Existence, uniqueness, continuous dependence on the data for the product of $n$-fractional integral equations in Orlicz spaces

1.   Introduction

Various fields of science depend on the use of fractional calculus, such as economics, biology, electrical circuits, engineering, physics, traffic models, viscoelasticity, earthquakes, electrochemistry, and fluid dynamics ^[1,2,3,4,5].

The study of fractional integration in Orlicz spaces was started by O'Neil in 1965 (see ^[6]), but there are a few results considering the Erdélyi-Kober ($\mathbf{\mathcal{EK}}$) operators in Orlicz spaces ^[7].

The Orlicz spaces weight on conditions that enable us to discuss ($\mathbf{\mathcal{EK}}$) fractional integral operators with singular kernels or operators with strong nonlinearities (for example, exponential growth), and then discontinuous solutions are expected ^[8,9]. This requires us to examine the solutions to the considered problem not in Lebesgue spaces but in specific Orlicz spaces. Moreover, these issues have important implications for the study of equivalent differential equations in Orlicz spaces or Sobolev-Orlicz spaces, which are closely related to these issues ^[10,11]. This may be inspired by statistical physics and physics models ^[12,13]. For example, the thermodynamics problem

has exponential nonlinearity ^[14].

The product of two or more than two integral equations through ($\mathbf{\mathcal{EK}}$) fractional operators can be applied effectively in neutron transport ^[15], the kinetic theory of gases ^[16], radiative transfer, and the traffic theory ^[17]. Consequently, it is worthwhile to investigate the product of more than two operators, so we propose to develop a mathematical basis for this theory, especially for fractional operators (see ^[18,19]).

The purpose of this paper is to analyze and demonstrate the solutions to the integral equation

for $\nu \in [0, \rho]$ in the Orlicz spaces $L_\varphi$.

It is generally impossible to determine the solutions to all nonlinear integral equations analytically, except using numerical methods. Consequently, indirect procedures should be used to obtain information about the qualitative behavior of integral equation solutions when there is no analytical expression for the solutions.

Therefore, we establish and present assumptions that allow us to solve and study the integral Eq (1.1) under general growth conditions in $L_\varphi$. As a result, we examine some qualitative properties of the solutions for the problem (1.1), such as existence, monotonicity, and uniqueness, as well as continuous dependence on the data in the spaces $L_\varphi$.

Our method covers and generalizes different types of fractional integrals that have been examined separately and encourages us to recall some of them.

In particular, the existence and the uniqueness theorems of continuous solutions of the SI models

were presented in ^[20]. As the model (1.2) shows the spread of diseases without permanent immunity and with discontinuous data functions, it is appropriate to examine it in Orlicz spaces "$L_\varphi$".

The authors in ^[19] generalized the model (1.2) and examined the existence and uniqueness theorems of the continuous solutions to the equation

The author in ^[21] demonstrated and presented some basic properties of the Riemann-Liouville type fractional integral operator and explored equation solutions

in Orlicz spaces $L_\varphi$.

The author in ^[22] demonstrated and studied fundamental features of the Hadamard-type fractional operator within $L_\varphi$-spaces and utilized them to solve the equation:

The authors in ^[7] showed the basic characteristics of the Erdélyi-Kober fractional operators in Lebesgue and Orlicz spaces and used them to analyze the problem:

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

In ^[23], the author demonstrated some fixed-point theorems and applied them in solving the equation

in some ideal spaces ($L_p, p > 1$ and Orlicz spaces $L_\varphi$).

In ^[24], the existence of solutions for the product of n-integral equations acting on distinct Orlicz spaces

were discussed in $L_\varphi(I)$, when the studied generating $N$-function verifies $\Delta', \Delta_2,$ and $\Delta_3$ conditions.

The existence and the uniqueness results for the abstract product of $n$-quadratic Hadamard-type integral equations

were discussed in arbitrary Orlicz spaces $L_\varphi$ ^[25], where $G_{j_i}, \; j = 1, 2, 3$ are known operators.

Furthermore, the noncompactness measure ($\mathbf{\mathcal{MNC}}$) and fixed-point hypothesis ($\mathbf{\mathcal{FPT}}$) were used to study different types of quadratic integral equations in Orlicz spaces $L_\varphi$ under various sets of assumptions ^[26,27].

The current manuscript is motivated and induced by the extension and generalization of the results introduced in the previous literature to prove some qualitative properties of the solutions for a product of quadratic ($\mathbf{\mathcal{EK}}$)-fractional integral Eq (1.1), including existence, monotonicity, uniqueness, as well as continuous dependence on the data in $L_\varphi$-spaces. We use the technique of ($\mathbf{\mathcal{MNC}}$) concerning the fixed-point hypothesis ($\mathbf{\mathcal{FPT}}$) and the theory of fractional calculus to obtain our findings. We present a few constructed examples that support and illustrate our findings.

2.   Preliminaries

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

and $q:{\mathbb{R}}^+ \to {\mathbb{R}}^+$ is neither identically zero nor infinite and an increasing and left-continuous function on ${\mathbb{R}}^+$. The pair $(P, Q)$ is said to be a complementary pair of YF if $Q(\psi) = \sup_{z \geq 0} (\psi z - P(\psi))$.

The function $\varphi$ is known as the $N$ function when it is finitely valued and verifies $\lim_{\nu \to \infty} \frac{ \varphi(\nu)}{\nu} = \infty$, $\lim_{\nu \to 0} \frac{ \varphi(\nu)}{\nu} = 0$, and $\Big(\varphi(\nu) > 0$ if $\nu > 0$, $\varphi(\nu) = 0 \iff \nu = 0 \Big)$.

The Orlicz space $L_\varphi = L_\varphi(\mathbb{I})$ is the space of all measurable functions $\psi : \mathbb{I} \to {\mathbb{R}}$ with the norm

It is important to recall that for any YF $\varphi$, we have $\varphi(\nu+s) \leq \varphi(\nu)+\varphi(s)$ and $\varphi(k \nu)\leq k \varphi(\nu)$, where $\nu, s \in \mathbb{R}$, and $k \in[0, 1]$.

Assume that $E_{\varphi}(\mathbb{I})$ is the set of all bounded functions in $L_\varphi(\mathbb{I})$ that contain absolutely continuous norms.

Moreover, we obtain $L_\varphi = E_{\varphi}$, if $\varphi$ satisfies the $\Delta_2$-condition, i.e.,

It should be noted that the classical Lebesgue spaces $L_\varphi(\mathbb{I})$ shall be considered as a particular case of Orlicz spaces $L_{ \varphi_p}(\mathbb{I})$ with the corresponding $N$-function ${\varphi}_p = s^p, \; p > 1,$ satisfying the above $\Delta_2$-condition.

Proposition 2.1. ^[28] Let $\varphi$ be a Young function; then for any $\alpha \in (0, 1)$ and $s \in {\mathbb{R}}^+$, the set

is an increasing and continuous function with $\mathbb{P}(0) = 0$.

Lemma 2.2. ^[12] Assume that the function $h(\nu, \psi) : \; \mathbb{I} \times {\mathbb{R}}\; \rightarrow\; {\mathbb{R}}$ verifies Carathéodory conditions i.e.,

(1) It is measurable in $\nu$ for any $\psi \in {\mathbb{R}}$.

(2) It is continuous in $\psi$ for almost all $\nu \in J$.

The superposition operator $F_h = h(\nu, \psi) :E_{\varphi_1} \to L_{\varphi} = E_{\varphi}$ is bounded and continuous if

where $b\geq 0$, $a \in L_{\varphi}$ and the $N$-function $\varphi(\psi)$ verifies the $\Delta_2$-condition.

Lemma 2.3. ^[29] Assume that $\varphi$, $\varphi_1$, and $\varphi_2$ are arbitrary different $N$-functions. The following given conditions are equivalent:

(1) For every function $\psi_1 \in L_{\varphi_1}$ and $\psi_2 \in L_{\varphi_2}$, $\psi_1 \cdot \psi_2 \in L_{\varphi}$.

(2) $\exists\; k > 0$ s.t. for all measurable $\psi_1, \psi_2$ on $\mathbb{I}$, we have $\| \psi_1 \psi_2\|_{\varphi} \leq k \| \psi_1\|_{\varphi_1} \| \psi_1\|_{\varphi_2}$.

(3) $\exists\; C > 0$, $s_0 \geq 0$ s.t. for all $v, s \geq s_0, \; \varphi \left(\frac{s v}{C} \right) \leq \varphi_1(s)+ \varphi_2(v)$.

(4) $\limsup_{v \to \infty} \frac{\varphi_1^{-1}(v) \varphi_2^{-1}(v)}{\varphi (v)} < \infty$.

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

is a complete space. Additionally, the convergence in measure on $\mathbb{I}$ is equivalent to the convergence regarding $d$ ^[30]. The compactness in $S$ is known as "compactness in measure".

Lemma 2.4. ^[26] Assume that $\Psi \subset L_\varphi(\mathbb{I})$ is a bounded set, and $\exists$ a family $(\omega_r)_{0\leq r\leq d} \subset \mathbb{I}$ s.t. meas $\omega_r = r$ for every $r \in [0, d]$, and for every $\psi \in \Psi$,

Then, $\Psi$ is compact in measure in $L_\varphi(\mathbb{I})$.

Definition 2.5. ^[31] The Hausdorff measure of noncompactness ($\mathbf{\mathcal{MNC}}$) $\mu_H(\Psi)$ for a bounded set $\emptyset \neq \Psi \subset L_\varphi$ is known as

where $\Omega_r = \{\Psi \in L_\varphi(\mathbb{I}): \| \psi\|_{\varphi} \leq r\}$ is the ball centered at the origin with radius $r$.

Remark 2.6. The above Hausdorff ($\mathbf{\mathcal{MNC}}$) $\mu_H(\Psi)$ is suited for studying our problem because it is related to the ball $\Omega_r$ and is equivalent to the following measure of equi-integrability in $L_\varphi(\mathbb{I})$. These are useful in employing Darbo's $\mathbf{\mathcal{FPT}}$ to get our results.

Denote a measure of equi-integrability $c$ of $\Psi \in L_\varphi(\mathbb{I})$ by :

where $\epsilon\; > \; 0$ and $\chi_A$ points to the characteristic function $A \subset \mathbb{I}$ (see ^[30,32]).

Lemma 2.7. ^[26,32] Assume that $\emptyset \neq \Psi \subset L_\varphi$ is a bounded set and compact in measure. Then, we get:

Theorem 2.8. ^[31] (Darbo's $\mathbf{\mathcal{FPT}}$) Assume that $\emptyset \neq \Omega \subset L_\varphi$ is a convex, bounded, and closed set and $T : \Omega \; \rightarrow \; \Omega$ is a continuous operator and satisfies the contraction condition, i.e.;

for any $\emptyset \neq \Psi \subset \Omega$. Then, the map $T$ has at least one fixed point in $\Omega$.

We give and present some concepts of the Erdélyi-Kober ($\mathbf{\mathcal{EK}}$) fractional operator in $L_\varphi$-spaces.

Definition 2.9 ^[1,4,33] The Erdélyi-Kober fractional ($\mathbf{\mathcal{EK}}$) integral operator $J_\beta^{ \alpha}, \; \alpha > 0, \beta > 0$, of a function $\psi(\nu)$ is known as

Remark 2.10. There are several special cases covered by Definition 2.9, which include the following:

● Put $\beta = 1$, then the ($\mathbf{\mathcal{EK}}$) operator (2.1) reduces to ($\mathbf{\mathcal{RL}}$) fractional operators that were discussed in ^[1,2]:

● If $\beta = 0$, the ($\mathbf{\mathcal{EK}}$) operator (2.1) reduces to the Hadamared operator

that was discussed in ^[22].

● If and $\beta = 1$, and $\alpha = 1$, then the ($\mathbf{\mathcal{EK}}$) operator (2.1) reduces to the Hardy-Littlewood (Cesaro) operator

that was discussed in ^[34].

● Put $\beta = 2$, then ($\mathbf{\mathcal{EK}}$) operator (2.1) reduces to the $\mathbf{\mathcal{EK}}$) fractional integral operator $J_2^{ \alpha}$ (Sneddon ^[35]):

Proposition 2.11. ^[7] For $\alpha > 0, \beta > 0,$ we have:

(1) The operator $J_\beta^{\alpha}$ maps nonnegative and therefore a.e. nondecreasing functions to functions with the same properties.

(2) If $\varphi$ is an $N$-function and $(P, Q)$ represents a complementary pair of $N$-functions, where $P$ verifies $\int_0^{\nu^\beta } P(t^{\alpha-1})\, dt < \infty, \; \alpha\in(0, 1), \; \beta > 0.$ Then, $J_\beta^{ \alpha} :L_Q(\mathbb{I}) \to L_{\varphi}(\mathbb{I})$ is continuous, satisfying

where

3.   Main results

Equation (1.1) can take the form:

where

such that $\nu^{\beta\alpha} J_\beta^{ \alpha}$ is defined in Definition 2.9 and $F_{f_i}, F_{h_{j_i}}, (j = 1, 2)$ are known as the superposition operators.

Next, we will demonstrate and study the existence theorems in $L_\varphi$.

3.1.   Existence of solutions.

The presented case permits us to utilize some general conditions for the studied operators.

Theorem 3.1. For $i = 1, \cdots, n$, assume that $\varphi_i, \varphi_{1_i}, \varphi_{2_i}, and\; \varphi$ are $N$-functions and $(P_i, Q_i)$ is a complementary pair of $N$-functions, in which $Q_i, \varphi_i, \varphi_{1_i}$ verify the $\Delta_2$ condition and $\int_0^{\nu^\beta } P_i(t^{\alpha-1})\, dt < \infty, \; \alpha_i \in(0, 1), \beta_i > 0,$ and that:

(G1) $\; \forall \; \psi_i \in L_{\varphi_i}, \; \exists K\geq 0, \; s.t.\; \Big\|\prod_{i = 1}^n \psi_i\Big\|_\varphi \leq K\prod_{i = 1}^n \| \psi_i\|_{\varphi_i}$.

(G2) $\exists\; \; k_{1_i} > 0$ s.t. for $\psi_1 \in L_{\varphi_{1_i}}(\mathbb{I})$ and $\psi_2 \in L_{\varphi_{2_i}}(\mathbb{I})$ we obtain $\| \psi_1 \psi_2\|_{\varphi_i} \leq k_{1_i} \| \psi_1\|_{\varphi_{1_i}}\| \psi_2\|_{\varphi_{2_i}}$.

(C1) $\; g \in E_{\varphi}(\mathbb{I})$ is a.e. nondecreasing on $\mathbb{I}$.

(C2) $h_{1_i}, h_{2_i}, f_i: \; \mathbb{I} \times{\mathbb{R}}\; \rightarrow\; {\mathbb{R}}$ satisfy Carathéodory conditions, and $(s, \psi)\to f_i(s, \psi), \; (s, \psi) \to h_{1_i}(s, \psi), \; \; (s, \psi) \to h_{2_i}(s, \psi)$ are nondecreasing.

(C3) $\exists\; e_i, d_{1_i}, \; d_{2_i}$ and functions $b_{1_i} \in E_{\varphi_{1_i}}(\mathbb{I}), \; b_{2_i} \in E_{Q_i}(\mathbb{I})$, and $a_i \in E_{\varphi_i}(\mathbb{I})$ s.t.

(C4) Assume that for a.e. $\nu \in \mathbb{I}$, $\; \exists\; \epsilon > 0$, s.t.

(C5) Assume that, $\exists\; r > 0$ on $I_0 = [0, \rho_0]\subset \mathbb{I}$ verifying

and

Then, there exists a.e. nondecreasing solution $\psi \in E_\varphi(I_0)$ of (1.1) on $I_0 \subset \mathbb{I}$.

Proof. Step Ⅰ. We will show that the operator $B$ is well defined on $E_{\varphi}$ i.e., $B:E_{\varphi}(\mathbb{I}) \to E_{\varphi}(\mathbb{I})$ is continuous.

For $i = 1, \cdots, n$, Lemma 2.2 and assumptions (C2), (C3) imply that $F_{h_{1_i}}: E_{\varphi}(\mathbb{I}))\to L_{\varphi_{1_i}}({\mathbb{I}}), \; F_{h_{2_i}}: E_{\varphi}(\mathbb{I}))\to L_{Q_i}({\mathbb{I}})$ and $F_{f_i}: E_{\varphi}(\mathbb{I})) \to E_{\varphi}(\mathbb{I}))$ are continuous. Proposition 2.11$_2$ gives us that $A_i = \nu^{\beta_i\alpha_i} J_{\beta_i}^{ \alpha_i} F_{h_{2_i}}:\; E_{\varphi}(\mathbb{I}) \to E_{\varphi_{2_i}}(\mathbb{I})$ and are continuous. Assumption (G2) implies that $U_i: E_{\varphi}(\mathbb{I}) \to E_{\varphi_i}(\mathbb{I})$ and by assumptions $(C1),$ and $(G1)$ $B:E_{\varphi}(\mathbb{I}) \to E_{\varphi}(\mathbb{I})$ and are continuous.

Step Ⅱ. We should construct the ball $\Omega_r(E_\varphi) = \{ \psi\in L_\varphi: \| \psi\|_\varphi \leq r \}$, where $r$ is given in assumption (C5) for the operator $B$ acts on.

For arbitrary $\psi \in \Omega_r(E_\varphi)$ and by using Proposition 2.11$_2$ and our assumptions, we get

where $\Big\|\varphi_{1_i}^{-1}\Big(\varphi(| \psi|)\Big)\Big\|_{\varphi_{1_i}}\leq \| \psi\|_\varphi \; \text{and}\; \bigg\| Q_i^{-1} \Big(\varphi(\psi) \Big)\bigg\|_{Q_i} \leq \| \psi\|_\varphi$. Recalling assumptions (G1) and (C1), we have

Therefore, assumption (C5) indicates that $B : \Omega_r(E_\varphi) \to E_\varphi$ is continuous.

Step Ⅲ. We should construct subset $\omega_r\subset \Omega_r$, and investigate the properties of $\omega_r$.

Assume that $\omega_r\subset \Omega_r$ contains all a.e. monotonic (nondecreasing) functions on $I_0$. The set $\emptyset \neq \omega_r$ is bounded, closed, compact in measure, and convex in $L_\varphi(I_0)$ ^[27].

Step Ⅳ. We shall check the monotonicity and continuity of the operator $B$ on $\omega_r$.

Take $\psi \; \in\; \omega_r$, then $\psi$ is a.e. nondecreasing on $I_0$ and, consequently, for $i = 1, 2, \cdots, n,$ the operators $F_{f_i}, F_{h_{1_i}}$ and $F_{h_{2_i}}$ are also a.e. nondecreasing on $I_0$. By Proposition 2.11$_1$, $A_i$ is a.e. nondecreasing on $I_0$, then $U_i = F_{h_{1_i}} A_i$ are also, a.e., nondecreasing. Using assumptions (C1) and (G1), we obtain $B:\omega_r \to \omega_r$ is continuous.

Step Ⅴ. Now, we show that $B$ satisfies the contraction condition with respect to $\mathbf{\mathcal{MNC}}$ $\mu_H$.

Suppose there is a set $D \subset I_0$, with meas $D \leq \varepsilon, \; \varepsilon > 0$. Therefore, for $\psi \in \Psi$ and $\emptyset \neq \Psi \subset \omega_{r}$, we have:

Therefore,

Since $g, a_i \in E_{\varphi}$, $b_{1_i} \in E_{\varphi_{1_i}}$, then we have

and

By using the formula of $c(Y)$, we obtain

Based on the previously established properties, we may apply Lemma 2.7 to obtain

The above inequality with $Kr^n \prod_{i = 1}^n \Bigg(\frac{2e_i k_{1_i} d_{1_i} }{\Gamma(\alpha_i)}\|k_i\|_{\varphi_{2_i}} \bigg(\| b_{2_i}\|_{Q_i}+ d_{2_i} \cdot r\bigg) \Bigg) < 1$ allows us to apply Theorem 2.8. That ends the proof. □

3.1.1.   Uniqueness of the solution

We may prove and discuss the uniqueness of the solutions of Eq (1.1).

Theorem 3.2. Assume the assumptions of Theorem 3.1 are verified but replace assumption (C3) with:

(C6) There exist positive constants $e_i, d_{1_i}, \; d_{2_i}$ and functions $a_i \in E_{\varphi_i}(\mathbb{I}), \; \; b_{1_i} \in E_{\varphi_{1_i}}(\mathbb{I})$, and $b_{2_i} \in E_{Q_i}(\mathbb{I})$, s.t.

where $\omega_r$ is as in Theorem 3.1 for $\; i = 1, \cdots, n,$ in addition, let

where $r$ is given in assumption (C5). Then (1.1) has a unique solution $\psi \in L_\varphi$ in $\omega_r$.

Proof. By applying assumption (C6), we obtain

Similarly, $|h_{2_i}(s, \psi)| \leq b_{2_i}(s) +d_{2_i} Q_i^{-1}\bigg(\varphi \big(\psi \big)\bigg)$ and $|f_i(s, \psi)| \leq a_i(s) + e_i\| \psi\|_\varphi$. Thus, Theorem 3.1 implies that there exists a.e. nondecreasing solution $\psi \in E_\varphi$ of (1.1) in $\omega_r$.

Next, let $\psi, \; z \in \omega_r$ be two distinct solutions of Eq (1.1); then by using assumption (C6), we obtain

Therefore,

To estimate inequality (3.2), we use Proposition 2.11, for $j = 1, \cdots, n,$ to calculate the following:

From assumption (C5), we have $\prod_{i = 1}^n\| B_i(\psi)\|_\varphi \leq \frac{r-\|g\|_{\varphi}}{K},$ and by substituting from (3.3) into (3.2), we obtain

The above estimate with inequality (3.1) concludes the proof. □

3.2.   Continuous dependence on the function g.

Here, we study how solutions of Eq (1.1) depend continuously on $g$.

Definition 3.3. A solution $\psi \in L_\varphi(\mathbb{I})$ of (1.1) is continuously dependent on the function $g$ if $\forall\; \epsilon > 0, \; \exists \delta > o$ such that $\|g-\bar{g}\|_\varphi \leq \delta$ implies that $\|\psi -\bar{\psi}\|_\varphi \leq \epsilon$, where

Theorem 3.4. Suppose that the assumptions of Theorem 3.2 are held. Then solutions $\psi \in L_\varphi(\mathbb{I})$ of Eq (1.1) depend continuously on the function $g$.

Proof. Let $\psi, \bar{ \psi}$ be any two solutions of (1.1); then, similarly as done in Theorem 3.2, we have

where $C$ is given by (3.1). Then, we obtain

Therefore, if $\|g-\bar{g}\|_\varphi \leq \delta(\epsilon)$, then $\| \psi-\bar{ \psi}\|_\varphi \leq \epsilon$, where

□

4.   Remarks and examples

Remark 4.1. It is important to note that our solutions are not necessarily continuous, as in many of the cases that have been examined and investigated previously ^[1,36] or in $L_p$-spaces ^[8,37]. So we do not assume that the operators studied map the spaces $C(I)$ or $L_p$ into themselves. Our results belong to the space $L_\varphi$, for more examples and assumptions related to the operators studied in $L_\varphi$ see ^[9,29].

Finally, we demonstrate and clarify some examples that support our outcomes.

Example 4.2. Considering the $N$-functions $P_i(\nu) = Q_i(\nu) = \nu^2$ and $\varphi_{2_i}(\nu) = \exp{|\nu|} - |\nu| - 1.$ We can examine that Proposition 2.11$_2$ is verified and the fractional operator $J^{\alpha_i}_{\beta_i} : L_{Q_i} (\mathbb{I}) \to L_{\varphi_{2_i}} (\mathbb{I})$ is continuous, where $\mathbb{I} = [0, \rho]$.

Therefore: For $i = 1, 2, \cdots, n,$ and any $\alpha_i, \beta_i > 0$ and $\nu \in \mathbb{I}$, we obtain

Moreover,

Then, Proposition 2.11$_2$ is verified. Therefore, for $\psi \in L_{Q_i}(\mathbb{I})$, we obtain $J^{\alpha_i}_{\beta_i} : L_{Q_i} (\mathbb{I}) \to L_{\varphi_{2_i}} (\mathbb{I})$ is continuous.

Remark 4.3. In the literature, many applications and significant results exist for $N$-functions that do not satisfy the global $\Delta_2$-condition. Such functions often appear in models of nonlinear elasticity, statistical physics, and image processing, where exponential or logarithmic growth is relevant. Regarding the examples in the manuscript:

(1) The functions $P_i(\nu) = Q_i(\nu) = \nu^2$ satisfy the $\Delta_2$-condition globally, as it is a simple power function with polynomial growth.

(2) The functions $\varphi_{2_i}(\nu) = \exp{|\nu|} - |\nu| - 1$ do not satisfy the $\Delta_2$-condition globally due to the exponential growth. However, for small values of $\nu$, they behave like the quadratic functions:

and satisfy the $\Delta_2$-condition locally in this regime.

Remark 4.4. For further particular cases verifying Example 4.2, see ^[7,21,22], and for more information and numerous examples of the functions $\varphi_{2_i}$ and $(P_i, Q_i)$ verifying Proposition 2.11$_2$, see ^[12].

Example 4.5. For $\alpha_i = \frac{1}{2}$, $\beta_i = \frac{1}{2}, i = 1, 2, \cdots, n,$ and $\nu \in \mathbb{I} = [0, 1]$, we have

This may be considered a particular case of Eq (1.1), where

with

Thus, assumptions (C2) and (C3) are satisfied with constants $e_i = d_{1_i} = d_{2_i} = \frac{1}{10}$ and for suitable forms of the functions $b_{1_i} \in E_{\varphi_{1_i}}(\mathbb{I}), \; b_{2_i} \in E_{Q_i}(\mathbb{I})$, and $a_i \in E_{\varphi_i}(\mathbb{I})$.

For assumption (C4), see Example 4.2.

Now for a suitable nondecreasing function $g$ satisfying assumption (C1), we can find $r > 0$ on $\mathbb{I}$, verifying

and

Therefore, we obtain our verifications and get that Eq

5.   Conclusions

The presented paper established existence, monotonicity, and uniqueness, as well as continuous dependence on the data of a product of $n$-quadratic Erdélyi-Kober ($\mathbf{\mathcal{EK}}$) fractional integral equation. The analysis used to obtain the results is the ($\mathbf{\mathcal{MNC}}$) measure of noncompactness, the ($\mathbf{\mathcal{FPT}}$) fixed-point theorem, as well as fractional calculus in the Orlicz spaces $L_\varphi$. Some examples are given to illustrate the hypothesis.

Open discussion and future possibilities:

(1) The authors could discuss potential extensions of their work to other fractional operators.

(2) The authors could apply their results to specific fields such as physics, engineering, or biology.

(3) The authors can focus on the corresponding problems of equivalent differential equations in Orlicz spaces or Sobolev-Orlicz spaces.

(4) The authors could check some numerical results for the considered problems.

Author contributions

Mohamed M. A. Metwali: methodology, validation, formal analysis, investigation, writing-original draft preparation, visualization; Abdulaziz M. Alotaibi: Methodology, validation, resources, investigation; writing-original draft preparation, and editing; Hala H.Taha: formal analysis, resources, investigation, writing-review and editing; Ravi P Agarwal: Conceptualization, validation, writing-review and editing, Supervision. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This work is supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R899), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446).

Conflicts of interest

The authors declare that there are no conflict of interest.