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

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## Abstract

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

$Δ ( r n Δ ( x n + p n x n − k ) ) + a n f( x n )=0,$

where $x: N 0 →R$, $a: N 0 →R$, $p,r: N 0 →R∖{0}$, $f:R→R$ 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

$Δ ( r n Δ ( x n + p n x n − k ) ) + a n f( x n )=0,$
(1)

where $x: N 0 →R$, $a: N 0 →R$, $p,r: N 0 →R∖{0}$, and $f:R→R$ is a continuous function. Here $N 0 :={0,1,2,…}$, $N k :={k,k+1,k+2,…}$, where k is a given positive integer and is a set of all real numbers. By a solution of equation (1), we mean a sequence $x: N 0 →R$ which satisfies (1) for every $n∈ N 0$.

Putting $f(x)= x γ$, where $γ<1$ is a quotient of two odd integers, $r n ≡1$ and $p n ≡p∈(0,∞)$, $p≠1$ in equation (1), we get an Emden-Fowler difference equation of the form

$Δ 2 ( x n +p x n − k )+ a n x n γ =0.$
(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 , Migda , Migda and Migda , Migda et al. , Musielak and Popenda , Popenda and Werbowski , Schmeidel , Schmeidel and Zba̧szyniak  and Thandapani et al. ).

Neutral difference equations were studied in many other papers by Grace and Lalli  and , Lalli and Zhang , Migda and Migda , Luo and Bainov , and Luo and Yu .

Some relevant results related to this topic can be found in papers by Baštinec et al. , Baštinec et al. , Berezansky et al. , Diblík and Hlavičková , and Diblík et al. .

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  as well as in those by Elaydi , Kocić and Ladas , or Kelley and Peterson .

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

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

Let $(E,∥⋅∥)$ be an infinite-dimensional Banach space. If X is a subset of E, then $X ¯$, ConvX denote the closure and the convex closure of X, respectively. Moreover, we denote by $M E$ the family of all nonempty and bounded subsets of E and by $N E$ the subfamily consisting of all relatively compact sets.

Definition 1 A mapping $μ: M E →[0,∞)$ is called a measure of noncompactness in E if it satisfies the following conditions:

1 $kerμ={X∈ M E :μ(X)=0}≠∅$ and $kerμ⊂ N E$,

2 $X⊂Y⇒μ(X)≤μ(Y)$,

3 $μ( X ¯ )=μ(X)=μ(ConvX)$,

4 $μ(αX+(1−α)Y)≤αμ(X)+(1−α)μ(Y)$ for $0≤α≤1$,

5 if $X n ∈ M E$, $X n + 1 ⊂ X n$, $X n = X ¯ n$ for $n=1,2,3,…$ and $lim n → ∞ μ( X n )=0$, then $⋂ n = 1 ∞ X n ≠∅$.

The following Darbo’s fixed point theorem given in  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 $T:M→M$ be a continuous operator such that $μ(T(X))≤kμ(X)$ for all nonempty subset X of M, where $k∈[0,1)$ is a constant. Then T has a fixed point in the subset M.

We consider the Banach space $l ∞$ of all real bounded sequences $x: N 0 →R$ equipped with the standard supremum norm, i.e.,

Let X be a nonempty, bounded subset of $l ∞$, $X n ={ x n :x∈X}$ (it means $X n$ is a set of n th terms of any sequence belonging to X), and let

$diam X n =sup { | x n − y n | : x , y ∈ X } .$

We use the following measure of noncompactness in the space $l ∞$ (see ):

$μ(X)= lim sup n → ∞ diam X n .$

## 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

$f:R→R be a continuous function ,$
(3)

and let there exist constants L and M such that for all $x∈R$,

$|f(x)|≤M|x|+L,$
(4)

the sequence $p: N 0 →R∖{0}$ satisfies the following condition:

$−1< lim inf n → ∞ p n ≤ lim sup n → ∞ p n <1,$
(5)

sequences $a: N 0 →R$, $r: N 0 →R∖{0}$ are such that

$∑ n = 0 ∞ | 1 r n | ∑ i = n ∞ | a i |<∞.$
(6)

Then there exists a bounded solution $x: N 0 →R$ of equation (1).

Proof Condition (5) implies that there exist $n 1 ∈ N 0$ and a constant $P∈[0,1)$ 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 $α n$ the remainder of series $∑ n = 0 ∞ | 1 r n | ∑ i = n ∞ | a i |$ so that

$α n = ∑ j = n ∞ | 1 r j | ∑ i = j ∞ | a i |.$
(8)

From (6), the remainder $α n$ tends to zero. Therefore, we can denote

$lim n → ∞ α n =0.$
(9)

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

$α n ≤C 1 − P 2 ( C M + L )$
(10)

for $n≥ n 2$.

We define a set B as follows:

(11)

where $N n 3 :={ n 3 , n 3 +1, n 3 +2,…}$ and $n 3 =max{ n 1 , n 2 }$.

It is not difficult to prove that B is a nonempty, bounded, convex, and closed subset $l ∞$.

Let us define a mapping $T:B→ l ∞$ as follows:

$( T x ) n =− p n x n − k − ∑ j = n ∞ 1 r j ∑ i = j ∞ a i f( x i )$
(12)

for any $n∈ N n 3$.

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

Firstly, we show that $T(B)⊂B$. Indeed, if $x∈B$, 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 $∥ x ( p ) −x∥→0$ as $p→∞$. Because of (3), we have $∥f( x ( p ) )−f(x)∥→0$. Since B is closed, $x∈B$. Now, utilizing (12), we get

$| ( T x ( p ) ) n − ( T x ) n |≤| p n || x n − k ( p ) − x n − k |+ ∑ j = n ∞ | 1 r j | ∑ i = j ∞ | a i ||f ( x i ( p ) ) −f( x i )|.$

Hence, by (7) and (8),

$| ( T x ( p ) ) n − ( T x ) n |≤P| x n − k ( p ) − x n − k |+ α n sup i ≥ n |f ( x i ( p ) ) −f( x i )|,n∈ N n 3 .$

Therefore, by (10),

$∥ T x ( p ) − T x ∥ ≤P ∥ x ( p ) − x ∥ +C 1 − P 2 ( C M + L ) ∥ f ( x i ( p ) ) − f ( x i ) ∥ →0$

and

$lim p → ∞ ∥ T x ( p ) − T x ∥ =0.$

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 $X⊂B$. For any sequences $x,y∈X$, we get

$| ( T x ) n − ( T y ) n |≤P| x n − y n |+CM α n ,n∈ N n 3 .$

Hence, we obtain

$diam ( T ( X ) ) n ≤kdiam X n +CM α n .$

This yields

$lim sup n → ∞ diam ( T ( X ) ) n ≤k lim sup n → ∞ diam X n .$

From the above, for any $X⊂B$, we have $μ(T(X))≤kμ(X)$, where $k= P + 1 2 ∈[0,1)$.

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

$x n =− p n x n − k + ∑ j = n ∞ 1 r j ∑ i = j ∞ a i f( x i ),n∈ N n 3$
(13)

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

$x n + p n x n − k = ∑ j = n ∞ 1 r j ∑ i = j ∞ a i f( x i ),$

which is obtained from (13). We find that

$Δ( x n + p n x n − k )=− 1 r n ∑ i = n ∞ a i f( x i ),n∈ N n 3 .$

Using again the operator Δ for both sides of the above equation, we get equation (1) for $n∈ N n 3$. The sequence x, which is a fixed point of the mapping T, is a bounded sequence which fulfills equation (1) for large n. If $n 3 ≤k$, 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 $x: N 0 →R$.

This completes the proof. □

Example 1

Let us consider the equation

$Δ ( ( − 1 ) n Δ ( x n + ( 1 2 + 1 2 n ) x n − 2 ) ) + 3 ( − 1 ) n + 1 2 n + 2 ( x n ) 1 3 =0.$

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

$p n ≡p∈(0,1)$
(14)

and

$∑ n = 0 ∞ ∑ i = n ∞ | a i |<∞$
(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 $f:R→R$ given by formula $f(x)= x γ$ is a continuous function, and $|f(x)|=| x γ |≤γ|x|+1−γ$. So, taking $M=γ$ and $L=1−γ$, 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 .

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

$x=Fx.$
(16)

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

$|x(t)−y(t)|≤ε.$

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

$|f(x)−f(y)|≤D|x−y|$
(17)

for any $x,y∈R$, and conditions (3)-(6) hold. Then equation (1) has at least one asymptotically stable solution $x: N 0 →R$.

Proof From Theorem 2, equation (1) has at least one bounded solution $x: N 0 →R$ which can be rewritten in the form

$x n = ( T x ) n ,$
(18)

where a mapping T is defined by (12).

Because of Definition 2, the sequence x is an asymptotically stable solution of the equation $x n = ( T x ) n$, which means that for any $ε>0$, there exists $n 4 ∈ N 0$ such that for every $n≥ n 4$ and for every other solution y of equation (1), the following inequality holds:

$| x n − y n |≤ε.$
(19)

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

$| ( T x ) n − ( T y ) n |≤P| x n − k − y n − k |+ ∑ j = n ∞ | 1 r j | ∑ i = j ∞ | a i ||f( x i )−f( y i )|$

for $n≥ n 3$. The above and (17) yield

$| ( T x ) n − ( T y ) n |≤P| x n − k − y n − k |+D ∑ j = n ∞ | 1 r j | ∑ i = j ∞ | a i || x i − y i |$

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

$| ( T x ) n − ( T y ) n |≤P| x n − k − y n − k |+D sup i ≥ n | x i − y i | α n$

for $n≥ n 5$. Thus, linking the above inequality and (18), we have

$| x n − y n |≤P| x n − k − y n − k |+D sup i ≥ n | x i − y i | α n .$
(20)

Let us denote

$lim sup n → ∞ | x n − y n |=l.$

Because of

$lim sup n → ∞ | x n − y n |= lim sup n → ∞ | x n − k − y n − k |,$

and (20), we get

$l ( 1 − P − D lim n → ∞ α n ) ≤0.$

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 $lim sup n → ∞ | x n − y n |=0$. 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 $f(x)=x$, $a n = Δ 2 p n$ and $∑ n = 0 ∞ ∑ i = n ∞ | a i |<∞$. Such an equation has infinitely many solutions of the form $x n ≡c$, 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 $|f(x)|≤M|x|$. This implies that $f(0)=0$. Hence, the sequence $x≡0$ 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.

## Author information

Correspondence to Ewa Schmeidel.

### Competing interests

The author declares that they have no competing interests.

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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) doi:10.1186/1687-1847-2013-91

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### Keywords

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