AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi 10.1155/2018/3905632 3905632 Research Article Stability for Linear Volterra Difference Equations in Banach Spaces http://orcid.org/0000-0002-1392-8049 Medina Rigoberto 1 Chun Changbum Departamento de Ciencias Exactas Universidad de Los Lagos Casilla 933 Osorno Chile ulagos.cl 2018 132018 2018 27 10 2017 23 12 2017 16 01 2018 132018 2018 Copyright © 2018 Rigoberto Medina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is devoted to studying the existence and stability of implicit Volterra difference equations in Banach spaces. The proofs of our results are carried out by using an appropriate extension of the freezing method to Volterra difference equations in Banach spaces. Besides, sharp explicit stability conditions are derived.

Dirección de Investigación NU06/16
1. Introduction

In this paper, we study the existence and stability of solutions for a class of abstract functional difference equations described in the form(1)xk=j=0kKk,k-jxj+fk,k1,in a Banach space (X,·), flZ+,X, the space of bounded sequences equipped with the norm · on l, and Kk,j is a function defined on 0jk<, whose values are bounded operators in X. In addition, for any fixed integer τ0, Kτ,· is summable and bounded on Z+, the set of nonnegative integers.

A solution of (1) is a sequence defined on Z+ and satisfying (1) for all finite k>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 f· and Kk,·, one can use the freezing method for abstract Volterra difference equations, so the difficulty is overcome.

Existence and uniqueness problems for the Volterra difference equations were discussed by some authors (e.g., see [1, 2]). Existence and stability of Volterra difference equations have been studied by many authors (Federson et al. , Murakami and Nagabuchi , Györi and Horvath , Mingarelli , Gonzalez et al. , Kolmanovskii et al. , and Song and Baker ).

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. , the Kurzweil-Henstok integral formalism is applied to establish the existence of solutions to integral equations of Volterra type. In Murakami and Nagabuchi , sufficient stability properties and the asymptotic almost periodicity for linear Volterra difference equations in Banach spaces are derived. Gonzalez et al.  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 , Volterra-Stieltjes integral equations are studied, which can be considered as generalized Volterra difference equations. In Banás and Sadarangani , 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 , 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. , stability and boundedness problems of some classes of scalar Volterra nonlinear difference equations are investigated. Their stability conditions are formulated in terms of the characteristic equations. In Song and Baker , 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 lpZ+,X, where X 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’ ), as well as on the concept of analytical pencils (analytic operator-valued functions of a complex argument). See, for example, . In Medina , 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  are absolutely different compared with the results of this article.

Consider an X-valued Volterra-Stieltjes equation of the form(2)xt=0tKt,s,xsdμs+ft,t0,where K:0,×0,×XX and f:Lμ,Lμ,. A solution of this equation is a function xLμ,, which is locally μ-integrable in the Riemann-Stieltjes sense.

If μt=k-1, k-1<tk, for k=1,2,, then we can restrict our attention to functions xLμ, which are piecewise constants with xt=xk for k-1<tk, k=1,2,. We can identify such a function x with a sequence x1,x2, in the space l. In this case, the Volterra-Stieltjes original equation is equivalent to a Volterra difference equation(3)xj=k=1jKj,k,xk+fj,j=2,3,.Consequently, [1, 2, 15, 16] are highly appropriate to our research.

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. ) and extended to difference systems by Gil’ and Medina .

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.

2. Preliminary Results

To prove our main results, it is convenient to establish some known qualitative results for a class of convolution Volterra difference equations (see [7, 8, 11, 1925]).

Let Ki,i=0,1,2,, be bounded linear operators in a Banach space X with norm ·.

Consider the convolution Volterra difference equation(4)xk=j=1kKk-jxj+hk;k=1,2,where K0=0, h=hii=1, and hiX is a given sequence.

Assume that limk¯Kkk< and limk¯hkk<.

To solve (4), put(5)Tz=j=1Kjzj,fz=j=1hjzj,zC.Consider the equation(6)yz=Tzyz+fz.In a neighborhood ω of zero, let I-Tz be boundedly invertible. Then(7)yz=I-Tz-1fz,zω.Hence it follows that yz is infinitely many times differentiable at zero.

Differentiating (6) j times, we get(8)yjz=i=0jCjiTj-izyiz+fjz.Since Ki=Ti0/i!, substituting z=0 into the later equality, we obtain the following relations:(9)bj=i=0jKj-ibi+hj;K0=0;j=1,2,,where bj=yj0/j!. Therefore, we arrive at (7). Hence, the sequence xk=bk is a solution to (7). According to (7), we obtain (10)xj=1j!djyzdzjz=0=1j!djdzj1-Tz-1fzz=0.Thanks to the Cauchy formula(11)xj=12πiγ1zj+11-Tz-1fzdz,j=1,2,,where γ is a smooth contour surrounding zero, provided that I-Tz is boundedly invertible and f is regular inside γ and on γ. Thus, the next result can be established.

Theorem 2 (see [<xref ref-type="bibr" rid="B15">19</xref>, <xref ref-type="bibr" rid="B23">25</xref>]).

Inside γ and on γ, let I-Tz 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 [<xref ref-type="bibr" rid="B7">7</xref>, <xref ref-type="bibr" rid="B8">8</xref>, <xref ref-type="bibr" rid="B19">22</xref>–<xref ref-type="bibr" rid="B21">24</xref>]).

We will say that (1) is stable if, for any flZ+,X, a solution x of (1) satisfies the inequality(12)xlc0fl,where the constant c0 does not depend on f.

Let H be a separable Hilbert space and A a linear compact operator in H. If ekk=1 is an orthogonal basis in H and the series k=1Aek,ek converges, then the sum of the series is called the trace of the operator A and is denoted by(13)TraceA=TrA=k=1Aek,ek.

Definition 5 (see [<xref ref-type="bibr" rid="B16">20</xref>, <xref ref-type="bibr" rid="B18">26</xref>]).

An operator A satisfying the relation TrAA< is said to be a Hilbert-Schmidt operator, where A is the adjoint operator of A.

The norm(14)N2A=NA=TrAAis called the Hilbert-Schmidt norm of A.

Definition 6 (see [<xref ref-type="bibr" rid="B16">20</xref>, <xref ref-type="bibr" rid="B18">26</xref>]).

A bounded linear operator A is said to be quasi-Hermitian if its imaginary component(15)AI=A-A2iis a Hilbert-Schmidt operator, where A is the adjoint operator of A.

Theorem 7 (see [<xref ref-type="bibr" rid="B14">17</xref>, <xref ref-type="bibr" rid="B15">19</xref>, <xref ref-type="bibr" rid="B18">26</xref>]).

Let V be a Hilbert-Schmidt completely continuous quasinilpotent (Volterra) operator acting in a separable Hilbert space H. Then the inequality(16)VkNpkVk!,for  anynaturalkis 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 τ0, Kτ,· is l1-summable and bounded on Z+. In addition, assume that there exists a nonnegative constant q such that(17)j=0kKk,j-Kτ,jqk-τ,q=const.;k,τ0.Under (17), the function Kτ,j, for a fixed integer τ, admits the Z-transform K~τz=j=0z-jKτ,j,zρ, where ρ is the radius of convergence of K~τz. Besides, it is assumed that the operator Wτz=1-K~τz is boundedly invertible for all z in a neighborhood ω of zero.

Introduce the Green function(18)Gτk=12π02πe-iykWτ-1eiydy.

Theorem 8.

Under assumption (17), let(19)qk=0ksupτ0Gτk<1.Then (1) is stable. Moreover, constant c0 in (12) is explicitly pointed below.

Proof.

Consider the convolution equation(20)xk=j=0kKτ,k-jxj+fk,k1,with a fixed integer τ0.

The solution of (20) may be written as(21)xk=j=1kGτk-jfj,whereby the discrete Green’s function Gk,j is defined.

Now, rewrite (20) in the form(22)xk-j=1kKτ,k-jxj=f0k,τ+fk,k0,where(23)f0k,τ=j=1kKk,k-j-Kτ,k-jxj.Hence, according to (21),(24)xk=j=1kGτk-jfj+f0j,k,(25)=Fk+j=1kGτk-jf0j,k,where Fk=j=1kGτk-jfj.

Denote(26)Mk=supτ0Gτk.Hence, we get(27)Flflsupkj=1kMk-jMlfl.Due to (22),(28)f0k,τj=1kKk,k-j-Kτ,k-jxjqxlk-τ.By (24), we have(29)xkMlfl+qxlj=1kMk-jj-τ.Take k=τ; then(30)xτMlfl+qsup1jτxjj=1ττ-jMτ-j.Hence(31)xτMlfl+qsup1jτxjj=1jMj.Therefore, for any k0>0,(32)supτk0xτMlfl+λsupτk0xτ,where λ=qj=1jMj.

On the other hand, condition qj=1jMj<1 implies(33)sup1kk0xkMlfl1-λ.Since the right hand side does not depend on k0, condition (12) follows with c0=Ml/1-λ.

The existence of solutions is due to the convergence of the Neumann series(34)x=j=0Vjf,where(35)Vxk=j=1kKk,jxj,provided that k=1Kk,τ< for any fixed integer τ0. In fact, (1) is rewritten in the operator form(36)x=Vx+fHence(37)x=I-V-1f=j=0Vjf.This yields(38)xlj=0Vjlfl.Since V is a quasinilpotent Hilbert-Schmidt operator, it follows by [19, 25] that(39)VjlNpVk!p<.Consequently, the Neumann series j=0Vjf is convergent.

Remark 9.

The stability theory of Volterra difference equations has been considered, for example, by Song and Baker , Mingarelli , and Gonzalez et al. . 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.

4. Example

To illustrate the main result, consider in X the equation(40)xk-Akj=0ke-k-jhxj=fk;h=const.>0,where Ak is a variable bounded operator in X satisfying(41)Ak-Aτq0k-τ;k,τ0.Take Kk,j=Ake-jh. Then, by (41),(42)j=0kKk,j-Kτ,jAk-Aτj=0ke-jhq0ehk-τ.We also have (43)K~τz=j=0Kτ,jzj=Aτj=0e-hjzj=Aτzz-e-h,z>1eh.This relation yields(44)Wτz=I-Aτzz-e-h.On the other hand,(45)kGτk=k12πiγzk-1Wτ-1zdz=12πiγzk-1Tτzdz,where(46)Tτz=-ddzWτ-1z.If z<b<1 and letting Tτz be regular and(47)λb=supτ012π02πTτeiy-bdy<,then(48)kGτkbk-112π02πTτeiy-bdybk-1λb.Consequently,(49)k=0ksupτ0Gτkλbb1-b.

Theorem 10.

Under conditions (42), for a positive b and all z, with z<b<1, let Tτz be regular and λb/b1-b<1. Then (40) is stable.

For example, if At=sintB, where B is a constant operator, then condition (41) holds with(50)qt=2Bsint2,since(51)sinα-sinβ=2sinα-β2cosα+β2,α,βR.Taking Kk,j=sintBe-jh. Then, by (41),(52)j=0kKk,j-Kτ,jAk-Aτj=0ke-jh2Behk-τ.Thus, every statement of Theorem 10 can be easily verified.

5. 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 Rn (a finite dimensional Euclidean space). However, with appropriate conditions on f· and Kj,·, one can use the freezing method for abstract difference equations, so the difficulty is overcome. In fact, considering the time j 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 .

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by Dirección de Investigación under Grant NU06/16.

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