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Application of measures of noncompactness to the infinite system of secondorder differential equations in \(\ell_{p}\) spaces
Advances in Difference Equations volume 2016, Article number: 317 (2016)
Abstract
In this article, we use the technique based upon measures of noncompactness in conjunction with a Darbotype fixed point theorem with a view to studying the existence of solutions of infinite systems of secondorder differential equations in the Banach sequence space \(\ell_{p}\). An illustrative example is also given in support of our existence result.
Introduction
Measures of noncompactness endow helpful information, which is extensively used in the theory of integral and integrodifferential equations. Besides, it is very helpful in the study of optimization, differential equations, functional equations, fixed point theory, etc. Some of the wellknown 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 secondorder differential equations. We use the Dardotype 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 abovementioned 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 \(\ell_{1}\).
Preliminaries and notation
Let ω denote the space of all complex sequences \(x=(x_{j})_{j=0}^{\infty}\) or, simply, \(x=(x_{j})\). Any vector subspace of ω is called a sequence space. We use the standard notation \(\ell_{\infty}\), c, and \(c_{0}\) to denote the set of all bounded, convergent, and null sequences of real numbers, respectively. By \(\mathbb{N}\), \(\mathbb{R}\), and \(\mathbb{C}\) we denote the sets of natural, real, and complex numbers, respectively. We recall that the notion of little o is used for comparison of growth of two arbitrary sequences \(x_{j}\)and \(y_{j}\) and is defined by
We introduce the space \(\ell_{p}\) of all absolutely psummable series as follows:
Clearly, \(\ell_{p}\) is a Banach space with norm
By \(e^{(j)}\) we denote the sequence with jth term 1 and all other terms zero \((j\in\mathbb{N)}\); we also denote \(e=(1,1,1,\ldots)\). For any sequence \(x=(x_{j})\), let its nsection be given by
A sequence space X is called a BK space if it is a Banach space with continuous coordinates \(p_{k}:X\to\mathbb{C}\) and \(p_{k}(x)=x_{k}\) for all \(x=(x_{j})\in X\) and \(k\in\mathbb{N}\). A BK space \(X\supset\psi\) (that is, the set of all finite sequences that terminate in zeros) is said to have AK if every sequence \(x=(x_{j})\in X\) has a unique representation
We denote by \(\mathcal{M}_{X}\), \((X,d)\), and \(B(x,r)\), respectively, the class of all bounded subsets of X, the metric space, and the open ball with center at x and radius r, that is,
Let \(F\in\mathcal{M}_{X}\). Then the Hausdorff measure of noncompactness of F is defined by
The function \(\chi:\mathcal{M}_{X}\to[0,\infty)\) is called the Hausdorff measure of noncompactness.
We now recall some basic properties of the Hausdorff measure of noncompactness. Let F, \(F_{1}\), and \(F_{2}\) be bounded subsets of the metric space \((X,d)\). Then

(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)
\(\chi(F_{1}\cap F_{2})=\min\{\chi(F_{1}),\chi(F_{2})\}\).
In the case of a normed space \((X,\\cdot\)\), the function χ has some additional properties connected with the linear structure. For example, we have
Theorem 1
see [19]
Let X be a BK space with a Schauder basis \((b_{j})_{j=0}^{\infty}\) and \(F\in\mathcal{M}_{X}\). Also, let \(P_{j}:X\rightarrow X\) (\(j\in\mathbb{N}\)) be the projector onto the linear span of \(\{e^{(1)},e^{(2)},\ldots,e^{(j)}\}\). Then
where I is the identity operator on X, and
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
Let F be a bounded subset of \(X=\ell_{p}\). Then
The following generalization of the Darbo fixed point theorem was established by Aghajani et al. [21] by using a control function.
Theorem 3
Let \({\mathcal{C}}\) be a nonempty, bounded, closed, and convex subset of a Banach space X, and let \(T:{\mathcal {C}}\to{\mathcal{C}}\) be a continuous function satisfying the inequality
for each \(F\subset{\mathcal{C}}\), where μ is an arbitrary measure of noncompactness, and \(\varphi:[0,\infty)\to[0,\infty)\) is an increasing (not necessarily continuous) function with
Then T has at least one fixed point in the set \({\mathcal{C}}\).
The notion of \((\alpha,\phi, \varphi)\)μcondensing operators and αadmissible operators were recently demonstrated by Aghajani and Pourhadi [18] by considering φ and ϕ as follows. We use the notation Ψ to denote the functions \(\varphi:[0,+\infty )\to[0,\infty)\) like
conferred that
where \((a_{n})_{n\in\mathbb{N}}\) is a nonnegative sequence. For \(\varphi\in\Psi\), let us consider a function \(\phi:[0,+\infty)\to[0,+\infty)\) that satisfies the following conditions:

(i)
ϕ is a lower semicontinuous function with \(\phi(t)=0\) if and only if \(t=0\);

(ii)
\(\liminf_{n\to\infty}\varphi(a_{n})<\phi(a)\), provided that \(\lim_{n\to\infty} \{a_{n} \}=a\).
We use the notation \(\Phi_{\varphi}\) to denote the class of all such functions. Throughout this paper, by ConvF we denote the convex hull of \(F\subset X\).
Let \(T:W\subseteq X\to X\) is an arbitrary mapping. Further, we state that T is \((\alpha,\varphi,\phi)\)μcondensing if the functions \(\alpha:{\mathcal{M}}_{X}\to[0,+\infty)\), \(\varphi\in\Psi\), and \(\phi=\Phi_{\varphi}\) are such that
where both F and its image TF belong to \({\mathcal{M}}_{X}\).
Let T and α be given mappings as before. Then T is αadmissible if
Remark 1
If T follows the Darbo condition with regard to a measure μ and a constant \(k\in[0,1)\), that is, if
then T is an \((\alpha,\varphi,\phi)\)μcondensing operator, where \(\alpha(F)=1\) for any set \(F\in W\) such that \(F\in{\mathcal{M}}_{X}\), ϕ is the identity mapping, and the function \(\varphi(t)=kt\), \(t\geqq0\). In this regard, T is a μcontraction.
Aghajani and Pourhadi [18] also established the following fixed point theorem by using αadmissible 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 \((\alpha,\varphi,\phi)\)μcondensing operator, where μ is an arbitrary measure of noncompactness. Moreover, T is αadmissible, and \(\alpha({\mathcal{C}})\geqq1\). Then T has at least one fixed point that pertains to kerμ.
Infinite systems of secondorder differential equations
Let us consider the following infinite system of secondorder differential equations:
with the initial conditions given by
The space of all continuous real functions on I with values in \(\mathbb{R}\) and the space of all functions with two continuous derivatives on the interval I are shown by the standard notations \(C(I,\mathbb{R})\) and \(C^{2}(I,\mathbb{R})\), respectively. It is evident that \(x\in C^{2}(I,\mathbb{R})\) is a solution of (4) if and only if \(x\in C(I,\mathbb{R})\) is a solution of the following system of integral equations:
where
and the Green function \(G(t,s)\) associated with (4) is given by
For more details of green functions, we refer to [22]. We can rewrite (5) with the help of (6) as follows:
Upon differentiating both sides of (7) with respect to t, we get
Again, by differentiating both sides of (8) with respect to t we obtain
We now investigate the existence result concerning the secondorder differential equations for the infinite system given by (4) in the Banach sequence space \(\ell_{p}\) (\(1\leqq p<\infty\)) with the help of measures of noncompactness. For this investigation, we consider the following hypotheses:

(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 \(I\times\ell_{p}\) as
$$(t,x)\mapsto(fx) (t)= \bigl(f_{1}(t,x),f_{2}(t,x),f_{3}(t,x), \ldots \bigr), $$which 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 \(\ell_{p}\).

(ii)
There are a nonnegative mapping \(g:I\to\mathbb{R}_{+}\), a function \(h:I\times\ell_{p}\to\mathbb{R}\), and a superadditive mapping \(\varphi:\mathbb{R}_{+}\to\mathbb{R}_{+}\), that is,
$$\varphi(s+t)\geqq\varphi(s)+\varphi(t) $$for all \(s,t\in\mathbb{R}_{+}\), such that
$$ h(t,x)\geqq0\quad \Longrightarrow\quad \bigl\vert f_{i} (t,x_{0},x_{1},x_{2},\ldots )\bigr\vert ^{p} \leqq g_{i}(t)\varphi\bigl(x_{i}^{p} \bigr), $$(9)where \(x=(x_{i})\in\ell_{p}\), \(t\in I\), and \(i\geqq k\) for some \(k\in \mathbb{N}_{0}\).

(iii)
The function \(G(t,s)g(s)\) is integrable on I and such that
$$g(s)=\limsup_{i\to\infty} \bigl\{ g_{i}(s) \bigr\} \quad (i \in\mathbb{N}_{0}) $$for any fixed element \(t\in I\). Additionally, if a nonnegative sequence \((y_{n})_{n\in\mathbb{N}}\) converges to some number ℓ, then
$$ \liminf_{n\to\infty}\varphi(y_{n})< \frac{\ell}{C} $$(10)such that
$$\sup_{t\in I} \biggl\{ \int_{0}^{T}\biglG(t,s)\bigr^{p}g(s) \,ds \biggr\} \leqq C $$for some positive constant C.

(iv)
There is a function x such that
$$ h \bigl(t,x(t) \bigr)\geqq0 \quad (\forall t\in I). $$(11)In addition, for \(t\in I\), we have
$$ h\bigl(t,y(t)\bigr)\geqq0\quad \Longrightarrow\quad h \biggl(t, \biggl( \int_{0}^{T} G(t,s)f_{i} \bigl(s,y_{0}(s),y_{1}(s),y_{2}(s),\ldots\bigr) \,ds \biggr) \biggr)\geqq0 $$(12)for all \(y(t)\in\ell_{p}\).
We are now prepared to formulate our main result.
Theorem 5
Under assumptions (i) to (iv), the infinite system of secondorder 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
Let us consider the operator \({\mathcal{F}}=({\mathcal{F}}_{i})\) defined on \(C(I,\ell_{p})\) by
where \(x(t)= (x_{i}(t) )\in\ell_{p}\), \(x_{i}\in C(I,\mathbb{R})\), and \(t\in I\). Taking into account assumption (i), it is clearly seen that \({\mathcal{F}}\) is continuous on \(C(I,\ell_{p})\). Obviously, the function \({\mathcal{F}}x\) is also continuous, and \(({\mathcal{F}}x )(t)\in\ell_{p}\) if \(x(t)= (x_{i}(t) )\in \ell_{p}\). In view of the fact that φ is superadditive, together with Eq. (9) and hypothesis (iii), it follows that
where \(p>1\) and \(1/p+1/p'=1\). We now consider the operator \({\mathcal{F}}=({\mathcal{F}}_{i})\) defined on a nonempty bounded set \(Q\in{\mathcal{M}}_{\ell_{p}}\) (where \({\mathcal{M}}_{\ell_{p}}\) denotes the family of all nonempty bounded subsets of \(\ell_{p}\)) including the functions \(x(t)= (x_{i}(t) )\in\ell_{p}\) with
for any fixed \(t\in I\). Then, clearly, Eq. (2) yields
This shows that
where \(\alpha:{\mathcal{M}}_{\ell_{p}}\to[0,\infty)\) is the mapping defined by
and
Obviously, \(\phi\in\Phi_{\varphi}\), and it satisfies (10). Interestingly, by hypothesis (iv) we conclude that the operator \({\mathcal{F}}\) is αadmissible and satisfies all of the conditions of Theorem 5. Therefore, \({\mathcal{F}}\) has at least one fixed point \(x=x(t)\) such that \(x(t)\in\ell_{p}\) for all \(t\in I\). Hereof, the function \(x=x(t)\) is a solution of the infinite system (4). □
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
Consider the following secondorder differential equations:
where \(q\in\mathbb{N}_{0}\) and \(t\in I=[0,T]\) (\(0< T<2\sqrt{2}\)). Obviously, the functions \(a_{qr}(t)\) given by
are continuous, and the series
is absolutely uniformly continuous on I. Since
is uniformly bounded on I, for any \(t\in I\) and \(q\in\mathbb{N}_{0}\), we consider
We note that, if \(x(t)= (x_{q}(t) )\in\ell_{p}\), then
because the norm
is finite. We have to demonstrate that the operator \((fx)(t)= ((f_{q}x)(t) )\) is uniformly continuous on \(\ell_{p}\). For this, we suppose to prove that the sequence \((f_{q}(x) )\) is equicontinuous. Let \(\epsilon>0\) be given, and \(x(t)= (x_{q}(t) )\in\ell_{p}\). By considering
with
it follows from (14) that, for any fixed q,
where \(p>1\) and \(1/p+1/p'=1\), which yields the continuity, as desired. Hence hypothesis (i) is satisfied. In order to verify hypotheses from (ii) to (iv), we reckon a function \(h:I\times\ell_{p}\to\mathbb{R}\) that occurs on nonnegative values if and only if
where \((x_{q}(t) )\) is a nonincreasing sequence in \(\mathbb{R}_{+}\) with
We thus find that
uniformly with regard to \(t\in(0,T)\). It is convenient to observe that
Let
Now, from the data
and (16) it follows that
for all \(q>r\), \(r\in\mathbb{N}_{0}\), and \(t\in I\). Thus, taking into account (15) and (17), we find that, for all \(q>r\) and \(t\in I\),
which yields
where
Since
we obtain
By considering \(\varphi(t)\) as a kind of identity mapping, we conclude that conditions (ii), (iii), and (11) are satisfied. It is now left to show that (12) holds. Indeed, if we assume that
and
then it follows from the term of h that \((x_{q}(t) )\) is a nonincreasing sequence in \(\mathbb{R}_{+}\). Therefore,
which shows that
for all \(t\in I\) and \(q\in\mathbb{N}_{0}\). Accordingly, we have
for all \(t\in I\) and \(q\in\mathbb{N}_{0}\). It only remains to demonstrate that
uniformly with respect to \(t\in(0,T)\). In order to verify (18), we have to show that
uniformly in \((0,T)\). By straightforward calculation we obtain
which converges uniformly to zero as \(q\to\infty\). This evidently proves (19), so that assumption (iv) is satisfied. Hence, in light of Theorem 5, Eq. (13) has a solution in the space \(\ell_{p}\).
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Acknowledgements
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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Mohiuddine, S.A., Srivastava, H.M. & Alotaibi, A. Application of measures of noncompactness to the infinite system of secondorder differential equations in \(\ell_{p}\) spaces. Adv Differ Equ 2016, 317 (2016). https://doi.org/10.1186/s136620161016y
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MSC
 47H10
 34A34
Keywords
 measure of noncompactness
 Darbotype fixed point theorem
 infinite system of secondorder differential equations
 sequence spaces