Boundedness and stability of discrete Volterra equations
- Małgorzata Migda^{1}Email author,
- Miroslava Růžičková^{2} and
- Ewa Schmeidel^{3}
https://doi.org/10.1186/s13662-015-0361-6
© Migda et al.; licensee Springer. 2015
Received: 16 October 2014
Accepted: 6 January 2015
Published: 20 February 2015
Abstract
Keywords
Volterra difference equation stability bounded solution periodic solution contraction mappingMSC
39A10 39A22 39A301 Introduction
Let ℕ, ℤ, ℝ denote the set of nonnegative integers, the set of all integers and the set of real numbers, respectively, and \(\mathbb{N}_{0}=\{n_{0},n_{0} +1,\dots\}\), where \(n_{0}\in\mathbb{N}\).
We introduce some notation and definitions that will be used in the paper. Hereafter, we denote the solution of equation (1) with the initial condition \(x(n_{0})=x_{0}\) by \(x(n,n_{0},x_{0})\).
Definition 1
Solutions of equation (1) are equi-bounded if for each constant \(M_{1} > 0\), there is \(M_{2} > 0\) such that \(\vert x_{0}\vert \leq M_{1}\) and \(n\geq n_{0}\) implies that \(\vert x(n, n_{0}, x_{0})\vert \leq M_{2}\).
Definition 2
The zero solution of equation (2) is stable if for every \(\varepsilon>0\) there exists \(\delta>0\) such that \(\vert x_{0}\vert \leq \delta\) implies \(\vert x(n, n_{0}, x_{0})\vert \leq\varepsilon\) for \(n\geq n_{0}\).
Definition 3
The zero solution of equation (2) is asymptotically stable if there exists \(\mu>0\) such that \(\vert x_{0}\vert \leq\mu\) implies \(\lim_{n\rightarrow\infty} x(n)=0\).
Denote the Banach space of all bounded real sequences \(x\colon\mathbb {N}_{0}\to\mathbb{R}\) equipped with the supremum norm \(\Vert x \Vert =\sup_{n\geq n_{0}}\vert x(n)\vert \) by BS.
Boundedness of solutions of linear and nonlinear discrete Volterra equations was also studied by Crisci et al. [7], Diblík and Schmeidel [8], Gronek and Schmeidel [9], Győri and Awwad [10], Győri and Horváth [11], Kolmanovskii and Shaikhet [1], Medina [12], Migda and Migda [13] or Migda and Morchało [14]. A survey of the fundamental results on the stability of linear Volterra difference equations, of both convolution and non-convolution type, can be found in Elaydi [15], see also Crisci et al. [16] and [17]. The problem of finding periodic and asymptotically periodic solutions of linear discrete Volterra equations of type (1) was investigated, for example, by Diblik et al. [8, 18–21], Elaydi [22], Gajda et al. [23], Győri and Reynolds [24], Migda and Migda [13] or Song and Baker [25].
Motivated by the results obtained in the papers by Islam and Yankson [26] and Raffoul [27], in this paper, we derive explicit sufficient conditions for the equi-boundedness of solutions of equation (1) and the asymptotic stability of the zero solution of equation (2).
We prove our main results using the variation of constants formula and the contraction mapping principle. We study necessary conditions for the existence of periodic solutions. Our results generalize certain results obtained in [9] and [1]. Moreover, they can be applied to equations for which the results obtained in some of the above mentioned papers could not be used (see Examples 1-3).
2 Boundedness and stability
Theorem 1
Proof
Theorem 1 extends Theorem 2.1 in [1] and Theorem 1 in [9].
The following example illustrates the result presented in Theorem 1.
Example 1
In the next example we present a Volterra difference equation for which condition (24) from [2] does not hold. Hence, Theorem 5.1 [2] is not applicable for this example whereas our Theorem 1 is.
Example 2
Remark 1
From Theorem 3.1 in [7] it follows that the boundedness of equation (2) is equivalent to the stability of its zero solution. Hence, and by Theorem 1, conditions (5) and (6) ensure the stability of the zero solution of equation (2).
Now, we provide conditions for the asymptotic stability of the zero solution of equation (2).
Theorem 2
Proof
Similarly as in the proof of Theorem 1, we get that T has a fixed point. This fixed point solves equation (1) and tends to zero. Hence, the zero solution of equation (2) is asymptotically stable. □
Example 3
Note that the result about asymptotic stability of the zero solution obtained in the section Scalar Equation (see Example 3.1) of [4] is not applicable here because assumption vi) \(\sum_{n=1}^{\infty} \vert K(n,i)\vert \leq C\), where C is a positive constant, for (15) is not satisfied. Similarly, Theorem 2.7 of [7] could not be applied for (15) since the kernel \(K(n,i)= \frac{1}{2n}\) does not satisfy condition (2.28).
In Proposition 3.2 of [3], the necessary and sufficient conditions for boundedness of all solutions of equations of type (2) are given. It is easy to see that for (15), assumption (17) of [3] does not hold. But, as it was shown above, each solution of this equation is bounded. Corollary 3.7 of [3] is not applicable here, too.
3 Existence of periodic solutions
In the next theorem we give necessary conditions for the existence of a periodic solution of equation (1).
Theorem 3
Let conditions (16) and (17) hold, and let \(A_{q}\) be the matrix defined by (19). If there exists a q-periodic solution of equation (1), then \(\det A_{q} =0\).
Proof
Remark 2
Example 4
The next example shows that the condition \(\det A_{q} =0\) is not sufficient for the existence of a q-periodic solution of (1).
Example 5
For some periodic difference equations and systems, see also, for example, [29, 30] or [31].
Declarations
Acknowledgements
The first author was supported by the project PB-43-081/14DS of Ministry of Science and Higher Education of Poland.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Kolmanovskii, V, Shaikhet, L: Some conditions for boundedness of solutions of difference Volterra equations. Appl. Math. Lett. 16, 857-862 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Appleby, JAD, Győri, I, Reynolds, DW: On exact convergence rates for solutions of linear systems of Volterra difference equations. J. Differ. Equ. Appl. 12, 1257-1275 (2006) View ArticleGoogle Scholar
- Győri, I, Reynolds, DW: Sharp conditions for boundedness in linear discrete Volterra equations. J. Differ. Equ. Appl. 15, 1151-1164 (2009) View ArticleGoogle Scholar
- Khandaker, TM, Raffoul, YN: Stability properties of linear Volterra discrete systems with nonlinear perturbation. J. Differ. Equ. Appl. 8, 857-874 (2002) View ArticleMATHMathSciNetGoogle Scholar
- Raffoul, YN: General theorems for stability and boundedness for nonlinear functional discrete systems. J. Math. Anal. Appl. 279(2), 639-650 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Yankson, E: Stability of Volterra difference delay equations. Electron. J. Qual. Theory Differ. Equ. 2006, Article ID 20 (2006) MathSciNetGoogle Scholar
- Crisci, MR, Kolmanovskii, VB, Russo, E, Vecchio, A: Boundedness of discrete Volterra equation. J. Math. Anal. Appl. 211, 106-130 (1997) View ArticleMATHMathSciNetGoogle Scholar
- Diblík, J, Schmeidel, E: On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence. Appl. Math. Comput. 218(18), 9310-9320 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Gronek, T, Schmeidel, E: Existence of bounded solution of Volterra difference equations via Darbo’s fixed-point theorem. J. Differ. Equ. Appl. 19(10), 1645-1653 (2013) View ArticleMATHMathSciNetGoogle Scholar
- Győri, I, Awwad, E: On the boundedness of the solutions in nonlinear discrete Volterra difference equations. Adv. Differ. Equ. 2012, Article ID 2 (2012). doi:10.1186/1687-1847-2012-2 View ArticleGoogle Scholar
- Győri, I, Horváth, L: Asymptotic representation of the solutions of linear Volterra difference equations. Adv. Differ. Equ. 2008, Article ID 932831 (2008) Google Scholar
- Medina, R: Asymptotic behavior of Volterra difference equations. Comput. Math. Appl. 41(5-6), 679-687 (2001) View ArticleMATHMathSciNetGoogle Scholar
- Migda, M, Migda, J: Bounded solutions of nonlinear discrete Volterra equations. Math. Slovaca (in press) Google Scholar
- Migda, M, Morchało, J: Asymptotic properties of solutions of difference equations with several delays and Volterra summation equations. Appl. Math. Comput. 220, 365-373 (2013) View ArticleMathSciNetGoogle Scholar
- Elaydi, S: Stability and asymptoticity of Volterra difference equations: a progress report. J. Comput. Appl. Math. 228(2), 504-513 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Crisci, MR, Jackiewicz, Z, Russo, E, Vecchio, A: Stability analysis of discrete recurrence equations of Volterra type with degenerate kernels. J. Math. Anal. Appl. 162, 49-62 (1991) View ArticleMATHMathSciNetGoogle Scholar
- Crisci, MR, Kolmanovskii, VB, Russo, E, Vecchio, A: Stability of difference Volterra equations: direct Liapunov method and numerical procedure. Comput. Math. Appl. 36(10-12), 77-97 (1998) View ArticleMATHMathSciNetGoogle Scholar
- Diblík, J, Růžičková, M, Schmeidel, E: Asymptotically periodic solutions of Volterra difference equations. Tatra Mt. Math. Publ. 43, 43-46 (2009) Google Scholar
- Diblík, J, Růžičková, M, Schmeidel, E: Existence of asymptotically periodic solutions of Volterra system difference equations. J. Differ. Equ. Appl. 15(11-12), 1165-1177 (2009) View ArticleMATHGoogle Scholar
- Diblík, J, Růžičková, M, Schmeidel, E: Asymptotically periodic solutions of Volterra systems of difference equations. Comput. Math. Appl. 59(8), 2854-2867 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Diblík, J, Růžičková, M, Schmeidel, E, Zbąszyniak, M: Weighted asymptotically periodic solutions of linear Volterra difference equations. Abstr. Appl. Anal. 2011, Article ID 37098 (2011). doi:10.1155/2011/370982 Google Scholar
- Elaydi, S: Periodicity and stability of linear Volterra difference systems. J. Math. Anal. Appl. 181, 483-492 (1994) View ArticleMATHMathSciNetGoogle Scholar
- Gajda, K, Gronek, T, Schmeidel, E: On the existence of a weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type. Discrete Contin. Dyn. Syst., Ser. B 19(8), 2681-2690 (2014) View ArticleMATHMathSciNetGoogle Scholar
- Győri, I, Reynolds, DW: On asymptotically periodic solutions of linear discrete Volterra equations. Fasc. Math. 44, 53-67 (2010) Google Scholar
- Song, Y, Baker, CTH: Perturbations of Volterra difference equations. J. Differ. Equ. Appl. 10, 379-397 (2004) View ArticleMATHMathSciNetGoogle Scholar
- Islam, MN, Yankson, E: Boundedness and stability in nonlinear delay difference equations employing fixed point theory. Electron. J. Qual. Theory Differ. Equ. 2005, Article ID 26 (2005) MathSciNetGoogle Scholar
- Raffoul, YN: Stability and periodicity in discrete delay equations. J. Math. Anal. Appl. 324, 1356-1362 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Raffoul, YN: Boundedness and periodicity of Volterra systems of difference equations. J. Differ. Equ. Appl. 4(4), 381-393 (1998) View ArticleMATHMathSciNetGoogle Scholar
- Diblík, J, Hlavičková, I: Asymptotic behavior of solutions of delayed difference equations. Abstr. Appl. Anal. 2011, Article ID 671967 (2011). doi:10.1155/2011/671967 Google Scholar
- Stevič, S, Diblík, J, Iričanin, B, Šmarda, Z: On a periodic system of difference equations. Abstr. Appl. Anal. 2012, Article ID 258718 (2012) Google Scholar
- Stevič, S, Diblík, J, Šmarda, Z: On periodic and solutions converging to zero of some systems of differential-difference equations. Appl. Math. Comput. 227, 43-49 (2014) View ArticleMathSciNetGoogle Scholar