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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 Equations volume 2013, Article number: 91 (2013)
The purpose of this paper is to investigate a nonlinear second-order neutral difference equation of the form
where , , , 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.
In presented paper we study a nonlinear second-order difference equation of the form
where , , , and is a continuous function. Here , , 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 which satisfies (1) for every .
Putting , where is a quotient of two odd integers, and , 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 , Migda , Migda and Migda , Migda et al. , Musielak and Popenda , Popenda and Werbowski , Schmeidel , Schmeidel and Zba̧szyniak  and Thandapani 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 be an infinite-dimensional Banach space. If X is a subset of E, then , ConvX denote the closure and the convex closure of X, respectively. Moreover, we denote by the family of all nonempty and bounded subsets of E and by the subfamily consisting of all relatively compact sets.
Definition 1 A mapping is called a measure of noncompactness in E if it satisfies the following conditions:
1∘ and ,
4∘ for ,
5∘ if , , for and , then .
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 be a continuous operator such that for all nonempty subset X of M, where is a constant. Then T has a fixed point in the subset M.
We consider the Banach space of all real bounded sequences equipped with the standard supremum norm, i.e.,
Let X be a nonempty, bounded subset of , (it means 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 (see ):
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 ,
the sequence satisfies the following condition:
sequences , are such that
Then there exists a bounded solution of equation (1).
Proof Condition (5) implies that there exist and a constant 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 the remainder of series so that
From (6), the remainder 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 such that
We define a set B as follows:
where and .
It is not difficult to prove that B is a nonempty, bounded, convex, and closed subset .
Let us define a mapping as follows:
for any .
We will prove that the mapping T has a fixed point in B.
Firstly, we show that . Indeed, if , then by (12), (7), (11), and (10), we have
Next, we prove that T is continuous. Let be a sequence in B such that as . Because of (3), we have . Since B is closed, . Now, utilizing (12), we get
Hence, by (7) and (8),
Therefore, by (10),
This means that T is continuous.
Now, we need to compare a measure of noncompactness of any subset X of B and . Let us take a nonempty set . For any sequences , we get
Hence, we obtain
From the above, for any , we have , where .
By virtue of Theorem 1, we conclude that T has a fixed point in the set B. It means that there exists such that . Thus
for any . To show that there exists a connection between the fixed point 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 . The sequence x, which is a fixed point of the mapping T, is a bounded sequence which fulfills equation (1) for large n. If , the proof is ended. If , then we find previous 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 .
This completes the proof. □
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 is such a solution.
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 given by formula is a continuous function, and . So, taking and , we obtain condition (4). The thesis follows directly from Theorem 2. □
Definition 2 Let x be a real function defined, bounded, and continuous on . The function x is an asymptotically stable solution of the equation
It means that for any , there exists such that for every 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 , and conditions (3)-(6) hold. Then equation (1) has at least one asymptotically stable solution .
Proof From Theorem 2, equation (1) has at least one bounded solution 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 , which means that for any , there exists such that for every and for every other solution y of equation (1), the following inequality holds:
From (12), by (7), we have
for . The above and (17) yield
for . Hence, by (8) and (19), we obtain
for . Thus, linking the above inequality and (18), we have
Let us denote
and (20), we get
From the above and (9), we obtain
Suppose to the contrary that . Thus, we obtain a contradiction with the fact that . Therefore we get . 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 , and . Such an equation has infinitely many solutions of the form , 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 in (4). Under conditions (3)-(6) and (17), if there exists a zero solution of equation (1), then it is asymptotically stable.
Proof If , then condition (4) takes the form . This implies that . Hence, the sequence is a bounded solution of equation (1). By Remark 2, the zero solution is asymptotically stable. □
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The author would like to thank the reviewers for their helpful comments and valuable suggestions.
The author declares that they have no competing interests.