psi }+{\parallel F_{h_1}\left(y\right)A\left(z\right)-F_{h_1}\left(z\right)A\left(z\right)\parallel }_{\psi }\hfill \\ & \le & k_1\parallel F_{h_1}{\left(y\right)\parallel }_{{\psi }_1}{\parallel A\left(y\right)-A\left(z\right)\parallel }_{{\psi }_2}+k_1\parallel F_{h_1}\left(y\right)-F_{h_1}{\left(z\right)\parallel }_{{\psi }_1}{\parallel A\left(z\right)\parallel }_{{\psi }_2}\hfill \\ & \le & k_1\parallel b_1+d_1{\psi }_1^{-1}\left(\psi \left(y\right)\right){\parallel }_{{\psi }_1}\parallel J_g^{\alpha }\left(F_{h_2}\left(y\right)-F_{h_2}\left(z\right)\right){\parallel }_{{\psi }_2}+k_1\parallel d_1{\psi }_1^{-1}\left(\psi \left(|y-z|\right)\right){\parallel }_{{\psi }_1}\parallel J_g^{\alpha }{F}_{h_2}\left(z\right){\parallel }_{{\psi }_2}\hfill \\ & \le & k_1\left(\parallel b_1{\parallel }_{{\psi }_1}+d_1·r\right){\displaystyle \frac{{2\parallel k\parallel }_{{\psi }_2}}{\Gamma \left(\alpha \right)}}\parallel F_{h_2}\left(y\right)-F_{h_2}\left(z\right){\parallel }_Q+d_1{k}_1{\parallel y-z\parallel }_{\psi }{\displaystyle \frac{{2\parallel k\parallel }_{{\psi }_2}}{\Gamma \left(\alpha \right)}}\parallel F_{h_2}\left(z\right){\parallel }_Q\hfill \\ & \le & {\displaystyle \frac{2k_1{\parallel k\parallel }_{{\psi }_2}}{\Gamma \left(\alpha \right)}}\left(\parallel b_1{\parallel }_{{\psi }_1}+d_1·r\right)\parallel d_2{Q}^{-1}\left(\psi \left(\right|y-z\left|\right)\right){\parallel }_Q+{\displaystyle \frac{2d_1{k}_1{\parallel k\parallel }_{{\psi }_2}}{\Gamma \left(\alpha \right)}}{\parallel y-z\parallel }_{\psi }\parallel b_2+d_2{Q}^{-1}\left(\psi \left(\right|z\left|\right)\right){\parallel }_Q\hfill \\ & \le & {\displaystyle \frac{2k_1{d}_2{\parallel k\parallel }_{{\psi }_2}}{\Gamma \left(\alpha \right)}}\left(\parallel b_1{\parallel }_{{\psi }_1}+d_1·r\right)\parallel y-z{\parallel }_{\psi }+{\displaystyle \frac{2d_1{k}_1{\parallel k\parallel }_{{\psi }_2}}{\Gamma \left(\alpha \right)}}\parallel y-z{\parallel }_{\psi }\left(\parallel b_2{\parallel }_{{\psi }_2}+d_2·r\right)\hfill \\ & =& {\displaystyle \frac{2k_1{\parallel k\parallel }_{{\psi }_2}}{\Gamma \left(\alpha \right)}}\left(d_2\left(\parallel b_1{\parallel }_{{\psi }_1}+d_1·r\right)+d_1\left(\parallel b_2{\parallel }_{{\psi }_2}+d_2·r\right)\right)\parallel y-z{\parallel }_{\psi }.\hfill \end{array}$$

The above estimate with the assumption (C7) concludes the proof.

□

5. Conclusions

In the article, we prove and illustrate several novel features of the g-fractional type operator, encompassing boundedness, monotonicity, and continuity within Orlicz spaces $L_{\psi }$. The g-fractional operator combines and unifies many forms of fractional operators such as the Hadamard, Riemann-Liouville, and Erdlyi–Kober fractional operators into one form.