1 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

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

$$x_{j}=o(y_{j})\quad \Longleftrightarrow\quad \lim _{j\to\infty} \frac{x_{j}}{y_{j}}=0 \quad (y_{j}\neq0 ). $$

We introduce the space \(\ell_{p}\) of all absolutely p-summable series as follows:

$$\ell_{p}= \Biggl\{ x\in\omega:\sum_{j=0}^{\infty}|x_{j}|^{p} < \infty \Biggr\} \quad (1\leqq p< \infty). $$

Clearly, \(\ell_{p}\) is a Banach space with norm

$$\|x\|_{p}= \Biggl(\sum_{j=0}^{\infty}|x_{j}|^{p} \Biggr)^{1/p}\quad (1\leqq p< \infty). $$

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 n-section be given by

$$x^{[n]}=\sum_{j=0}^{n}x_{j}e^{(j)}. $$

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

$$x=\sum_{j=0}^{\infty}x_{j}e^{(j)}. $$

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,

$$B(x,r)=\bigl\{ y\in X:d(x,y)< r\bigr\} . $$

Let \(F\in\mathcal{M}_{X}\). Then the Hausdorff measure of noncompactness of F is defined by

$$ \chi(F)=\inf \Biggl\{ \epsilon>0:F\subset\bigcup_{j=1}^{n} B(x_{j},r_{j}), x_{j}\in X, r_{j}< \epsilon \ (1\leqq j\leqq n; n\in\mathbb{N}) \Biggr\} . $$

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

  1. (i)

    \(\chi(F)=0\) if and only if F is totally bounded;

  2. (ii)

    \(\chi(F)=\chi(\bar{F})\), where F̄ denotes the closure of F;

  3. (iii)

    \(F_{1}\subset F_{2}\) implies that \(\chi(F_{1})\leqq\chi(F_{2})\);

  4. (iv)

    \(\chi(F_{1}\cup F_{2})=\max\{\chi(F_{1}),\chi(F_{2})\}\);

  5. (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

$$\begin{aligned}& \chi(F_{1}+F_{2})\leqq\chi(F_{1})+ \chi(F_{2}), \\& \chi(F+x)=\chi(F) \quad \mbox{for all } x\in X, \\& \chi(\alpha F)=|\alpha|\chi(F)\quad \mbox{for all } \alpha\in\mathbb{C}. \end{aligned}$$

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

$$ \frac{1}{a}\limsup_{j\to\infty} \Bigl\{ \sup _{x\in F}\bigl\Vert (I-P_{j}) (x)\bigr\Vert \Bigr\} \leqq\chi(F)\leqq\limsup_{j\to\infty} \Bigl\{ \sup_{x\in F} \bigl\Vert (I-P_{j}) (x)\bigr\Vert \Bigr\} , $$
(1)

where I is the identity operator on X, and

$$a=\limsup_{j\to\infty}\|I-P_{j}\|. $$

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

$$ \chi(F)=\lim_{k\to\infty}\sup_{x\in F} \biggl\{ \biggl(\sum_{j\geqq k}|x_{j}|^{p} \biggr)^{1/p} \biggr\} . $$
(2)

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

$$ \mu\bigl(T(F)\bigr)\leqq\varphi\bigl(\mu(F)\bigr) $$
(3)

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

$$\lim_{n\to\infty}\varphi^{n}(t)=0. $$

Then T has at least one fixed point in the set \({\mathcal{C}}\).

The notion of \((\alpha,\phi, \varphi)\)-μ-condensing operators and α-admissible 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

$$\liminf_{n\to\infty}\varphi(a_{n})=0, $$

conferred that

$$\lim_{n\to\infty}a_{n}=0, $$

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:

  1. (i)

    ϕ is a lower semi-continuous function with \(\phi(t)=0\) if and only if \(t=0\);

  2. (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 \((\alpha,\varphi,\phi)\)-μ-condensing if the functions \(\alpha:{\mathcal{M}}_{X}\to[0,+\infty)\), \(\varphi\in\Psi\), and \(\phi=\Phi_{\varphi}\) are such that

$$ \alpha(F)\phi \bigl(\mu(TF) \bigr)=\varphi \bigl(\mu(F) \bigr)\quad (F\in W), $$

where both F and its image TF belong to \({\mathcal{M}}_{X}\).

Let T and α be given mappings as before. Then T is α-admissible if

$$ \alpha(F)\geqq1\quad \Longrightarrow\quad \alpha(\operatorname{Conv}TF)\geqq1 \quad (F\in W; F,TF\in{\mathcal{M}}_{X} ). $$

Remark 1

If T follows the Darbo condition with regard to a measure μ and a constant \(k\in[0,1)\), that is, if

$$ \mu(TF)=k\mu(F) \quad (F\in W; F,TF\in{\mathcal{M}}_{X} ), $$

then T is an \((\alpha,\varphi,\phi)\)-μ-condensing 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 μ-contraction.

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

Let us consider the following infinite system of second-order differential equations:

$$ -\frac{d^{2}x_{i}}{dt^{2}}=f_{i} (t,x_{0}, x_{1},x_{2}, \ldots ) $$
(4)

with the initial conditions given by

$$x_{i}(0)=x_{i}(T)=0\quad \bigl(i\in\mathbb{N}_{0}:= \mathbb{N}\cup\{0\}=\{0,1,2,\ldots\}; t\in I=[0,T] \bigr). $$

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:

$$ x_{i}(t)= \int_{0}^{T}G(t,s)f_{i} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s), \ldots \bigr) \,ds\quad (t\in I), $$
(5)

where

$$f_{i} (t,x_{0},x_{1},x_{2},\ldots ) \in C(I,\mathbb{R}) \quad (i\in\mathbb{N}_{0}), $$

and the Green function \(G(t,s)\) associated with (4) is given by

$$ G(t,s)=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{t}{T}(T-s) & (0\leqq t\leqq s\leqq T), \\ \frac{s}{T}(T-t) & (0\leqq s\leqq t\leqq T). \end{array}\displaystyle \right . $$
(6)

For more details of green functions, we refer to [22]. We can rewrite (5) with the help of (6) as follows:

$$\begin{aligned} x_{i}(t) =& \int_{0}^{t}\frac{s}{T}(T-t)f_{i} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \,ds \\ &{} + \int_{t}^{T} \frac{t}{T}(T-s)f_{i} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \, ds. \end{aligned}$$
(7)

Upon differentiating both sides of (7) with respect to t, we get

$$\begin{aligned} \frac{d}{dt} \bigl\{ x_{i}(t) \bigr\} =&-\frac{1}{T} \int_{0}^{t}sf_{i} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s), \ldots \bigr) \,ds \\ &{}+\frac{1}{T} \int_{t}^{T}(T-s)f_{i} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \,ds. \end{aligned}$$
(8)

Again, by differentiating both sides of (8) with respect to t we obtain

$$\begin{aligned} \frac{d^{2}}{dt^{2}} \bigl\{ x_{i}(t) \bigr\} =&-\frac{1}{T} tf_{i} \bigl(t,x_{0}(t),x_{1}(t),x_{2}(t), \ldots \bigr) \\ &{} +\frac{1}{T}(t-T)f_{i} \bigl(t,x_{0}(t),x_{1}(t),x_{2}(t), \ldots \bigr) \\ =&-f_{i} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s), \ldots \bigr). \end{aligned}$$

We now investigate the existence result concerning the second-order 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:

  1. (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 \(\ell_{p}\).

  2. (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,

    $$\varphi(s+t)\geqq\varphi(s)+\varphi(t) $$

    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 \(k\in \mathbb{N}_{0}\).

  3. (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

    $$\sup_{t\in I} \biggl\{ \int_{0}^{T}\bigl|G(t,s)\bigr|^{p}g(s) \,ds \biggr\} \leqq C $$

    for some positive constant C.

  4. (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 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

Let us consider the operator \({\mathcal{F}}=({\mathcal{F}}_{i})\) defined on \(C(I,\ell_{p})\) by

$$({\mathcal{F}}x) (t)= \bigl(({\mathcal{F}}_{i}x) (t) \bigr) = \biggl( \int_{0}^{T}G(t,s)f_{i} \bigl(s,x_{0}(t),x_{1}(t),x_{2}(t),\ldots \bigr) \biggr), $$

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

$$\begin{aligned} \bigl\Vert ({\mathcal{F}}x) (t)\bigr\Vert _{p}^{p} & =\sum_{i=0}^{\infty} \biggl\vert \int_{0}^{T}G(t,s)f_{i} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \,ds\biggr\vert ^{p} \\ &\leqq\sum_{i=0}^{\infty} \biggl\vert \biggl(\int_{0}^{T}\bigl(G(t,s)f_{i} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr)\bigr)^{p}\,ds\biggl)^{1/p} \biggl(\int_{0}^{T}ds\biggr)^{1/p'}\biggr\vert ^{p} \\ &\leqq T^{p/p'}\sum_{i=0}^{\infty}\int_{0}^{T}\bigl|G(t,s)\bigr|^{p}\bigl|f_{i} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr)\bigr|^{p}\,ds \\ &\leqq\frac{T^{p/p'}T^{p}}{4^{p}} \int_{0}^{T}\bigl\Vert (fx) (s)\bigr\Vert _{p}^{p} \,ds< \infty, \end{aligned}$$

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

$$h \bigl(t,x(t) \bigr)\geqq0 $$

for any fixed \(t\in I\). Then, clearly, Eq. (2) yields

$$\begin{aligned} \chi({\mathcal{F}}Q)&=\lim_{n\to\infty}\sup_{x(t)\in Q} \biggl\{ \biggl(\sum_{j\geqq n}\biggl\vert \int_{0}^{T}G(t,s)f_{j} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \,ds\biggr\vert ^{p} \biggr)^{1/p}\biggr\} \\ &\leqq\lim_{n\to\infty}\sup_{x(t)\in Q} \biggl\{ \biggl(T^{p/p'}\sum_{j\geqq n} \int_{0}^{T}\bigl|G(t,s)\bigr|^{p}\bigl|f_{j} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr)\bigr|^{p} \,ds\biggr)^{1/p}\biggr\} \\ &\leqq\lim_{k\to\infty}\sup_{x(t)\in Q}\biggl\{ \biggl( T^{p/p'}\sum_{j\geqq k} \int_{0}^{T}\bigl|G(t,s)\bigr|^{p}g_{j}(s)\varphi\bigl( \bigl\vert x_{j}(s)\bigr\vert ^{p}\bigr) \,ds \biggr)^{1/p}\biggr\} \\ &\leqq\lim_{k\to\infty}\sup_{x\in Q}\biggl\{ \biggl( T^{p/p'}\sum_{j\geqq k} C\varphi\bigl(\vert x_{j} \vert ^{p}\bigr)\biggr)^{1/p}\biggr\} \\ &\leqq C'\lim_{k\to\infty}\sup_{x\in Q} \biggl\{ \varphi\biggl(\sum_{j\geqq k}\vert x_{j}\vert ^{p}\biggr)^{1/p}\biggr\} \\ &=C' \lim_{k\to\infty}\biggl\{ \varphi\biggl(\sup _{x\in Q} \biggl\{ \biggl(\sum_{j\geqq k} \vert x_{j}\vert ^{p}\biggr)^{1/p} \biggr\} \biggr) \biggr\} \\ &\leqq C'\varphi \bigl(\chi(Q) \bigr). \end{aligned}$$

This shows that

$$\alpha(Q)\phi \bigl(\chi({\mathcal{F}}Q) \bigr)\leqq\varphi \bigl(\chi(Q) \bigr), $$

where \(\alpha:{\mathcal{M}}_{\ell_{p}}\to[0,\infty)\) is the mapping defined by

$$ \alpha(Q)=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 & (h (t,x(t) )\geqq0; x\in Q; t\in I ), \\ 0 & (\mbox{otherwise}), \end{array}\displaystyle \right . $$

and

$$\phi(b)=\frac{b}{C'}\quad (b\in\mathbb{R}_{+}). $$

Obviously, \(\phi\in\Phi_{\varphi}\), and it satisfies (10). Interestingly, by hypothesis (iv) we conclude that the operator \({\mathcal{F}}\) is α-admissible 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 second-order differential equations:

$$ -\frac{d^{2}}{dt^{2}}\{x_{q}\}=\frac{t(T-t)e^{-qt}}{(q+1)^{4}} +\sum _{r=q}^{\infty}\frac{x_{r}(t)\sqrt{t}}{(1+q^{2})(r+1)^{2}}, $$
(13)

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

$$a_{qr}(t)=\frac{\sqrt{t}}{(1+q^{2})(r+1)^{2}} $$

are continuous, and the series

$$\sum_{r=q}^{\infty}\bigl\vert a_{qr}(t)\bigr\vert ^{p} $$

is absolutely uniformly continuous on I. Since

$$a_{q}(t):=\sum_{r=q}^{\infty}\bigl\vert a_{qr}(t)\bigr\vert ^{p} $$

is uniformly bounded on I, for any \(t\in I\) and \(q\in\mathbb{N}_{0}\), we consider

$$ B=\sup \bigl\{ a_{q}(t) \bigr\} < \infty. $$
(14)

We note that, if \(x(t)= (x_{q}(t) )\in\ell_{p}\), then

$$\begin{aligned} (fx) (t)&= \bigl(f_{q} (t,x_{0},x_{1},x_{2}, \ldots ) \bigr) \\ &= \Biggl(\frac{t(T-t)e^{-qt}}{(q+1)^{4}} +\sum_{r=q}^{\infty} \frac{x_{r}(t)\sqrt{t}}{(1+q^{2})(r+1)^{2}} \Biggr)\in\ell_{p} \end{aligned}$$
(15)

because the norm

$$\begin{aligned} \bigl\Vert (fx) (t)\bigr\Vert _{p}^{p} \leqq&2^{p}\sum _{q=0}^{\infty} \biggl\vert \frac{t(T-t)e^{-qt}}{(q+1)^{4}} \biggr\vert ^{p} \\ &{} +2^{p}\sum_{q=0}^{\infty}\biggl\vert \sum _{r=q}^{\infty} \frac{x_{r}(t)\sqrt{t}}{(1+q^{2})(r+1)^{2}}\biggr\vert ^{p} \end{aligned}$$

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

$$x'(t)= \bigl(x'_{q}(t) \bigr)\in \ell_{p} $$

with

$$\bigl\Vert x(t)-x'(t)\bigr\Vert _{p}^{p} \leqq\delta(\epsilon)=\epsilon B^{-1} $$

it follows from (14) that, for any fixed q,

$$\begin{aligned} \bigl\vert (f_{q}x)(t)-\bigl(f_{q}x' \bigr)(t)\bigr\vert ^{p} &=\Biggl\vert \sum _{r=q}^{\infty}\frac{\bigl(x_{r}(t)-x'_{r}(t)\bigr)\sqrt{t}}{(1+q^{2})(r+1)^{2}}\Biggr\vert ^{p} \\ &\leqq\Biggl\vert \Biggl(\sum _{r=q}^{\infty}\bigl(x_{r}(t)-x'_{r}(t)\bigr)^{p}\Biggr)^{1/p}\Biggl(\sum _{r=q}^{\infty}\biggl(\frac{\sqrt{t}}{(1+q^{2})(r+1)^{2}}\biggr)^{p'}\Biggr)^{1/p'}\Biggr\vert ^{p} \\ &\leqq\sum _{r=q}^{\infty}\bigl|x_{r}(t)-x'_{r}(t)\bigr|^{p}\sum _{r=q}^{\infty}\biggl\vert \frac{\sqrt{t}}{(1+q^{2})(r+1)^{2}}\biggr\vert ^{p} \\ &\leqq B\bigl\Vert x(t)-x'(t)\bigr\Vert _{p}^{p}\leqq B\epsilon B^{-1}=\epsilon, \end{aligned}$$

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

$$x(t)= \bigl(x_{q}(t) \bigr)\in\ell_{p}, $$

where \((x_{q}(t) )\) is a nonincreasing sequence in \(\mathbb{R}_{+}\) with

$$x(0)=0=x(T). $$

We thus find that

$$ \frac{t(T-t)e^{-qt}}{(q+1)^{4}}=o \bigl(x_{q}(t) \bigr) $$
(16)

uniformly with regard to \(t\in(0,T)\). It is convenient to observe that

$$\bigl\{ x\in\ell_{p}:h \bigl(t,x(t) \bigr)\geqq0\ (t\in I) \bigr\} \neq \emptyset. $$

Let

$$h \bigl(t,x(t) \bigr)\geqq0. $$

Now, from the data

$$x(0)=x(T)=0 $$

and (16) it follows that

$$ \frac{t(T-t)e^{-qt}}{(q+1)^{4}}\leqq x_{q}(t) $$
(17)

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 \(t\in I\),

$$\begin{aligned} \bigl|(f_{q}x) (t)\bigr|^{p}&\leqq\biggl|x_{q}(t)+ \frac{\sqrt{t}}{1+q^{2}}\sum _{k=q}^{\infty}\frac{x_{k}(t)}{(k+1)^{2}}\biggr|^{p} \\ &\leqq 2^{p}\Biggl\{ \bigl|x_{q}(t)\bigr|^{p}+\frac{t^{p/2}}{(1+q^{2})^{p}}\Biggl|\sum _{k=q}^{\infty}\frac{x_{k}(t)}{(k+1)^{2}}\Biggr|^{p}\Biggr\} \\ &\leqq 2^{p}\Biggl\{ \bigl|x_{q}(t)\bigr|^{p}+\frac{t^{p/2}}{(1+q^{2})^{p}}\Biggl|\Biggl(\sum _{k=q}^{\infty}\bigl(x_{k}(t)\bigr)^{p}\Biggr)^{1/p}\Biggl(\sum _{k=q}^{\infty}\Biggl(\frac{1}{(k+1)^{2}}\Biggr)^{p'}\Biggr)^{1/p'}\Biggr|^{p}\Biggr\} \\ &\leqq 2^{p}\Biggl\{ \bigl|x_{q}(t)\bigr|^{p}+\frac{\pi^{2p}t^{p/2}}{6^{p}(1+q^{2})^{p}}\sum _{k=q}^{\infty}\bigl|x_{k}(t)\bigr|^{p}\Biggr\} \\ &\leqq 2^{p}\biggl(1+\frac{\pi^{2p}t^{p/2}}{6^{p}(1+q^{2})^{p}} \biggr)\bigl|x_{q}(t)\bigr|^{p}, \end{aligned}$$

which yields

$$\bigl\vert (f_{q}x) (t)\bigr\vert ^{p}\leqq g_{q}(t)\bigl\vert x_{q}(t)\bigr\vert ^{p}, $$

where

$$g_{q}(t)= 2^{p}\biggl(1+\frac{\pi^{2p}t^{p/2}}{6^{p}(1+q^{2})^{p}} \biggr). $$

Since

$$g(t)=\limsup_{q\to\infty} \bigl\{ g_{q}(t) \bigr\} =2^{p}, $$

we obtain

$$\sup_{t\in I} \biggl\{ \int_{0}^{T}\bigl|G(t,s)\bigr|^{p}g(s) \,ds \biggr\} \leqq\frac{T^{p+1}}{2^{p}}=C. $$

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

$$h \bigl(t,x(t) \bigr)\geqq0\quad (t\in I) $$

and

$$x(t)= \bigl(x_{q}(t) \bigr)\in\ell_{p}, $$

then it follows from the term of h that \((x_{q}(t) )\) is a nonincreasing sequence in \(\mathbb{R}_{+}\). Therefore,

$$\begin{aligned} f_{q+1} \bigl(t,x_{0}(t),x_{1}(t),x_{2}(t), \ldots \bigr) &=\frac{t(T-t)e^{-(q+1)t}}{(q+2)^{4}}+\sum_{r=q+1}^{\infty} \frac{x_{r}(t)\sqrt{t}}{(1+(q+1)^{2})(r+1)^{2}} \\ &\leqq\frac{t(T-t)e^{-qt}}{(q+1)^{4}}+\sum_{r=q}^{\infty} \frac{x_{r}(t)\sqrt{t}}{(1+q^{2})(r+1)^{2}}, \end{aligned}$$

which shows that

$$0\leqq f_{q+1} \bigl(t,x_{0}(t),x_{1}(t),x_{2}(t), \ldots \bigr) \leqq f_{q} \bigl(t,x_{0}(t),x_{1}(t),x_{2}(t), \ldots \bigr) $$

for all \(t\in I\) and \(q\in\mathbb{N}_{0}\). Accordingly, we have

$$\begin{aligned} 0&\leqq \int_{0}^{T}G(t,s)f_{q+1} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \,ds \\ &\leqq \int_{0}^{T}G(t,s)f_{q} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \,ds \end{aligned}$$

for all \(t\in I\) and \(q\in\mathbb{N}_{0}\). It only remains to demonstrate that

$$ \frac{t(T-t)e^{-qt}}{(q+1)^{4}}=o \biggl( \int_{0}^{T}G(t,s)f_{q} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \,ds \biggr) $$
(18)

uniformly with respect to \(t\in(0,T)\). In order to verify (18), we have to show that

$$ \frac{(q+1)^{4}}{t(T-t)} \int_{0}^{T}G(t,s)e^{qt}f_{q} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \,ds\rightarrow\infty\quad (q\rightarrow\infty) $$
(19)

uniformly in \((0,T)\). By straightforward calculation we obtain

$$\begin{aligned}& \frac{(q+1)^{4}}{t(T-t)} \int_{0}^{T}G(t,s)e^{qt}f_{q} \bigl(s,x_{0}(s),x_{1}(s),x_{2}(s),\ldots \bigr) \,ds \\& \quad \geqq\frac{(q+1)^{4}}{tT} \int_{0}^{t/2}s^{2}(T-s) e^{ (q(t-s) )} \,ds \\& \quad \geqq\frac{(q+1)^{4}e^{qt/2}}{T^{2}}\biggl(\frac{2 (-\frac{3}{q} +T )e^{qt/2}}{q^{3}}-\frac{t^{2} (-\frac{3}{q}+T )}{4q} - \frac{t (-\frac{3}{q}+T )}{q^{2}} -\frac{2 (-\frac{3}{q}+T )}{q^{3}}\biggr) \\& \quad \geqq\frac{(q+1)^{4}}{T^{2}}\biggl(\frac{2 (-\frac{3}{q}+T )}{q^{3}} -\frac{T^{2} (-\frac{3}{q}+T )}{4q}- \frac{T (-\frac {3}{q}+T )}{q^{2}} -\frac{2 (-\frac{3}{q}+T )}{q^{3}}\biggr) \quad \biggl(q>\frac {3}{T}\biggr), \end{aligned}$$

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 \(\ell_{p}\).