Stability for Linear Volterra Difference Equations in Banach Spaces

and Applied Analysis 3 where b(j) = y(j)(0)/j!.Therefore, we arrive at (7). Hence, the sequence x(k) = b(k) is a solution to (7). According to (7), we obtain x (j) = 1 j! d jy (z) dzj 󵄨󵄨󵄨󵄨󵄨󵄨󵄨z=0 = 1 j! d j dzj (1 − T (z))−1 f (z) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨z=0 . (10) Thanks to the Cauchy formula x (j) = 1 2πi ∫γ 1 zj+1 (1 − T (z))−1 f (z) dz, (j = 1, 2, . . .) , (11) where γ is a smooth contour surrounding zero, provided that I − T(z) is boundedly invertible and f is regular inside γ and on γ. Thus, the next result can be established. Theorem 2 (see [19, 25]). Inside γ and on γ, let I − T(z) be boundedly invertible and f be regular. Then a solution of (4) is given by formula (11). Remarks 3. Theorem 2 will play a fundamental role to establish the existence and stability of the solution of nonconvolution equations of kind (1). In doing so, we will use the freezing method. Definition 4 (see [7, 8, 22–24]). We will say that (1) is stable if, for any f ∈ l∞(Z+, X), a solution x of (1) satisfies the inequality ‖x‖l∞ ≤ c0 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩l∞ , (12) where the constant c0 does not depend on f. LetH be a separable Hilbert space andA a linear compact operator in H. If {ek}∞k=1 is an orthogonal basis in H and the series ∑∞k=1(Aek, ek) converges, then the sum of the series is called the trace of the operator A and is denoted by Trace (A) = Tr (A) = ∞ ∑ k=1 (Aek, ek) . (13) Definition 5 (see [20, 26]). An operator A satisfying the relationTr(A∗A) < ∞ is said to be aHilbert-Schmidt operator, where A∗ is the adjoint operator of A. The norm N2 (A) = N (A) = √Tr (A∗A) (14) is called the Hilbert-Schmidt norm of A. Definition 6 (see [20, 26]). A bounded linear operator A is said to be quasi-Hermitian if its imaginary component AI = A − A∗ 2i (15) is a Hilbert-Schmidt operator, where A∗ is the adjoint operator of A. Theorem 7 (see [17, 19, 26]). LetV be a Hilbert-Schmidt completely continuous quasinilpotent (Volterra) operator acting in a separable Hilbert spaceH. Then the inequality 󵄩󵄩󵄩󵄩󵄩Vk󵄩󵄩󵄩󵄩󵄩 ≤ N k p (V) √k! , for any natural k (16) is true. 3. Main Results Now, we are in a position to establish sufficient conditions on the existence and stability of solutions of (1). Assume that, for any fixed integer τ ≥ 0, K(τ, ⋅) is l1summable and bounded onZ+. In addition, assume that there exists a nonnegative constant q such that k ∑ j=0 󵄩󵄩󵄩󵄩K (k, j) − K (τ, j)󵄩󵄩󵄩󵄩 ≤ q |k − τ| , (q = const.; k, τ ≥ 0) . (17) Under (17), the function K(τ, j), for a fixed integer τ, admits the Z-transform ?̃?τ(z) = ∑∞j=0 z−jK(τ, j), |z| ≥ ρ, where ρ is the radius of convergence of ?̃?τ(z). Besides, it is assumed that the operator Wτ(z) = 1 − ?̃?τ(z) is boundedly invertible for all z in a neighborhood ω of zero. Introduce the Green function Gτ (k) = 1 2π ∫ 2π 0 e−iykW−1 τ (eiy) dy. (18) Theorem 8. Under assumption (17), let q∞ ∑ k=0 k sup τ≥0 󵄩󵄩󵄩󵄩Gτ (k)󵄩󵄩󵄩󵄩 < 1. (19) Then (1) is stable. Moreover, constant c0 in (12) is explicitly pointed below. Proof. Consider the convolution equation


Introduction
In this paper, we study the existence and stability of solutions for a class of abstract functional difference equations described in the form in a Banach space (, ‖ ⋅ ‖),  ∈  ∞ ( + , ), the space of bounded sequences equipped with the norm ‖ ⋅ ‖ on  ∞ , and (, ) is a function defined on 0 ≤  ≤  < ∞, whose values are bounded operators in .In addition, for any fixed integer  ≥ 0, (, ⋅) is summable and bounded on  + , the set of nonnegative integers.A solution of ( 1) is a sequence defined on  + and satisfying (1) for all finite  > 0. The study of existence and stability of solutions for implicit Volterra difference equations of nonconvolution type, defined in abstract spaces, is a complicated problem.However, with appropriate conditions on (⋅) and (, ⋅), one can use the freezing method for abstract Volterra difference equations, so the difficulty is overcome.
The main technique in the theory of stability and boundedness of Volterra difference equations is the direct Lyapunov method and its variants.In contrast, many alternative methods to Lyapunov's function have been successfully applied to the stability analysis of Volterra difference equations; for example, in Federson et al. [3], the Kurzweil-Henstok integral formalism is applied to establish the existence of solutions to integral equations of Volterra type.In Murakami and Nagabuchi [4], sufficient stability properties and the asymptotic almost periodicity for linear Volterra difference equations in Banach spaces are derived.Gonzalez et al. [6] considered an implicit Volterra difference equation in a Hilbert space and obtained sufficient conditions so that the solutions exist and have a bounded behavior.The coefficients of the considered equations are sequences of real numbers.In Mingarelli [1], Volterra-Stieltjes integral equations are studied, which can be considered as generalized Volterra difference equations.In Banás and Sadarangani [2], a class of operator-integral equations of Volterra-Stieltjes type which create a generalization of numerous integral equations appearing in mathematical literature is studied.In Györi and Horvath [5], sufficient conditions are presented under which the solutions to a linear nonconvolution Volterra difference equation converge to limits, which are given by a limit formula.In Kolmanovskii et al. [7], stability and boundedness problems of some classes of scalar Volterra 2 Abstract and Applied Analysis nonlinear difference equations are investigated.Their stability conditions are formulated in terms of the characteristic equations.In Song and Baker [8], the fixed point theory is used to establish sufficient conditions to ensure the stability of the zero solution of an implicit nonlinear Volterra difference equation.However, in the above-mentioned articles, Volterra equations with convolution kernels are mainly considered.
In this paper, formulating the Volterra discrete equations in the phase space   ( + , ), where  is an appropriate Hilbert space, and assuming that the kernel operator is completely continuous, we obtain sufficient conditions for the existence and uniqueness problem.The suggested approach is based on the "freezing" method to abstract difference equations (Medina and Gil' [9]), as well as on the concept of analytical pencils (analytic operator-valued functions of a complex argument).See, for example, [10][11][12][13].In Medina [14], a class of nonlinear discrete-time Volterra equations in Banach spaces is considered.Using a linearization method, sufficient conditions of existence and boundedness are established.In fact, assuming that the kernels are Causal Operators, the existence and boundedness of solutions are derived.Consequently, the methodology and the corresponding results obtained in [14] are absolutely different compared with the results of this article.
Consider an -valued Volterra-Stieltjes equation of the form where A solution of this equation is a function  ∈  ,∞ , which is locally -integrable in the Riemann-Stieltjes sense.
Remark 1.We want to point out that the freezing method was introduced by V. M. Alekseev for linear ordinary differential equations (see Bylov et al. [17]) and extended to difference systems by Gil' and Medina [18].
Our aim in this paper is to make new contributions to the development of the theory of existence and qualitative properties of solutions for the nonconvolution Volterra difference equations described by Volterra operators in Banach spaces.
The remainder of this article is organized as follows: In Section 2, we establish a preliminary result to a class of convolution Volterra difference equations which will be fundamental to formulating the corresponding nonconvolution problem in Banach spaces.In Section 3, sufficient conditions on the existence and stability of solutions of nonconvolution Volterra difference equations are established.In Section 4, we illustrate the main result studying an interesting problem.Finally, Section 5 is devoted to the discussion of our results.
Assume that lim To solve (4), put Consider the equation In a neighborhood  of zero, let  − () be boundedly invertible.Then Hence it follows that () is infinitely many times differentiable at zero.Differentiating (6)  times, we get Since () =  () (0)/!, substituting  = 0 into the later equality, we obtain the following relations: where () =  () (0)/!.Therefore, we arrive at (7).Hence, the sequence () = () is a solution to (7).According to (7), we obtain Thanks to the Cauchy formula where  is a smooth contour surrounding zero, provided that  − () is boundedly invertible and  is regular inside  and on .Thus, the next result can be established.
Theorem 2 (see [19,25]).Inside  and on , let  − () be boundedly invertible and  be regular.Then a solution of ( 4) is given by formula (11).Remarks 3. Theorem 2 will play a fundamental role to establish the existence and stability of the solution of nonconvolution equations of kind (1).In doing so, we will use the freezing method.
Let  be a separable Hilbert space and  a linear compact operator in .If {  } ∞ =1 is an orthogonal basis in  and the series ∑ ∞ =1 (  ,   ) converges, then the sum of the series is called the trace of the operator  and is denoted by Definition 5 (see [20,26]).An operator  satisfying the relation Tr( * ) < ∞ is said to be a Hilbert-Schmidt operator, where  * is the adjoint operator of .The norm is called the Hilbert-Schmidt norm of .
Definition 6 (see [20,26]).A bounded linear operator  is said to be quasi-Hermitian if its imaginary component is a Hilbert-Schmidt operator, where  * is the adjoint operator of .

Main Results
Now, we are in a position to establish sufficient conditions on the existence and stability of solutions of (1).Assume that, for any fixed integer  ≥ 0, (, ⋅) is  1summable and bounded on  + .In addition, assume that there exists a nonnegative constant  such that Under ( 17), the function (, ), for a fixed integer , admits the -transform K () = ∑ ∞ =0  − (, ), || ≥ , where  is the radius of convergence of K ().Besides, it is assumed that the operator   () = 1 − K () is boundedly invertible for all  in a neighborhood  of zero.
Proof.Consider the convolution equation with a fixed integer  ≥ 0.
The existence of solutions is due to the convergence of the Neumann series where provided that ∑ ∞ =1 ‖(, )‖ < ∞ for any fixed integer  ≥ 0. In fact, (1) is rewritten in the operator form Hence This yields Since  is a quasinilpotent Hilbert-Schmidt operator, it follows by [19,25] that Consequently, the Neumann series ∑ ∞ =0    is convergent.
Remark 9.The stability theory of Volterra difference equations has been considered, for example, by Song and Baker [8], Mingarelli [1], and Gonzalez et al. [6].However, the "freezing" method has not been used previously to study qualitative properties of Volterra difference systems in Banach spaces.Consequently, the theoretical contributions of this paper are significantly new.

Example
To illustrate the main result, consider in  the equation where () is a variable bounded operator in  satisfying We also have This relation yields On the other hand, where If || <  < 1 and letting   () be regular and  (52) Thus, every statement of Theorem 10 can be easily verified.

Concluding Remarks
The stability problem for Volterra difference equations of nonconvolution type in an infinite dimensional Hilbert space is more complicated than that for equations in   (a finite dimensional Euclidean space).However, with appropriate conditions on (⋅) and (, ⋅), one can use the freezing method for abstract difference equations, so the difficulty is overcome.In fact, considering the time  as a parameter, we obtain an infinite family of convolution Volterra difference equations.Thus, using the freezing method, we deduce the qualitative properties corresponding to the nonconvolution Volterra difference equations and to the convolution original equation.On the other hand, the study of existence of solutions of this kind of implicit Volterra difference equations is a complicated problem.Our proof of the existence of solutions is carried out using the convergence of Neumann series of quasinilpotent Hilbert-Schmidt operators [27].