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# An application of measures of noncompactness in the investigation of boundedness of solutions of second-order neutral difference equations

*Advances in Difference Equations***volume 2013**, Article number: 91 (2013)

## Abstract

The purpose of this paper is to investigate a nonlinear second-order neutral difference equation of the form

where $x:{\mathbb{N}}_{0}\to \mathbb{R}$, $a:{\mathbb{N}}_{0}\to \mathbb{R}$, $p,r:{\mathbb{N}}_{0}\to \mathbb{R}\setminus \{0\}$, $f:\mathbb{R}\to \mathbb{R}$ is a continuous function, and *k* is a given positive integer. Sufficient conditions for the existence of a bounded solution of this equation are obtained. Also, stability and asymptotic stability of this equation are studied. Additionally, the Emden-Fowler difference equation is considered as a special case of the above equation. The obtained results are illustrated by examples.

**MSC:**39A10, 39A22, 39A30.

## 1 Introduction

In presented paper we study a nonlinear second-order difference equation of the form

where $x:{\mathbb{N}}_{0}\to \mathbb{R}$, $a:{\mathbb{N}}_{0}\to \mathbb{R}$, $p,r:{\mathbb{N}}_{0}\to \mathbb{R}\setminus \{0\}$, and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function. Here ${\mathbb{N}}_{0}:=\{0,1,2,\dots \}$, ${\mathbb{N}}_{k}:=\{k,k+1,k+2,\dots \}$, where *k* is a given positive integer and ℝ is a set of all real numbers. By a solution of equation (1), we mean a sequence $x:{\mathbb{N}}_{0}\to \mathbb{R}$ which satisfies (1) for every $n\in {\mathbb{N}}_{0}$.

Putting $f(x)={x}^{\gamma}$, where $\gamma <1$ is a quotient of two odd integers, ${r}_{n}\equiv 1$ and ${p}_{n}\equiv p\in (0,\mathrm{\infty})$, $p\ne 1$ in equation (1), we get an Emden-Fowler difference equation of the form

In the last years many authors have been interested in studying the asymptotic behavior of solutions of difference equations, in particular, second-order difference equations (see, for example, papers of Medina and Pinto [1], Migda [2], Migda and Migda [3], Migda *et al.* [4], Musielak and Popenda [5], Popenda and Werbowski [6], Schmeidel [7], Schmeidel and Zba̧szyniak [8] and Thandapani *et al.* [9]).

Neutral difference equations were studied in many other papers by Grace and Lalli [10] and [11], Lalli and Zhang [12], Migda and Migda [13], Luo and Bainov [14], and Luo and Yu [15].

Some relevant results related to this topic can be found in papers by Baštinec *et al.* [16], Baštinec *et al.* [17], Berezansky *et al.* [18], Diblík and Hlavičková [19], and Diblík *et al.* [20].

For the reader’s convenience, we note that the background for difference equations theory can be found, *e.g.*, in the well-known monograph by Agarwal [21] as well as in those by Elaydi [22], Kocić and Ladas [23], or Kelley and Peterson [24].

The theory of measures of noncompactness can be found in the book of Akhmerov *et al.* [25] and in the book of Banaś and Goebel [26]. In our paper, we used axiomatically defined measures of noncompactness as presented in paper [27] by Banaś and Rzepka.

## 2 Measures of noncompactness and Darbo’s fixed point theorem

Let $(E,\parallel \cdot \parallel )$ be an infinite-dimensional Banach space. If *X* is a subset of *E*, then $\overline{X}$, Conv*X* denote the closure and the convex closure of *X*, respectively. Moreover, we denote by ${\mathcal{M}}_{E}$ the family of all nonempty and bounded subsets of *E* and by ${\mathcal{N}}_{E}$ the subfamily consisting of all relatively compact sets.

**Definition 1** A mapping $\mu :{\mathcal{M}}_{E}\to [0,\mathrm{\infty})$ is called a measure of noncompactness in *E* if it satisfies the following conditions:

1^{∘} $ker\mu =\{X\in {\mathcal{M}}_{E}:\mu (X)=0\}\ne \mathrm{\varnothing}$ and $ker\mu \subset {\mathcal{N}}_{E}$,

2^{∘} $X\subset Y\Rightarrow \mu (X)\le \mu (Y)$,

3^{∘} $\mu (\overline{X})=\mu (X)=\mu (ConvX)$,

4^{∘} $\mu (\alpha X+(1-\alpha )Y)\le \alpha \mu (X)+(1-\alpha )\mu (Y)$ for $0\le \alpha \le 1$,

5^{∘} if ${X}_{n}\in {\mathcal{M}}_{E}$, ${X}_{n+1}\subset {X}_{n}$, ${X}_{n}={\overline{X}}_{n}$ for $n=1,2,3,\dots $ and ${lim}_{n\to \mathrm{\infty}}\mu ({X}_{n})=0$, then ${\bigcap}_{n=1}^{\mathrm{\infty}}{X}_{n}\ne \mathrm{\varnothing}$.

The following Darbo’s fixed point theorem given in [27] is used in the proof of the main result.

**Theorem 1** *Let* *M* *be a nonempty*, *bounded*, *convex*, *and closed subset of the space* *E*, *and let* $T:M\to M$ *be a continuous operator such that* $\mu (T(X))\le k\mu (X)$ *for all nonempty subset* *X* *of* *M*, *where* $k\in [0,1)$ *is a constant*. *Then* *T* *has a fixed point in the subset* *M*.

We consider the Banach space ${l}^{\mathrm{\infty}}$ of all real bounded sequences $x:{\mathbb{N}}_{0}\to \mathbb{R}$ equipped with the standard supremum norm, *i.e.*,

Let *X* be a nonempty, bounded subset of ${l}^{\mathrm{\infty}}$, ${X}_{n}=\{{x}_{n}:x\in X\}$ (it means ${X}_{n}$ is a set of *n* th terms of any sequence belonging to *X*), and let

We use the following measure of noncompactness in the space ${l}^{\mathrm{\infty}}$ (see [26]):

## 3 Main result

In this section, sufficient conditions for the existence of a bounded solution of equation (1) are derived. Further, stable solutions of (1) are studied. We start with the following theorem.

**Theorem 2**
*Let*

*and let there exist constants* *L* *and* *M* *such that for all* $x\in \mathbb{R}$,

*the sequence* $p:{\mathbb{N}}_{0}\to \mathbb{R}\setminus \{0\}$ *satisfies the following condition*:

*sequences* $a:{\mathbb{N}}_{0}\to \mathbb{R}$, $r:{\mathbb{N}}_{0}\to \mathbb{R}\setminus \{0\}$ *are such that*

*Then there exists a bounded solution* $x:{\mathbb{N}}_{0}\to \mathbb{R}$ *of equation* (1).

*Proof* Condition (5) implies that there exist ${n}_{1}\in {\mathbb{N}}_{0}$ and a constant $P\in [0,1)$ such that

The remainder of a series is the difference between the *n* th partial sum and the sum of a series. Let us denote by ${\alpha}_{n}$ the remainder of series ${\sum}_{n=0}^{\mathrm{\infty}}|\frac{1}{{r}_{n}}|{\sum}_{i=n}^{\mathrm{\infty}}|{a}_{i}|$ so that

From (6), the remainder ${\alpha}_{n}$ tends to zero. Therefore, we can denote

Let us denote that *C* is a given positive constant. Condition (6) implies that there exists a positive integer ${n}_{2}$ such that

for $n\ge {n}_{2}$.

We define a set *B* as follows:

where ${\mathbb{N}}_{{n}_{3}}:=\{{n}_{3},{n}_{3}+1,{n}_{3}+2,\dots \}$ and ${n}_{3}=max\{{n}_{1},{n}_{2}\}$.

It is not difficult to prove that *B* is a nonempty, bounded, convex, and closed subset ${l}^{\mathrm{\infty}}$.

Let us define a mapping $T:B\to {l}^{\mathrm{\infty}}$ as follows:

for any $n\in {\mathbb{N}}_{{n}_{3}}$.

We will prove that the mapping *T* has a fixed point in *B*.

Firstly, we show that $T(B)\subset B$. Indeed, if $x\in B$, then by (12), (7), (11), and (10), we have

Next, we prove that *T* is continuous. Let ${x}^{(p)}$ be a sequence in *B* such that $\parallel {x}^{(p)}-x\parallel \to 0$ as $p\to \mathrm{\infty}$. Because of (3), we have $\parallel f({x}^{(p)})-f(x)\parallel \to 0$. Since *B* is closed, $x\in B$. Now, utilizing (12), we get

Hence, by (7) and (8),

Therefore, by (10),

and

This means that *T* is continuous.

Now, we need to compare a measure of noncompactness of any subset *X* of *B* and $T(X)$. Let us take a nonempty set $X\subset B$. For any sequences $x,y\in X$, we get

Hence, we obtain

This yields

From the above, for any $X\subset B$, we have $\mu (T(X))\le k\mu (X)$, where $k=\frac{P+1}{2}\in [0,1)$.

By virtue of Theorem 1, we conclude that *T* has a fixed point in the set *B*. It means that there exists $x\in B$ such that ${x}_{n}={(Tx)}_{n}$. Thus

for any $n\in {\mathbb{N}}_{{n}_{3}}$. To show that there exists a connection between the fixed point $x\in B$ and the existence of a solution of equation (1), we use the operator Δ for both sides of the following equation:

which is obtained from (13). We find that

Using again the operator Δ for both sides of the above equation, we get equation (1) for $n\in {\mathbb{N}}_{{n}_{3}}$. The sequence *x*, which is a fixed point of the mapping *T*, is a bounded sequence which fulfills equation (1) for large *n*. If ${n}_{3}\le k$, the proof is ended. If ${n}_{3}>k$, then we find previous ${n}_{3}-k+1$ terms of the sequence *x* by the formula

the results of which follow directly from (1). It means that equation (1) has at least one bounded solution $x:{\mathbb{N}}_{0}\to \mathbb{R}$.

This completes the proof. □

**Example 1**

Let us consider the equation

All the assumptions of Theorem 2 are fulfilled. Then there exists a bounded solution *x* of the above equation. So, the sequence ${x}_{n}={(-1)}^{n}$ is such a solution.

**Remark 1**

Assume that

and

in an Emden-Fowler difference equation of the form (2). Then there exists a bounded solution of equation (2).

*Proof* Here all the assumptions of Theorem 2 are satisfied, *e.g.*, the function $f:\mathbb{R}\to \mathbb{R}$ given by formula $f(x)={x}^{\gamma}$ is a continuous function, and $|f(x)|=|{x}^{\gamma}|\le \gamma |x|+1-\gamma $. So, taking $M=\gamma $ and $L=1-\gamma $, we obtain condition (4). The thesis follows directly from Theorem 2. □

Finally, sufficient conditions for the existence of an asymptotically stable solution of equation (1) will be presented. We recall the following definition which can be found in [27].

**Definition 2** Let *x* be a real function defined, bounded, and continuous on $[0,\mathrm{\infty})$. The function *x* is an asymptotically stable solution of the equation

It means that for any $\epsilon >0$, there exists $T>0$ such that for every $t\ge T$ and for every other solution *y* of equation (16), the following inequality holds:

**Theorem 3**
*Assume that there exists a positive constant*
*D*
*such that*

*for any* $x,y\in \mathbb{R}$, *and conditions* (3)-(6) *hold*. *Then equation* (1) *has at least one asymptotically stable solution* $x:{\mathbb{N}}_{0}\to \mathbb{R}$.

*Proof* From Theorem 2, equation (1) has at least one bounded solution $x:{\mathbb{N}}_{0}\to \mathbb{R}$ which can be rewritten in the form

where a mapping *T* is defined by (12).

Because of Definition 2, the sequence *x* is an asymptotically stable solution of the equation ${x}_{n}={(Tx)}_{n}$, which means that for any $\epsilon >0$, there exists ${n}_{4}\in {\mathbb{N}}_{0}$ such that for every $n\ge {n}_{4}$ and for every other solution *y* of equation (1), the following inequality holds:

From (12), by (7), we have

for $n\ge {n}_{3}$. The above and (17) yield

for $n\ge {n}_{5}=max\{{n}_{3},{n}_{4}\}$. Hence, by (8) and (19), we obtain

for $n\ge {n}_{5}$. Thus, linking the above inequality and (18), we have

Let us denote

Because of

and (20), we get

From the above and (9), we obtain

Suppose to the contrary that $l>0$. Thus, we obtain a contradiction with the fact that $0<P<1$. Therefore we get ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}|{x}_{n}-{y}_{n}|=0$. This completes the proof. □

**Remark 2** Under conditions (3)-(6) and (17), any bounded solution of equation (1) is asymptotically stable.

*Proof* If boundedness of a solution of equation (1) is assumed, then by virtue of the same arguments as in Theorem 3, the thesis of the above remark is obtained. □

**Example 2** Let us consider equation (1) with $f(x)=x$, ${a}_{n}={\mathrm{\Delta}}^{2}{p}_{n}$ and ${\sum}_{n=0}^{\mathrm{\infty}}{\sum}_{i=n}^{\mathrm{\infty}}|{a}_{i}|<\mathrm{\infty}$. Such an equation has infinitely many solutions of the form ${x}_{n}\equiv c$, where *c* is a real constant. All the assumptions of Theorem 3 are fulfilled, then each of such solutions is asymptotically stable.

**Theorem 4** *Assume that* $L=0$ *in* (4). *Under conditions* (3)-(6) *and* (17), *if there exists a zero solution of equation* (1), *then it is asymptotically stable*.

*Proof* If $L=0$, then condition (4) takes the form $|f(x)|\le M|x|$. This implies that $f(0)=0$. Hence, the sequence $x\equiv 0$ is a bounded solution of equation (1). By Remark 2, the zero solution is asymptotically stable. □

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## Acknowledgements

The author would like to thank the reviewers for their helpful comments and valuable suggestions.

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### Keywords

- difference equation
- measures of noncompactness
- Darbo’s fixed point theorem
- boundedness
- stability
- Emden-Fowler equation