- Open Access
An application of measures of noncompactness in the investigation of boundedness of solutions of second-order neutral difference equations
© Schmeidel; licensee Springer. 2013
- Received: 24 August 2012
- Accepted: 14 March 2013
- Published: 4 April 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.
- difference equation
- measures of noncompactness
- Darbo’s fixed point theorem
- Emden-Fowler equation
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 .
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.
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.
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.
Then there exists a bounded solution of equation (1).
where and .
It is not difficult to prove that B is a nonempty, bounded, convex, and closed subset .
for any .
We will prove that the mapping T has a fixed point in B.
This means that T is continuous.
From the above, for any , we have , where .
the results of which follow directly from (1). It means that equation (1) has at least one bounded solution .
This completes the proof. □
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. □
for any , and conditions (3)-(6) hold. Then equation (1) has at least one asymptotically stable solution .
where a mapping T is defined by (12).
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. □
The author would like to thank the reviewers for their helpful comments and valuable suggestions.
- Medina R, Pinto M: Asymptotic behavior of solutions of second order nonlinear difference equations. Nonlinear Anal. 1992, 19: 187–195. 10.1016/0362-546X(92)90119-YMathSciNetView ArticleGoogle Scholar
- Migda M: Asymptotic behavior of solutions of nonlinear delay difference equations. Fasc. Math. 2001, 31: 57–62.MathSciNetGoogle Scholar
- Migda J, Migda M: Asymptotic properties of the solutions of second order difference equation. Arch. Math. 1998, 34: 467–476.MathSciNetGoogle Scholar
- Migda M, Schmeidel E, Zba̧szyniak M: On the existence of solutions of some second order nonlinear difference equations. Arch. Math. 2005, 42: 379–388.MathSciNetGoogle Scholar
- Musielak R, Popenda J: The periodic solutions of the second order nonlinear difference equation. Publ. Mat. 1988, 32: 49–56.MathSciNetView ArticleGoogle Scholar
- Popenda J, Werbowski J: On the asymptotic behavior of the solutions of difference equations of second order. Ann. Pol. Math. 1980, 22: 135–142.MathSciNetGoogle Scholar
- Schmeidel E: Asymptotic behaviour of solutions of the second order difference equations. Demonstr. Math. 1993, 25: 811–819.MathSciNetGoogle Scholar
- Schmeidel E, Zba̧szyniak Z: An application of Darbo’s fixed point theorem in investigation of periodicity of solutions of difference equations. Comput. Math. Appl. 2012. doi:10.1016/j.camwa.2011.12.025Google Scholar
- Thandapani E, Arul R, Graef JR, Spikes PW: Asymptotic behavior of solutions of second order difference equations with summable coefficients. Bull. Inst. Math. Acad. Sin. 1999, 27: 1–22.MathSciNetGoogle Scholar
- Grace SR, Lalli BS: Oscillatory and asymptotic behavior of solutions of nonlinear neutral-type difference equations. J. Aust. Math. Soc. Ser. B 1996, 38: 163–171. 10.1017/S0334270000000552View ArticleGoogle Scholar
- Lalli BS, Grace SR: Oscillation theorems for second order neutral difference equations. Appl. Math. Comput. 1994, 62: 47–60. 10.1016/0096-3003(94)90132-5MathSciNetView ArticleGoogle Scholar
- Lalli BS, Zhang BG: On existence of positive solutions and bounded oscillations for neutral difference equations. J. Math. Anal. Appl. 1992, 166: 272–287. 10.1016/0022-247X(92)90342-BMathSciNetView ArticleGoogle Scholar
- Migda J, Migda M: Asymptotic properties of solutions of second-order neutral difference equations. Nonlinear Anal. 2005, 63: e789-e799. 10.1016/j.na.2005.02.005View ArticleGoogle Scholar
- Luo JW, Bainov DD: Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima. J. Comput. Appl. Math. 2001, 131: 333–341. 10.1016/S0377-0427(00)00264-8MathSciNetView ArticleGoogle Scholar
- Luo J, Yu Y: Asymptotic behavior of solutions of second order neutral difference equations with ‘maxima’. Demonstr. Math. 2001, 35: 83–89.MathSciNetGoogle Scholar
- Baštinec J, Berezansky L, Diblík J, Šmarda Z:A final result on the oscillation of solutions of the linear discrete delayed equation with a positive coefficient. Abstr. Appl. Anal. 2011., 2011: Article ID 586328Google Scholar
- Baštinec J, Diblík J, Šmarda Z: Existence of positive solutions of discrete linear equations with a single delay. J. Differ. Equ. Appl. 2010, 16: 1047–1056. 10.1080/10236190902718026View ArticleGoogle Scholar
- Berezansky L, Diblík J, Růžičková M, Šutá Z: Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case. Abstr. Appl. Anal. 2011., 2011: Article ID 709427Google Scholar
- Diblík J, Hlavičková I: Asymptotic behavior of solutions of delayed difference equations. Abstr. Appl. Anal. 2011., 2011: Article ID 671967Google Scholar
- Diblík J, Růžičková M, Šutá Z: Asymptotic convergence of the solutions of a discrete equation with several delays. Appl. Math. Comput. 2012, 218: 5391–5401. 10.1016/j.amc.2011.11.023MathSciNetView ArticleGoogle Scholar
- Agarwal RP Monographs and Textbooks in Pure and Applied Mathematics 228. In Difference Equations and Inequalities. Theory, Methods, and Applications. 2nd edition. Dekker, New York; 2000.Google Scholar
- Elaydi SN Undergraduate Texts in Mathematics. In An Introduction to Difference Equations. 3rd edition. Springer, New York; 2005.Google Scholar
- Kocić VL, Ladas G Mathematics and Its Applications 256. In Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht; 1993.Google Scholar
- Kelley WG, Peterson AC: Difference Equations: An Introduction with Applications. Academic Press, San Diego; 2001.Google Scholar
- Akhmerov RR, Kamenskij MI, Potapov AS, Rodkina AS, Sadovskij BN Operator Theory: Advances and Applications 55. In Measures of Noncompactness and Condensing Operators. Birkhäuser, Basel; 1992. Translated from the Russian by A. IacobView ArticleGoogle Scholar
- Banaś J, Goebel K Lecture Notes in Pure and Applied Mathematics 60. In Measures of Noncompactness in Banach Spaces. Dekker, New York; 1980.Google Scholar
- Banaś J, Rzepka B: An application of measure of noncompactness in study of asymptotic stability. Appl. Math. Lett. 2003, 16: 1–6. 10.1016/S0893-9659(02)00136-2MathSciNetView ArticleGoogle Scholar
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.