Theory and Modern Applications

# Solvability of second order linear differential equations in the sequence space $$n(\phi)$$

## Abstract

We apply the concept of measure of noncompactness to study the existence of solution of second order differential equations with initial conditions in the sequence space $$n(\phi)$$.

## Introduction and preliminaries

In recent years, the notion of measure of noncompactness has been effectively utilized in sequence spaces for different classes of differential equations (see [4, 5, 8, 1115]). By applying this notion, Aghajani and Pourhadi  investigated the infinite system of second-order differential equations in an $$\ell_{1}$$-space. Since then Mohiuddine et al.  and Banaś et al.  focused on this system in the sequence space $$\ell_{p}$$.

A measure of noncompactness is a nonnegative real-valued map defined on a collection of bounded subsets of a normed (metric) space which maps the class of relatively compact sets (known as kernel) to zero, while other sets are mapped to a positive value. There are several ways to define this notion on a given space. The widely used approach is the axiomatic one, introduced in , which is given below.

Let $$\mathfrak{M}_{E}$$ denote the family of all nonempty bounded subsets of a Banach space E and $$\mathfrak{N}_{E}$$ be its subfamily consisting of all relatively compact sets. Let $$B(x,r)$$ denote the closed ball centered at x with radius r and $$B_{r}=$$ $$B(\theta,r)$$.

We recall the following definition given in .

### Definition 1.1

([3, Definition 3.1.3])

A mapping $$\mu \colon \mathfrak{M} _{E}\longrightarrow \mathbb{R}^{+}$$ is called a measure of noncompactness (MNC for short) if

1. (i)

kerμ is nonempty and a subset of $$\mathfrak{N}_{E}$$.

2. (ii)

$$\mu (X)\leq \mu (Y)$$ for $$X\subset Y$$.

3. (iii)

$$\mu (\overline{X})=\mu (X)$$.

4. (iv)

$$\mu (\operatorname{Conv}X)=\mu (X)$$.

5. (v)

For all $$\lambda \in {}[ 0,1]$$,

$$\mu \bigl(\lambda X+(1-\lambda) Y\bigr)\leq \lambda \mu (X)+(1-\lambda) \mu (Y).$$
6. (vi)

If $$(X_{n})_{n\in \mathbb{N}}$$ is a sequence of closed sets from $$\mathfrak{M}_{E}$$ satisfying

$$X_{n+1}\subset X_{n}\quad \text{for all }n\in \mathbb{N}\quad \text{and}\quad \mu (X _{n})\rightarrow 0\quad \text{as }n\rightarrow \infty,$$

then

$$X_{\infty }=\bigcap_{n=1}^{\infty }X_{n} \neq \varnothing .$$

### Definition 1.2

([5, Definition 3.1.3])

For a measure of noncompactness μ in E, the mapping $$G\colon B\subseteq E\longrightarrow E$$ is said to be a $$\mu_{E}$$-contraction if there exists a constant $$0< k<1$$ such that

$$\mu \bigl(G(Y)\bigr)\leq k \mu (Y)$$
(1.1)

for any bounded closed subset $$Y\subseteq B$$.

Darbo  used the idea of measure of noncompactness to obtain a new fixed point theorem which generalizes the Banach contraction principle and assures the existence of a fixed point concerning the so-called condensing operators.

### Theorem 1.1

()

Let be a nonempty, closed, bounded, and convex subset of a Banach space E, and let $$\mathcal{G}:\complement \mapsto \complement$$ be a continuous mapping such that there exists a constant $$\theta \in {}[ 0,1)$$ with the property $$\mu ( \mathcal{G}(\complement))\leq \theta \mu (\complement)$$. Then $$\mathcal{G}$$ has a fixed point in .

The following definition was given in  which is a generalization of Meir–Keeler contraction (MKC) given in .

### Definition 1.3

()

For an arbitrary measure of noncompactness μ on a Banach space X, we say that an operator $$\mathfrak{T}:B \mapsto B$$ is a Meir–Keeler condensing operator if for any $$\epsilon >0$$ there exists $$\delta >0$$ such that

$$\epsilon \leq \mu (E)< \epsilon +\delta\quad \Longrightarrow\quad \mu \bigl( \mathfrak{T}(E)\bigr)< \epsilon$$
(1.2)

for any bounded subset E of B; where B is a nonempty subset of X.

Now we state the following theorem for Meir–Keeler condensing operators which will be applied in our main results.

### Theorem 1.2

()

Let μ be an arbitrary measure of noncompactness on a Banach space X. If $$\mathfrak{T}:B\mapsto B$$ is a continuous and Meir–Keeler condensing operator, then $$\mathfrak{T}$$ has at least one fixed point and the set of all fixed points of $$\mathfrak{T}$$ in B is compact, where B is a nonempty, bounded, closed, and convex subset of X.

## The sequence space $$n(\phi )$$

We denote by $$\mathcal{C}$$ the space of finite sets of distinct positive integers. For any $$\sigma \in \mathcal{C}$$, we define $$\alpha (\sigma)=\{\alpha_{n}(\sigma)\}$$ such that $$\alpha_{n}(\sigma)$$ is 1 if n is in σ; and 0 elsewhere. Write

$$\mathcal{C}_{r}= \Biggl\{ \sigma \in \mathcal{C}:\sum _{n=1}^{\infty } {}\alpha_{n}(\sigma)\leq r \Biggr\} ,$$

and define

$$\Phi = \bigl\{ \phi =(\phi_{k}):0< \phi_{1}\leq \phi_{n}\leq \phi_{n+1} \text{ and } (n+1)\phi_{n} \geq n\phi_{n+1} \bigr\} .$$

Sargent  defined the following sequence spaces which were further studied in . Write $$S(x)$$ for the set of all sequences that are rearrangements of x. For $$\phi \in \Phi$$,

\begin{aligned}& m(\phi)= \biggl\{ x=(x_{k}):\Vert x\Vert _{m(\phi)}=\sup _{r \geq 1}{}\sup_{\sigma \in \mathcal{C}_{r}} \biggl( \frac{1}{\phi_{r}} \sum_{k\in \sigma } \vert x_{k} \vert \biggr) < \infty \biggr\} , \\& n(\phi)= \Biggl\{ x=(x_{k}):\Vert x\Vert _{n(\phi)}= \sup _{u\in S(x)}{} \Biggl( \sum_{k=1}^{\infty } \vert u_{k} \vert \Delta \phi_{k} \Biggr) < \infty \Biggr\} , \end{aligned}

where $$\Delta \phi_{k}=\phi_{k}-\phi_{k-1}$$. Note that, for all $$n\in \mathbb{N=}\{1,2,3,\ldots\}$$, $$m(\phi)=\ell_{1}$$, $$n(\phi)= \ell_{\infty }$$ if $$\phi_{n}=1$$; and $$m(\phi)=\ell_{\infty }$$, $$n(\phi)=\ell_{1}$$ if $$\phi_{n}=n$$.

We have the following important result.

### Theorem 2.1

()

For any bounded subset $$\mathcal{Q}$$ of $$n(\phi)$$, we have

$$\chi (\mathcal{Q})=\lim_{k\rightarrow \infty }\sup_{x\in Q} \Biggl( \sup_{u\in S(x)}{} \Biggl( \sum_{n=k}^{\infty } \vert u_{n} \vert \Delta \phi_{n} \Biggr) \Biggr),$$

where $$\chi (Q)$$ denotes the Hausdorff measure of noncompactness of the set Q which is defined by

$$\chi (Q):=\inf \Biggl\{ \epsilon >0:Q \subset \bigcup _{i=1}^{n}B(x_{i},r_{i}), x_{i}\in X, r _{i}< \epsilon (i=1,2,\ldots) \Biggr\} .$$

## Infinite system of second order differential equations in $$n( \phi )$$

We study the following infinite system:

$$\frac{d^{2}u_{i}}{dt^{2}}=-f_{i}\bigl(t,u_{1}(t),u_{2}(t),u_{3}(t), \ldots\bigr);\quad u_{i}(0)=u_{i}(T)=0, t\in {}[ 0,T], i=1, 2, 3\dots$$
(3.1)

Let $$C(I,\mathbb{R})$$ be the space of all continuous real functions on the interval $$I=[a,b]$$ and $$C^{2}(I,\mathbb{R})$$ be the class of functions with the second continuous derivative on I. A function $$u=(u_{i})\in C^{2}(I,\mathbb{R})$$ is a solution of (3.1) if and only if $$u\in C(I,\mathbb{R})$$ is a solution of the system of integral equations

$$u_{i}(t) = \int_{0}^{T}\mathfrak{G}(t,s)f_{i} \bigl(s,u(s)\bigr)\,ds \quad \text{for } t\in I,$$
(3.2)

where $$f_{i}(t,u)\in C(I\times \mathbb{R}^{\infty },\mathbb{R})$$, $$i=1, 2, 3,\dots$$; and the Green’s function associated with (3.1) is given by

$$\mathfrak{G}(t,s)= \textstyle\begin{cases} \frac{t}{T }(T -s),& 0\leq t\leq s\leq T, \\ \frac{s}{T }(T -t),& 0\leq s\leq t\leq T. \end{cases}$$
(3.3)

From (3.2) and (3.3)

$$u_{i}(t)= \int_{0}^{t}\frac{s}{T }(T -t)f _{i} \bigl(s,u(s)\bigr)\,ds+ \int_{t}^{T }\frac{t}{T }( T -s)f_{i} \bigl(s,u(s)\bigr)\,ds.$$

Now compute

$$\frac{d}{dt}u_{i}(t)=-\frac{1}{T } \int_{0}^{t}sf_{i}\bigl(s,u(s)\bigr)\,ds+ \frac{1}{ T } \int_{t}^{T }(T -s)f_{i}\bigl(s,u(s)\bigr) \,ds.$$

Again differentiating we get

$$\frac{d^{2}u_{i}(t)}{dt^{2}}=-\frac{1}{T }\bigl(tf_{i}\bigl(t,u(t)\bigr) \bigr)+\frac{1}{ T }(t-T )f_{i}\bigl(t,u(t)\bigr))=-f_{i} \bigl(t,u(t)\bigr)).$$

The solution of the infinite system (3.1) in the sequence space $$\ell_{1}$$ was discussed by Aghajani and Pourhadi  by establishing a generalization of Darbo type fixed point theorem using the concept of α-admissibility function and Schauder’s fixed point theorem. Here, we determine the solvability of system (3.1) in Banach sequence spaces $$n(\phi)$$. Our result is more general than that of .

Assume that

1. (i)

The functions $$f_{i}$$ are defined on the set $$I\times \mathbb{R} ^{\infty }$$ and take real values. The operator f defined on the space $$I\times n(\phi)$$ into $$n(\phi)$$ as

$$(t,u)\rightarrow (fu) (t)=\bigl(f_{1}\bigl(t,u(t) \bigr),f_{2}\bigl(t,u(t)\bigr),f_{3}\bigl(t,u(t)\bigr),\ldots \bigr)$$

is such that the class of all functions $$((fu)(t))_{t\in I}$$ is equicontinuous at every point of the space $$n(\phi)$$.

2. (ii)

The following inequality holds:

$$\bigl\vert f_{n}\bigl(t,u_{1}(t),u_{2}(t),u_{3}(t), \ldots\bigr) \bigr\vert \leq g_{n}(t)+h_{n}(t) \bigl\vert u_{n}(t) \bigr\vert ,$$

where $$g_{n}(t)$$ and $$h_{n}(t)$$ are real functions defined and continuous on I such that $$\sum_{k=1}^{\infty }g_{k}(t)\Delta \phi _{k}$$ converges uniformly on I and the sequence $$(h_{n}(t))$$ is equibounded on I.

Write

$$G=\sup_{t\in I}\sum_{k=1}^{\infty }g_{k}(t) \Delta \phi_{k}$$

and

$$H=\sup_{n\in \mathbb{N}, t\in I}h_{n}(t).$$

### Theorem 3.1

Let conditions (i)(ii) hold. Then system (3.1) has at least one solution $$u(t)=(u_{i}(t))\in n(\phi)$$ for all $$t\in {}[ 0,T]$$.

### Proof

Let $$S(u(t))$$ denote the set of all sequences that are rearrangements of $$u(t)$$. If $$v(t)\in S(u(t))$$, then $$\sum_{k=1}^{ \infty }\vert v_{k}(t)\vert \Delta \phi_{k}\leq M$$, where M is a finite positive real number for all $$u(t)=(u_{i}(t))\in n(\phi)$$ for all $$t\in I$$. Using (3.2) and (ii), we have, for all $$t\in I$$,

\begin{aligned}& \bigl\Vert u(t)\bigr\Vert _{n(\phi)} \\& \quad =\sup_{v\in S(u(t))} \Biggl( \sum_{k=1}^{\infty } \biggl\vert \int_{0}^{T }\mathfrak{G}(t,s)f _{k} \bigl(s,u(s)\bigr)\,ds \biggr\vert \Delta \phi_{k} \Biggr) \\& \quad \leq \sup_{v\in S(u(t))} \Biggl( \sum _{k=1}^{\infty } \int_{0}^{ T } \bigl\vert \mathfrak{G}(t,s)f_{k} \bigl(s,u(s)\bigr) \bigr\vert \,ds\Delta \phi _{k} \Biggr) \\& \quad \leq \sup_{v\in S(u(t))} \Biggl( \sum _{k=1}^{\infty } \int_{0}^{ T } \bigl\vert \mathfrak{G}(t,s) \bigr\vert \bigl(g_{k}(t)+h_{k}(t) \bigl\vert v _{k}(t) \bigr\vert \bigr)\,ds\Delta \phi_{k} \Biggr) \\& \quad = \sup_{v\in S(u(t))} \Biggl( \sum_{k=1}^{\infty } \int_{0}^{ T }\mathfrak{G}(t,s)g_{k}(t) \Delta \phi_{k}\,ds+\sum_{k=1} ^{\infty } \int_{0}^{T }\mathfrak{G}(t,s) \bigl\vert v_{k}(t) \bigr\vert \Delta \phi_{k}\,ds \Biggr) \\& \quad \leq \sup_{v\in S(u(t))} \Biggl( \int_{0}^{T } \mathfrak{G}(t,s)\Biggl\{ \sum _{k=1}^{\infty }g_{k}(t)\Delta \phi_{k}\Biggr\} \,ds+H \int_{0}^{T }\mathfrak{G}(t,s)\Biggl\{ \sum _{k=1}^{\infty } \bigl\vert u_{k}(t) \bigr\vert \Delta \phi_{k}\Biggr\} \,ds \Biggr) \\& \quad \leq G\sup_{v\in S(u(t))} \int_{0}^{T }\mathfrak{G}(t,s)\,ds+H \sup _{v\in S(u(t))} \int_{0}^{T }\mathfrak{G}(t,s)M\,ds \\& \quad \leq \frac{GT ^{2}}{8}+\frac{HMT ^{2}}{8}=R, \end{aligned}

say.

Let $$u^{0}(t)=(u_{i}^{0}(t))$$ where $$u_{i}^{0}(t)=0$$ for all $$t\in I$$.

Consider the closed ball $$\bar{B}=\bar{B}(u^{0},r_{1})$$ centered at $$u^{0}$$ and of radius $$r_{1}\leq r$$ which is of course a nonempty, bounded, closed, and convex subset of $$n(\phi)$$. Consider the operator $$\mathcal{F}=(\mathcal{F}_{i})$$ on $$C(I,\bar{B})$$ defined as follows. For $$t\in I$$,

$$(\mathcal{F}u) (t)=\bigl\{ (\mathcal{F}_{i}u) (t)\bigr\} = \biggl\{ \int_{0}^{T} \mathfrak{G}(t,s)f_{i} \bigl(s,u(s)\bigr)\,ds \biggr\} ,$$

where $$u(t)=(u_{i}(t))$$ and $$u_{i}(t)\in C(I,\mathbb{R})$$.

We have $$(\mathcal{F}u)(t)=\{(\mathcal{F}_{i}u)(t)\}\in n(\phi)$$ for each $$t\in I$$. Since $$(f_{i}(t,u(t)))\in n(\phi)$$ for each $$t\in I$$, we have

$$\sup_{v\in S(u(t))} \Biggl( \sum_{k=1}^{\infty } \bigl\vert (\mathcal{F}_{k}u) (t) \bigr\vert \Delta \phi_{k}\}\,ds \Biggr) \leq R< \infty.$$

Also since $$(\mathcal{F}_{i}u)(t)$$ satisfies the boundary conditions, we have

$$(\mathcal{F}_{i}u) (0)= \int_{0}^{T }\mathfrak{G}(0,s)f _{i} \bigl(s,u(s)\bigr)\,ds=0$$

and

$$(\mathcal{F}_{i}u) (T)= \int_{0}^{T }\mathfrak{G}( T ,s)f_{i} \bigl(s,u(s)\bigr)\,ds=0.$$

Since $$\Vert (\mathcal{F}u)(t)-u^{0}(t)\Vert _{n(\phi)}\leq R$$, $$\mathcal{F}$$ is an operator on .

The operator $$\mathcal{F}$$ is continuous on $$C(I,\bar{B})$$ by assumption (i). Now, we shall show that $$\mathcal{F}$$ is a Meir–Keeler condensing operator. For $$\varepsilon >0$$, we have to find $$\delta >0$$ such that $$\varepsilon \leq \chi (\bar{B})<\varepsilon +\delta \Rightarrow \chi (\mathcal{F}\bar{B})<\varepsilon$$. Now

\begin{aligned}& \lim_{k\rightarrow \infty } \Biggl\{ \sup_{u(t)\in \bar{B}} \Biggl( \sup _{v\in S(u(t))} \Biggl( \sum_{n=k}^{\infty } \biggl\vert \int_{0}^{ T }\mathfrak{G}(t,s)f_{n} \bigl(s,v(s)\bigr)\,ds \biggr\vert \Delta \phi_{n} \Biggr) \Biggr) \Biggr\} \\& \quad \leq\lim_{k\rightarrow \infty } \Biggl\{ \sup_{u(t)\in \bar{B}} \Biggl( \sup_{v\in S(u(t))} \Biggl( \sum_{n=k}^{\infty } \int_{0}^{ T } \bigl\vert \mathfrak{G}(t,s)f_{n} \bigl(s,v(s)\bigr) \bigr\vert \Delta \phi _{n}\,ds \Biggr) \Biggr) \Biggr\} \\& \quad \leq \lim_{k\rightarrow \infty } \Biggl\{ \sup_{u(t)\in \bar{B}} \Biggl( \sup_{v\in S(u(t))} \Biggl( \sum_{n=k}^{\infty } \int_{0}^{ T }\mathfrak{G}(t,s)g_{n}(s) \Delta \phi_{n}\,ds \Biggr) \Biggr) \Biggr\} \\& \qquad {}+\lim_{k\rightarrow \infty } \Biggl\{ \sup_{u(t)\in \bar{B}} \Biggl( \sup_{v\in S(u(t))} \Biggl( \sum_{n=k}^{\infty } \int_{0}^{ T }\mathfrak{G}(t,s)h_{n}(s) \bigl\vert v_{n}(s) \bigr\vert \Delta \phi _{n}\,ds \Biggr) \Biggr) \Biggr\} \\& \quad \leq \lim_{k\rightarrow \infty } \Biggl\{ \sup_{u(t)\in \bar{B}} \Biggl( \sup_{v\in S(u(t))} \Biggl( \int_{0}^{T }\mathfrak{G}(t,s) \Biggl( \sum _{n=k}^{\infty }g_{n}(s)\Delta \phi_{n} \Biggr)\,ds \Biggr) \Biggr) \Biggr\} \\& \qquad {}+H\lim_{k\rightarrow \infty } \Biggl\{ \sup_{u(t)\in \bar{B}} \Biggl( \sup_{v\in S(u(t))} \int_{0}^{T }\mathfrak{G}(t,s) \Biggl( \sum _{n=k}^{\infty } \bigl\vert v_{n}(s) \bigr\vert \Delta \phi_{n} \Biggr)\,ds \Biggr) \Biggr\} \\& \quad \leq H\chi (\bar{B}) \int_{0}^{T }\mathfrak{G}(t,s)\,ds \leq \frac{HT ^{2}}{8}\chi (\bar{B}). \end{aligned}

Hence $$\chi (\mathcal{F}\bar{B})<\frac{HT ^{2}}{8}\chi ( \bar{B})<\varepsilon \Rightarrow \chi (\bar{B})<\frac{8\varepsilon }{H T ^{2}}$$.

Taking $$\delta =\frac{\varepsilon }{HT ^{2}}(8-HT ^{2})$$, we get $$\varepsilon \leq \chi (\bar{B})<\varepsilon +\delta$$. Therefore, $$\mathcal{F}$$ is a Meir–Keeler condensing operator defined on the set $$\bar{B}\subset n(\phi)$$. So $$\mathcal{F}$$ satisfies all the conditions of Theorem 1.2 which implies that $$\mathcal{F}$$ has a fixed point in , which is a required solution of system (3.1). □

### Remark 3.1

For $$\phi_{n}=n$$, for all $$n\in \mathbb{N}$$, the above result is reduced to that of Aghajani and Pourhadi  but our proof is quite different.

## Example

In order to illustrate the above result, we provide the following example.

### Example 4.1

Let us consider the system of second order differential equations

$$-\frac{d^{2}u_{j}(t)}{dt^{2}}=\frac{\sqrt[j]{t}}{j^{4}}+\sum_{i=j} ^{\infty }\frac{t\cos (t)u_{i}(t)}{i^{4}}, \quad j\in \mathbb{N}, t\in I=[0,T].$$
(4.1)

Here $$f_{i}(t,u_{1}(t),u_{2}(t),u_{3}(t),\ldots)=\frac{\sqrt[j]{t}}{j ^{4}}+\sum_{i=j}^{\infty }\frac{t\cos (t)u_{i}(t)}{i^{4}}$$, and so (4.1) is a special case of the considered system (3.1). Clearly $$\frac{ \sqrt[j]{t}}{j^{4}}$$ and $$\sum_{i=j}^{\infty } \frac{t\cos (t)u_{i}(t)}{i4}$$ are continuous on I for each $$n\in \mathbb{N}$$.

Notice that, for any $$t\in I=[0,T]$$, $$(f_{k}(t,u(t)))\in n(\phi)$$ if $$(u_{k}(t))\in n(\phi)$$. Moreover, we have

\begin{aligned} \sum_{k=1}^{\infty } \bigl\vert f_{k}\bigl(t,u(t)\bigr) \bigr\vert =&\sum _{k=1}^{\infty } \Biggl\vert \frac{ \sqrt[k]{t}}{k^{4}}+\sum _{i=k}^{\infty }\frac{t\cos (t)u_{i}(t)}{i ^{4}} \Biggr\vert \\ \leq &\sum_{k=1}^{\infty }\frac{\sqrt[k]{t}}{k^{4}}+ \sum_{k=1}^{ \infty }\sum _{i=k}^{\infty } \biggl\vert \frac{t\cos (t)u_{i}(t)}{i^{4}} \biggr\vert \\ \leq &\frac{T\pi^{4}}{90}+\sum_{k=1}^{\infty } \sum_{i=k}^{\infty }\frac{t}{i ^{4}} \bigl\vert u_{i}(t) \bigr\vert \\ \leq &\frac{T\pi^{4}}{90}+T \bigl\Vert u(t) \bigr\Vert _{n(\phi)}< \infty. \end{aligned}

We will show that assumption (i) is satisfied. Let us fix $$\epsilon >0$$ arbitrarily and $$u(t)=(u_{k}(t))\in n(\phi)$$. Then, taking $$v(t)=(v_{k}(t))\in n(\phi)$$ with $$\|u(t)-v(t)\|\leq \delta (\epsilon)$$:= $$\frac{\epsilon }{T}$$, we have

\begin{aligned} \bigl\vert f\bigl(t,u(t)\bigr)-f\bigl(t,v(t)\bigr) \bigr\vert =&\sum _{i=j}^{\infty }\frac{t(u_{i}(t)-v_{i}(t))}{i ^{4}} \\ \leq &T \bigl\Vert u(t)-v(t) \bigr\Vert _{n(\phi)} \\ \leq &T\delta < \epsilon, \end{aligned}

which implies continuity as in assumption (i). Now, we show that assumption (ii) is satisfied.

\begin{aligned} \bigl\vert f_{j}\bigl(t,u(t)\bigr) \bigr\vert =& \Biggl\vert \frac{\sqrt[j]{t}}{j^{4}}+\sum_{i=j}^{\infty } \frac{t \cos (t)u_{i}(t)}{i^{4}} \Biggr\vert \\ \leq &\frac{\sqrt{t}}{j^{4}}+\sum_{i=j}^{\infty } \frac{t}{i^{4}} \bigl\vert u _{i}(t) \bigr\vert \\ \leq &g_{j}(t)+h_{j}(t) \bigl\vert u_{j}(t) \bigr\vert . \end{aligned}

The function $$g_{j}(t)=\frac{\sqrt{t}}{j^{4}}$$ is continuous and $$\sum_{j\geq 1}g_{j}(t)$$ converges uniformly to $$\frac{\sqrt{t}\pi ^{4}}{90}$$, also $$h_{j}(t)=\frac{t\pi^{4}}{90}$$ is continuous and the sequence $$(h_{j}(t))$$ is equibounded on I by $$H=\frac{T\pi^{4}}{80}$$. Also $$\frac{HT^{2}}{8}<1$$ is satisfied by taking $$T=1.2$$, which gives $$H\approx 1.9739$$ and $$G\approx 1.9739$$.

Thus, from Theorem 3.1, for a suitable value of $$r_{1}$$ (as discussed in Theorem 3.1) the operator $$\mathcal{F}$$ as defined in Theorem 3.1 on $$\bar{B}(u^{0},r_{1})$$ has a fixed point $$u(t)=((u_{i}(t))\in n(\phi)$$, which is a solution of system (4.1).

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## Funding

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant No. RG-18-130-37. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

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### Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to M. Mursaleen.

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