A Study of an IBVP of Fractional Differential Equations in Banach Space via the Measure of Noncompactness

Abstract

In this article, we are concerned with a very general integral boundary value problem of Riemann–Liouville derivatives. We will study the problem in Banach space. To be more specific, we are interested in proving the existence of a solution to our problem via the measure of noncompactness and Mönch fixed-point theorem. Our study in Banach space contains two nonlinear terms and two different orders of derivatives, $\varsigma$ and $\tau$, such that $\varsigma \in \left(1,2\right]$ and $\tau \in \left(0,\varsigma \right]$. Our paper ends with a conclusion.

1. Introduction

The concept of measures of noncompactness was created by Kuratowski [1] in 1930. This measure has many uses in mathematical study [2,3]. Darbo [4] was the first researcher that employed the measure of noncompactness to study the relationship between that of compact and contraction mappings. On the other hand, Darbo’s, Sadovski’s [5], and Mönch’s [6,7] fixed-point theorems were considered effective tools for studying the existence of solutions of several classes and types of differential equations, especially for fractional differential equations (see [8,9,10,11,12,13,14,15,16,17,18] and references therein).

The goal of this article is to study an integral boundary value problem of fractional differential equations, given as follows:

$$D^{\varsigma }\nu \left(\iota \right)+\phi \left(\iota ,\nu \left(\iota \right)\right)=D^{\tau }\psi \left(\iota ,\nu \left(\iota \right)\right),\iota \in \left(0,1\right),$$

(1)

$$\nu \left(0\right)=0,\nu \left(1\right)={\int }_0^1\Psi \left(1,r\right)\psi \left(r,\nu \left(r\right)\right)dr,$$

(2)

where $\varsigma \in \left(1,2\right]$, $\tau \in \left(0,\varsigma \right]$, $D^{\varsigma }$, and $D^{\tau }$ are the standard Riemann–Liouville derivatives, and

$$\Psi \left(\iota ,r\right)=\frac{1}{\Gamma \left(\varsigma -\tau \right)}{\left(\iota -r\right)}^{\varsigma -\tau -1}.$$

The functions $\phi ,\psi :\left[0,1\right]\times \mathcal{Y}\to \mathcal{Y}$ will be specified later, and $\mathcal{Y}$ is a Banach space supplied with the norm $∥·∥$ in the case of $\mathcal{Y}=\mathbb{R}$.

In the literature, Xu and Han in [19] investigated and proved the existence and uniqueness of a positive solution of (1) and (2); they used the method of the upper and lower solutions and the Schauder and Banach fixed-point theorems. In [20], Xu and Sun studied problems (1) and (2) when $\Psi \left(1,r\right)\equiv 0$, which means $\tau =\varsigma -1$, and they gave the existence and uniqueness of a positive solution using the same method in [19]. On the other hand, Lachouri, Ardjouni, and Djoudi [10] generalized problems (1) and (2) in the case of $\tau =\varsigma -1$ (see [20]) and gave interesting results for the existence of a solution in Banach space by using the Mönch fixed-point theorem and the measure of noncompactness. In view of the above works, we take $\varsigma \in \left(1,2\right]$, $\tau \in \left(0,\varsigma \right]$ and study problem roblems (1) and (2) in Banach space.

We move from this section to the second section to give some of the notations, definitions, and results that we will use in the study. Then, we use the material from the previous section to give and prove the existence result for problems (1) and (2). In Section 4, we present an example to illustrate the main theorem. Finally, we outline the contributions and generalizations of our research in the conclusion.

2. Preliminaries

Let $C\left(\Lambda ,\mathcal{Y}\right)$ be a Banach space of continuous functions from $\Lambda =\left[0,1\right]$ to $\mathcal{Y}$ supplied with the norm

$${∥\nu ∥}_{\infty }=\underset{\iota \in \left[0,1\right]}{sup}∥\nu \left(\iota \right)∥.$$

Let $L^1\left(\Lambda ,\mathcal{Y}\right)$ be the Banach space of measurable functions from $\Lambda$ to $\mathcal{Y}$ that are Lebesgue integrable with the norm

$${∥\nu ∥}_{L^1}={\int }_0^1∥\nu \left(\iota \right)∥d\iota .$$

Definition 1

([21,22]). The fractional integral of order $\varsigma >0$ of a function $\nu :\Lambda \to \mathcal{Y}$ is given by

$$I^{\varsigma }\nu \left(\iota \right)=\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^{\iota }{\left(\iota -r\right)}^{\varsigma -1}\nu \left(r\right)dr,$$

where the right side is pointwise defined on Λ.

Let $AC\left(\Lambda ,\mathcal{Y}\right)$ be the space of the valued functions that are absolutely continuous on $\Lambda$, and set

$$A{C}^n=\left\{\nu \in C\left(\Lambda ,\mathcal{Y}\right):{\nu }^{\prime },{\nu }^{″},\dots ,{\nu }^{\left(n-1\right)}\in C\left(\Lambda ,\mathcal{Y}\right)\mathrm{and}{\nu }^{\left(n\right)}\in AC\left(\Lambda ,\mathcal{Y}\right)\right\}.$$

Definition 2

([21,22]). For a function $\nu \in A{C}^n\left(\Lambda \right)$, the Riemann–Liouville fractional order derivative of order ς of νis defined by

$$D^{\varsigma }\nu \left(\iota \right)=\frac{1}{\Gamma \left(n-\varsigma \right)}{\left(\frac{d}{d\iota }\right)}^n{\int }_0^{\iota }{\left(\iota -r\right)}^{n-\varsigma -1}\nu \left(r\right)dr,$$

where $n=\left[\varsigma \right]+1$ and $\left[\varsigma \right]$ denotes the integer part of real number ς.

For more information about fractional order derivatives and integrals, the reader can see the references in [22,23]. Throughout of this paper, we need the following materials in our proof and study.

Lemma 1

([19]). Let $\nu \in L^1\left(\Lambda ,\mathcal{Y}\right)$ and $I^{2-\varsigma }\nu \in A{C}^2\left(\Lambda ,\mathcal{Y}\right)$ with $1<\varsigma \le 2$. Then, ν is a solution of boundary value problems (1) and (2) if and only if

$$\nu \left(\iota \right)={\int }_0^1\Phi \left(\iota ,r\right)\phi \left(r,\nu \left(r\right)\right)dr+{\int }_0^{\iota }\Psi \left(\iota ,r\right)\psi \left(r,\nu \left(r\right)\right)dr,$$

(3)

where

$$\Phi \left(\iota ,r\right)=\left\{\begin{array}{c}\frac{\iota {\left(1-r\right)}^{\varsigma -1}-{\left(\iota -r\right)}^{\varsigma -1}}{\Gamma \left(\varsigma \right)},0\le r\le \iota \le 1,\\ \frac{\iota {\left(1-r\right)}^{\varsigma -1}}{\Gamma \left(\varsigma \right)},0\le \iota \le r\le 1.\end{array}\right.$$

(4)

Proof. 

The proof is given in [19]. □

Lemma 2

([24]). The function $\Phi \left(\iota ,r\right)$, defined by (4), satisfies the following:

(a) $\Phi \left(\iota ,r\right)>0$ for $\iota ,r\in \left(0,1\right)$.

(b) ${max}_{\iota \in \Lambda }\Phi \left(\iota ,r\right)=\Phi \left(r,r\right)$, $r\in \left(0,1\right)$.

Now, we define the set W of functions $w:\Lambda \to \mathcal{Y}$, which we denote as

$$W\left(\iota \right)=\left\{w\left(\iota \right):w\in W\right\},\iota \in \Lambda ,$$

and recall some information and properties of the measure of noncompactness $\mu$ (see [25,26]). Let $convK$ and $\overline{K}$ be the convex hull and the closure of the bounded set K, respectively.

(1)

$\mu \left(K\right)=0\iff$$\overline{K}$ is compact (K is relatively compact).

(2)

$\mu \left(K\right)=\mu \left(\overline{K}\right)$.

(3)

$\mu \left(K\right)=\mu \left(convK\right)$.

(4)

$K_1\subset K_2\Rightarrow \mu \left(K_1\right)\le \mu \left(K_2\right)$.

(5)

$\mu \left(K_1+K_2\right)\le \mu \left(K_1\right)+\mu \left(K_2\right)$.

(6)

$\mu \left(cK\right)=\left|c\right|\mu \left(K\right),$ for $c\in \mathbb{R}.$

Definition 3.

A map $\phi :\Lambda \times \mathcal{Y}\to \mathcal{Y}$ is said to be Caratheodory if the following is true:

(i) $\iota \to \phi \left(\iota ,\nu \right)$ is measurable for each $\nu \in \mathcal{Y}$.

(ii) $\iota \to \phi \left(\iota ,\nu \right)$ is continuous for almost all $\iota \in \Lambda$.

Lemma 3

([8]). Let Σ be a bounded, closed, and convex subset of the Banach space $C\left(\Lambda ,\mathcal{Y}\right)$. Let Φ be a continuous function on $\Lambda \times \Lambda$ and φ a function from $\Lambda \times \mathcal{Y}$ to $\mathcal{Y}$, which satisfies the Caratheodory conditions, and assume there exists $\xi \in L^1\left(\Lambda ,{\mathbb{R}}^{+}\right)$ such that, for each $\iota \in \Lambda$ and each bounded set $K\subset \mathcal{Y}$, we have

$$\underset{ϵ\to 0^{+}}{lim}\mu \left(\phi \left({\Lambda }_{ϵ}\times K\right)\right)\le \lambda \left(\iota \right)\mu \left(K\right),$$

with ${\Lambda }_{ϵ}=\left[\iota -ϵ,\iota \right]\cap \Lambda$. If W is an equicontinuous subset of Σ, then

$$\mu \left(\left\{{\int }_{\Lambda }\Phi \left(\iota ,r\right)\phi \left(r,\nu \left(r\right)\right)dr,\nu \in W\right\}\right)\le {\int }_{\Lambda }\left|\Phi \left(\iota ,r\right)\right|\xi \left(r\right)\mu \left(W\left(r\right)\right)dr.$$

Theorem 1

(Mönch [7]). Let Σ be a closed, bounded, and convex subset of a Banach space such that $0\in \Sigma$, and let $\mathcal{M}$ be a continuous mapping of Σ into itself. Moreover, if

$$W=\overline{conv}\left(\left(\mathcal{M}W\right)\right)\mathrm{or}W=\overline{conv}\left(\left(\mathcal{M}W\right)\right)\cup \left\{0\right\}\Rightarrow \mu \left(W\right)=0$$

(5)

holds for every W of Σ, then $\mathcal{M}$ has a fixed point.

Theorem 2

(Arzela–Ascoli’s Theorem [27]). A bounded, uniformly Cauchy subset K of $\mathcal{Y}$ is relatively compact.

In the section below, we will prove the existence results for boundary value problems (1) and (2) by using the Mönch fixed-point Theorem 1. For this purpose, we prepare at the end of this section all the necessary conditions.

(C1)

The functions $\phi ;\psi :\Lambda \times \mathcal{Y}\to \mathcal{Y}$ satisfy the Caratheodory conditions.

(C2)

There exist $\xi ,\eta \in L^1\left(\Lambda ,{\mathbb{R}}^{+}\right)\cap \left(\Lambda ,{\mathbb{R}}^{+}\right)$ such that the following is true:

$$∥\phi \left(\iota ,\nu \left(\iota \right)\right)∥\le \xi \left(\iota \right)∥\nu ∥,\mathrm{for}\iota \in \Lambda \mathrm{and}\mathrm{each}\nu \in \mathcal{Y}.$$

$$∥\psi \left(\iota ,\nu \left(\iota \right)\right)∥\le \eta \left(\iota \right)∥\nu ∥,\mathrm{for}\iota \in \Lambda \mathrm{and}\mathrm{each}\nu \in \mathcal{Y}.$$

and

$$\frac{{∥\xi ∥}_{\infty }}{\Gamma \left(\varsigma \right)}+\frac{{∥\eta ∥}_{\infty }}{\Gamma \left(\varsigma -\tau \right)}<1.$$

(C3)

Let ${\Lambda }_{ϵ}=\left[\iota -ϵ,\iota \right]\cap \Lambda$. Then, for each $\iota \in \Lambda$ and any bounded set $K\subset \mathcal{Y}$, we have the following:

$$\underset{ϵ\to 0^{+}}{lim}\mu \left(\phi \left({\Lambda }_{ϵ}\times K\right)\right)\le \xi \left(\iota \right)\mu \left(K\right).$$

$$\underset{ϵ\to 0^{+}}{lim}\mu \left(\psi \left({\Lambda }_{ϵ}\times K\right)\right)\le \eta \left(\iota \right)\mu \left(K\right).$$

3. Main Results

Now, we would like to apply the tools and methods in the previous section to problems (1) and (2). So, according to Lemma 1, we define the operator $\mathcal{M}$ from $C\left(\Lambda ,\mathcal{Y}\right)$ into itself as

$$\left(\mathcal{M}\nu \right)\left(\iota \right)={\int }_0^1\Phi \left(\iota ,r\right)\phi \left(r,\nu \left(r\right)\right)dr+{\int }_0^{\iota }\Psi \left(\iota ,r\right)\psi \left(r,\nu \left(r\right)\right)dr,$$

(6)

where its fixed points are the solution of problems (1) and (2). For this purpose, for $L>0$ we define $\Sigma$ the closed, bounded, and convex subset of $C\left(\Lambda ,\mathcal{Y}\right)$ as

$$\Sigma =\left\{\nu \in C\left(\Lambda ,\mathcal{Y}\right):{∥\nu ∥}_{\infty }\le L\right\}.$$

To prove the principle results, we divide the proof into the following lemmas.

Lemma 4.

Assume that condition (C2) holds. Then, $\mathcal{M}$ maps Σ into itself. Moreover, $\mathcal{M}\left(\Sigma \right)$ is bounded and equicontinuous.

Proof. 

First, let $\iota \in \Lambda$ and $\nu \in \Sigma$. By using condition (C2), we find the following:

$$\begin{array}{cc}\hfill ∥\left(\mathcal{M}\nu \right)\left(\iota \right)∥& \le ∥{\int }_0^1\Phi \left(r,r\right)\phi \left(r,\nu \left(r\right)\right)dr∥+∥{\int }_0^{\iota }\Psi \left(\iota ,r\right)\psi \left(r,\nu \left(r\right)\right)dr∥\hfill \\ \hfill & \le \frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^{1}r{\left(1-r\right)}^{\varsigma -1}∥\phi \left(r,\nu \left(r\right)\right)∥dr\hfill \\ \hfill & +\frac{1}{\Gamma \left(\varsigma -\tau \right)}{\int }_0^{\iota }{\left(\iota -r\right)}^{\varsigma -\tau -1}∥\psi \left(r,\nu \left(r\right)\right)∥dr\hfill \\ \hfill & \le \left(\frac{{∥\xi ∥}_{\infty }}{\Gamma \left(\varsigma \right)}+\frac{{∥\eta ∥}_{\infty }}{\Gamma \left(\varsigma -\tau \right)}\right)L\hfill \\ \hfill & \le L,\hfill \end{array}$$

which implies that $\mathcal{M}$ maps $\Sigma$ into itself and $\mathcal{M}\left(\Sigma \right)$ is bounded.

Next, let ${\iota }_1,{\iota }_2\in \Lambda ,{\iota }_1<{\iota }_2$, and $\nu \in \Sigma$. Then,

$$\begin{array}{cc}\hfill & ∥\left(\mathcal{M}\nu \right)\left({\iota }_2\right)-\left(\mathcal{M}\nu \right)\left({\iota }_1\right)∥\hfill \\ \hfill & \le ∥{\int }_0^1\Phi \left({\iota }_2,r\right)\phi \left(r,\nu \left(r\right)\right)dr-{\int }_0^1\Phi \left({\iota }_1,r\right)\phi \left(r,\nu \left(r\right)\right)dr∥\hfill \\ \hfill & +∥{\int }_0^{{\iota }_2}\Psi \left({\iota }_2,r\right)\psi \left(r,\nu \left(r\right)\right)dr-{\int }_0^{{\iota }_1}\Psi \left({\iota }_1,r\right)\psi \left(r,\nu \left(r\right)\right)dr∥\hfill \\ \hfill & =I_1+I_2\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill I_1& =∥\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^1{\iota }_2{\left(1-r\right)}^{\varsigma -1}\phi \left(r,\nu \left(r\right)\right)dr+\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\iota }_2}{\left({\iota }_2-r\right)}^{\varsigma -1}\phi \left(r,\nu \left(r\right)\right)dr\hfill \\ \hfill & -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^1{\iota }_1{\left(1-r\right)}^{\varsigma -1}\phi \left(r,\nu \left(r\right)\right)dr-\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\iota }_1}{\left({\iota }_1-r\right)}^{\varsigma -1}\phi \left(r,\nu \left(r\right)\right)dr∥\hfill \\ \hfill & \le ∥\frac{{\iota }_2-{\iota }_1}{\Gamma \left(\varsigma \right)}{\int }_0^1{\left(1-r\right)}^{\varsigma -1}\phi \left(r,\nu \left(r\right)\right)dr\hfill \\ \hfill & +\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\iota }_2}\left({\left({\iota }_2-r\right)}^{\varsigma -1}-{\left({\iota }_1-r\right)}^{\varsigma -1}\right)\phi \left(r,\nu \left(r\right)\right)dr\hfill \\ \hfill & +\frac{1}{\Gamma \left(\varsigma \right)}{\int }_{{\iota }_2}^{{\iota }_1}{\left({\iota }_1-r\right)}^{\varsigma -1}\phi \left(r,\nu \left(r\right)\right)dr∥\hfill \\ \hfill & \le \left|{\iota }_2-{\iota }_1+{\iota }_2^{\varsigma -1}-{\iota }_1^{\varsigma -1}\right|\frac{{∥\xi ∥}_{\infty }}{\Gamma \left(\varsigma +1\right)},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill I_2& =∥\frac{1}{\Gamma \left(\varsigma -\tau \right)}{\int }_0^{{\iota }_2}{\left({\iota }_2-r\right)}^{\varsigma -\tau -1}\psi \left(r,\nu \left(r\right)\right)dr\hfill \\ \hfill & -\frac{1}{\Gamma \left(\varsigma -\tau \right)}{\int }_0^{{\iota }_1}{\left({\iota }_1-r\right)}^{\varsigma -\tau -1}\psi \left(r,\nu \left(r\right)\right)dr∥\hfill \\ \hfill & \le ∥\frac{1}{\Gamma \left(\varsigma -\tau \right)}{\int }_0^{{\iota }_2}\left({\left({\iota }_2-r\right)}^{\varsigma -1}-{\left({\iota }_1-r\right)}^{\varsigma -1}\right)\psi \left(r,\nu \left(r\right)\right)dr\hfill \\ \hfill & +\frac{1}{\Gamma \left(\varsigma -\tau \right)}{\int }_{{\iota }_2}^{{\iota }_1}{\left({\iota }_1-r\right)}^{\varsigma -1}\psi \left(r,\nu \left(r\right)\right)dr∥\hfill \\ \hfill & \le \left|{\iota }_2^{\varsigma -\tau }-{\iota }_1^{\varsigma -\tau }\right|{\frac{{∥\eta ∥}_{\infty }}{\Gamma \left(\varsigma -\tau +1\right)}}_{\infty },\hfill \end{array}$$

as ${\iota }_1\to {\iota }_2$, the sum $I_1+I_2$ tends to zero, which means that $\mathcal{M}\left(\Sigma \right)$ is equicontinuous. □

Lemma 5.

Assume that conditions (C1) and (C2) hold. Then, $\mathcal{M}$ is continuous.

Proof. 

Consider the sequence $\left\{{\nu }_n\right\}$ such that ${\nu }_n\to \nu \in C\left(\Lambda ,\mathcal{Y}\right)$. So, for each $\iota \in \Lambda$,

$$\begin{array}{cc}\hfill & ∥\left(\mathcal{M}{\nu }_n\right)\left(\iota \right)-\left(\mathcal{M}\nu \right)\left(\iota \right)∥\hfill \\ \hfill & \le {\int }_0^1\Phi \left(r,r\right)∥\phi \left(r,{\nu }_n\left(r\right)\right)-\phi \left(r,\nu \left(r\right)\right)∥dr\hfill \\ \hfill & +{\int }_0^{\iota }\Psi \left(\iota ,r\right)∥\psi \left(r,{\nu }_n\left(r\right)\right)-\psi \left(r,\nu \left(r\right)\right)∥dr.\hfill \end{array}$$

Then, we conclude using the Lebesgue dominated convergence theorem that

$$∥\left(\mathcal{M}{\nu }_n\right)\left(\iota \right)-\left(\mathcal{M}\nu \right)\left(\iota \right)∥\to 0\mathrm{as}n\to \infty ,$$

which means that $\left(\mathcal{M}{\nu }_n\right)$ converges to $\mathcal{M}\nu$ on $\Lambda$. On the other hand, as shown in Lemma 4, the sequence $\left(\mathcal{M}{\nu }_n\right)$ is equicontinuous. Consequently, $\left(\mathcal{M}{\nu }_n\right)$ converges uniformly to $\mathcal{M}\nu$ and, hence, $\mathcal{M}$ is continuous. □

Theorem 3.

Assume that conditions (C1)–(C3) hold. Then, IBVPs (1) and (2) have at least one solution.

Proof. 

The first part of the proof was proved in Lemmas 4 and 5. So, to achieve the proof, it suffices to prove (5). Let W be the subset of $\Sigma$ such that $W\subset \overline{conv}\left(\left(\mathcal{M}W\right)\cup \left\{0\right\}\right)$. Since W is bounded and equicontinuous, then the function $w\to w\left(\iota \right)=\mu \left(w\left(\iota \right)\right)$ is continuous on $\Lambda$. By condition (C3) and Lemma 3, we have, for each each $\iota \in \Lambda$,

$$\begin{array}{ccc}\hfill w\left(\iota \right)& \le & \mu \left(\left(\mathcal{M}W\right)\cup \left\{0\right\}\right)\le \mu \left(\left(\mathcal{M}W\right)\right)\hfill \\ & \le & {\int }_0^1\Phi \left(\iota ,r\right)\xi \left(r\right)\mu \left(w\left(r\right)\right)dr+{\int }_0^{\iota }\Psi \left(\iota ,r\right)\eta \left(r\right)\mu \left(w\left(r\right)\right)dr\hfill \\ & \le & {∥w∥}_{\infty }\left(\frac{{∥\xi ∥}_{\infty }}{\Gamma \left(\varsigma \right)}+\frac{{∥\eta ∥}_{\infty }}{\Gamma \left(\varsigma -\tau \right)}\right).\hfill \end{array}$$

Then, we obtain

$${∥w∥}_{\infty }\left(1-\frac{{∥\xi ∥}_{\infty }}{\Gamma \left(\varsigma \right)}-\frac{{∥\eta ∥}_{\infty }}{\Gamma \left(\varsigma -\tau \right)}\right)\le 0.$$

Since $\frac{{∥\xi ∥}_{\infty }}{\Gamma \left(\varsigma \right)}+\frac{{∥\eta ∥}_{\infty }}{\Gamma \left(\varsigma -\tau \right)}<1$, then ${∥w∥}_{\infty }=0$, that is, $w\left(\iota \right)=0$ for each $\iota \in \Lambda$. Hence, W is relatively compact in $\mathcal{Y}$. Furthermore, W is relatively compact in $\Sigma$ due to the Ascoli–Arzela theorem. Finally, as a conclusion of this proof, we deduce by Theorem 1 that $\mathcal{M}$ has a fixed point, which represents the solution of problem (1). □

4. An Example

In this part, based on the examples given into the papers [10,19], we give an example to make Theorem 3 more clear to the reader. Therefore, we propose the following problem:

$$D^{\frac{5}{4}}\nu \left(\iota \right)+\frac{\nu \left(\iota \right)}{4+e^{\iota }}=D^{\frac{1}{8}}\frac{\nu \left(\iota \right)}{6+e^{{\iota }^2}},\iota \in \left(0,1\right),$$

(7)

$$\nu \left(0\right)=0,\nu \left(1\right)=\frac{1}{\Gamma \left(\frac{9}{8}\right)}{\int }_0^1{\left(\iota -r\right)}^{\frac{1}{8}}\frac{\nu \left(r\right)}{6+e^{r^2}}dr,$$

(8)

where $\nu =\left({\nu }_1,{\nu }_2,\dots ,{\nu }_i,\dots \right)$, $\phi =\left({\phi }_1,{\phi }_2,\dots ,{\phi }_i,\dots \right)$, and $\psi =\left({\psi }_1,{\psi }_2,\dots ,{\psi }_i,\dots \right)$ such that

$${\phi }_i\left(\iota ,{\nu }_i\left(\iota \right)\right)=\frac{{\nu }_i\left(\iota \right)}{4+e^{\iota }}\mathrm{and}{\psi }_i\left(\iota ,{\nu }_i\left(\iota \right)\right)=\frac{{\nu }_i\left(\iota \right)}{6+e^{{\iota }^2}}\mathrm{for}\iota \in \Lambda ,$$

and let the space

$$\mathcal{Y}=l^1=\left\{\nu =\left({\nu }_1,{\nu }_2,\dots ,{\nu }_i,\dots \right)\mathrm{such}\mathrm{that}\sum _{i=1}^{\infty }\left|{\nu }_i\right|<\infty \right\},$$

be supplied with the norm

$${∥\nu ∥}_{l^1}=\sum _{i=1}^{\infty }\left|{\nu }_i\right|.$$

Then, for each ${\nu }_i$ and $\iota \in \Lambda$, we have

$$\left|{\phi }_i\left(\iota ,{\nu }_i\left(\iota \right)\right)\right|\le \frac{1}{4+e^{\iota }}\left|{\nu }_i\left(\iota \right)\right|.$$

(9)

$$\left|{\psi }_i\left(\iota ,{\nu }_i\left(\iota \right)\right)\right|\le \frac{1}{6+e^{{\iota }^2}}\left|{\nu }_i\left(\iota \right)\right|.$$

(10)

So, we conclude that conditions (C1) and (C2) are satisfied with $\xi \left(\iota \right)=\frac{1}{4+e^{\iota }},\left({∥\xi ∥}_{\infty }=\frac{1}{5}\right)$ and $\eta \left(\iota \right)=\frac{1}{6+e^{{\iota }^2}},\left({∥\xi ∥}_{\infty }=\frac{1}{7}\right)$. In addition, condition (C3) is satisfied because, for each $\iota \in \Lambda$ and each bounded set $K\subset \mathcal{Y}$, we have

$$\mu \left(\phi \left(\iota ,K\right)\right)\le \frac{1}{4+e^{\iota }}\mu \left(K\right),$$

$$\mu \left(\psi \left(\iota ,K\right)\right)\le \frac{1}{6+e^{{\iota }^2}}\mu \left(K\right).$$

Consequently, we deduce by Theorem 3 that Equation (7) with condition (8) has a solution defined on $\Lambda =\left[0,1\right]$.

5. Conclusions

In this research paper, we find sufficient conditions to ensure the existence of a solution for the Riemann–Liouville integral boundary value problem. The study tools that we used in this research paper are Mönch fixed-point theorem and the measure of noncompactness. The importance of this research lies in the fact that the study was in the Banach space with two fractional orders of derivatives. Then, our existence results extend the main results in [19]. In addition, if $\tau =\varsigma -1$ our problem and study reduces to the problem studied in [10], which is a generalization of [20].