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
MSC
1 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
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