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Solvability of second order linear differential equations in the sequence space \(n(\phi)\)
Advances in Difference Equations volume 2018, Article number: 377 (2018)
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, 11–15]). By applying this notion, Aghajani and Pourhadi [2] investigated the infinite system of secondorder 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 realvalued 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

(i)
kerμ is nonempty and a subset of \(\mathfrak{N}_{E}\).

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

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

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

(v)
For all \(\lambda \in {}[ 0,1]\),
$$ \mu \bigl(\lambda X+(1\lambda) Y\bigr)\leq \lambda \mu (X)+(1\lambda) \mu (Y). $$ 
(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
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 socalled 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
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
and define
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 \(\phi \in \Phi\),
where \(\Delta \phi_{k}=\phi_{k}\phi_{k1}\). 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
where \(\chi (Q)\) denotes the Hausdorff measure of noncompactness of the set Q which is defined by
Infinite system of second order differential equations in \(n( \phi )\)
We study the following infinite system:
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
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
Now compute
Again differentiating we get
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 αadmissibility 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

(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 \(n(\phi)\).

(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
and
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 \(t\in I\),
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 \(t\in I\),
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
Also since \((\mathcal{F}_{i}u)(t)\) satisfies the boundary conditions, we have
and
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
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 \(\delta =\frac{\varepsilon }{HT ^{2}}(8HT ^{2})\), 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.
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
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
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
which implies continuity as in assumption (i). Now, we show that assumption (ii) is satisfied.
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. RG1813037. The authors, therefore, acknowledge with thanks DSR for technical and financial support.
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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/s1366201818109
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DOI: https://doi.org/10.1186/s1366201818109
MSC
 47H09
 47H10
 34A34
Keywords
 Measure of noncompactness
 Darbo’s fixed point theorem
 Sequence spaces
 Infinite system of second order differential equations