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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 Equations20132013:91

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

  • Received: 24 August 2012
  • Accepted: 14 March 2013
  • Published:

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.

Keywords

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

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 [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 , ) 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 ker μ = { X M E : μ ( X ) = 0 } and ker μ N E ,

2 X Y μ ( X ) μ ( Y ) ,

3 μ ( X ¯ ) = μ ( X ) = μ ( Conv X ) ,

4 μ ( α X + ( 1 α ) Y ) α μ ( 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 [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 M be a continuous operator such that μ ( T ( X ) ) k μ ( 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.,
x = sup n N 0 | x n | for  x l .
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 [26]):
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
| p n | P < 1 for  n 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 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:
B : = { ( x n ) n = 0 : | x n | C  for  n N n 3 } ,
(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
| ( T x ) n | | p n | | x n k | + j = n | 1 r j | i = j | a i | | f ( x i ) | P C + j = n | 1 r j | i = j | a i | ( M | x i | + L ) C P + ( M C + L ) j = n | 1 r j | i = j | a i | C P + ( C M + L ) α n = C P + 1 2 C for  n N n 3 .
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 | + C M α n , n N n 3 .
Hence, we obtain
diam ( T ( X ) ) n k diam X n + C M 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 ) ) k μ ( 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
x n k + l = 1 p n + l ( x n + l + j = n + l 1 r j i = j a i f ( x i ) ) , where  l { 0 , 1 , 2 , , k 1 } ,

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 [27].

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 = F x .
(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
l ( 1 P ) 0 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 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. □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Institute of Mathematics, University of Białystok, Białystok, Poland

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© Schmeidel; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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