Solvability of product of $n$-quadratic Hadamard-type fractional integral equations in Orlicz spaces

1.   Introduction

The theory of fractional integral and differential equations has a fundamental role in several branches of science, such as economics, biology, engineering, physics, electrical circuits, electro-chemistry, earthquakes, fluid dynamics, traffic models, and viscoelasticity (cf. ^[1,2,3]).

Hadamard fractional integral operators were defined by Hadamard in 1892 ^[4]. These operators have a kernel of logarithmic function of arbitrary order, which is not of convolution type. Consequently, they should be examined separately from the more well-known Caputo and Riemann-Liouville fractional operators. These types of operators have been studied by several researchers in numerous function spaces. (cf. ^[5,6,7]).

The present work investigates and establishes the existence theorem as well as the uniqueness of the solution to a general and abstract form of a product of $n$-quadratic fractional integral equations of Hadamard-type in Orlicz spaces $L_\varphi$, which has the form

in arbitrary Orlicz spaces $L_\varphi$, where $G_{j_i}, j = 1, 2, 3$ are general operators.

The theory of fractional calculus in Orlicz spaces was studied by O'Neill in 1965 ^[8], and, subsequently, several interesting articles were published on this topic (see, for example, ^[9,10,11]).

Orlicz spaces $L_\varphi$ are suitable spaces for studying operators with strong nonlinearities (e.g., exponential growth) rather than polynomial growth in Lebesgue spaces $L_p, \; p \geq 1,$ (see ^[12,13]). These are motivated by some problems in statistical physics and mathematical physics (see ^[14,15]). In particular, the thermodynamics problem

contains exponential nonlinearity (cf. ^[16]).

Moreover, quadratic integral equations have been applied in astrophysics, radiative transfer theory, or neutron transport ^[17,18,19]. It should be noted that several kinds of quadratic integral equations have been investigated in $L_p$ spaces ^[20,21,22] and in $L_\varphi$-spaces ^[12,13,23] using the measure of non-compactness analysis associated with Darbo's fixed-point hypothesis via different sets of assumptions.

It is useful to study the product of two or more than two operators, as mentioned by Medve$\check{\rm{d}}$ and Brestovanská in ^[24,25]; however, they consider the Banach algebras of continuous functions, which have a different technique in the proof. Since Orlicz spaces are not Banach algebras, we use the methods given in ^[26,27] to obtain our results.

In ^[26], the author proved some fixed point theorems and employed them in examining the solution of the equation

in some types of ideal spaces like $L_p, p > 1$ and Orlicz spaces $L_\varphi(I), \; I = [a, b]$, where $\varphi$ verifies the $\Delta_2$-condition.

In ^[27], the existence theorems for the product of $n$-integral equations operating on $n$-distinct Orlicz spaces

were discussed in Orlicz spaces $L_\varphi([a, b])$, for $n\geq 2$, when the function $\varphi$ verifies the so-called $\Delta', \; \Delta_3,$ and $\Delta_2$-conditions.

The author in ^[28] demonstrated and proved some basic theorems for the Riemann-Liouville fractional integral operator and investigated the existence theorems in $L_\varphi$-spaces for the equation

In ^[29], some basic theorems were demonstrated and proved for the Hadamard fractional order integral operator, and the existence theorems were also investigated for the equation:

in Orlicz spaces $L_\varphi$.

Basic theorems for the Erdélyi-Kober fractional order integral operator can be found, both demonstrated and proved, in ^[30], where the existence theorems were also investigated for the following equation:

where $0 < \alpha < 1$ and $\beta > 0$ in both $L_p$ and $L_\varphi$ spaces.

This paper is motivated by studying monotonic solutions for a general and abstract form of a product of $n$-quadratic fractional integral equations of Hadamard-type in Orlicz spaces $L_\varphi$. We provide two existence theorems, namely (the existence and the uniqueness of) the solutions for Eq (1.1). The measure of non-compactness and Darbo's fixed point theorem are our main tools for examining the obtained results.

2.   Preliminaries

Let ${\mathbb{R}}^+ = [0, \infty) \subset {\mathbb{R}} = (-\infty, \infty)$ and $I = [1, e], \; e\approx 2.718$. A function $M:[0, \infty) \to [0, \infty)$ points to a Young function if

where $u:[0, \infty) \to [0, \infty)$ is a left-continuous-increasing function and is neither equal to infinite, nor zero on ${\mathbb{R}}^+$. The functions $N$ and $M$ are referred to the complementary Young functions, if $M(y) = \sup_{z \geq 0} (yz - N(y))$. Furthermore, if $M$ is finite-valued with $\lim_{\tau \to 0} \frac{M(\tau)}{\tau} = 0$, $\lim_{\tau \to \infty} \frac{M(\tau)}{\tau} = \infty$, and $M(\tau) > 0$ if $\tau > 0$ ($M(\tau) = 0 \iff \tau = 0$), then $M$ is said to be an $N$-function.

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

Let $E_M = E_M(I)$ contain the set of all bounded functions of $L_M$ and have absolutely continuous norms.

Definition 2.1. ^[31] The Hadamard-type fractional integral of an integrable function $y$ of order $\alpha > 0$ is given by

where $\Gamma(\alpha) = \int_0^\infty e^{-s} s^{\alpha -1}\, ds$.

Proposition 2.1. ^[5] The operator $J^\alpha$ maps a.e. nondecreasing and nonnegative functions to functions of similar types.

Lemma 2.1. ^[29] Assume, that $M$ and $N$ are complementary $N$-functions with $\int_0^s M(\tau^{\alpha-1})\, d\tau < \infty, \; \alpha \in(0, 1)$. Moreover, suppose that $\varphi$ is $N$-function, where

for a.e. $\tau \in I$ and $\epsilon > 0$, then the operator $J^\alpha :L_N \to L_{\varphi}$ is continuous and verifying

The following lemma characterizes the product of the operators in $L_\varphi$:

Lemma 2.2. ([32, Theorem 1]) Let $n\geq 2$. If $\varphi$ and $\varphi_i, \; i = 1, \cdots n$ are arbitrary $N$-functions, then the following conditions are equivalent:

(1) For every $u_i \in L_{\varphi_i}$, $\prod_{i = 1}^n u_i \in L_\varphi$.

(2) There exists a constant $K > 0$ s.t.

for every $u_i \in L_{\varphi_i}, i = 1, 2, \cdots n$.

(3) There exists a constant $C > 0$ s.t.

for every $s \geq 0$.

(4) There exists a constant $C > 0$ s.t. $\forall\; s_i \geq 0, i = 1, \cdots n,$

Let $S = S(I)$ refer to all Lebesgue measurable functions on the interval $I$. The set $S$ concerning the metric

becomes a complete space, where $"meas"$ points to the Lebesgue measure in ${\mathbb{R}}$. The convergence w.r. to $d$ is identical to the convergence in measure on $I$ (cf. Proposition 2.14 in ^[34]). We call the compactness in $S$ by "compactness in measure".

Lemma 2.3. ^[23] Let $Y \subset L_M$ be a bounded set, and there is a family $(\Omega_c)_{0\leq c\leq e-1} \subset I$ s.t. meas $\Omega_c = c$ for every $c \in [1, e]$, and for every $y \in Y$,

Thus, $Y$ represents a compact in measure set in $L_M$.

Definition 2.2. ^[23] Let $\emptyset \neq Y\subset L_M$ be bounded, then

is called the Hausdorff measure of non-compactness (MNC), where $B_r = \{m \in L_M:\; \|m\|_M \leq r\}$.

The measure of equi-integrability $c$ of the set $Y \in L_M$ is given by

where $\epsilon\; > \; 0$ and $\chi_D$ is the characteristic function of $D \subset I$ (cf. ^[33] or ^[34]).

Lemma 2.4. ^[23,33] Let $\emptyset \neq Y \subset E_M$ provide a bounded and compact in measure set, then we have

3.   Main results

Rewrite Eq (1.1) as

where

s.t. $J_i^{\alpha_i}$ is as in Definition 2.1 and $G_{j_i}(y)$ are general operators that act on some different Orlicz spaces for $j = 1, 2, 3$ and $i = 1, \cdots, n$.

Next, we discuss the existence of $L_\varphi$ solutions for Eq (1.1).

3.1.   The existence of $L_\varphi$-solutions

For $i = 1, \cdots, n$, suppose that $\varphi, \varphi_i, \varphi_{1_i}, \varphi_{2_i}$ are $N$-functions and that $N_i, \; M_i$ are complementary $N$-functions with $\int_0^s M_i(\tau^{\alpha_i-1})\, d\tau < \infty, \; \alpha_i \in(0, 1)$, and consider the assumptions:

$({\rm{N1}})$ There exists a constant $K > 0$ s.t. for every $u_i \in L_{\varphi_i}$, and we have $\|\prod_{i = 1}^n u_i\|_{\varphi} \leq K\prod_{i = 1}^n \|u_i\|_{\varphi_i}$.

$({\rm{N2}})$ There exists a constant $k_{1_i} > 0$ such that for every $u_1 \in L_{\varphi_{1_i}}$ and $u_2 \in L_{\varphi_{2_i}},$ we get $\|u_1 u_2\|_{\varphi_i} \leq k_{1_i} \|u_1\|_{\varphi_{1_i}} \|u_2\|_{\varphi_{2_i}}$.

$({\rm{N3}})$ The functions $h_i \in E_{\varphi_i}$ are a.e. nondecreasing on the interval $I$.

$({\rm{N4}})$ $G_{1_i} : L_{\varphi} \to L_{\varphi_{1_i}}$ take continuously $E_{\varphi} \to E_{\varphi_{1_i}}$, the operators $G_{2_i} : L_{\varphi} \to L_{\varphi_i}$ take continuously $E_{\varphi} \to E_{\varphi_i},$ and the operators $G_{3_i} : L_{\varphi} \to L_{N_i}$ take continuously $E_{\varphi} \to E_{N_i}$.

$({\rm{N5}})$ There exist positive functions $g_{1_i} \in L_{\varphi_{1_i}}, \; g_{2_i} \in L_{\varphi_i}, \; g_{3_i} \in L_{N_i}$ s.t. for $s \in I$, $| G_{j_i}(y)(s) | \leq g_{j_i}(s) \|y\|_{\varphi};$ and $G_{j_i}, \; j = 1, 2, 3,$ takes the set of all a.e. nondecreasing functions to functions of similar properties. Moreover, suppose that for any $y \in E_{\varphi}$, we have $G_{1_i}(y) \in E_{\varphi_{1_i}}, \; G_{2_i}(y) \in E_{\varphi_i},$ and $G_{3_i}(y) \in E_{N_i}$.

$({\rm{N6}})$ Assume that $k_i(s) = \frac{1}{\epsilon^{\frac{1}{1-\alpha_i}}}\int_0^ {s \epsilon^{\frac{1}{1-\alpha_i}}}M_i(\tau^{\alpha_i-1})\, d\tau \in E_{\varphi_{2_i}}$ for $\epsilon > 0$ and $s \in I$.

$({\rm{N7}})$ Suppose that $\exists\; r > 0$ and $L_i > 0$ verify

and

Theorem 3.1. Let the assumptions (N1)–(N7) be verified, then there exists a solution $y \in E_\varphi$ of (1.1) that is a.e. nondecreasing on $I$.

Proof. Ⅰ. In what follows, put $i = 1, \cdots, n$. First, Lemma 2.1 implies that each $J_i^\alpha: L_{N_i} \to L_{\varphi_{2_i}}$ is continuous. By assumption (N4), we have that the operators $G_{1_i}:\; E_{\varphi} \to E_{\varphi_{1_i}}, \; G_{2_i}:\; E_{\varphi} \to E_{\varphi_{i}},$ and $G_{3_i}:\; E_{\varphi} \to E_{N_i}$ are continuous, then $A_i = J_i^{\alpha_i} G_{3_i}:\; E_{\varphi} \to E_{\varphi_{2_i}}$ are continuous. By assumption (N2) and the Hölder inequality, we get that $U_i = G_{1_i}\cdot A_i:E_{\varphi} \to E_{\varphi_i}$, and they are continuous. By using assumptions (N3), we have the operators $B_i: E_{\varphi} \to E_{\varphi_i}$. Finally, assumption (N1) and the Hölder inequality give us that $B = \prod_{i = 1}^n B_i:\; E_{\varphi} \to E_{\varphi}$ is continuous.

Ⅱ. We shall establish the ball $B_r(E_{\varphi}) = \{y\in L_\varphi: \|y\|_\varphi \leq r\}$, where $r$ is defined in assumption (N7).

Let $y \in B_r(E_{\varphi}),$ and by recalling Lemma 2.1, we have

Therefore, utilizing assumption (N1), we have

By using assumption (N7), we have that $B : B_r(E_{\varphi}) \rightarrow E_{\varphi}$ is continuous.

Ⅲ. Let $Q_r \subset B_r(E_{\varphi})$ contain the a.e. nondecreasing functions of $I$. The set $Q_r$ is a closed, nonempty, bounded, and convex set in $L_{\varphi};$ see ^[23]. Furthermore, $Q_r$ is compact in measure (thanks to Lemma 2.3).

Ⅳ. Next, we discuss the monotonicity for the operator $B$. Take $y \; \in\; Q_r$, then $y$ is a.e. nondecreasing on $I$. By assumption (N5), the operators $G_{j_i}(y), \; j = 1, 2, 3$ are a.e. nondecreasing on $I$, by Proposition, 2.1 the operator $A_i$ is of the same type, then the operators $U_i(y) = G_{1_i}(y)\cdot A_i(y)$ are a.e. nondecreasing on $I$, and by using assumption (N3), we have that $B: Q_r \to Q_r$ is continuous.

Ⅴ. We will demonstrate that $B$ is a contraction w.r. to the MNC. Suppose that $\emptyset \neq Y \subset Q_r$. For $y \in Y$ and for a set $D \subset I, \; \epsilon > 0$, meas$D \leq \epsilon$. By assumption (N4), we have

and, similarly,

then we have

Therefore,

Since $h_i \in E_{\varphi_i}$, we obtain

From the definition of $c(y)$, we have

where $\|y\cdot \chi_D \|^n_\varphi = \|y\cdot \chi_D \|^{n-1}_\varphi \|y\cdot \chi_D \|_\varphi \leq r^{n} \|y\cdot \chi_D \|_\varphi.$

Since $\emptyset \neq Y \subset Q_r$ is a bounded and compact in measure subset of $E_\varphi$, we can employ Lemma 2.4 to get

Since $\prod_{i = 1}^n \bigg(\|g_{2_i}\|_{\varphi_i}+ \frac{2 k_{1_i}\|k_i \|_{\varphi_{2_i}} \cdot r}{\Gamma(\alpha_i)}\|g_{1_i}\|_{\varphi_{1_i}}\| g_{3_i}\|_{N_i} \bigg) < \frac{1}{ r^{n}K},$ we have finished (cf. ^[26]). □

Remark 3.1. If the $N$-functions $N_i, i = 1, \cdots, n$ verify the $\Delta'$-condition, then Theorem 3.1 is valid on the unite balls $B_1(E_{\varphi}) = \{y \in L_\varphi: \|y\|_\varphi \leq 1 \}$. Furthermore, if they verify the $\Delta_3$ or $\Delta_2$-conditions, then Theorem 3.1 is valid on the whole $E_\varphi$ (cf. ^[13,23]).

3.1.1.   Uniqueness of the solution

Now, we discuss the uniqueness of Eq (1.1).

Theorem 3.2. Let assumption (N1)–(N7) be verified. If

where $r$ and $L_i$ are defined in assumption (N7), then Eq (1.1) has a unique solution $y \in L_\varphi$ in $Q_r$.

Proof. Let $y$ and $z$ be any two different solutions of Eq (1.1), then we obtain

Therefore,

To calculate the above inequality, we need the following estimation. For $j = 1, \cdots, n,$ and by using Lemma 2.1, we have

By substituting from (3.1) and (3.3) in (3.2), we obtain

Since $C < 1,$ we get $y = z$ (a.e.), and we have finished. □

4.   Examples

We need to provide some examples to demonstrate our results.

Example 4.1. Put the $N$-functions $M_i(u) = N_i(u) = u^2$ and $\varphi_{2_i}(u) = \exp{|u|} - |u| - 1.$ We shall show that $J_i^{\alpha_i} : L_{N_i} \to L_{\varphi_{2_i}}, \; i = 1, \cdots, n$ are continuous, and Lemma 2.1 is verified.

Indeed: For $s \in [1, e]$ and any $\alpha_i \in(0, 1)$, we have

Moreover,

Thus for $y\in L_{N_i}$, we get that $J_i^{\alpha_i} : L_{N_i} \to L_{\varphi_{2_i}}$ is continuous.

Remark 4.1. For more details and information about the acting and continuity assumptions of $G_i(y) = g_i(s)\cdot y(s)$, (see our assumption (N5) and [15, Theorem 18.2]).

Example 4.2. Let $G_{j_i}(y)(s) = g_i(s) \cdot y(s), \; j = 1, 2, 3$, and $i = 1, \cdots n$, then we have

which provides a special case of Eq (1.1).

5.   Conclusions

The current study demonstrates and studies two existence theorems, namely, (the existence and the uniqueness) the monotonic solutions for a general and abstract form of a product of $n$-quadratic Hadamard-type fractional integral equations in Orlicz spaces $L_\varphi$. The measure of non-compactness associated with Darbo's fixed-point theorem and fractional calculus are the main tools used to obtain our results in $L_\varphi$-spaces. For the upcoming work in this direction, we will look for some numerical solutions for similar problems in different function spaces.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1445).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this article.