## 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.

### Similar content being viewed by others

## 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. □

## References

Medina R, Pinto M: Asymptotic behavior of solutions of second order nonlinear difference equations.

*Nonlinear Anal.*1992, 19: 187–195. 10.1016/0362-546X(92)90119-YMigda M: Asymptotic behavior of solutions of nonlinear delay difference equations.

*Fasc. Math.*2001, 31: 57–62.Migda J, Migda M: Asymptotic properties of the solutions of second order difference equation.

*Arch. Math.*1998, 34: 467–476.Migda M, Schmeidel E, Zba̧szyniak M: On the existence of solutions of some second order nonlinear difference equations.

*Arch. Math.*2005, 42: 379–388.Musielak R, Popenda J: The periodic solutions of the second order nonlinear difference equation.

*Publ. Mat.*1988, 32: 49–56.Popenda J, Werbowski J: On the asymptotic behavior of the solutions of difference equations of second order.

*Ann. Pol. Math.*1980, 22: 135–142.Schmeidel E: Asymptotic behaviour of solutions of the second order difference equations.

*Demonstr. Math.*1993, 25: 811–819.Schmeidel E, Zba̧szyniak Z: An application of Darbo’s fixed point theorem in investigation of periodicity of solutions of difference equations.

*Comput. Math. Appl.*2012. doi:10.1016/j.camwa.2011.12.025Thandapani E, Arul R, Graef JR, Spikes PW: Asymptotic behavior of solutions of second order difference equations with summable coefficients.

*Bull. Inst. Math. Acad. Sin.*1999, 27: 1–22.Grace SR, Lalli BS: Oscillatory and asymptotic behavior of solutions of nonlinear neutral-type difference equations.

*J. Aust. Math. Soc. Ser. B*1996, 38: 163–171. 10.1017/S0334270000000552Lalli BS, Grace SR: Oscillation theorems for second order neutral difference equations.

*Appl. Math. Comput.*1994, 62: 47–60. 10.1016/0096-3003(94)90132-5Lalli BS, Zhang BG: On existence of positive solutions and bounded oscillations for neutral difference equations.

*J. Math. Anal. Appl.*1992, 166: 272–287. 10.1016/0022-247X(92)90342-BMigda J, Migda M: Asymptotic properties of solutions of second-order neutral difference equations.

*Nonlinear Anal.*2005, 63: e789-e799. 10.1016/j.na.2005.02.005Luo JW, Bainov DD: Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima.

*J. Comput. Appl. Math.*2001, 131: 333–341. 10.1016/S0377-0427(00)00264-8Luo J, Yu Y: Asymptotic behavior of solutions of second order neutral difference equations with ‘maxima’.

*Demonstr. Math.*2001, 35: 83–89.Baštinec J, Berezansky L, Diblík J, Šmarda Z:A final result on the oscillation of solutions of the linear discrete delayed equation \mathrm{\Delta}x(n)=-p(n)x(n-k) with a positive coefficient.

*Abstr. Appl. Anal.*2011., 2011: Article ID 586328Baštinec J, Diblík J, Šmarda Z: Existence of positive solutions of discrete linear equations with a single delay.

*J. Differ. Equ. Appl.*2010, 16: 1047–1056. 10.1080/10236190902718026Berezansky L, Diblík J, Růžičková M, Šutá Z: Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case.

*Abstr. Appl. Anal.*2011., 2011: Article ID 709427Diblík J, Hlavičková I: Asymptotic behavior of solutions of delayed difference equations.

*Abstr. Appl. Anal.*2011., 2011: Article ID 671967Diblík J, Růžičková M, Šutá Z: Asymptotic convergence of the solutions of a discrete equation with several delays.

*Appl. Math. Comput.*2012, 218: 5391–5401. 10.1016/j.amc.2011.11.023Agarwal RP Monographs and Textbooks in Pure and Applied Mathematics 228. In

*Difference Equations and Inequalities. Theory, Methods, and Applications*. 2nd edition. Dekker, New York; 2000.Elaydi SN Undergraduate Texts in Mathematics. In

*An Introduction to Difference Equations*. 3rd edition. Springer, New York; 2005.Kocić VL, Ladas G Mathematics and Its Applications 256. In

*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*. Kluwer Academic, Dordrecht; 1993.Kelley WG, Peterson AC:

*Difference Equations: An Introduction with Applications*. Academic Press, San Diego; 2001.Akhmerov RR, Kamenskij MI, Potapov AS, Rodkina AS, Sadovskij BN Operator Theory: Advances and Applications 55. In

*Measures of Noncompactness and Condensing Operators*. Birkhäuser, Basel; 1992. Translated from the Russian by A. IacobBanaś J, Goebel K Lecture Notes in Pure and Applied Mathematics 60. In

*Measures of Noncompactness in Banach Spaces*. Dekker, New York; 1980.Banaś J, Rzepka B: An application of measure of noncompactness in study of asymptotic stability.

*Appl. Math. Lett.*2003, 16: 1–6. 10.1016/S0893-9659(02)00136-2

## Acknowledgements

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

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Competing interests

The author declares that they have no competing interests.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Schmeidel, E. An application of measures of noncompactness in the investigation of boundedness of solutions of second-order neutral difference equations.
*Adv Differ Equ* **2013**, 91 (2013). https://doi.org/10.1186/1687-1847-2013-91

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/1687-1847-2013-91