Application of measures of noncompactness to the infinite system of second-order differential equations in \(\ell_{p}\) spaces
- Syed Abdul Mohiuddine^{1}Email authorView ORCID ID profile,
- Hari M Srivastava^{2, 3} and
- Abdullah Alotaibi^{1}
https://doi.org/10.1186/s13662-016-1016-y
© Mohiuddine et al. 2016
Received: 8 June 2016
Accepted: 31 October 2016
Published: 5 December 2016
Abstract
In this article, we use the technique based upon measures of noncompactness in conjunction with a Darbo-type fixed point theorem with a view to studying the existence of solutions of infinite systems of second-order differential equations in the Banach sequence space \(\ell_{p}\). An illustrative example is also given in support of our existence result.
Keywords
measure of noncompactness Darbo-type fixed point theorem infinite system of second-order differential equations sequence spacesMSC
47H10 34A341 Introduction
Measures of noncompactness endow helpful information, which is extensively used in the theory of integral and integro-differential equations. Besides, it is very helpful in the study of optimization, differential equations, functional equations, fixed point theory, etc. Some of the well-known measures of noncompactness are the Kuratowski measure (α), the Hausdorff measure (χ), and the Istrăţescu measure (β), which were introduced by Kuratowski [1], Goldenštein et al. [2] (also studied by Goldenštein and Markus [3]), and Istrăţescu [4], respectively. Darbo [5] was the first who presented a fixed point theorem by using the idea of Kuratowski measures of noncompactness, the function α, which is popularly called the Darbo fixed point theorem. This fixed point theorem generalized two very important and famous fixed point theorems, namely, (i) the classical Schauder fixed point theorem and (ii) special variant of the Banach fixed point theorem. The Darbo fixed point theorem has been generalized in many different directions. In fact, there is a vast amount of literature dealing with extensions and/or generalizations of this remarkable theorem. Recently, Aghajani et al. [6] presented a generalization of the Darbo fixed point theorem and used it to investigate the existence result concerning a general system of nonlinear integral equations. For some other recent works related to these concepts, we refer the interested reader to (for example) [7–12], and [13]. We also refer to the recent work by Srivastava et al. [14] for some applications of fixed point theorems to fractional differential equations (for details, see [15]).
Mursaleen and Mohiuddine [16] earlier reported the existence theorems in the classical sequence space \(\ell_{p}\) for an infinite system of differential equations. On the other hand, existence theorems for infinite systems of linear equations in \(\ell_{1}\) and \(\ell_{p}\) were given by Alotaibi et al. [17]. Our main object in this sequel is to determine sufficient conditions for the solvability of an infinite system of second-order differential equations. We use the Dardo-type fixed point theorem given by Aghajani and Pourhadi [18] for a new type of condensing operator and the method based upon the measures of noncompactness to establish the existence theorem for the above-mentioned infinite systems in the Banach sequence space \(\ell_{p}\) with \(1\leqq p<\infty\). Our existence theorem is an extension of those obtained by Aghajani and Pourhadi [18] in the sequence space \(\ell_{1}\).
2 Preliminaries and notation
- (i)
\(\chi(F)=0\) if and only if F is totally bounded;
- (ii)
\(\chi(F)=\chi(\bar{F})\), where F̄ denotes the closure of F;
- (iii)
\(F_{1}\subset F_{2}\) implies that \(\chi(F_{1})\leqq\chi(F_{2})\);
- (iv)
\(\chi(F_{1}\cup F_{2})=\max\{\chi(F_{1}),\chi(F_{2})\}\);
- (v)
\(\chi(F_{1}\cap F_{2})=\min\{\chi(F_{1}),\chi(F_{2})\}\).
Theorem 1
see [19]
It is known that \(\ell_{p}\) (\(1\leqq p<\infty\)) is a BK space with AK with respect to its usual norm \(\|\cdot\|_{p}\). Additionally, \(\{e^{(1)},e^{(2)},\ldots\}\) as depicted from a Schauder basis for \(\ell _{p}\), in view of (1), the following result is derivable by using Theorem 1 (see [19] and [20]).
Theorem 2
The following generalization of the Darbo fixed point theorem was established by Aghajani et al. [21] by using a control function.
Theorem 3
- (i)
ϕ is a lower semi-continuous function with \(\phi(t)=0\) if and only if \(t=0\);
- (ii)
\(\liminf_{n\to\infty}\varphi(a_{n})<\phi(a)\), provided that \(\lim_{n\to\infty} \{a_{n} \}=a\).
Remark 1
Aghajani and Pourhadi [18] also established the following fixed point theorem by using α-admissible and \((\alpha,\phi, \varphi)\mbox{-}\mu\)-condensing operators.
Theorem 4
Let \({\mathcal{C}}\in{\mathcal{M}}_{X}\) be a closed convex subset of a Banach space X, and let \(T:{\mathcal{C}}\to{\mathcal{C}}\) be a continuous \((\alpha,\varphi,\phi)\)-μ-condensing operator, where μ is an arbitrary measure of noncompactness. Moreover, T is α-admissible, and \(\alpha({\mathcal{C}})\geqq1\). Then T has at least one fixed point that pertains to kerμ.
3 Infinite systems of second-order differential equations
- (i)The functions \(f_{i}\) (\(i\in\mathbb{N}_{0}\)) are defined on \(I\times\mathbb{R}^{\infty}\) and take real values. Furthermore, the operator f is shown on the space \(I\times\ell_{p}\) aswhich represent the space of maps from \(I\times\ell_{p}\) into \(\ell _{p}\); it is found that the class of all functions \(\{(fx)(t)\}_{t\in I}\) is equicontinuous at every point of \(\ell_{p}\).$$(t,x)\mapsto(fx) (t)= \bigl(f_{1}(t,x),f_{2}(t,x),f_{3}(t,x), \ldots \bigr), $$
- (ii)There are a nonnegative mapping \(g:I\to\mathbb{R}_{+}\), a function \(h:I\times\ell_{p}\to\mathbb{R}\), and a super-additive mapping \(\varphi:\mathbb{R}_{+}\to\mathbb{R}_{+}\), that is,for all \(s,t\in\mathbb{R}_{+}\), such that$$\varphi(s+t)\geqq\varphi(s)+\varphi(t) $$where \(x=(x_{i})\in\ell_{p}\), \(t\in I\), and \(i\geqq k\) for some \(k\in \mathbb{N}_{0}\).$$ h(t,x)\geqq0\quad \Longrightarrow\quad \bigl\vert f_{i} (t,x_{0},x_{1},x_{2},\ldots )\bigr\vert ^{p} \leqq g_{i}(t)\varphi\bigl(|x_{i}|^{p} \bigr), $$(9)
- (iii)The function \(G(t,s)g(s)\) is integrable on I and such thatfor any fixed element \(t\in I\). Additionally, if a nonnegative sequence \((y_{n})_{n\in\mathbb{N}}\) converges to some number ℓ, then$$g(s)=\limsup_{i\to\infty} \bigl\{ g_{i}(s) \bigr\} \quad (i \in\mathbb{N}_{0}) $$such that$$ \liminf_{n\to\infty}\varphi(y_{n})< \frac{\ell}{C} $$(10)for some positive constant C.$$\sup_{t\in I} \biggl\{ \int_{0}^{T}\bigl|G(t,s)\bigr|^{p}g(s) \,ds \biggr\} \leqq C $$
- (iv)There is a function x such thatIn addition, for \(t\in I\), we have$$ h \bigl(t,x(t) \bigr)\geqq0 \quad (\forall t\in I). $$(11)for all \(y(t)\in\ell_{p}\).$$ h\bigl(t,y(t)\bigr)\geqq0\quad \Longrightarrow\quad h \biggl(t, \biggl( \int_{0}^{T} G(t,s)f_{i} \bigl(s,y_{0}(s),y_{1}(s),y_{2}(s),\ldots\bigr) \,ds \biggr) \biggr)\geqq0 $$(12)
We are now prepared to formulate our main result.
Theorem 5
Under assumptions (i) to (iv), the infinite system of second-order differential equations (4) has at least one solution \(x(t)= (x_{i}(t) )\) such that \(x(t)\in\ell_{p}\) for each \(t\in I\).
Proof
Remark 2
Our existence theorem (Theorem 5) is more general than that proved earlier by Aghajani and Pourhadi [18]. Indeed, if we set \(p=1\) in the sequence space \(\ell_{p}\), then it reduces to the sequence space \(\ell_{1}\), and so Theorem 4.1 of Aghajani and Pourhadi [18] is a particular case of our Theorem 5.
We now present an interesting illustrative example in support of our result.
Example
Declarations
Acknowledgements
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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.
Authors’ Affiliations
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