lp-solutions and stability analysis of difference equations using the Kummer’s test

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Abstract

We address the p-summability and asymptotic stability properties in nonautonomous linear difference equations. We focus our discussion on two kind of difference equations. The first one is a first order system of linear nonautonomous difference equations, and our discussion involves the use of Kummer’s convergence test. The second one is a linear nonautonomous scalar higher order difference equation. In this case our discussion is based on a recently introduced transformation of a higher order system into a first-step recursion, where the companion matrices are well treatable from our point of view. We give insight on our ideas that are behind our methods, prove new results, and show applications.

Introduction

Difference equations are the appropriate representation for discrete processes which have special importance in such areas as population dynamics, control theory, economics and numerical analysis (see the monographs [1], [3] and the references in them).

The primary goal in this work is to give easily applicable explicit sufficient and some necessary conditions for the solutions to be p-summable.

Our results will be about the s-dimensional system of the first order linear difference equationsxn=Anxn-1,n0and the sth order scalar linear difference equationy(n)=j=n-sn-1B(n,j)y(j),n0,where AnRs×s(n0), and B(n,j)R(n0,n-sjn-1). Rs×s is the space of all s × s real matrices.

By a solution of Eqs. (1), (2) we mean a sequence x(xn)n-1inRs(y(y(n))n-sinR) which satisfies Eq. (1), (2) for all nonnegative integer n. We say that a solution x of (1) (a solution y of (2)) is p-summable with a p  1, if xlsp(yl1p), where the spaces lsp and l1p are defined in the following way:

Definition 1

Let p  1.

  • (a)

    The real Banach space lsp consists of all sequences v  (v(n))n⩾0 in Rs for which n=0v(n)p< with some norm ∥·∥ on Rs.

  • (b)

    The real Banach space l1p consists of all sequences u  (u(n))n⩾0 in R for which n=0u(n)p<.

The p-summability results, especially about Eq. (2), are very scare in the nonautonomous case. Gordon [7] obtained criteria for p-summable solutions in terms of Lyapunov function, Petropoulou and Siafarikas [10] considered essentially the space of square summable sequences, and Ey and Pötzsche [5] used fixed point theorems for nonlinear one-step recursions. The summability property in Volterra difference equations was discussed by Gil’ and Medina [6] for nonlinear and by Elaydi [4] for linear equations. Also, Győri and Horváth [8] examined the p-summability properties of solutions of higher order difference equations with time dependent coefficients. The results in [8] are based on a newly invented transformation of the higher order system into a one-step recursion, where the companion matrices are well treatable. Examples show that the obtained conditions in [8] are sharp in some cases. However, these observations don’t provide a practical way of determining the p-summability of the solutions in the general case.

In this paper we provide explicit sufficient and also explicit necessary conditions for the p-summability of the solutions of Eqs. (1), (2).

The main tool in this study is the Kummer’s convergence test. Kummer’s test gives a very powerful necessary and sufficient condition for the convergence or the divergence of a positive series and it is the source of many other tests. For instance, D’Alambert’s test, Raabe’s test, Bertrand’s test and Gauss’ test are all special cases of Kummer’s test obtained by choosing special sequences pk’s.

Theorem 2 Kummer’s test (see Tong [11])

Let ak be a positive series.

  • (a)

    ak is convergent if and only if there is a positive series pk and a real number c > 0, such that pkakak+1-pk+1c.

  • (b)

    ak is divergent if and only if there is a positive series pk, such that 1pk diverges and pkakak+1-pk+10.

Section 2 is devoted to the main results and it is subdivided into three parts. Sub Section 2.1 contains very powerful sufficient conditions for the solutions of Eq. (1) to be or not to be p-summable with some given p  1. The proofs of these results are based on the suitable application of Kummer’s test and one can find them in Section 4. The main statements of Sub Section 2.1 are utilized in Sub Sections 2.2 Main results for scalar linear difference equations of higher order via on specially constructed companion matrices, 2.3 Main results for scalar linear difference equations of higher order via the coefficients which contain summability results for Eq. (2) via its companion matrices and via the coefficient sequences, respectively. In Section 3 some illustrative examples are given to show the effectiveness of our method. Section 4 is devoted to the proofs of the results stated in Section 2.

The novelty of our method is that it provides p-summability and asymptotic stability results for difference equations with sign changing, unbounded and time varying coefficients. From that point of view our results are compared to some existing asymptotic stability theorems given by Berezansky and Braverman [2] and Wei and Shen [12].

We need the following definitions.

Definition 3

  • (a)

    O and I mean the zero matrix and the identity matrix in Rs×s, respectively.

  • (b)

    The following norm on Rs will be used: for a(a1,,as)TRsamax1isai.

The spectral radius of a matrix ARs×s is denoted by ρ(A), and tr(A) means the sum of the diagonal elements of A.

The elements of Rs are considered as column vectors. Rs is a lattice under the canonical ordering defined by: a  b means that ai  bi for every 1  i  s, where a = (ai) and b = (bi).

If aR, then [a] denotes the largest integer that does not exceed a.

Section snippets

Main results

In the next three subsections we state our main results on the p-summability and asymptotic stability of Eqs. (1), (2), respectively. This statement will be proved in Section 4.

Discussion, applications and examples

To illustrate our results for scalar linear difference equations of higher order we consider the equationy(n)=K(n)y(n-1)+L(n)y(n-2),n0with the initial conditiony(i)=φ(i),-2i-1,where K(n),L(n)R(n0), and φ(φ(-2),φ(-1))TR2. The Eq. (23) belongs to the family (20).

Considering (23), Lemma 20 insures thatC(k)=L(2k-2)K(2k-2)K(2k-1)L(2k-2)K(2k-2)K(2k-1)+L(2k-1),k1.The sharpness of our results is illustrated by the next two examples. We construct difference equations whose every solution is p

Some lemmas and the proofs of the main results

To prove Theorem 6, we require a lemma.

Lemma 33

Assume that the conditions of Theorem 6 are satisfied, and let k0 = 0. Let q1  q0 + d, and we introducezk(i)l=-1kqlql+1+d1pvk+1(i),k-1,1is.Then

  • (a)

    For each 1  i  sz-1(i)=v0(i),zk(i)>0,andzk+1(i)zk(i)=qk+1qk+2+d1pvk+2(i)vk+1(i),k-1.

  • (b)

    k=0(zk(i))p<,1is.

  • (c)

    j=1s|Ak(i,j)|zk-1(j)zk(i),k0,1is.

Proof

  • (a)

    These are obvious.

  • (b)

    Let 1  i  s be fixed. (a) implies that(zk+1(i))p(zk(i))p=qk+1qk+2+dvk+2(i)vk+1(i)p=qk+1(vk+1(i))pqk+2(vk+2(i))p+d(vk+2(i))p,k0.By (6)vk(i)ei,k0for some ei >

Acknowledgement

Supported by Hungarian National Foundations for Scientific Research Grant No. K73274.

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