Skip to main content

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 [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.

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

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

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 .

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 [2] 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).

References

  1. Aghajani, A., Mursaleen, M., Haghighi, A.S.: A fixed point theorem for Meir–Keeler condensing operator via measure of noncompactness. Acta Math. Sci. 35B(3), 552–556 (2015)

    MathSciNet  Article  Google Scholar 

  2. Aghajani, A., Pourhadi, E.: Application of measure of noncompactness to \(\ell_{1}\)-solvability of infinite systems of second order differential equations. Bull. Belg. Math. Soc. Simon Stevin 22(1), 105–118 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Banaś, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Dekker, New York (1980)

    MATH  Google Scholar 

  4. Banaś, J., Lecko, M.: Solvability of infinite systems of differential equations in Banach sequence spaces. J. Comput. Appl. Math. 137, 363–375 (2001)

    MathSciNet  Article  Google Scholar 

  5. Banaś, J., Mursaleen, M.: Sequence Spaces and Measure of Noncompactness with Applications to Differential and Integral Equation. Springer, India (2014)

    Book  Google Scholar 

  6. Banaś, J., Mursaleen, M., Rizvi, S.M.H.: Existence of solutions to a boundary-value problem for an infinite system of differential equations. Electron. J. Differ. Equ. 2017, 262 (2017)

    MathSciNet  Article  Google Scholar 

  7. Darbo, G.: Punti uniti in transformazioni a condominio non compatto. Rend. Semin. Mat. Univ. Padova 24, 84–92 (1955)

    MATH  Google Scholar 

  8. Das, A., Hazarika, B., Arab, R., Mursaleen, M.: Solvability of the infinite system of integral equations in two variables in the sequence spaces \(c_{0}\) and \(\ell_{1}\). J. Comput. Appl. Math. 326, 183–192 (2017)

    MathSciNet  Article  Google Scholar 

  9. Meir, A., Keeler, E.: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)

    MathSciNet  Article  Google Scholar 

  10. Mohiuddine, S.A., Srivastava, H.M., Alotaibi, A.: Application of measures of noncompactness to the infinite system of second-order differential equations in \(\ell_{p}\) spaces. Adv. Differ. Equ. 2016, 317 (2016)

    MathSciNet  Article  Google Scholar 

  11. Mursaleen, M.: On some geometric properties of a sequence space related to \(l_{p}\). Bull. Aust. Math. Soc. 67, 343–347 (2003)

    MathSciNet  Article  Google Scholar 

  12. Mursaleen, M.: Application of measure of noncompactness to infinite system of differential equations. Can. Math. Bull. 56, 388–394 (2013)

    MathSciNet  Article  Google Scholar 

  13. Mursaleen, M., Bilalov, B., Rizvi, S.M.H.: Applications of measures of noncompactness to infinite system of fractional differential equations. Filomat 31(11), 3421–3432 (2017)

    MathSciNet  Article  Google Scholar 

  14. Mursaleen, M., Mohiuddine, S.A.: Applications of measure of noncompactness to the infinite system of differential equations in \(\ell_{p}\) spaces. Nonlinear Anal. 75, 2111–2115 (2012)

    MathSciNet  Article  Google Scholar 

  15. Mursaleen, M., Rizvi, S.M.H.: Solvability of infinite system of second order differential equations in \(c_{0}\) and \(\ell_{1}\) by Meir–Keeler condensing operator. Proc. Am. Math. Soc. 144(10), 4279–4289 (2016)

    Article  Google Scholar 

  16. Sargent, W.L.C.: Some sequence spaces related to the \(l_{p}\) spaces. J. Lond. Math. Soc. 35, 161–171 (1960)

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

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.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alotaibi, A., Mursaleen, M. & Alamri, B.A.S. Solvability of second order linear differential equations in the sequence space \(n(\phi)\). Adv Differ Equ 2018, 377 (2018). https://doi.org/10.1186/s13662-018-1810-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-018-1810-9

MSC

  • 47H09
  • 47H10
  • 34A34

Keywords

  • Measure of noncompactness
  • Darbo’s fixed point theorem
  • Sequence spaces
  • Infinite system of second order differential equations