On Erdélyi–Kober Fractional Operator and Quadratic Integral Equations in Orlicz Spaces

Abstract

We provide and prove some new fundamental properties of the Erdélyi–Kober ($\mathcal{EK}$) fractional operator, including monotonicity, boundedness, acting, and continuity in both Lebesgue spaces ($L_p$) and Orlicz spaces ($L_{\phi }$). We employ these properties with the concept of the measure of noncompactness ($\mathcal{MNC}$) associated with the fixed-point hypothesis ($\mathcal{FPT}$) in solving a quadratic integral equation of fractional order in $L_p,p\ge 1$ and $L_{\phi }$. Finally, we provide a few examples to support our findings. Our suppositions can be successfully applied to various fractional problems.

1. Introduction

Fractional integral and differential operators perform an essential role in various fields of science, including biology, economics, physics, engineering, mathematical physics, electrical circuits, viscoelasticity, earthquakes, traffic models, fluid dynamics, and electro-chemistry (cf. [1,2,3,4,5]).

The advantage of studying fractional integrals and derivatives compared with classical (integer-order) ones is that they provide the description of memory and hereditary characteristics associated with numerous materials and processes.

Fractional operators can take a variety of forms, including ($\mathcal{RL}$) Riemann–Liouville, Caputo, Hadamard, ($\mathcal{EK}$) Erdélyi–Kober, or Riesz fractional operators that have been proposed in a wide number of manuscripts (see [6,7,8,9,10,11]. Compared to ($\mathcal{RL}$) operators, the Erdélyi–Kober fractional integral is a generalization and modification of the ($\mathcal{RL}$) fractional integral and it is more capable of representing the memory property.

Quadratic integral equations, including ($\mathcal{EK}$) Erdélyi–Kober operators, should be used more effectively in the kinetic theory of gases [12], the neutron transport [13], radiative transfer, and in the traffic theory [14].

The goal of this paper is to present and prove some fundamental properties of the ($\mathcal{EK}$) Erdélyi–Kober fractional operator, including monotonicity, boundedness, acting, and continuity, in both Lebesgue spaces $L_p,p\ge 1$ and Orlicz spaces $L_{\phi }$. These properties will then be applied to analyze the integral equation

$$z\left(\nu \right)=g\left(\nu \right)+f_1(\nu ,z\left(\nu \right))+f_2\left(\nu ,\frac{\beta h_1(\nu ,z\left(\nu \right))}{\Gamma \left(\alpha \right)}·{\int }_0^{\nu }\frac{{\tau }^{\beta -1}{h}_2(\tau ,z\left(\tau \right))}{{({\nu }^{\beta }-{\tau }^{\beta })}^{1-\alpha }}d\tau \right),\nu \in [0,\rho ],$$

(1)

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

We concentrate on establishing assumptions that enable us to analyze the singular integral Equation (1) with exponential growth in addition to polynomial growth. As a result, we consider our problem in the $L_p,p\ge 1$ and $L_{\phi }$ spaces.

These are motivated by statistical physics and physics models (cf. [15,16]). For instance, the integral equation produced by the thermodynamics problem

$$z\left(\nu \right)+{\int }_{I}k(\nu ,\tau )·e^{z\left(\tau \right)}d\tau =0$$

involves exponential nonlinearities (cf. [17]).

Recall that the quadratic integral equations were inspected in Orlicz spaces in [18,19,20] and in $L_p$ spaces [9,21,22,23] by utilizing the measure of noncompactness ($\mathcal{MNC}$) associated with Darbo’s fixed-point hypothesis ($\mathcal{FPT}$) across various sets of assumptions.

The author in [24] provided some fundamental properties of ($\mathcal{RL}$) fractional integral operator and discussed the solutions of the equation

$$z\left(\nu \right)=g\left(\nu \right)+G\left(z\right)\left(\nu \right){\int }_0^{\nu }\frac{{(\nu -\tau )}^{\alpha -1}}{\Gamma \left(\alpha \right)}f(\tau ,z\left(\tau \right))d\tau ,\nu \in [0,d],0<\alpha <1$$

in Orlicz spaces $L_{\phi }$.

In [25], the author provided some fundamental properties of the Hadamard fractional operator in Orlicz spaces and employed them in solving the following equation:

$$z\left(\nu \right)=G_2\left(z\right)\left(\nu \right)+\frac{G_1\left(z\right)\left(\nu \right)}{\Gamma \left(\alpha \right)}{\int }_1^{\nu }{\left(log\frac{\nu }{\tau }\right)}^{\alpha -1}{G}_2\left(z\right)\left(\tau \right)d\tau ,\nu \in [1,e],0<\alpha <1.$$

This article is motivated by illustrating and proving some fundamental properties of the ($\mathcal{EK}$) Erdélyi–Kober fractional operator, including monotonicity, boundedness, acting, and continuity, in both Lebesgue and Orlicz spaces. We employ these properties with the help of the measure of noncompactness ($\mathcal{MNC}$) associated with Darbo’s fixed-point hypothesis ($\mathcal{FPT}$) to exhibit the solvability of the integral equation of non-integer order (1) in the investigated spaces. We provide a few examples that support our theorems.

2. Preliminaries

Let $\mathbb{I}=[0,\rho ]\subset [0,\infty )={\mathbb{R}}^{+}$ and $\mathbb{R}=(-\infty ,\infty )$. The function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a Young function if

$$\psi \left(\nu \right)={\int }_0^{\nu }h\left(\tau \right)d\tau ,for\nu \ge 0,$$

where $h:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a left-continuous and increasing function that is neither identically infinite nor zero on ${\mathbb{R}}^{+}$. The pair $(\psi ,N)$ is called a complementary pair of Young functions if $N\left(z\right)={sup}_{y\ge 0}(zy-\psi \left(z\right))$.

The function $\psi$ is said to be an N-function if it has a finite-valued with ${lim}_{\nu \to 0}\frac{\psi \left(\nu \right)}{\nu }=0$, ${lim}_{\nu \to \infty }\frac{\psi \left(\nu \right)}{\nu }=\infty$, and $\psi \left(\nu \right)>0$ if $\nu >0$ ($\psi \left(\nu \right)=0\iff \nu =0$).

The Orlicz space $L_{\psi }=L_{\psi }\left(\mathbb{I}\right)$ as the space of all measurable functions $z:\mathbb{I}\to \mathbb{R}$ s.t.

$${\parallel z\parallel }_{\psi }=\underset{\lambda >0}{inf}\left\{{\int }_{\mathbb{I}}\psi \left(\frac{z\left(\tau \right)}{\lambda }\right)d\tau \le 1\right\}$$

is finite. It is worth recalling that for any Young function $\psi$, we have $\psi (\nu +\tau )\le \psi \left(\nu \right)+\psi \left(\tau \right)$ and $\psi \left(k\nu \right)\le \rho \psi \left(\nu \right)$, which hold for any $\nu ,\tau \in \mathbb{R}$, and $k\in [0,1]$. For more properties of Orlicz spaces and Young functions, see, e.g., [15,24,26].

Let $E_{\psi }\left(\mathbb{I}\right)$ be the set of all bounded functions in $L_{\psi }\left(\mathbb{I}\right)$ with absolutely continuous norms.

Furthermore, we obtain $L_{\psi }=E_{\psi }$ if $\psi$ verifies the ${\Delta }_2$ condition, i.e.:

$$\left({\Delta }_2\right)\mathrm{there\; exists}\omega ,{\nu }_0\ge 0s.t.\psi \left(2\nu \right)\le \omega \psi \left(\nu \right),\nu \ge {\nu }_0.$$

Note that the Lebesgue spaces $L_p\left(\mathbb{I}\right)$ can be employed as Orlicz spaces $L_{{\psi }_p}\left(\mathbb{I}\right)$ with N-function ${\psi }_p={\tau }^p,p>1$ verifies the ${\Delta }_2$ condition.

Proposition 1

([26]). Let ψ be a Young function, then for any $\alpha \in (0,1)$ and $\tau \in {\mathbb{R}}^{+}$, the set

$$\Psi \left(\tau \right)=\left\{ϵ>0:\frac{1}{\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}{\int }_0^{{\tau }^{\beta }{\sigma }^{\frac{1}{1-\alpha }}}\psi \left(u^{\alpha -1}\right)du\le {\sigma }^{\frac{1}{1-\alpha }}\right\}\ne \varnothing ,\sigma >0,\beta >0$$

is continuous and increases functions with $\Psi \left(0\right)=0$.

Lemma 1

([15] Theorem 17.6). Let the function $f(\nu ,z):\mathbb{I}\times \mathbb{R}\to \mathbb{R}$ verify Carathéodory conditions (i.e., it is measurable in ν for any $z\in \mathbb{R}$ and continuous in z for almost all $\nu \in \mathbb{I}$). The superposition operator $F_f=f(\nu ,z):E_{{\psi }_1}\to L_{\psi }=E_{\psi }$ is continuous and bounded if

$$\left|f(\nu ,z)\right|\le m\left(\nu \right)+b{\psi }^{-1}\left({\psi }_1\left(z\right)\right),\nu \in \mathbb{I},z\in \mathbb{R}$$

and $F_f:E_{\psi }\to E_{\psi }$ is continuous and bounded if

$$\left|f\right(\nu ,z\left)\right|\le m\left(\nu \right)+b\left|z\right|,$$

where $b\ge 0$, $m\in L_{\psi }$ and the N-function $\psi \left(z\right)$ verifies the ${\Delta }_2$ condition.

Lemma 2

([27] Theorem 10.2). Let φ, ${\phi }_1$, and ${\phi }_2$ be arbitrary N-functions. The next statements are equivalent:

1. 

For every functions $z_1\in L_{{\phi }_1}$ and $z_2\in L_{{\phi }_2}$, $z_1·z_2\in L_{\phi }$.

2. 

$\exists k>0$ s.t. for all measurable $z_1,z_2$ on $\mathbb{I}$, we have $\parallel z_1{z}_2{\parallel }_{\phi }\le k\parallel z_1{\parallel }_{{\phi }_1}{\parallel z_1\parallel }_{{\phi }_2}$.

3. 

$\exists C>0$, $t_0\ge 0$ s.t. for all $\nu ,\tau \ge t_0,\phi \left(\frac{\tau \nu }{C}\right)\le {\phi }_1\left(\tau \right)+{\phi }_2\left(\nu \right)$.

4. 

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

Let “$meas$” be the Lebesgue measure in $\mathbb{R}$ and $S=S\left(\mathbb{I}\right)$ be the set of Lebesgue measurable functions on $\left(\mathbb{I}\right)$. The set S combined with the metric

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

represents a complete space. The convergence concerning d is identical to the convergence in measure on $\mathbb{I}$ (cf. Proposition 2.14 in [28]). The compactness in S is defined as “compactness in measure”.

Lemma 3

( [18]). Suppose that for a bounded set $Z\subset L_{\psi }\left(\mathbb{I}\right)$, there exists a family ${\left({\Omega }_c\right)}_{0\le c\le \rho }\subset \mathbb{I}$ s.t. meas ${\Omega }_c=c$ for every $c\in [0,\rho ]$, and for every $z\in Z$,

$$z\left({\tau }_1\right)\ge z\left({\tau }_2\right),({\tau }_1\in {\Omega }_c,{\tau }_2\notin {\Omega }_c).$$

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

Definition 1

([29]). Let $\varnothing \ne Z\subset L_{\psi }$ be bounded. The Hausdorff measure of noncompactness ($\mathcal{MNC}$) ${\beta }_H\left(X\right)$ is given by

$${\beta }_H\left(Z\right)=inf\{r>0:\exists Y\subset L_{\psi }\mathrm{such\; that}Z\subset Y+B_r\},$$

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

For any $ϵ>0$, let c be a measure of equi-integrability of the set $Z\in L_{\psi }\left(\mathbb{I}\right)$ (cf. Definition 3.9 in [28] or [30]):

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

where ${\chi }_D$ refers to the characteristic function $D\subset \mathbb{I}$.

Lemma 4

([18,30]). Let $\varnothing \ne Z\subset L_{\psi }$ be a bounded set and compact in measure. Then:

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

Theorem 1

([29]). Let $\varnothing \ne W\subset L_{\psi }$ be a closed, convex, and bounded set and let $T:W\to W$ be continuous and a contraction concerning ${\beta }_H$, i.e.:

$${\beta }_H\left(T\left(Z\right)\right)\le k{\beta }_H\left(Z\right),k\in [0,1)$$

for any $\varnothing \ne Z\subset W$. Then, T has at least one fixed point in W.

3. Erdélyi–Kober Fractional Integral Operator

We recall and establish some concepts and properties of the ($\mathcal{EK}$) Erdélyi–Kober fractional integral operator in $L_{\psi }\left(\mathbb{I}\right)$ and in $L_p[0,1]$.

Definition 2

([1,4,31]). The ($\mathcal{EK}$) Erdélyi–Kober fractional integral operator $J_{\beta }^{\alpha },\alpha >0,\beta >0$, with a well-defined function $z\left(\nu \right)$ takes the form

$$J_{\beta }^{\alpha }z\left(\nu \right)=\frac{\beta {\nu }^{-\beta \alpha }}{\Gamma \left(\alpha \right)}{\int }_0^{\nu }\frac{{\tau }^{\beta -1}z\left(\tau \right)}{{({\nu }^{\beta }-{\tau }^{\beta })}^{1-\alpha }}d\tau =\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^1{(1-\tau )}^{\alpha -1}z\left(\nu {\tau }^{\frac{1}{\beta }}\right)d\tau$$

(2)

or the form

$${\nu }^{\beta \alpha }{J}_{\beta }^{\alpha }z\left(\nu \right)=\frac{\beta }{\Gamma \left(\alpha \right)}{\int }_0^{\nu }\frac{{\tau }^{\beta -1}z\left(\tau \right)}{{({\nu }^{\beta }-{\tau }^{\beta })}^{1-\alpha }}d\tau .$$

Remark 1.

• If $\beta =1$, the operator (2) refers to Riemann–Liouville ($\mathcal{RL}$) fractional operator, which has been analyzed (cf. [1,2]):

  $${\nu }^{\alpha }{J}_1^{\alpha }z\left(\nu \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\nu }\frac{z\left(\tau \right)}{{(\nu -\tau )}^{1-\alpha }}d\tau .$$
• If $\alpha =1$ and $\beta =1$, the operator (2) refers to the Hardy–Littlewood (Cesaro) operator

  $$J_1^{1}z\left(\nu \right)=\frac{1}{\nu }{\int }_0^{\nu }z\left(\tau \right)d\tau ,$$

  which has been analyzed in (cf. [32]).
• If $\beta =2$, the operator (2) refers to the Erdélyi–Kober’s ($\mathcal{EK}$) fractional integral operator $J_2^{\alpha }$ (Sneddon [33]):

  $$J_2^{\alpha }z\left(\nu \right)=\frac{2{\nu }^{-2\alpha }}{\Gamma \left(\alpha \right)}{\int }_0^{\nu }\frac{z\left(\tau \right)}{{({\nu }^2-{\tau }^2)}^{1-\alpha }}\tau d\tau .$$

Next, we discuss the monotonicity property of the ($\mathcal{EK}$) operator (2).

Lemma 5.

The operator $J_{\beta }^{\alpha },\alpha >0,\beta >0$ transforms non-negative and, consequently, a.e. nondecreasing functions into functions have the same properties.

Proof. 

Let ${\nu }_1,{\nu }_2\in \mathbb{I},{\nu }_1\le {\nu }_2$, and z be a.e. a nonnegative, nondecreasing function; then, we can obtain:

$$\begin{array}{ccc}\hfill J_{\beta }^{\alpha }z\left({\nu }_1\right)& =& \frac{\beta {\nu }_1^{-\beta \alpha }}{\Gamma \left(\alpha \right)}{\int }_0^{{\nu }_1}\frac{{\tau }^{\beta -1}z\left(\tau \right)}{{({\nu }_1^{\beta }-{\tau }^{\beta })}^{1-\alpha }}d\tau \hfill \\ & =& \frac{1}{\Gamma \left(\alpha \right)}{\int }_0^1{(1-\tau )}^{\alpha -1}z({\nu }_1{\tau }^{\frac{1}{\beta }})d\tau \hfill \\ & \le & \frac{1}{\Gamma \left(\alpha \right)}{\int }_0^1{(1-\tau )}^{\alpha -1}z({\nu }_2{\tau }^{\frac{1}{\beta }})d\tau =J_{\beta }^{\alpha }z\left({\nu }_2\right).\hfill \end{array}$$

Therefore, $0\le J_{\beta }^{\alpha }z\left({\nu }_1\right)\le J_{\beta }^{\alpha }z\left({\nu }_2\right)$ for ${\nu }_1\le {\nu }_2$. □

Lemma 6.

Let φ be an N-function and the pair $(\psi ,N)$ represent a complementary pair of N-functions, where ψ satisfies ${\int }_0^{{\nu }^{\beta }}\psi \left(u^{\alpha -1}\right)du<\infty ,\alpha \in (0,1),\beta >0.$ Furthermore, suppose that

$$k\left(\nu \right)=\frac{{\sigma }^{\frac{1}{\alpha -1}}}{\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}{\int }_0^{{\nu }^{\beta }{\sigma }^{\frac{1}{1-\alpha }}}\psi \left(u^{\alpha -1}\right)du\in E_{\phi }\left(\mathbb{I}\right),\sigma =\frac{ϵ}{\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}$$

for a.e. $\nu \in \mathbb{I}$ and $ϵ>0$, then $J_{\beta }^{\alpha }:L_N\left(\mathbb{I}\right)\to L_{\phi }\left(\mathbb{I}\right)$ is continuous.

Proof. 

Assume that

$$K(\nu ,\tau )=\left\{\begin{array}{c}\frac{{({\nu }^{\beta }-{\tau }^{\beta })}^{\alpha -1}\beta {\tau }^{\beta -1}}{\Gamma \left(\alpha \right)}\mathrm{if}\tau \in [0,\nu ],\nu >0,\\ 0\mathrm{otherwise}.\end{array}\right.$$

Therefore, by Hölder inequality and for $z\in L_N\left(\mathbb{I}\right)$, we obtain:

$$\begin{array}{ccc}\hfill |{\nu }^{\beta \alpha }{J}_{\beta }^{\alpha }z\left(\nu \right)|& =& \left|{\int }_0^{\infty }K(\nu ,\tau )z\left(\tau \right)d\tau \right|\hfill \\ & \le & {2\parallel K(\nu ,·)\parallel }_{\psi }{\parallel z\parallel }_N\hfill \\ & \le & \frac{2}{\Gamma \left(\alpha \right)}\underset{ϵ>0}{inf}\left\{{\int }_{\mathbb{I}}\psi \left(\frac{{({\nu }^{\beta }-{\tau }^{\beta })}^{\alpha -1}\beta {\tau }^{\beta -1}}{ϵ}\right)d\tau \le 1\right\}{\parallel z\parallel }_N\hfill \\ & =& \frac{2}{\Gamma \left(\alpha \right)}\underset{ϵ>0}{inf}\left\{{\int }_{\mathbb{I}}\psi \left(\frac{{({\nu }^{\beta }-{\tau }^{\beta })}^{\alpha -1}\beta {\tau }^{\beta -1}\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}{ϵ·\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}\right)d\tau \le 1\right\}{\parallel z\parallel }_N\hfill \\ & \le & \frac{2}{\Gamma \left(\alpha \right)}\underset{ϵ>0}{inf}\left\{{\int }_{\mathbb{I}}\psi \left(\frac{{({\nu }^{\beta }-{\tau }^{\beta })}^{\alpha -1}\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}{ϵ}\right)\frac{\beta {\tau }^{\beta -1}}{\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}d\tau \le 1\right\}{\parallel z\parallel }_N.\hfill \end{array}$$

Put $u=\left({\nu }^{\beta }-{\tau }^{\beta }\right){\sigma }^{\frac{1}{1-\alpha }}$, where $\sigma =\frac{ϵ}{\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}$, we have

$$\begin{array}{ccc}\hfill \parallel {\nu }^{\beta \alpha }{J}_{\beta }^{\alpha }{z\parallel }_{\phi }& \le & \frac{2}{\Gamma \left(\alpha \right)}{∥\underset{ϵ>0}{inf}\left\{\frac{1}{\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}{\int }_0^{{\nu }^{\beta }{\sigma }^{\frac{1}{1-\alpha }}}\psi \left(u^{\alpha -1}\right)du\le {\sigma }^{\frac{1}{1-\alpha }}\right\}∥}_{\phi }{\parallel z\parallel }_N\hfill \\ & \le & \frac{2}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{\phi }{\parallel z\parallel }_N,\hfill \end{array}$$

where $k\left(\nu \right)\in E_{\phi }\left(\mathbb{I}\right)$. Then, by utilizing [15] Lemma 16.3 and Proposition 1, we have $J_{\beta }^{\alpha }:L_N\left(\mathbb{I}\right)\to L_{\phi }\left(\mathbb{I}\right)$ and is continuous. □

For the acting condition of the operator $J_{\beta }^{\alpha }$ in $L_p[0,1]$, we shall rewrite [1] Lemma 2.28 in the form:

Lemma 7.

Let $\alpha >0,\beta >\frac{1}{p},p\ge 1$, then the operator $J_{\beta }^{\alpha }:L_p[0,1]\to L_p[0,1]$ continuously with

$$\parallel J_{\beta }^{\alpha }{z\parallel }_p\le \frac{\Gamma (1-\frac{1}{p\beta })}{\Gamma (\alpha +1-\frac{1}{p\beta })}{\parallel z\parallel }_p.$$

Proof. 

Let $\alpha >0,\beta >\frac{1}{p},p\ge 1,$ and by using the generalized Minkowsky inequality, we have

$$\begin{array}{cc}\hfill \Gamma \left(\alpha \right)\parallel J_{\beta }^{\alpha }z{\parallel }_p& =\parallel \beta {\nu }^{-\beta \alpha }{\int }_0^{\nu }\frac{{\tau }^{\beta -1}z\left(\tau \right)}{{({\nu }^{\beta }-{\tau }^{\beta })}^{1-\alpha }}d\tau {\parallel }_p=\parallel {\int }_0^1{(1-\tau )}^{\alpha -1}z(\nu {\tau }^{\frac{1}{\beta }})d\tau {\parallel }_p\hfill \\ \hfill & \le {\int }_0^1{(1-\tau )}^{\alpha -1}{\left({\int }_0^1|z(\nu {\tau }^{\frac{1}{\beta }}){|}^{p}d\nu \right)}^{\frac{1}{p}}d\tau \hfill \\ \hfill & \le {\int }_0^1{(1-\tau )}^{\alpha -1}{\left({\int }_0^1|z\left(u\right){|}^p\frac{du}{{\tau }^{\frac{1}{\beta }}}\right)}^{\frac{1}{p}}d\tau \hfill \\ \hfill & ={\int }_0^1{(1-\tau )}^{\alpha -1}{\tau }^{\frac{-1}{p\beta }}d\tau ·{\parallel z\parallel }_p\hfill \\ \hfill & =\frac{\Gamma \left(\alpha \right)\Gamma (1-\frac{1}{p\beta })}{\Gamma (\alpha +1-\frac{1}{p\beta })}·{\parallel z\parallel }_p.\hfill \end{array}$$

Then, $J_{\beta }^{\alpha }:L_p[0,1]\to L_p[0,1]$ and is continuous. □

4. Main Results

Let $\mathbb{I}=[0,\rho ]$ and Equation (1) should be written in the following format:

$$z=B\left(z\right)=g+F_{f_1}\left(z\right)+F_{f_2}U\left(z\right),$$

where

$$U\left(z\right)=F_{h_1}\left(z\right)·A\left(z\right),A\left(z\right)\left(\nu \right)={\nu }^{\beta \alpha }{J}_{\beta }^{\alpha }{F}_{h_2}\left(z\right)\left(\nu \right),$$

such that $F_{f_i},F_{h_i},(i=1,2)$ are the superposition operators and ${\nu }^{\beta \alpha }{J}_{\beta }^{\alpha }$ is as in Definition 2. We will describe our existence theorems in the spaces $L_{\phi }$ and $L_p$.

4.1. The Case of Orlicz Spaces

This case allows us to use more general growth conditions for the presented functions.

Theorem 2.

Suppose that ${\phi }_1,{\phi }_2,and\phi$ are N-functions and the pair $(\psi ,N)$ is a complementary pair of N-functions, in which $N,\phi ,{\phi }_1$ verify the ${\Delta }_2$ condition and ${\int }_0^{{\nu }^{\beta }}\psi \left(u^{\alpha -1}\right)du<\infty ,\alpha \in (0,1),\beta >0$ and that:

(G1) 

$\exists k_1>0$ s.t. for $z_1\in L_{{\phi }_1}\left(\mathbb{I}\right)$ and $z_2\in L_{{\phi }_2}\left(\mathbb{I}\right)$ we obtain $\parallel z_1{z}_2{\parallel }_{\phi }\le k_1\parallel z_1{\parallel }_{{\phi }_1}{\parallel z_2\parallel }_{{\phi }_2}$.

(C1) 

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

(C2) 

$f_i,h_i:\mathbb{I}\times \mathbb{R}\to \mathbb{R}$ verify Carathéodory conditions and $(\tau ,z)\to f_i(\tau ,z),(\tau ,z)\to h_i(\tau ,z),i=1,2$ are nondecreasing.

(C3) 

$\exists e_i,d_i\ge 0$ and functions $a_i\in E_{\phi }\left(\mathbb{I}\right),b_1\in E_{{\phi }_1}\left(\mathbb{I}\right)$, and $b_2\in E_N\left(\mathbb{I}\right)$, such that

$$|h_1(\tau ,z)|\le b_1\left(\tau \right)+d_1{\phi }_1^{-1}(\phi (z)),\left|h_2(\tau ,z)\right|\le b_2\left(\tau \right)+d_2{N}^{-1}(\phi (z)),$$

and

$$|f_i(\tau ,z)|\le a_i\left(\tau \right)+e_i{\parallel z\parallel }_{\phi },i=1,2.$$

(C4) 

Suppose that for a.e. $\nu \in \mathbb{I}$, there exists $ϵ>0$, such that

$$k\left(\nu \right)=\frac{{\sigma }^{\frac{1}{\alpha -1}}}{\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}{\int }_0^{{\nu }^{\beta }{\sigma }^{\frac{1}{1-\alpha }}}\psi \left(u^{\alpha -1}\right)du\in E_{\phi }\left(\mathbb{I}\right),\sigma =\frac{ϵ}{\parallel \beta {\tau }^{\beta -1}{\parallel }_{\psi }}.$$

(C5) 

Suppose that, there is $r>0$ on the interval $I_0=[0,{\rho }_0]\subset \mathbb{I}$ satisfying

$${\int }_{I_0}\phi \left(\left|g\left(\nu \right)\right|+|a_1\left(\nu \right)|+|a_2\left(\nu \right)|+e_{1}r+\frac{2e_2{k}_1}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}\left(\parallel b_1{\parallel }_{{\phi }_1}+d_{1}r\right)(\parallel b_2{\parallel }_N+d_{2}r)\right)d\nu \le r$$

and

$$\left(e_1+\frac{2e_2{k}_1{d}_1}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}(\parallel b_2{\parallel }_N+d_2·r)\right)<1.$$

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

Proof. 

Step I. Assumptions (C2), (C3), and Lemma 1 imply that $F_{f_i}:E_{\phi }\left(\mathbb{I}\right))\to E_{\phi }\left(\mathbb{I}\right)),i=1,2$ and $F_{h_1}:E_{\phi }\left(\mathbb{I}\right))\to {Ł}_{{\phi }_1}\left(\mathbb{I}\right),F_{h_2}:E_{\phi }\left(\mathbb{I}\right))\to {Ł}_N\left(\mathbb{I}\right)$ are continuous. The operator $A={\nu }^{\beta \alpha }{J}_{\beta }^{\alpha }{F}_{h_2}:E_{\phi }\left(\mathbb{I}\right)\to E_{{\phi }_2}\left(\mathbb{I}\right)$ is continuous (thanks to Lemma 6). By assumption (N1), the operator $U:E_{\phi }\left(\mathbb{I}\right)\to E_{\phi }\left(\mathbb{I}\right)$, and by $\left(C1\right)$, $B:E_{\phi }\left(\mathbb{I}\right)\to E_{\phi }\left(\mathbb{I}\right)$, and they are continuous.

Step II. We shall now demonstrate that the operator B is bounded in $E_{\phi }\left(\mathbb{I}\right)$.

Let W denote the closure of the set $\{z\in E_{\phi }\left(I_0\right):{\int }_0^{{\rho }_0}\phi \left(\right|z\left(\tau \right)\left|\right)d\tau \le r-1\}$. Obviously, W does not represent a ball in $E_{\phi }\left(I_0\right)$, but $W\subset B_r\left(E_{\phi }\left(I_0\right)\right)$ (cf. [15] p. 222) and the set $\overline{W}$ is a closed, convex, and bounded subset of $E_{\phi }\left(I_0\right)$.

For arbitrary $z\in W$ and $\nu \in I_0$, and by utilizing Theorem 10.5 with k = 1 [15], we obtain:

$$\parallel {\phi }_1^{-1}\left(\phi \left(\right|z\left|\right)\right){\parallel }_{{\phi }_1}\le {\parallel z\parallel }_{\phi }=1+{\int }_{I_0}\phi \left(z\left(\tau \right)\right)d\tau \mathrm{and}\parallel N^{-1}\left(\phi \left(z\right)\right){\parallel }_N\le {\parallel z\parallel }_{\phi }=1+{\int }_{I_0}\phi \left(z\left(\tau \right)\right)d\tau .$$

Therefore, by utilizing Lemma 6 and our assumptions, we have

$$\begin{array}{cc}\hfill \left|B\right(z\left)\right(\nu \left)\right|& \le \left|g\left(\nu \right)\right|+|F_{f_1}\left(z\right)\left(\nu \right)|+|F_{f_2}U\left(z\right)\left(\nu \right)|\hfill \\ \hfill & \le \left|g\left(\nu \right)\right|+|a_1\left(\nu \right)|+e_1{\parallel z\parallel }_{\phi }+|a_2\left(\nu \right)|+e_2{\parallel U\left(z\right)\parallel }_{\phi }\hfill \\ \hfill & \le \left|g\left(\nu \right)\right|+|a_1\left(\nu \right)|+|a_2\left(\nu \right)|+e_1{\parallel z\parallel }_{\phi }+e_2{k}_1\parallel F_{h_1}{\parallel }_{{\phi }_1}{\parallel A\left(z\right)\parallel }_{{\phi }_2}\hfill \\ \hfill & \le \left|g\left(\nu \right)\right|+|a_1\left(\nu \right)|+|a_2\left(\nu \right)|+e_1{\parallel z\parallel }_{\phi }+e_2{k}_1\parallel b_1+d_1{\phi }_1^{-1}(\phi (|z|)){\parallel }_{{\phi }_1}·{\parallel {\nu }^{\beta \alpha }{J}_{\beta }^{\alpha }{F}_{h_2}\left(z\right)\parallel }_{{\phi }_2}\hfill \\ \hfill & \le \left|g\left(\nu \right)\right|+|a_1\left(\nu \right)|+|a_2\left(\nu \right)|+e_1{\parallel z\parallel }_{\phi }\hfill \\ \hfill & +e_2{k}_1\left(\parallel b_1{\parallel }_{{\phi }_1}+d_1\parallel {\phi }_1^{-1}(\phi (|z|)){\parallel }_{{\phi }_1}\right)\frac{2}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}(\parallel b_2{\parallel }_N+d_2\parallel N^{-1}(\phi (|z|)\left){\parallel }_N\right)\hfill \\ \hfill & \le \left|g\left(\nu \right)\right|+|a_1\left(\nu \right)|+|a_2\left(\nu \right)|+e_1+e_1{\int }_{I_0}\phi \left(z\left(\tau \right)\right)d\tau \hfill \\ \hfill & +\frac{2e_2{k}_1}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}\left(\parallel b_1{\parallel }_{{\phi }_1}+d_1+d_1{\int }_{I_0}\phi \left(z\left(\tau \right)\right)d\tau \right)\left(\parallel b_2{\parallel }_N+d_2+d_2{\int }_{I_0}\phi \left(z\left(\tau \right)\right)d\tau \right)\hfill \\ \hfill & \le \left|g\left(\nu \right)\right|+|a_1\left(\nu \right)|+|a_2\left(\nu \right)|+e_1+e_1(r-1)\hfill \\ \hfill & +\frac{2e_2{k}_1}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}\left(\parallel b_1{\parallel }_{{\phi }_1}+d_1+d_1(r-1)\right)(\parallel b_2{\parallel }_N+d_2+d_2(r-1)).\hfill \end{array}$$

Therefore, by assumption (C5), we obtain

$$\begin{array}{cc}\hfill {\int }_{I_0}\phi \left(B\left(z\right)\left(\nu \right)\right)d\nu & \le {\int }_{I_0}\phi (|g(\nu )|+|a_1(\nu )|+|a_2(\nu )|+e_{1}r\hfill \\ \hfill & +\frac{2e_2{k}_1}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}\left(\parallel b_1{\parallel }_{{\phi }_1}+d_{1}r\right)(\parallel b_2{\parallel }_N+d_{2}r))d\nu \le r,\hfill \end{array}$$

then, $B\left(W\right)\subset W$ and, consequently, $B\left(\overline{W}\right)\subset \overline{B\left(W\right)}\subset \overline{W}=W$. Then, $B:W\to W$ is continuous on $W\subset B_r\left(E_{\phi }\left(I_0\right)\right)$.

Step III. Let $Q_r\subset W$ collect all a.e. nondecreasing functions on $I_0$. The set $Q_r\ne \varnothing$ is closed, convex, bounded, and compact in measure in $L_{\phi }\left(I_0\right)$ (cf. [19]).

Step IV. The operator B maintains the monotonicity of functions.

Take $z\in Q_r$, then z is a.e. nondecreasing on $I_0$ and, consequently, each $F_{f_i},F_{h_i}$ is also of the same type in virtue of assumption (C2). Further, A is a.e. nondecreasing on $I_0$ (see Lemma 5), then $U=F_{h_1}A$ is also of the same type. Assumption (C1) implies that $B:Q_r\to Q_r$ is continuous.

Step V. Next, we demonstrate that B is a contraction concerning ${\beta }_H$.

Assume that $\epsilon >0$ and a set $D\subset I_0$, with meas $D\le \epsilon$. Then, for $z\in Z$ and $\varnothing \ne Z\subset Q_r$, we obtain:

$$\begin{array}{cc}\hfill \parallel B\left(z\right)·{\chi }_D{\parallel }_{\phi }& \le \parallel g·{\chi }_D{\parallel }_{\phi }+\parallel a_1·{\chi }_D{\parallel }_{\phi }+e_1\parallel z·{\chi }_D{\parallel }_{\phi }+\parallel a_2·{\chi }_D{\parallel }_{\phi }+e_2{\parallel F_{h_1}A\left(z\right)·{\chi }_D\parallel }_{\phi }\hfill \\ \hfill & \le \parallel g·{\chi }_D{\parallel }_{\phi }+\parallel a_1·{\chi }_D{\parallel }_{\phi }+e_1\parallel z·{\chi }_D{\parallel }_{\phi }+\parallel a_2·{\chi }_D{\parallel }_{\phi }+e_2{k}_1\parallel F_{h_1}\left(z\right)·{\chi }_D{\parallel }_{{\phi }_1}{\parallel A\left(z\right)·{\chi }_D\parallel }_{{\phi }_2}\hfill \\ \hfill & \le \parallel g·{\chi }_D{\parallel }_{\phi }+\parallel a_1·{\chi }_D{\parallel }_{\phi }+e_1\parallel z·{\chi }_D{\parallel }_{\phi }+{\parallel a_2·{\chi }_D\parallel }_{\phi }\hfill \\ \hfill & +e_2{k}_1\parallel (b_1+d_1{\phi }_1^{-1}(\phi (|z|)\left)\right)·{\chi }_D{\parallel }_{{\phi }_1}·\parallel {\nu }^{\beta \alpha }{J}_{\beta }^{\alpha }{F}_{h_2}{\parallel }_{{\phi }_2}\hfill \\ \hfill & \le \parallel g·{\chi }_D{\parallel }_{\phi }+\parallel a_1·{\chi }_D{\parallel }_{\phi }+e_1\parallel z·{\chi }_D{\parallel }_{\phi }+{\parallel a_2·{\chi }_D\parallel }_{\phi }\hfill \\ \hfill & +e_2{k}_1(\parallel b_1·{\chi }_D{\parallel }_{{\phi }_1}+d_1\parallel {\phi }_1^{-1}(\phi (|z|))·{\chi }_D{\parallel }_{{\phi }_1})\hfill \\ \hfill & \times \frac{2}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}(\parallel b_2{\parallel }_N+d_2\parallel N^{-1}(\phi (|z|)\left){\parallel }_N\right)\hfill \\ \hfill & \le \parallel g·{\chi }_D{\parallel }_{\phi }+\parallel a_1·{\chi }_D{\parallel }_{\phi }+e_1\parallel z·{\chi }_D{\parallel }_{\phi }+{\parallel a_2·{\chi }_D\parallel }_{\phi }\hfill \\ \hfill & +\frac{2e_2{k}_1}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}(\parallel b_1·{\chi }_D{\parallel }_{{\phi }_1}+d_1{\parallel z·{\chi }_D\parallel }_{\phi })(\parallel b_2{\parallel }_N+d_2·r).\hfill \end{array}$$

Hence, taking into consideration that $g,a_1,a_2\in E_{\phi }$, $b_1\in E_{{\phi }_1}$, then

$$\underset{\epsilon \to 0}{lim}\{\underset{measD\le \epsilon }{sup}[\underset{z\in Z}{sup}\{\parallel g·{\chi }_D{\parallel }_{\phi }+\parallel a_1·{\chi }_D{\parallel }_{\phi }+\parallel a_2·{\chi }_D{\parallel }_{\phi }\left\}\right]\}=0$$

and

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

By definition of $c\left(Z\right)$, we obtain

$$c\left(B\left(Z\right)\right)\le \left(e_1+\frac{2e_2{k}_1{d}_1}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}(\parallel b_2{\parallel }_N+d_2·r)\right)c\left(Z\right).$$

From the properties established above, we can use Lemma 4 and obtain

$${\beta }_H\left(B\left(Z\right)\right)\le \left(e_1+\frac{2e_2{k}_1{d}_1}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}(\parallel b_2{\parallel }_N+d_2·r)\right){\beta }_H\left(Z\right).$$

The above inequality with $\left(e_1+\frac{2e_2{k}_1{d}_1}{\Gamma \left(\alpha \right)}{\parallel k\parallel }_{{\phi }_2}(\parallel b_2{\parallel }_N+d_2·r)\right)<1$ enables us to employ Theorem 1, which ends the proof. □

4.2. The Case of Lebesgue Spaces

The Lebesgue spaces $L_p\left(I\right),I=[0,1]$ are interesting Banach spaces with the usual norm ${\parallel z\parallel }_p={\left({\int }_0^1{\left|z\left(\tau \right)\right|}^{p}d\tau \right)}^{\frac{1}{p}},$ which demonstrate important and more general solutions to Equation (1) than the former ones. Here, we will consider Equation (1) with the operator A in the form

$$A\left(z\right)\left(\nu \right)=J_{\beta }^{\alpha }{F}_{h_2}\left(z\right)\left(\nu \right)=\frac{\beta {\nu }^{-\beta \alpha }}{\Gamma \left(\alpha \right)}{\int }_0^{\nu }\frac{{\tau }^{\beta -1}{h}_2(\tau ,z\left(\tau \right))}{{({\nu }^{\beta }-{\tau }^{\beta })}^{1-\alpha }}d\tau .$$

Theorem 3.

Assume that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2},p\ge 1$ and that:

(i) 

$g\in L_p\left(I\right)$ be a.e. nondecreasing functions on I.

(ii) 

$h_i,f_i:I\times \mathbb{R}\to \mathbb{R}$ verify Carathéodory conditions and $(\tau ,z)\to f_i(\tau ,z),(\tau ,z)\to h_i(\tau ,z),i=1,2$ are nondecreasing.

(iii) 

There exist $d_i,e_i\ge 0$ and functions $a_i\in L_p\left(I\right)$ and $b_i\in L_{p_i}\left(I\right)$ such that

$$|f_i(\tau ,z)|\le a_i\left(\tau \right)+e_i\left|z\right|and|h_i(\tau ,z)|\le b_i\left(\tau \right)+d_i{\left|z\right|}^{\frac{p}{p_i}},i=1,2.$$

(iv) 

For $\alpha \in (0,1),\beta >\frac{1}{p}$, there exists $R\ge 0$, fulfilling

$${\parallel g\parallel }_p+\parallel a_1{\parallel }_p+e_1·R+{\parallel a_2\parallel }_p+\frac{e_2\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta })}\left(\parallel b_1{\parallel }_{p_1}+d_1·R^{\frac{p}{p_1}}\right)\left(\parallel b_2{\parallel }_{p_2}+d_2·R^{\frac{p}{p_2}}\right)\le R$$

and

$$\mathbb{L}=\left(e_1+\frac{e_2{d}_1\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta })}\left(\parallel b_2{\parallel }_{p_2}+d_2{R}^{\frac{p}{p_2}}\right)R^{\frac{p}{p_1}}\right)<1.$$

Then, Equation (1) has a.e. nondecreasing solution $z\in L_p\left(I\right)$ on I.

Proof. 

Step I’. Assumption (ii) and (iii) imply that $F_{f_i}:L_p\left(I\right)\to L_p\left(I\right)$ and $F_{h_i}:L_p\left(I\right)\to L_{p_i}\left(I\right)$ and are continuous for $i=1,2$. The operator $A:L_p\left(I\right)\to L_{p_2}\left(I\right)$ is continuous (thanks to Lemma 7). Through the Hölder inequality, the operator $U:L_p\left(I\right)\to L_p\left(I\right)$. Assumption (i) implies that $B:L_p\left(I\right)\to L_p\left(I\right)$ is continuous.

Step II’. We shall establish the ball $B_R$ that the operator B operates on.

For $z\in L_p\left(I\right)$ and by utilizing assumptions (i)–(iii) with $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$, we have

$$\begin{array}{cc}\hfill {\parallel B\left(z\right)\parallel }_p& \le {\parallel g\parallel }_p+\parallel F_{f_1}{\left(z\right)\parallel }_p+{\parallel F_{f_2}U\left(z\right)\parallel }_p\hfill \\ \hfill & \le {\parallel g\parallel }_p+\parallel a_1{\parallel }_p+e_1{\parallel z\parallel }_p+\parallel a_2{\parallel }_p+e_2\parallel F_{h_1}{\left(z\right)\parallel }_{p_1}{\parallel A\left(z\right)\parallel }_{p_2}\hfill \\ \hfill & \le {\parallel g\parallel }_p+\parallel a_1{\parallel }_p+e_1{\parallel z\parallel }_p+\parallel a_2{\parallel }_p+e_2\parallel b_1+d_1·{\left|z\right|}^{\frac{p}{p_1}}{\parallel }_{p_1}\frac{\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta }))}{\parallel F_{h_2}\left(z\right)\parallel }_{p_2}\hfill \\ \hfill & \le {\parallel g\parallel }_p+\parallel a_1{\parallel }_p+e_1{\parallel z\parallel }_p+{\parallel a_2\parallel }_p\hfill \\ \hfill & +e_2\left(\parallel b_1{\parallel }_{p_1}+d_1·\parallel z^{\frac{p}{p_1}}{\parallel }_{p_1}\right)\frac{\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta }))}\left(\parallel b_2{\parallel }_{p_2}+d_2\parallel z^{\frac{p}{p_2}}{\parallel }_{p_2}\right)\hfill \\ \hfill & \le {\parallel g\parallel }_p+\parallel a_1{\parallel }_p+e_1{\parallel z\parallel }_p+{\parallel a_2\parallel }_p\hfill \\ \hfill & +\frac{e_2\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta })}\left(\parallel b_1{\parallel }_{p_1}+d_1·{\parallel z\parallel }_p^{\frac{p}{p_1}}\right)\left(\parallel b_2{\parallel }_{p_2}+d_2{\parallel z\parallel }_p^{\frac{p}{p_2}}\right)\hfill \\ \hfill & \le {\parallel g\parallel }_p+\parallel a_1{\parallel }_p+e_1·R+{\parallel a_2\parallel }_p\hfill \\ \hfill & +\frac{e_2\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta })}\left(\parallel b_1{\parallel }_{p_1}+d_1·R^{\frac{p}{p_1}}\right)\left(\parallel b_2{\parallel }_{p_2}+d_2·R^{\frac{p}{p_2}}\right)\le R,\hfill \end{array}$$

where $\parallel z^{\frac{p}{p_i}}{\parallel }_{p_i}={\parallel z\parallel }_p^{\frac{p}{p_i}},i=1,2$. Then, we can deduce that $H:B_R\to B_R$ is continuous.

Step III’ and Step IV’ are identical to those in Theorem 2 for $Q_R\subset B_R\left(L_p\right)$.

Step V’. Assume that $\epsilon >0$ and a set $D\subset I_0$, with meas $D\le \epsilon$. Then, for $z\in Z$ and $\varnothing \ne Z\subset Q_R$, we obtain

$$\begin{array}{cc}\hfill \parallel B\left(z\right)·{\chi }_D{\parallel }_p& \le \parallel g·{\chi }_D{\parallel }_p+\parallel a_1·{\chi }_D{\parallel }_p+e_1\parallel z·{\chi }_D{\parallel }_p+\parallel a_2·{\chi }_D{\parallel }_p+e_2\parallel F_{h_1}\left(z\right)·{\chi }_D{\parallel }_{p_1}{\parallel A\left(z\right)\parallel }_{p_2}\hfill \\ \hfill & \le \parallel g·{\chi }_D{\parallel }_p+\parallel a_1·{\chi }_D{\parallel }_p+e_1\parallel z·{\chi }_D{\parallel }_p+{\parallel a_2·{\chi }_D\parallel }_p\hfill \\ \hfill & +e_2\parallel (b_1·+d_1·{\left|z\right|}^{\frac{p}{p_1}})·{\chi }_D{\parallel }_{p_1}\frac{\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta }))}{\parallel F_{h_2}\left(z\right)\parallel }_{p_2}\hfill \\ \hfill & \le \parallel g·{\chi }_D{\parallel }_p+\parallel a_1·{\chi }_D{\parallel }_p+e_1\parallel z·{\chi }_D{\parallel }_p+{\parallel a_2·{\chi }_D\parallel }_p\hfill \\ \hfill & +\frac{e_2\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta })}\left(\parallel b_1·{\chi }_D{\parallel }_{p_1}+d_1·{\parallel z·{\chi }_D\parallel }_p^{\frac{p}{p_1}}\right)\left(\parallel b_2{\parallel }_{p_2}+d_2{R}^{\frac{p}{p_2}}\right).\hfill \end{array}$$

Taking into consideration that

$$\parallel z·{\chi }_D{\parallel }_p^{\frac{p}{p_1}}=\parallel z·{\chi }_D{\parallel }_p^{\frac{p}{p_1}-1}\parallel z·{\chi }_D{\parallel }_p\le R^{\frac{p}{p_1}}{\parallel z·{\chi }_D\parallel }_p.$$

Since $g,a_1,a_2\in L_p$, $b_1\in L_{p_1}$, then

$$\underset{\epsilon \to 0}{lim}\{\underset{measD\le \epsilon }{sup}[\underset{z\in Z}{sup}\{\parallel g·{\chi }_D{\parallel }_p+\parallel a_1·{\chi }_D{\parallel }_p+\parallel a_2·{\chi }_D{\parallel }_p\left\}\right]\}=0,\underset{\epsilon \to 0}{lim}\{\underset{measD\le \epsilon }{sup}[\underset{z\in Z}{sup}\{\parallel b_1{\chi }_D{\parallel }_{p_1}\left\}\right]\}=0.$$

By definition of $c\left(Z\right)$, we obtain

$$c\left(B\left(Z\right)\right)\le \mathbb{L}=\left(e_1+\frac{e_2{d}_1\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta })}\left(\parallel b_2{\parallel }_{p_2}+d_2{R}^{\frac{p}{p_2}}\right)R^{\frac{p}{p_1}}\right)c\left(Z\right).$$

From the properties of $Z\subset Q_R$ established above, we can apply Lemma 4 to obtain

$${\beta }_H\left(B\left(Z\right)\right)\le \mathbb{L}=\left(e_1+\frac{e_2{d}_1\Gamma (1-\frac{1}{p_2\beta })}{\Gamma (\alpha +1-\frac{1}{p_2\beta })}\left(\parallel b_2{\parallel }_{p_2}+d_2{R}^{\frac{p}{p_2}}\right)R^{\frac{p}{p_1}}\right){\beta }_H\left(Z\right).$$

The above inequality with $\mathbb{L}<1$ enables us to employ Theorem 1, which ends the proof. □

5. Example

We conclude by presenting and illuminating a few examples that verify our lemmas and theorems.

Example 1.

Consider the N-functions $\psi \left(u\right)=N\left(u\right)=u^2$ and ${\phi }_2\left(u\right)=exp\left|u\right|-\left|u\right|-1.$ We need to demonstrate that Lemma 6 is fulfilled and the operator $J_{\beta }^{\alpha }:L_N\left(\mathbb{I}\right)\to L_{{\phi }_2}\left(\mathbb{I}\right),\mathbb{I}=[0,\rho ]$ is continuous.

Indeed: For $\nu \in \mathbb{I}$ and any $\alpha ,\beta >0$, we have

$$k\left(\nu \right)={\int }_0^{{\nu }^{\beta }}\psi \left(u^{\alpha -1}\right)du={\int }_0^{{\nu }^{\beta }}{u}^{2\alpha -2}du=\frac{{\nu }^{\beta (2\alpha -1)}}{2\alpha -1}.$$

Moreover,

$${\int }_0^{\rho }{\phi }_2\left(k\left(\nu \right)\right)d\nu ={\int }_0^{\rho }\left(e^{\frac{{\nu }^{\beta (2\alpha -1)}}{2\alpha -1}}-\frac{{\nu }^{\beta (2\alpha -1)}}{2\alpha -1}-1\right)d\nu <\infty .$$

Then, Lemma 6 is fulfilled. Therefore, for $z\in L_N\left(\mathbb{I}\right)$, we obtain $J_{\beta }^{\alpha }:L_N\left(\mathbb{I}\right)\to L_{{\phi }_2}\left(\mathbb{I}\right)$ is continuous.

For further information and various examples of the function ${\phi }_2$ and the pair $(\psi ,N)$ fulfilling Lemma 6, see [15], Theorem 15.4.

Example 2.

For $\nu \in [0,1],\alpha =\frac{1}{2}$, and $\beta =\frac{1}{2}$, we have

$$\begin{array}{cc}\hfill z\left(\nu \right)& =g\left(\nu \right)+(a_1\left(\nu \right)+e_{1}z\left(\nu \right))+a_2\left(\nu \right)\hfill \\ \hfill & +\frac{e_2}{\sqrt{\pi }}(b_1\left(\nu \right)+d_1{\phi }_1^{-1}\left(\phi \left(z\left(\nu \right)\right)\right)){\int }_0^{\nu }\frac{b_2\left(\tau \right)+d_2{N}^{-1}(\phi (z(\tau )))}{2\sqrt{\tau }{(\sqrt{\nu }-\sqrt{\tau })}^{\frac{1}{2}}}d\tau ,\hfill \end{array}$$

which shall be regarded as a special form of Equation (1).

Example 3.

For $\nu \in I=[0,1],\alpha =\frac{1}{2}$, and $\beta =\frac{1}{3}$, consider the following equations in $L_2\left(I\right)$:

$$z\left(\nu \right)=e^{\frac{\nu }{2}}+\left({\nu }^{\frac{1}{2}}+\frac{1}{10}z\left(\nu \right)\right)+\frac{1}{3}{e}^{\frac{\nu }{6}}+\frac{{\nu }^{\frac{-1}{6}}}{100\Gamma \left(\frac{1}{2}\right)}\left({\nu }^{\frac{1}{4}}+\frac{1}{10}{\left|z\left(\nu \right)\right|}^{\frac{2}{4}}\right){\int }_0^{\nu }\frac{e^{\frac{\tau }{4}}+\frac{1}{10}{\left|z\left(\tau \right)\right|}^{\frac{2}{4}}}{{({\nu }^{\frac{1}{3}}-{\tau }^{\frac{1}{3}})}^{\frac{1}{2}}}\frac{d\tau }{3\sqrt[3]{{\tau }^2}}.$$

(3)

Let $p_1=p_2=4$, then Equation (3) is a special case of (1), with:

• $g\left(\nu \right)=e^{\frac{\nu }{2}}$ with ${\parallel g\parallel }_2=\sqrt{e-1}$.
• $|f_1(\nu ,z)|\le {\nu }^{\frac{1}{2}}+\frac{1}{10}\left|z\right|$, where $a_1\left(\nu \right)={\nu }^{\frac{1}{2}},e_1=\frac{1}{10}$, and $\parallel a_1{\parallel }_2=\sqrt{\frac{1}{2}}.$
• $|f_2(\nu ,z)|\le \frac{1}{3}{e}^{\frac{\nu }{6}}+\frac{1}{100}\left|z\right|$, where $a_2\left(\nu \right)=\frac{1}{3}{e}^{\frac{\nu }{6}},e_2=\frac{1}{100}$, and $\parallel a_2{\parallel }_2=\sqrt{\frac{\sqrt[3]{e}-1}{3}}.$
• $|h_1(\nu ,z)|\le {\nu }^{\frac{1}{4}}+\frac{1}{10}{\left|z\right|}^{\frac{2}{4}}$, where $b_1\left(\nu \right)={\nu }^{\frac{1}{4}},d_1=\frac{1}{10}$, and $\parallel b_1{\parallel }_4=\sqrt[4]{\frac{1}{2}}.$
• $|h_2(\nu ,z)|\le e^{\frac{\nu }{4}}+\frac{1}{10}{\left|z\right|}^{\frac{2}{4}}$, where $b_2\left(\nu \right)=e^{\frac{\nu }{4}},d_2=\frac{1}{10}$, and $\parallel b_2{\parallel }_4=\sqrt[4]{e-1}.$
• For $R=2.69$, we have

  $$\begin{array}{cc}\hfill & {\parallel g\parallel }_2+\parallel a_1{\parallel }_2+e_1·R+{\parallel a_2\parallel }_2+\frac{e_2\Gamma (1-\frac{3}{4})}{\Gamma (\alpha +1-\frac{3}{4})}\left(\parallel b_1{\parallel }_4+d_1·R^{\frac{2}{4}}\right)\left(\parallel b_2{\parallel }_4+d_2·R^{\frac{2}{4}}\right)\hfill \\ \hfill & \le \sqrt{e-1}+\sqrt{\frac{1}{2}}+\frac{\sqrt{2.69}}{10}+\sqrt{\frac{\sqrt[3]{e}-1}{3}}+\frac{\Gamma \left(\frac{1}{4}\right)}{100\Gamma \left(\frac{3}{4}\right)}\left(\sqrt[4]{\frac{1}{2}}+\frac{1}{10}\sqrt{2.69}\right)\left(\sqrt[4]{e-1}+\frac{\sqrt{2.69}}{10}\right)\hfill \\ \hfill & \le 2.69=R\hfill \end{array}$$

  and

  $$\begin{array}{cc}\hfill \mathbb{L}& =\left(e_1+\frac{e_2{d}_1\Gamma \left(\frac{1}{4}\right)}{\Gamma \left(\frac{3}{4}\right)}\left(\parallel b_2{\parallel }_4+d_2\sqrt{R}\right)\sqrt{R}\right)\hfill \\ \hfill & \le \left(\frac{1}{10}+\frac{\Gamma \left(\frac{1}{4}\right)}{1000\Gamma \left(\frac{3}{4}\right)}\left(\sqrt[4]{e-1}+\frac{1}{10}\sqrt{2.69}\right)\sqrt{2.69}\right)\le 0.10635<1.\hfill \end{array}$$

Hence, Theorem 3 proves that Equation (3) has a.e. nondecreasing solution $z\in L_2\left(I\right)$ on I.

6. Conclusions

In the presented manuscript, we demonstrate and prove some new properties of the Erdélyi–Kober ($\mathcal{EK}$) fractional operator, including monotonicity, boundedness acting, and continuity in Lebesgue spaces and Orlicz spaces. We discuss the existence of theorems for a quadratic integral equation of fractional order in the mentioned spaces with the help of these properties and the concept of the ($\mathcal{MNC}$) related to ($\mathcal{FPT}$). We provide a few examples that support our theorems.

In future research within this area, we shall study some fundamental properties for different types of fractional operators, such as the generalized fractional (or g-fractional) integral operator and the tempered fractional integral operators (cf. [1,4]) in various function spaces and employ these properties in solving different types of fractional problems. Additionally, some numerical results should be applied for solving such types of problems.