## 1 Introduction

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

$\mathrm{\Delta }\left({r}_{n}\mathrm{\Delta }\left({x}_{n}+{p}_{n}{x}_{n-k}\right)\right)+{a}_{n}f\left({x}_{n}\right)=0,$
(1)

where $x:{\mathbb{N}}_{0}\to \mathbb{R}$, $a:{\mathbb{N}}_{0}\to \mathbb{R}$, $p,r:{\mathbb{N}}_{0}\to \mathbb{R}\setminus \left\{0\right\}$, and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function. Here ${\mathbb{N}}_{0}:=\left\{0,1,2,\dots \right\}$, ${\mathbb{N}}_{k}:=\left\{k,k+1,k+2,\dots \right\}$, 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 $f\left(x\right)={x}^{\gamma }$, where $\gamma <1$ is a quotient of two odd integers, ${r}_{n}\equiv 1$ and ${p}_{n}\equiv p\in \left(0,\mathrm{\infty }\right)$, $p\ne 1$ in equation (1), we get an Emden-Fowler difference equation of the form

${\mathrm{\Delta }}^{2}\left({x}_{n}+p{x}_{n-k}\right)+{a}_{n}{x}_{n}^{\gamma }=0.$
(2)

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 $\left(E,\parallel \cdot \parallel \right)$ be an infinite-dimensional Banach space. If X is a subset of E, then $\overline{X}$, ConvX 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 $\mu :{\mathcal{M}}_{E}\to \left[0,\mathrm{\infty }\right)$ is called a measure of noncompactness in E if it satisfies the following conditions:

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

2 $X\subset Y⇒\mu \left(X\right)\le \mu \left(Y\right)$,

3 $\mu \left(\overline{X}\right)=\mu \left(X\right)=\mu \left(ConvX\right)$,

4 $\mu \left(\alpha X+\left(1-\alpha \right)Y\right)\le \alpha \mu \left(X\right)+\left(1-\alpha \right)\mu \left(Y\right)$ 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 ${lim}_{n\to \mathrm{\infty }}\mu \left({X}_{n}\right)=0$, 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 $\mu \left(T\left(X\right)\right)\le k\mu \left(X\right)$ for all nonempty subset X of M, where $k\in \left[0,1\right)$ 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 ${l}^{\mathrm{\infty }}$, ${X}_{n}=\left\{{x}_{n}:x\in X\right\}$ (it means ${X}_{n}$ is a set of n th terms of any sequence belonging to X), and let

$diam{X}_{n}=sup\left\{|{x}_{n}-{y}_{n}|:x,y\in X\right\}.$

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

$\mu \left(X\right)=\underset{n\to \mathrm{\infty }}{lim sup}diam{X}_{n}.$

## 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

$f:\mathbb{R}\to \mathbb{R}\mathit{\text{be a continuous function}},$
(3)

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

$|f\left(x\right)|\le M|x|+L,$
(4)

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

$-1<\underset{n\to \mathrm{\infty }}{lim inf}{p}_{n}\le \underset{n\to \mathrm{\infty }}{lim sup}{p}_{n}<1,$
(5)

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

$\sum _{n=0}^{\mathrm{\infty }}|\frac{1}{{r}_{n}}|\sum _{i=n}^{\mathrm{\infty }}|{a}_{i}|<\mathrm{\infty }.$
(6)

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 $P\in \left[0,1\right)$ such that

(7)

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

${\alpha }_{n}=\sum _{j=n}^{\mathrm{\infty }}|\frac{1}{{r}_{j}}|\sum _{i=j}^{\mathrm{\infty }}|{a}_{i}|.$
(8)

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

$\underset{n\to \mathrm{\infty }}{lim}{\alpha }_{n}=0.$
(9)

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

${\alpha }_{n}\le C\frac{1-P}{2\left(CM+L\right)}$
(10)

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

We define a set B as follows:

(11)

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

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:

${\left(Tx\right)}_{n}=-{p}_{n}{x}_{n-k}-\sum _{j=n}^{\mathrm{\infty }}\frac{1}{{r}_{j}}\sum _{i=j}^{\mathrm{\infty }}{a}_{i}f\left({x}_{i}\right)$
(12)

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

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

$|{\left(T{x}^{\left(p\right)}\right)}_{n}-{\left(Tx\right)}_{n}|\le |{p}_{n}||{x}_{n-k}^{\left(p\right)}-{x}_{n-k}|+\sum _{j=n}^{\mathrm{\infty }}|\frac{1}{{r}_{j}}|\sum _{i=j}^{\mathrm{\infty }}|{a}_{i}||f\left({x}_{i}^{\left(p\right)}\right)-f\left({x}_{i}\right)|.$

Hence, by (7) and (8),

$|{\left(T{x}^{\left(p\right)}\right)}_{n}-{\left(Tx\right)}_{n}|\le P|{x}_{n-k}^{\left(p\right)}-{x}_{n-k}|+{\alpha }_{n}\underset{i\ge n}{sup}|f\left({x}_{i}^{\left(p\right)}\right)-f\left({x}_{i}\right)|,\phantom{\rule{1em}{0ex}}n\in {\mathbb{N}}_{{n}_{3}}.$

Therefore, by (10),

$\parallel T{x}^{\left(p\right)}-Tx\parallel \le P\parallel {x}^{\left(p\right)}-x\parallel +C\frac{1-P}{2\left(CM+L\right)}\parallel f\left({x}_{i}^{\left(p\right)}\right)-f\left({x}_{i}\right)\parallel \to 0$

and

$\underset{p\to \mathrm{\infty }}{lim}\parallel T{x}^{\left(p\right)}-Tx\parallel =0.$

This means that T is continuous.

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

$|{\left(Tx\right)}_{n}-{\left(Ty\right)}_{n}|\le P|{x}_{n}-{y}_{n}|+CM{\alpha }_{n},\phantom{\rule{1em}{0ex}}n\in {\mathbb{N}}_{{n}_{3}}.$

Hence, we obtain

$diam{\left(T\left(X\right)\right)}_{n}\le kdiam{X}_{n}+CM{\alpha }_{n}.$

This yields

$\underset{n\to \mathrm{\infty }}{lim sup}diam{\left(T\left(X\right)\right)}_{n}\le k\underset{n\to \mathrm{\infty }}{lim sup}diam{X}_{n}.$

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

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 ${x}_{n}={\left(Tx\right)}_{n}$. Thus

${x}_{n}=-{p}_{n}{x}_{n-k}+\sum _{j=n}^{\mathrm{\infty }}\frac{1}{{r}_{j}}\sum _{i=j}^{\mathrm{\infty }}{a}_{i}f\left({x}_{i}\right),\phantom{\rule{1em}{0ex}}n\in {\mathbb{N}}_{{n}_{3}}$
(13)

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:

${x}_{n}+{p}_{n}{x}_{n-k}=\sum _{j=n}^{\mathrm{\infty }}\frac{1}{{r}_{j}}\sum _{i=j}^{\mathrm{\infty }}{a}_{i}f\left({x}_{i}\right),$

which is obtained from (13). We find that

$\mathrm{\Delta }\left({x}_{n}+{p}_{n}{x}_{n-k}\right)=-\frac{1}{{r}_{n}}\sum _{i=n}^{\mathrm{\infty }}{a}_{i}f\left({x}_{i}\right),\phantom{\rule{1em}{0ex}}n\in {\mathbb{N}}_{{n}_{3}}.$

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

$\mathrm{\Delta }\left({\left(-1\right)}^{n}\mathrm{\Delta }\left({x}_{n}+\left(\frac{1}{2}+\frac{1}{{2}^{n}}\right){x}_{n-2}\right)\right)+\frac{3{\left(-1\right)}^{n+1}}{{2}^{n+2}}{\left({x}_{n}\right)}^{\frac{1}{3}}=0.$

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

Remark 1

Assume that

${p}_{n}\equiv p\in \left(0,1\right)$
(14)

and

$\sum _{n=0}^{\mathrm{\infty }}\sum _{i=n}^{\mathrm{\infty }}|{a}_{i}|<\mathrm{\infty }$
(15)

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 $f\left(x\right)={x}^{\gamma }$ is a continuous function, and $|f\left(x\right)|=|{x}^{\gamma }|\le \gamma |x|+1-\gamma$. 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 $\left[0,\mathrm{\infty }\right)$. The function x is an asymptotically stable solution of the equation

$x=Fx.$
(16)

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:

$|x\left(t\right)-y\left(t\right)|\le \epsilon .$

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

$|f\left(x\right)-f\left(y\right)|\le D|x-y|$
(17)

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

${x}_{n}={\left(Tx\right)}_{n},$
(18)

where a mapping T is defined by (12).

Because of Definition 2, the sequence x is an asymptotically stable solution of the equation ${x}_{n}={\left(Tx\right)}_{n}$, 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:

$|{x}_{n}-{y}_{n}|\le \epsilon .$
(19)

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

$|{\left(Tx\right)}_{n}-{\left(Ty\right)}_{n}|\le P|{x}_{n-k}-{y}_{n-k}|+\sum _{j=n}^{\mathrm{\infty }}|\frac{1}{{r}_{j}}|\sum _{i=j}^{\mathrm{\infty }}|{a}_{i}||f\left({x}_{i}\right)-f\left({y}_{i}\right)|$

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

$|{\left(Tx\right)}_{n}-{\left(Ty\right)}_{n}|\le P|{x}_{n-k}-{y}_{n-k}|+D\sum _{j=n}^{\mathrm{\infty }}|\frac{1}{{r}_{j}}|\sum _{i=j}^{\mathrm{\infty }}|{a}_{i}||{x}_{i}-{y}_{i}|$

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

$|{\left(Tx\right)}_{n}-{\left(Ty\right)}_{n}|\le P|{x}_{n-k}-{y}_{n-k}|+D\underset{i\ge n}{sup}|{x}_{i}-{y}_{i}|{\alpha }_{n}$

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

$|{x}_{n}-{y}_{n}|\le P|{x}_{n-k}-{y}_{n-k}|+D\underset{i\ge n}{sup}|{x}_{i}-{y}_{i}|{\alpha }_{n}.$
(20)

Let us denote

$\underset{n\to \mathrm{\infty }}{lim sup}|{x}_{n}-{y}_{n}|=l.$

Because of

$\underset{n\to \mathrm{\infty }}{lim sup}|{x}_{n}-{y}_{n}|=\underset{n\to \mathrm{\infty }}{lim sup}|{x}_{n-k}-{y}_{n-k}|,$

and (20), we get

$l\left(1-P-D\underset{n\to \mathrm{\infty }}{lim}{\alpha }_{n}\right)\le 0.$

From the above and (9), we obtain

Suppose to the contrary that $l>0$. Thus, we obtain a contradiction with the fact that $0. Therefore we get ${lim sup}_{n\to \mathrm{\infty }}|{x}_{n}-{y}_{n}|=0$. 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 $f\left(x\right)=x$, ${a}_{n}={\mathrm{\Delta }}^{2}{p}_{n}$ 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 $|f\left(x\right)|\le M|x|$. This implies that $f\left(0\right)=0$. Hence, the sequence $x\equiv 0$ is a bounded solution of equation (1). By Remark 2, the zero solution is asymptotically stable. □