Examining Nonlinear Fredholm Equations in Lebesgue Spaces with Variable Exponents

Abstract

We investigate the existence of solutions for the Fredholm integral equation $\mathsf{\Phi }\left(\vartheta \right)=\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))+{\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta ,$ for $\vartheta \in [0,1]$, in the setting of the modular function spaces $L_{\rho }$. We also derive an application of this research within the framework of variable exponent Lebesgue spaces $L^{p(·)}$ subject to specific conditions imposed on the exponent function $p(·)$ and the functions $\mathcal{F}$ and $\mathcal{G}$.

1. Introduction

This work is devoted to the study of the Fredholm equation

$$\mathsf{\Phi }\left(\vartheta \right)=\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))+{\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta ,$$

for $\vartheta \in [0,1]$. Our main result is Theorem 2, in which it is proved that, under certain conditions, the equation has a solution in a suitable space.

As we shall see, the problem under consideration is modular in nature; therefore, in order to properly formulate it a brief discussion of the modular space setting is indispensable. One of the most conspicuous modular spaces in analysis due to their proximity to the classical Lebesgue spaces, are the variable-exponent $L^p$ spaces. For this reason, after analyzing the general modular case, we place special emphasis on the variable exponent case in the last section of our paper. The modular approach has the advantage of avoiding the use of the Luxemburg norm and potentially, it is more friendly for the implementation of numerical algorithms.

Modular function spaces constitute a foundational concept in the area of mathematics known as functional analysis [1,2,3]. They provide a flexible framework for investigating the characteristics of functions and their interactions with diverse mathematical and physical structures [4,5]. Within these spaces lies a rich assortment of tools and methodologies, which serve as valuable resources for comprehending functions defined across various domains; see for example [6,7]. This paves the path towards attaining deeper insights into the essence of functions, their inherent properties, and their intricate connections with other mathematical entities. In this introductory exploration, we shall delve into the core essence of modular function spaces, underscore their profound significance, and illuminate their pivotal role across multiple branches of mathematics, encompassing areas such as analysis, topology, and functional analysis. Throughout, we will unravel the salient features and practical applications of these spaces, thus shedding light on how they empower mathematicians to confront intricate challenges, construct rigorous proofs, and cultivate a heightened comprehension of the intricate universe of functions and their associated spaces.

Modular function spaces, in particular variable exponent Lebesgue spaces, are two intriguing and complementary facets of modern functional analysis. Modular function spaces, in general, provide a structured environment for studying functions and their interactions with mathematical structures, offering a versatile toolkit for analysis across different domains. On the other hand, variable exponent Lebesgue spaces introduce adaptability by allowing the exponent in the Lebesgue norm to vary, enabling the study of functions with varying regularity. Together, these two concepts represent a dynamic duo in mathematical research, offering a comprehensive framework that combines versatility with adaptability. They find applications in a wide array of fields, including partial differential equations, image processing, and functional analysis, providing researchers and practitioners with powerful tools to explore complex and changing phenomena in both theoretical and practical contexts; see [4,8,9,10] with more references therein.

The aim of this study is to investigate the existence of solutions for Fredholm equations within the domain of continuous functions that take values in modular function spaces. Numerous studies have previously addressed integral equations for $\rho$-continuous functions from the interval $[0,1]$ into $L_{\rho }$ employing methodologies like degree theory for condensing mappings and the Brouwer fixed-point theorem for continuous functions, as indicated in the references [11]. Significantly, the Banach fixed-point theorem for the contraction mappings has been utilized by various authors [12,13,14,15] to address integral equations within modular function spaces, while considering the condition ${\Delta }_2$. We will conclude our work by extending the main result to the case of variable exponents spaces.

2. Modular Spaces

Fredholm integral equations arise in functional analysis from the need to describe a wide range of physical phenomena as well as mathematical problems. They are named after the Swedish mathematician Erik Ivar Fredholm [16], who made significant contributions to the theory in the late 19th and early 20th centuries.

The difficulty in dealing with Fredholm integral equations arises from the fact that the unknown function is affected by an integral operator; the primary goal is to find a function that satisfies the equation in general Banach spaces. Our primary emphasis in the present work is on investigating solutions within modular spaces and variable exponent Lebesgue spaces for a particular Fredholm integral equation:

$$\mathsf{\Phi }\left(\vartheta \right)=\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))+{\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta .$$

(1)

In a general context, it is important to note that this equation may not possess a solution. For further insights, we refer to [10]. For foundational definitions and essential properties related to modular function spaces, we direct the reader to [4,6,10]. Throughout, we employ the notation ${\mathcal{M}}_{\infty }$ to denote the set of Lebesgue measurable functions defined on $[0,1]$.

In an effort to address potential concerns related to redundancy and to provide a solid groundwork in the realm of modular spaces, we refer the readers to the comprehensive works authored by Khamsi [6] and Diening [4]. These sources serve as foundational references. In what follows, we will introduce and elaborate in detail on two fundamental definitions central to our study: the concept of a modular function and the essential ${\Delta }_2$-type condition. Equipped with these two tools we will then undertake the analysis of the Fredholm equation under consideration.

Definition 1

([6,10]). We define a convex regular modular function as a mapping $\rho :{\mathcal{M}}_{\infty }\to [0,\infty ]$ that satisfies the following conditions:

(1) 

$\rho \left(\varphi \right)=0$ if and only if $\varphi =0$.

(2) 

$\rho \left(a\varphi \right)=\rho \left(\varphi \right)$, for $\left|a\right|=1$.

(3) 

$\rho (a\varphi +(1-a\left)\psi \right)\le a\rho \left(\varphi \right)+(1-a)\rho \left(\psi \right)$, for any $a\in [0,1]$,

where $\varphi ,\psi \in {\mathcal{M}}_{\infty }$.

For a comprehensive understanding of the topological definition and properties of the convex regular modular function $\rho :{\mathcal{M}}_{\infty }\to [0,\infty ]$, we suggest consulting the following references [6,7,10]. An important property in modular functional spaces, referred to as ${\Delta }_2$-type condition, is defined below.

Definition 2

([10]). We say that the modular ρ satisfies the ${\Delta }_2$-type condition if there is a positive constant $K>0$ such that for all $\varphi \in L_{\rho }$, the inequality $\rho \left(2\varphi \right)\le K\rho \left(\varphi \right)$ holds true.

This specific property holds great significance in the analysis of modular function space. It is important to emphasize that when the modular function $\rho$ adheres to the ${\Delta }_2$-type condition, then:

$${\omega }_{\rho }\left(2\right)=inf\{K:\rho \left(2\varphi \right)\le K\rho \left(\varphi \right)\mathrm{where}\varphi \in L_{\rho }\}<\infty .$$

Under these circumstances, it is easily noted that

$$\rho (\varphi +\psi )\le \frac{{\omega }_{\rho }\left(2\right)}{2}\left(\rho \left(\varphi \right)+\rho \left(\psi \right)\right),$$

(2)

for any $\varphi ,\psi \in L_{\rho }$.

In the upcoming section, we will undertake the study of the solvability of nonlinear Fredholm equations. We will consider solutions within the space of $L_{\rho }$-valued $\rho$-continuous functions defined over the interval [0, 1]. This exploration will also involve the introduction of a Poincaré operation within an appropriate mathematical framework, substantiated by rigorous functional analytic proofs.

3. Solvability of Nonlinear Fredholm Equations on ${\mathcal{C}}_{\mathit{\rho }}\mathbf{(}\mathbf{[}\mathbf{0},\mathbf{1}\mathbf{]},{\mathit{L}}_{\mathit{\rho }}\mathbf{)}$

Since, as was highlightted before, Fredholm equations arise from practical considerations while studying concrete problems in various scientific disciplines, such as physics, engineering, economics, and biology (see for example [17,18]), it is not surprising that their solvability is of the utmost interest both in mathematics and applications. In particular, nonlinear Fredholm equations involve nonlinear operators, making their analysis more elusive and, in cases where they can be solved, significantly more challenging. The study of their solvability requires advanced mathematical techniques, such as fixed-point theory, functional analysis, and integral operators. Several authors have developed various numerical methods and analytical tools to tackle these equations, providing valuable insights into the behavior of their solutions [10,11]. Understanding the solvability of nonlinear Fredholm equations is crucial for modeling complex phenomena and making informed decisions in diverse fields [19].

Throughout this section, we denote by $\mathsf{\Omega }=[0,1]\times [0,1]\times L_{\rho }$ and ${\mathsf{\Omega }}^{\ast }=[0,1]\times L_{\rho }$. This section deals with the solvability of the Fredholm integral equation within the framework of modular function spaces $L_{\rho }$:

$$\mathsf{\Phi }\left(\vartheta \right)=\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))+{\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta ,$$

where $\mathsf{\Phi }\in L_{\rho }$, $\mathcal{G}:{\mathsf{\Omega }}^{\ast }\to L_{\rho },$ and $\mathcal{F}:\mathsf{\Omega }\to \mathbb{R}$.

We start by revisiting the concept of $\rho$-continuity as defined in references [10,14,15] and establishing some notation. Let $\mathcal{C}\rho \left(\right[0,1],L\rho )$ represent the set of functions that are continuous with respect to $\rho$, mapping from the interval $[0,1]$ to $L_{\rho }$. Furthermore, we define ${\rho }_{{\mathcal{C}}_{\rho }}:{\mathcal{C}}_{\rho }([0,1],L_{\rho })\to [0,\infty ]$, by

$${\rho }_{{\mathcal{C}}_{\rho }}\left(\mathsf{\Phi }\right)=\underset{\vartheta \in [0,1]}{sup}\rho \left(\mathsf{\Phi }\left(\vartheta \right)\right),$$

where $\mathsf{\Phi }\in {\mathcal{C}}_{\rho }([0,1],L_{\rho })$.

For any nonempty subset $B\subset L_{\rho }$, we denote by $\mathcal{C}\rho \left(\right[0,1],B)$ the set of functions $\mathsf{\Phi }\in \mathcal{C}\rho ([0,1],L_{\rho })$, for which $\mathsf{\Phi }\left(\right[0,1\left]\right)\subset B$. As established in [15], it becomes evident that ${\rho }_{\mathcal{C}\rho }$ is a convex modular with both the Fatou property and the ${\Delta }_2$-type condition. Furthermore, $\mathcal{C}\rho ([0,1],L_{\rho })$ is ${\rho }_{\mathcal{C}\rho }$-complete, and $\mathcal{C}\rho \left(\right[0,1],B)$, assuming $\rho$ satisfies the ${\Delta }_2$-type condition and B is a nonempty convex subset of $L_{\rho }$ that is also $\rho$-closed, forms a ${\rho }_{\mathcal{C}\rho }$-closed, convex subset of $\mathcal{C}\rho ([0,1],L_{\rho })$.

Now, we can proceed to define the Luxemburg norm for $\mathcal{C}\rho \left(\right[0,1],L\rho )$ as follows:

$${\parallel \mathsf{\Phi }\parallel }_{{\mathcal{C}}_{\rho }}=inf\left\{t>0;{\rho }_{{\mathcal{C}}_{\rho }}\left(\frac{1}{t}\mathsf{\Phi }\right)\le 1\right\}.$$

Given our assumption that $\rho$ adheres to the ${\Delta }_2$-type condition, we can establish the supremum norm for an element $\mathsf{\Phi }\in \mathcal{C}\rho \left(\right[0,1],L\rho )$ as follows:

$${\parallel \mathsf{\Phi }\parallel }_{\infty }=\underset{\vartheta \in [0,1]}{sup}{\parallel \mathsf{\Phi }\left(\vartheta \right)\parallel }_{\rho }.$$

Prior to presenting the primary theorem, we will introduce a crucial lemma regarding the equivalence of the convergence in the Luxemburg norm ${\parallel ·\parallel }_{{\mathcal{C}}_{\rho }}$, the supremum norm ${\parallel ·\parallel }_{\infty }$, and the modular function ${\rho }_{{\mathcal{C}}_{\rho }}(·)$. For further details, we refer to [14].

Lemma 1

([14]). Assuming ρ is a convex, regular modular function that meets the ${\Delta }_2$-type condition, and considering a sequence ${\mathsf{\Phi }}_n$ in $\mathcal{C}\rho \left(\right[0,1],L\rho )$ along with $\mathsf{\Phi }\in \mathcal{C}\rho \left(\right[0,1],L\rho )$, the following statements are equivalent:

(a) 

$\underset{n\to \infty }{lim}{\rho }_{{\mathcal{C}}_{\rho }}({\mathsf{\Phi }}_n-\mathsf{\Phi })=0$,

(b) 

$\underset{n\to \infty }{lim}{\parallel {\mathsf{\Phi }}_n-\mathsf{\Phi }\parallel }_{{\mathcal{C}}_{\rho }}=0$,

(c) 

$\underset{n\to \infty }{lim}{\parallel {\mathsf{\Phi }}_n-\mathsf{\Phi }\parallel }_{\infty }=0$.

Let $\mathcal{F}:\mathsf{\Omega }\to \mathbb{R}$, $\mathsf{\Phi }\in L_{\rho }$, and $\mathcal{G}:{\mathsf{\Omega }}^{\ast }\to L_{\rho },$ such that for each $\vartheta \in [0,1]$, $\mathcal{F}(\vartheta ,·,\mathsf{\Phi }(·))\in L_{\rho }$, and

(H1)

$\zeta \to \mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }(\zeta \left)\right)$ is Lebesgue measurable over [0,1];

(H2)

$\vartheta \to {\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta$ is in ${\mathcal{C}}_{\rho }([0,1],L_{\rho })$;

(H3)

$\mathcal{F}$ and $\mathcal{G}$ are $\rho$-strongly continuous with respect to the first variable, i.e.,

$$\left\{\begin{array}{ccc}\underset{\vartheta \to {\vartheta }_0}{lim}sup\rho \left(\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\zeta \right))-\mathcal{G}({\vartheta }_0,\mathsf{\Phi }\left(\zeta \right))\right)\hfill & =\hfill & 0,\hfill \\ \underset{\vartheta \to {\vartheta }_0}{lim}sup\rho \left(\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left({\zeta }^{\ast }\right))-\mathcal{F}({\vartheta }_0,\zeta ,\mathsf{\Phi }\left({\zeta }^{\ast }\right))\right)\hfill & =\hfill & 0,\hfill \end{array}\right.$$

where the supremum is taken for $\zeta$ and ${\zeta }^{\ast }$ in $[0,1]$ and $\mathsf{\Phi }\in {\mathcal{C}}_{\rho }([0,1],B)$, where B is any nonempty $\rho$-bounded subset of $L_{\rho }$.

In the context of the Fredholm integral Equation (1) within the modular function space $L_{\rho }$, we introduce the integral operator $\mathcal{J}:\mathcal{C}\rho \left(\right[0,1],L\rho )\to \mathcal{C}\rho \left(\right[0,1],L\rho )$, which is defined as follows:

$$\mathcal{J}\left(\mathsf{\Phi }\right)(·)=g(\mathcal{G}(·,\mathsf{\Phi }(·))+{\int }_0^1\mathcal{F}(·,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta .$$

(3)

It is evident that the solutions to (3) correspond precisely to the fixed points of $\mathcal{J}$, i.e., to those functions $\mathsf{\Phi }$ such that $\mathcal{J}\left(\mathsf{\Phi }\right)=\mathsf{\Phi }$. The proof of the main result requires the notion of complete continuity for mappings [20] and the use of Schaeffer’s fixed-point theorem, which we recall.

Theorem 1

([21,22,23]). Consider a normed space X and a continuous mapping T from X to itself, such that the closure of $T\left(B\right)$ is compact for any bounded subset B of X. In this case, one of the following options $\left(i\right)$ or $\left(ii\right)$ will necessarily hold, where:

(i) 

The equation $x=\lambda Tx$ has a solution for $\lambda =1$,

(ii) 

The set of all such solutions x of $T\left(x\right)=\lambda x$, for some $0<\lambda <1$, is unbounded.

Now, we are prepared to present the principal result of our research.

Theorem 2.

Consider ρ a convex, regular modular function that satisfies the ${\Delta }_2$-type condition. Suppose that $\mathcal{G}\in \mathcal{C}\rho \left(\right[0,1],L\rho )$, and that $\mathcal{F}:\mathsf{\Omega }\to \mathbb{R}$ possesses the following properties:

(SC) 

$\mathcal{G}$ and $\mathcal{G}$ satisfy $\left(H1\right)$, $\left(H2\right)$, and $\left(H3\right)$,

(LC) 

There exist non-negative numbers $L_{\mathcal{G}}$ and $L_{\mathcal{F}}$ such that

$$\left\{\begin{array}{ccc}\rho \left(\mathcal{F}(\vartheta ,·,\mathsf{\Phi }(·\left)\right)-\mathcal{F}(\vartheta ,·,\mathsf{\Psi }(·\left)\right)\right)& \le \hfill & L_{\mathcal{F}}\rho (\mathsf{\Phi }(·)-\mathsf{\Psi }(·)),\hfill \\ \rho \left(\mathcal{G}(·,\mathsf{\Phi }(·\left)\right)-\mathcal{G}(·,\mathsf{\Psi }(·\left)\right)\right)& \le \hfill & L_{\mathcal{G}}\rho (\mathsf{\Phi }(·)-\mathsf{\Psi }(·)),\hfill \end{array}\right.$$

(BC) 

there exist non-negative numbers ${\overline{M}}_{\mathcal{F}}$, ${\overline{M}}_{\mathcal{G}}$, ${\overline{N}}_{\mathcal{F}}$, and ${\overline{N}}_{\mathcal{G}}$ such that

$$\left\{\begin{array}{ccc}\rho \left(f(\vartheta ,·,\mathsf{\Phi }(·\left)\right)\right)& \le \hfill & {\overline{M}}_{\mathcal{F}}{\rho }_{{\mathcal{C}}_{\rho }}\left(\mathsf{\Phi }\right)+{\overline{N}}_{\mathcal{F}},\hfill \\ \rho \left(g(·,\mathsf{\Phi }(·\left)\right)\right)& \le \hfill & {\overline{M}}_{\mathcal{G}}{\rho }_{{\mathcal{C}}_{\rho }}\left(\mathsf{\Phi }\right)+{\overline{N}}_{\mathcal{G}},\hfill \end{array}\right.$$

for any $\mathsf{\Phi },\mathsf{\Psi }\in {\mathcal{C}}_{\rho }([0,1],L_{\rho })$ and $\vartheta \in [0,1]$. Assume that ${\overline{M}}_{\mathcal{F}}+{\overline{M}}_{\mathcal{G}}<2/{\omega }_{\rho }\left(2\right)$. Then, (1) has a solution in ${\mathcal{C}}_{\rho }([0,1],L_{\rho })$.

Proof. 

Along the same lines as in the proof of Theorem 5.2 in [10], it becomes evident that the operator $\mathcal{J}$ maps $\mathcal{C}\rho \left(\right[0,1],L\rho )$ into $\mathcal{C}\rho \left(\right[0,1],L\rho )$. In the first part of the proof, we show that $\mathcal{J}$ is completely continuous with respect to the modular ${\rho }_{{\mathcal{C}}_{\rho }}$. To that effect, we choose the mesh points

$${\zeta }_i^n=\frac{i}{n},i=0,\cdots ,n,$$

where $n=1,2,\cdots$. Using $\left(LC\right)$ assumptions, Fatou property, the ${\Delta }_2$-type condition, and the property (2), we obtain

$$\begin{array}{ccc}\rho \left(\right(\mathcal{J}\mathsf{\Phi }\left)\right(\vartheta )-(\mathcal{J}\mathsf{\Psi }\left)\right(\vartheta \left)\right)\hfill & \le \hfill & {\displaystyle \frac{{\omega }_{\rho }\left(2\right)}{2}(L_{\mathcal{G}}\rho (\mathsf{\Phi }\left(\vartheta \right)-\mathsf{\Psi }\left(\vartheta \right))}\hfill \\ & & {\displaystyle +L_{\mathcal{F}}\underset{n\to \infty }{lim\; inf}\sum _{i=0}^{n-1}\frac{1}{n}\rho (\mathsf{\Phi }\left({\zeta }_i^n\right)-\mathsf{\Psi }\left({\zeta }_i^n\right)))}\hfill \\ & \le \hfill & {\displaystyle \frac{{\omega }_{\rho }\left(2\right)}{2}\left(L_{\mathcal{G}}+L_{\mathcal{F}}\underset{n\to \infty }{lim\; inf}\sum _{i=0}^{n-1}\frac{1}{n}\right){\rho }_{{\mathcal{C}}_{\rho }}(\mathsf{\Phi }-\mathsf{\Psi })}\hfill \\ & =\hfill & {\displaystyle \frac{{\omega }_{\rho }\left(2\right)}{2}(L_{\mathcal{G}}+L_{\mathcal{F}}){\rho }_{{\mathcal{C}}_{\rho }}(\mathsf{\Phi }-\mathsf{\Psi }),}\hfill \end{array}$$

for any $\vartheta \in [0,1]$, which implies

$${\rho }_{{\mathcal{C}}_{\rho }}(\mathcal{J}\left(\mathsf{\Phi }\right)-\mathcal{J}\left(\mathsf{\Psi }\right))\le \frac{{\omega }_{\rho }\left(2\right)}{2}(L_{\mathcal{G}}+L_{\mathcal{F}}){\rho }_{{\mathcal{C}}_{\rho }}(\mathsf{\Phi }-\mathsf{\Psi }),$$

for any $\mathsf{\Phi },\mathsf{\Psi }\in {\mathcal{C}}_{\rho }([0,1],L_{\rho })$. This will imply that the operator $\mathcal{J}$ is ${\rho }_{{\mathcal{C}}_{\rho }}$-continuous and that the image of any nonempty, ${\rho }_{{\mathcal{C}}_{\rho }}$-bounded subset of ${\mathcal{C}}_{\rho }([0,1],L_{\rho })$ is ${\rho }_{{\mathcal{C}}_{\rho }}$-bounded. Consider a $\rho$-bounded nonempty subset $B\subset L_{\rho }$. Let us prove that $\mathcal{J}\left({\mathcal{C}}_{\rho }([0,1],B)\right)$ is ${\rho }_{{\mathcal{C}}_{\rho }}$-relatively compact. Using the property (2), we have

$$\begin{array}{cc}\rho \left(\left(\mathcal{J}\mathsf{\Phi }\right)\left(\vartheta \right)-\left(\mathcal{J}\mathsf{\Phi }\right)\left(\tilde{\vartheta }\right)\right)\hfill & {\displaystyle \le \frac{{\omega }_{\rho }\left(2\right)}{2}\rho \left(\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))-\mathcal{G}(\tilde{\vartheta },\mathsf{\Phi }\left(\tilde{\vartheta }\right))\right)}\hfill \\ & {\displaystyle +\frac{{\omega }_{\rho }\left(2\right)}{2}\rho \left({\int }_0^1\left[\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))-\mathcal{F}(\tilde{\vartheta },\zeta ,\mathsf{\Phi }\left(\zeta \right))\right]d\zeta \right)}\hfill \\ & {\displaystyle \le \frac{{\omega }_{\rho }\left(2\right)}{2}\rho \left(\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))-\mathcal{G}(\tilde{\vartheta },\mathsf{\Phi }\left(\tilde{\vartheta }\right))\right)}\hfill \\ & {\displaystyle +\frac{{\omega }_{\rho }\left(2\right)}{2}\underset{\zeta \in [0,1]}{sup}\rho (\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))-\mathcal{F}(\tilde{\vartheta },\zeta ,\mathsf{\Phi }\left(\zeta \right)))\sum _{i=0}^{n-1}\frac{1}{n}}\hfill \\ & {\displaystyle =\frac{{\omega }_{\rho }\left(2\right)}{2}\rho \left(\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))-\mathcal{G}(\tilde{\vartheta },\mathsf{\Phi }\left(\tilde{\vartheta }\right))\right)}\hfill \\ & {\displaystyle +\frac{{\omega }_{\rho }\left(2\right)}{2}\underset{\zeta \in [0,1]}{sup}\rho \left(\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))-\mathcal{F}(\tilde{\vartheta },\zeta ,\mathsf{\Phi }\left(\zeta \right))\right),}\hfill \end{array}$$

for any $\vartheta ,\tilde{\vartheta }$ in $[0,1]$ and $\mathsf{\Phi }\in {\mathcal{C}}_{\rho }([0,1],B)$. Since $\mathcal{F}$ and $\mathcal{G}$ satisfy $\left(H3\right)$, $\rho$ satisfies the ${\Delta }_2$-type condition, Lemma 1 allows us to conclude that the family $\mathcal{J}\left({\mathcal{C}}_{\rho }([0,1],B)\right)$ is equicontinuous with respect to the Luxemburg norm ${\parallel ·\parallel }_{{\mathcal{C}}_{\rho }}$. Using the Arzelà–Ascoli theorem, we conclude that $\mathcal{J}\left({\mathcal{C}}_{\rho }([0,1],B)\right)$ is relatively compact with respect to the norm ${\parallel ·\parallel }_{{\mathcal{C}}_{\rho }}$. Therefore, the operator $\mathcal{J}$ is completely continuous with respect to ${\rho }_{\mathcal{C}\rho }$. Next, we prove the existence of a fixed point for $\mathcal{J}$. Consider the set

$$S=\{\mathsf{\Phi }\in {\mathcal{C}}_{\rho }([0,1],L_{\rho }):\mathsf{\Phi }=\nu \mathcal{J}\left(\mathsf{\Phi }\right),\mathrm{for}\mathrm{some}\nu \mathrm{in}[0,1]\}.$$

We claim that S is ${\rho }_{{\mathcal{C}}_{\rho }}$-bounded. Indeed, the set S is nonempty, since it contains the zero function. Next, let $\mathsf{\Phi }\in S$. We have $\mathsf{\Phi }=\nu \mathcal{J}\left(\mathsf{\Phi }\right)$, for some $\nu \in [0,1]$, which implies

$$\begin{array}{cc}\hfill \rho \left(\mathsf{\Phi }\right(\vartheta \left)\right)& =\rho \left(\nu \mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))+\nu {\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta \right)\hfill \\ \hfill & \le \nu \rho \left(\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))+{\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta \right)\hfill \\ \hfill & \le \nu \frac{{\omega }_{\rho }\left(2\right)}{2}\left(\rho \left(\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))\right)+\rho \left({\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta \right)\right),\hfill \end{array}$$

for any $\vartheta \in [0,1]$. By virtue of the Fatou property, one has

$$\begin{array}{cc}\hfill \rho \left(\mathsf{\Phi }\right(\vartheta \left)\right)& \le \frac{{\omega }_{\rho }\left(2\right)}{2}\left[\rho \left(\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))\right)+\underset{n\to \infty }{lim\; inf}\sum _{i=0}^{n-1}({\zeta }_{i+1}^n-{\zeta }_i^n)\rho (\mathcal{F}(\vartheta ,{\zeta }_i^n,\mathsf{\Phi }\left({\zeta }_i^n\right))\right],\hfill \\ \hfill & \le \frac{{\omega }_{\rho }\left(2\right)}{2}[({\overline{M}}_{\mathcal{G}}{\rho }_{{\mathcal{C}}_{\rho }}\left(\mathsf{\Phi }\right)+{\overline{N}}_{\mathcal{G}})+\hfill \\ \hfill & ({\overline{M}}_{\mathcal{F}}{\rho }_{{\mathcal{C}}_{\rho }}\left(\mathsf{\Phi }\right)+{\overline{N}}_{\mathcal{F}})\underset{n\to \infty }{lim\; inf}\sum _{i=0}^{n-1}({\zeta }_{i+1}^n-{\zeta }_i^n)],\hfill \\ \hfill & =\frac{{\omega }_{\rho }\left(2\right)}{2}\left(({\overline{M}}_{\mathcal{G}}+{\overline{M}}_{\mathcal{F}}){\rho }_{{\mathcal{C}}_{\rho }}\left(\mathsf{\Phi }\right)+{\overline{N}}_{\mathcal{G}}+{\overline{N}}_{\mathcal{F}}\right),\hfill \end{array}$$

for any $\vartheta \in [0,1]$, which implies

$${\rho }_{{\mathcal{C}}_{\rho }}\left(\mathsf{\Phi }\right)\left(1-\frac{{\omega }_{\rho }\left(2\right)}{2}({\overline{M}}_{\mathcal{G}}+{\overline{M}}_{\mathcal{F}})\right)\le \frac{{\omega }_{\rho }\left(2\right)}{2}({\overline{N}}_{\mathcal{G}}+{\overline{N}}_{\mathcal{F}}),$$

Since ${\overline{M}}_{\mathcal{G}}+{\overline{M}}_{\mathcal{F}}<2/{\omega }_{\rho }\left(2\right)$, we obtain

$${\rho }_{{\mathcal{C}}_{\rho }}\left(\mathsf{\Phi }\right)\le \frac{{\omega }_{\rho }\left(2\right)}{2}\frac{{\overline{N}}_{\mathcal{G}}+{\overline{N}}_{\mathcal{F}}}{1-\frac{{\omega }_{\rho }\left(2\right)}{2}({\overline{M}}_{\mathcal{G}}+{\overline{M}}_{\mathcal{F}})}·$$

Therefore, S is bounded with respect to ${\rho }_{\mathcal{C}\rho }$. The ${\Delta }_2$-type condition forces S to be bounded for the associated Luxemburg norm. Schaeffer’s theorem will yield the existence of a fixed point of $\mathcal{J}$, which is a solution of the equation $\left(FIE\right)$ in $\mathcal{C}\rho ([0,1],L_{\rho })$. □

In the forthcoming section, we will explore the precise application of these theoretical concepts to variable exponent Lebesgue spaces. This examination will entail a detailed and systematic analysis of their implications and potential outcomes within the predefined context. Through a rigorous evaluation, we aim to illuminate the practical significance and consequences of applying these concepts to variable exponent Lebesgue spaces, thereby contributing valuable insights to the broader discourse in this field of study.

4. Application to Variable Exponent Lebesgue Spaces

In this section, we discuss the implications of Theorem 2 in the case of variable exponent Lebesgue spaces $L^{p(·)}$, whose definitions and basic properties we succinctly present next. For a more detailed study of these spaces, we suggest the references [4,24].

Let $p:[0,1]\to [1,\infty ]$ be a Lebesgue measurable function finite almost everywhere. Define

$$p^{-}:=p_{[0,1]}^{-}:={ess\; inf}_{x\in [0,1]}p\left(x\right)and{p}^{+}:=p_{[0,1]}^{+}:={ess\; sup}_{x\in [0,1]}p\left(x\right).$$

The variable exponent Lebesgue space $L^{p(·)}\left([0,1]\right)$ is defined as

$$L^{p(·)}:=L^{p(·)}\left([0,1]\right)=\{\varphi \in L^0\left([0,1]\right):{\varrho }_{L^{p(·)}}\left(\lambda \varphi \right)<\infty \mathrm{for}\mathrm{some}\lambda >0\},$$

where $L^0\left([0,1]\right)$ denotes the space of all $\mathbb{R}$-valued, Lebesgue measurable functions on $[0,1]$ and

$${\varrho }_{L^{p(·)}}\left(\varphi \right)={\int }_0^1\mid \varphi \left(x\right){\mid }^{p\left(x\right)}dx.$$

According to Definition 1, the functional ${\varrho }_{L^{p(·)}}$ defines a convex modular. The vector space $L^{p(·)}\left([0,1]\right)$ may be endowed with the Luxemburg norm defined as:

$${\parallel \varphi \parallel }_{L^{p(·)}}=inf\left\{t>0;{\varrho }_{L^{p(·)}}\left(\frac{1}{t}\varphi \right)\le 1\right\}.$$

Most of the nice properties, whether geometric or topological, of the Luxemburg norm necessitate the condition $1<p^{-}<p^{+}<\infty$, which is equivalent to the ${\Delta }_2$ condition, as pointed out below. For a more detailed discussion of this condition and its implications, the reader may consult [4,25]. For example, the Luxemburg norm ${\parallel ·\parallel }_{L^{p(·)}}$ is uniformly convex if and only if $1<p^{-}<p^{+}<\infty$. Moreover, it is not hard to see that the ${\Delta }_2$-type condition holds for $L^{p(·)}$ if and only if $p^{+}<\infty$. Indeed, we have

$${\varrho }_{L^{p(·)}}\left(2\varphi \right)={\int }_0^1{2}^{p\left(x\right)}{\left|\varphi \left(x\right)\right|}^{p\left(x\right)}dx\le 2^{p^{+}}{\varrho }_{L^{p(·)}}\left(\varphi \right),$$

for any $\varphi \in L^{p(·)}$, which implies ${\omega }_{{\varrho }_{L^{p(·)}}}\left(2\right)\le 2^{p^{+}}$. Next, we consider the Fredholm integral equation

$$\mathsf{\Phi }\left(\vartheta \right)=\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))+{\int }_0^1\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }\left(\zeta \right))d\zeta ,$$

(4)

for $\vartheta \in [0,1]$, $\mathsf{\Phi }\in L^{p(·)}$, $\mathcal{G}:{\mathsf{\Omega }}^{\ast }\to L^{p(·)},$ and $\mathcal{F}:\mathsf{\Omega }\to \mathbb{R}$. The existence of a solution $\mathsf{\Phi }$ of Equation (4) will follow along the same lines as those of the investigation developed in the previous section. First, we assume that the following conditions hold:

(GC)

There exists a constant $C_{\mathcal{G}}\ge 0$ and a non-negative function $N_{\mathcal{G}}\in L^{p(·)}$, such that for almost every $\vartheta \in [0,1]$ and all $\varphi \in L^{p(·)}$, we have

$$\left|\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))\right|\le N_{\mathcal{G}}\left(\vartheta \right)+C_{\mathcal{G}}\left|\varphi \left(\vartheta \right)\right|;$$

(FC)

There exists a constant $C_{\mathcal{F}}\ge 0$ and a non-negative function $N_{\mathcal{F}}(\vartheta ,·)\in L^{p(·)}$, such that for each $\vartheta \in [0,1]$ and all $\varphi \in L^{p(·)}$, we have

$$\left|\mathcal{F}(\vartheta ,\zeta ,\varphi \left(\zeta \right))\right|\le N_{\mathcal{F}}(\vartheta ,\zeta )+C_{\mathcal{F}}\left|\varphi \left(\zeta \right)\right|.$$

Let $\mathsf{\Psi },\mathsf{\Phi }\in {\mathcal{C}}_{{\varrho }_{L^{p(·)}}}([0,1],L^{p(·)})$. Set

$$\varphi :=\mathsf{\Phi }(·)\in L^{p(·)},\psi :=\mathsf{\Psi }(·)\in L^{p(·)}.$$

We assume

$$\left|\mathcal{F}\right(\vartheta ,·,\mathsf{\Phi }(·))-\mathcal{F}(\vartheta ,·,\mathsf{\Psi }(·)\left)\right|\le \lambda \left|\mathsf{\Phi }\right(·)-\mathsf{\Psi }(·\left)\right|,$$

for some $\lambda \ge 0$, which implies

$${\varrho }_{L^{p(·)}}\left(\mathcal{F}(\vartheta ,·,\mathsf{\Phi }(·\left)\right)-\mathcal{F}(\vartheta ,·,\mathsf{\Psi }(·\left)\right)\right)\le L{\varrho }_{L^{p(·)}}(\mathsf{\Phi }(·)-\mathsf{\Psi }(·)),$$

for any $\vartheta \in [0,1]$, where $L=max\{{\lambda }^{p^{-}},{\lambda }^{p^{+}}\}$. Moreover,

$$\begin{array}{cc}\hfill {\varrho }_{L^{p(·)}}\left(\mathcal{F}(\vartheta ,·,\mathsf{\Phi }(·))\right)& ={\int }_0^1{\left|\mathcal{F}(\vartheta ,\zeta ,\varphi \left(\zeta \right))\right|}^{p\left(\zeta \right)}d\zeta \hfill \\ \hfill & \le {\int }_0^1{\left(N_{\mathcal{F}}(\vartheta ,\zeta )+C_{\mathcal{F}}\left|\varphi \left(\zeta \right)\right|\right)}^{p\left(\zeta \right)}d\zeta \hfill \\ \hfill & \le 2^{p^{+}-1}\left({\int }_0^1{\left(N_{\mathcal{F}}(\vartheta ,\zeta )\right)}^{p\left(\zeta \right)}+(C_{\mathcal{F}}{\left|\varphi \left(\zeta \right)\right|)}^{p\left(\zeta \right)}\right)d\zeta \hfill \\ \hfill & \le 2^{p^{+}-1}\left({\int }_0^1{\left(N_{\mathcal{F}}(\vartheta ,\zeta )\right)}^{p\left(\zeta \right)}+\overline{C_{\mathcal{F}}}{\left(\right|\varphi \left(\zeta \right)\left|\right)}^{p\left(\zeta \right)}\right)d\zeta \hfill \\ \hfill & ={\overline{M}}_{\mathcal{F}}{\rho }_{{\mathcal{C}}_{{\varrho }_{L^{p(·)}}}}\left(\mathsf{\Phi }\right)+{\overline{N}}_{\mathcal{F}},\hfill \end{array}$$

where $\overline{C_{\mathcal{F}}}=max\{C_{\mathcal{F}}^{p^{-}},C_{\mathcal{F}}^{p^{+}}\}$, ${\varrho }_{{\mathcal{C}}_{{\varrho }_{L^{p(·)}}}}\left(\mathsf{\Phi }\right)={sup}_{\vartheta \in [0,1]}{\varrho }_{L^{p(·)}}\left(\mathsf{\Phi }\left(\vartheta \right)\right)$, ${\overline{M}}_{\mathcal{F}}=2^{p^{+}-1}\overline{C_{\mathcal{F}}}$, and ${\overline{N}}_{\mathcal{F}}=2^{p^{+}-1}{\varrho }_{{\mathcal{C}}_{{\varrho }_{L^{p(·)}}}}\left(N_{\mathcal{F}}\right)$. Also,

$$\begin{array}{cc}\hfill {\varrho }_{L^{p(·)}}\left(\mathcal{G}(\vartheta ,\mathsf{\Phi }\left(\vartheta \right))\right)& \le 2^{p^{+}-1}\left({\int }_0^1{\left(N_{\mathcal{G}}\left(\vartheta \right)\right)}^{p\left(\vartheta \right)}+\overline{C_{\mathcal{G}}}{\left(\right|\varphi \left(\vartheta \right)\left|\right)}^{p\left(\vartheta \right)}\right)d\vartheta \hfill \\ \hfill & ={\overline{M}}_{\mathcal{G}}{\rho }_{{\mathcal{C}}_{{\varrho }_{L^{p(·)}}}}\left(\mathsf{\Phi }\right)+{\overline{N}}_{\mathcal{G}},\hfill \end{array}$$

where $\overline{C_{\mathcal{G}}}=max\{C_{\mathcal{G}}^{p^{-}},C_{\mathcal{G}}^{p^{+}}\}$, ${\overline{M}}_{\mathcal{G}}=2^{p^{+}-1}\overline{C_{\mathcal{G}}}$, and ${\overline{N}}_{\mathcal{G}}=2^{p^{+}-1}{\varrho }_{L^{p(·)}}\left(N_{\mathcal{G}}\right)$, which implies

$${\overline{M}}_{\mathcal{F}}+{\overline{M}}_{\mathcal{G}}=2^{p^{+}-1}\left(\overline{C_{\mathcal{F}}}+\overline{C_{\mathcal{G}}}\right).$$

Hence, if we have $\overline{C_{\mathcal{F}}}+\overline{C_{\mathcal{G}}}<2^{2(1-p^{+})}$, Theorem 2 will imply that (4) has a solution in ${\mathcal{C}}_{{\varrho }_{L^{p(·)}}}([0,1],L^{p(·)})$.

Note that the assumption $\overline{C_{\mathcal{F}}}+\overline{C_{\mathcal{G}}}<2^{2(1-p^{+})}$ will force $C_{\mathcal{F}}<1$ and $C_{\mathcal{G}}<1$. In this case, we have

$$\overline{C_{\mathcal{F}}}=C_{\mathcal{F}}^{p^{-}}and\overline{C_{\mathcal{G}}}=C_{\mathcal{G}}^{p^{-}}.$$

In the upcoming remarks, we will elucidate the critical significance of the size of the constant and the existence of the solution. Through our detailed analysis, we will underscore the profound impact these elements have on the overall outcomes of the study. This discussion will shed light on the nuanced complexities involved, offering a comprehensive understanding of the role played by the constant’s size $C_{\mathcal{F}}$ and the presence of the solutions in shaping the theoretical framework of our work.

Remark 1.

Let us suppose that

$$\mathcal{G}(\vartheta ,\mathsf{\Phi }(\vartheta \left)\right)=1and\mathcal{F}(\vartheta ,\zeta ,\mathsf{\Phi }(\zeta \left)\right)=\lambda \mathsf{\Phi }\left(\zeta \right),$$

where $\lambda \ge 0$. We are looking at the existence of a non-negative solution of the following equation

$$\mathsf{\Phi }\left(\vartheta \right)=1+\lambda {\int }_0^1\mathsf{\Phi }\left(\zeta \right)d\zeta ,$$

(5)

in ${\mathcal{C}}_{{\varrho }_{L^{p(·)}}}([0,1],L^{p(·)})$. In this case, we have $C_{\mathcal{G}}=0,N_{\mathcal{G}}\left(\vartheta \right)=1,N_{\mathcal{F}}(\vartheta ,\zeta )=0$ and $C_{\mathcal{F}}=\lambda$ is a positive number. It is clear that Φ is a non-negative constant solution of (5) in ${\mathcal{C}}_{{\varrho }_{L^{p(·)}}}([0,1],L^{p(·)})$. Equation (5) can be rewritten as

$$\mathsf{\Phi }=1+\lambda \mathsf{\Phi }.$$

It is clear this equation will have no solution if $\lambda \ge 1$ and one solution if $\lambda <1$.

5. Discussion

In the context of variable exponent spaces, our paper delves into a specific area of study, focusing on the theoretical aspects within the space of ${\mathcal{C}}_{{\varrho }_{L^{p(·)}}}([0,1],L^{p(·)})$. The exploration of this particular space has allowed us to uncover significant insights into solving a diverse range of functional differential equations. As we move forward, our research trajectory aims to expand upon these findings. In our future endeavors, we aspire to broaden the scope of our investigations, leveraging the general theory of semigroups. By incorporating this advanced theoretical framework, we anticipate addressing an even larger class of functional differential equations. This progression in our research not only signifies a natural evolution of our current work but also represents a crucial step toward comprehensively understanding the complexities inherent in variable exponent spaces. Our ongoing commitment to exploring these avenues underscores our dedication to advancing the broader knowledge base within this specialized domain.