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

Abstract

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

$$\mathrm{\Delta }(r_n\mathrm{\Delta }(x_n+p_n{x}_{n-k}))+a_{n}f(x_n)=0,$$

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

$$\mathrm{\Delta }(r_n\mathrm{\Delta }(x_n+p_n{x}_{n-k}))+a_{n}f(x_n)=0,$$

(1)

where $x:{\mathbb{N}}_0\to \mathbb{R}$, $a:{\mathbb{N}}_0\to \mathbb{R}$, $p,r:{\mathbb{N}}_0\to \mathbb{R}\setminus \{0\}$, 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

$${\mathrm{\Delta }}^2(x_n+p{x}_{n-k})+a_n{x}_n^{\gamma }=0.$$

(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 $(E,\parallel \cdot \parallel )$ 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 $\mu :{\mathcal{M}}_E\to [0,\mathrm{\infty })$ 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.,

$$\parallel x\parallel =\underset{n\in {\mathbb{N}}_0}{sup}|x_n|\text{for }x\in l^{\mathrm{\infty }}.$$

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

$$diam{X}_n=sup\{|x_n-y_n|:x,y\in X\}.$$

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

$$\mu (X)=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}diam{X}_n.$$

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 $x\in \mathbb{R}$,

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

(4)

the sequence $p:{\mathbb{N}}_0\to \mathbb{R}\setminus \{0\}$ satisfies the following condition:

$$-1<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{p}_n\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{p}_n<1,$$

(5)

sequences $a:{\mathbb{N}}_0\to \mathbb{R}$, $r:{\mathbb{N}}_0\to \mathbb{R}\setminus \{0\}$ are such that

$$\sum _{n=0}^{\mathrm{\infty }}\left|\frac{1}{r_n}\right|\sum _{i=n}^{\mathrm{\infty }}|a_i|<\mathrm{\infty }.$$

(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 $P\in [0,1)$ such that

$$|p_n|\le P<1\text{for }n\ge n_1.$$

(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

$${\alpha }_n=\sum _{j=n}^{\mathrm{\infty }}\left|\frac{1}{r_j}\right|\sum _{i=j}^{\mathrm{\infty }}|a_i|.$$

(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

$${\alpha }_n\le C\frac{1-P}{2(CM+L)}$$

(10)

for $n\ge n_2$.

We define a set B as follows:

$$B:=\{{(x_n)}_{n=0}^{\mathrm{\infty }}:|x_n|\le C\text{ for }n\in {\mathbb{N}}_{n_3}\},$$

(11)

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:

$${(Tx)}_n=-p_n{x}_{n-k}-\sum _{j=n}^{\mathrm{\infty }}\frac{1}{r_j}\sum _{i=j}^{\mathrm{\infty }}{a}_{i}f(x_i)$$

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

$$\begin{array}{rcl}\left|{(Tx)}_n\right|& \le & |p_n||x_{n-k}|+\sum _{j=n}^{\mathrm{\infty }}\left|\frac{1}{r_j}\right|\sum _{i=j}^{\mathrm{\infty }}|a_i||f(x_i)|\\ \le & PC+\sum _{j=n}^{\mathrm{\infty }}\left|\frac{1}{r_j}\right|\sum _{i=j}^{\mathrm{\infty }}|a_i|(M|x_i|+L)\\ \le & CP+(MC+L)\sum _{j=n}^{\mathrm{\infty }}\left|\frac{1}{r_j}\right|\sum _{i=j}^{\mathrm{\infty }}|a_i|\\ \le & CP+(CM+L){\alpha }_n=C\frac{P+1}{2}\le C\text{for }n\in {\mathbb{N}}_{n_3}.\end{array}$$

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

$$|{\left(T{x}^{(p)}\right)}_n-{(Tx)}_n|\le |p_n||x_{n-k}^{(p)}-x_{n-k}|+\sum _{j=n}^{\mathrm{\infty }}\left|\frac{1}{r_j}\right|\sum _{i=j}^{\mathrm{\infty }}|a_i||f\left(x_i^{(p)}\right)-f(x_i)|.$$

Hence, by (7) and (8),

$$|{\left(T{x}^{(p)}\right)}_n-{(Tx)}_n|\le P|x_{n-k}^{(p)}-x_{n-k}|+{\alpha }_n\underset{i\ge n}{sup}|f\left(x_i^{(p)}\right)-f(x_i)|,n\in {\mathbb{N}}_{n_3}.$$

Therefore, by (10),

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

and

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

This means that T is continuous.

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

$$|{(Tx)}_n-{(Ty)}_n|\le P|x_n-y_n|+CM{\alpha }_n,n\in {\mathbb{N}}_{n_3}.$$

Hence, we obtain

$$diam{(T(X))}_n\le kdiam{X}_n+CM{\alpha }_n.$$

This yields

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}diam{(T(X))}_n\le k\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}diam{X}_n.$$

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

$$x_n=-p_n{x}_{n-k}+\sum _{j=n}^{\mathrm{\infty }}\frac{1}{r_j}\sum _{i=j}^{\mathrm{\infty }}{a}_{i}f(x_i),n\in {\mathbb{N}}_{n_3}$$

(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:

$$x_n+p_n{x}_{n-k}=\sum _{j=n}^{\mathrm{\infty }}\frac{1}{r_j}\sum _{i=j}^{\mathrm{\infty }}{a}_{i}f(x_i),$$

which is obtained from (13). We find that

$$\mathrm{\Delta }(x_n+p_n{x}_{n-k})=-\frac{1}{r_n}\sum _{i=n}^{\mathrm{\infty }}{a}_{i}f(x_i),n\in {\mathbb{N}}_{n_3}.$$

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

$$x_{n-k+l}=\frac{1}{p_{n+l}}(-x_{n+l}+\sum _{j=n+l}^{\mathrm{\infty }}\frac{1}{r_j}\sum _{i=j}^{\mathrm{\infty }}{a}_{i}f(x_i)),\text{where }l\in \{0,1,2,\dots ,k-1\},$$

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

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

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

Remark 1

Assume that

$$p_n\equiv p\in (0,1)$$

(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 $f(x)=x^{\gamma }$ 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

$$x=Fx.$$

(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:

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

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

$$|f(x)-f(y)|\le D|x-y|$$

(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

$$x_n={(Tx)}_n,$$

(18)

where a mapping T is defined by (12).

Because of Definition 2, the sequence x is an asymptotically stable solution of the equation $x_n={(Tx)}_n$, 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:

$$|x_n-y_n|\le \epsilon .$$

(19)

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

$$|{(Tx)}_n-{(Ty)}_n|\le P|x_{n-k}-y_{n-k}|+\sum _{j=n}^{\mathrm{\infty }}\left|\frac{1}{r_j}\right|\sum _{i=j}^{\mathrm{\infty }}|a_i||f(x_i)-f(y_i)|$$

for $n\ge n_3$. The above and (17) yield

$$|{(Tx)}_n-{(Ty)}_n|\le P|x_{n-k}-y_{n-k}|+D\sum _{j=n}^{\mathrm{\infty }}\left|\frac{1}{r_j}\right|\sum _{i=j}^{\mathrm{\infty }}|a_i||x_i-y_i|$$

for $n\ge n_5=max\{n_3,n_4\}$. Hence, by (8) and (19), we obtain

$$|{(Tx)}_n-{(Ty)}_n|\le P|x_{n-k}-y_{n-k}|+D\underset{i\ge n}{sup}|x_i-y_i|{\alpha }_n$$

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

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}|x_n-y_n|=l.$$

Because of

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}|x_n-y_n|=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}|x_{n-k}-y_{n-k}|,$$

and (20), we get

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

From the above and (9), we obtain

$$l(1-P)\le 0\text{for enough large }n.$$

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. □