1 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, 11–15]). By applying this notion, Aghajani and Pourhadi [2] investigated the infinite system of second-order differential equations in an \(\ell_{1}\)-space. Since then Mohiuddine et al. [10] and Banaś et al. [6] 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 [3], 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 [3].

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 [7] 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

([7])

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 [1] which is a generalization of Meir–Keeler contraction (MKC) given in [9].

Definition 1.3

([1])

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

([1])

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.

2 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 [16] defined the following sequence spaces which were further studied in [11]. 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

([12])

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

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

3 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 [2] 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 [2].

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 B̄.

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 B̄, 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 [2] but our proof is quite different.

4 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).