Stability in discrete equations with variable delays

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 8, 1-7; Stability in discrete equations with variable delays ERNEST YANKSON Department of Mathematics and Statistics University of Cape Coast Ghana. Abstract In this paper we study the stability of the zero solution of difference equations with variable delays. In particular we consider the scalar delay equation and its generalization xn = anxn τn xn = a j nxn τ j n. Fixed point theorems are used in the analysis. Key words: Fixed point; Contraction mapping; Asymptotic stability Mathematics subject classifications: 34K13,34C25, 34G20. 1 Introduction Let R denote the real numbers, R = [0,, Z the integers, Z the negative integers, and Z = {x Z x 0}. In this paper we study the asymptotic stability of the zero solution of the scalar delay equation and its generalization xn = anxn τn 1.1 xn = a j nxn τ j n. 1.2 where a, a j : Z R and τ, τ j : Z Z with n τn as n. For each n 0, define m j n 0 = inf{s τ j s : s n 0 }, mn 0 = min{m j n 0 : 1 j N}. Note that 1.2 becomes 1.1 for N = 1. Recently, in [11], Raffoul studied the stability of the zero solution of 1.1 when EJQTDE, 2009 No. 8, p. 1

2 τn = r. Our objective in this research is to generalize the stability results in [11] to 1.2 for variable τ j n s. For more on stability using fixed point theory we refer to [1],[7],[9],[11],[12] and for basic results on difference calculus we refer to [2] and [8]. We also refer to [3],[4],[5],[6] and [10] for other results on stability for difference equations. Remark 1.1 In [7], the author and Islam showed that the zero solution of the equation xn 1 = bnxn anxn τn is asymptotically stable with one of the assumptions being that bs 0 as n. 1.3 s=0 However, as pointed out in [11], condition 1.3 cannot hold for 1.2 since bn = 1, for all n Z. The results we obtain in this paper overcome the requirement of 1.3. Let Dn 0 denote the set of bounded sequences ψ : [mn 0, n 0 ] R with the maximum norm.. Also, let B,. be the Banach space of bounded sequences ϕ : [mn 0, R with the maximum norm. Define the inverse of n τ i n by g i n if it exists and set Qn = bg j n, where bg j n = 1 ag j n. For each n 0, ψ Z Dn 0, a solution of 1.2 through n 0, ψ is a function x : [mn 0, n 0 α R n for some positive constant α > 0 such that xt satisfies 1.2 on [n 0, n 0 α and xn = ψn for n [mn 0, n 0 ]. We denote such a solution by xn = xn, n 0, ψ. For a fixed n 0, we define ψ = max{ ψn : mn 0 n n 0 }. EJQTDE, 2009 No. 8, p. 2

3 2 Stability In this section we obtain conditions for the zero solution of 1.2 to be asymptotically stable. We begin by rewriting 1.2 as xn = N a j g j nxn n s=nτ j n a j g j sxs 2.1 where n represents that the difference is with respect to n. If we let b j g j n = 1 then 2.1 is equivalent to a j g j n, xn 1 = N b j g j nxn n s=nτ j n a j g j sxs 2.2 Lemma 2.1 Suppose that Qn 0 for all n Z and the inverse function g j n of n τ j n exists. Then xn is a solution of 2.2 if and only if xn = xn 0 n 0 1 τ j n 0 a j g j sxs Qs s=nτ j n a j g j sxs [1 Qs] Qk s1 u=sτ j s a j g j uxu, n n 0. Proof. By the variation of parameters formula we obtain xn = xn 0 Qs k=0 N Qs k s=k k1 s=kτ j k a j g j sxs.2.3 EJQTDE, 2009 No. 8, p. 3

4 Using the summation by parts formula we obtain k=0 = N Qs k s=k s=nτ j n Qs k1 s=kτ j k a j g j sxs n 0 1 τ j n 0 [1 Qs] a j g j sxs a j g j sxs Qk s1 u=sτ j s a j g j uxu. 2.4 Substituting 2.5 into 2.3 gives the desired result. This completes the proof of Lemma 2.1. We next state and prove our main results. Theorem 2.1 Suppose that the inverse function g j n of n τ j n exists, and assume there exists a constant α 0, 1 such that s=nτ j n a j g j s [1 Qs] Qk s1 u=sτ j s Moreover, assume that there exists a positive constant M such that Qs M. Then the zero solution of 1.2 is stable. a j g j u α. 2.5 Proof. Let ɛ > 0 be given. Choose δ > 0 such that M Mαδ αɛ ɛ. Let ψ Dn 0 such that ψn δ. Define S = {ϕ B : ϕn = ψn if n [mn 0, n 0 ], ϕ ɛ}. EJQTDE, 2009 No. 8, p. 4

5 Then S,. is a complete metric space where,. is the maximum norm. and Define the mapping P : S S by Pϕ n = ψn for n [mn0, n 0 ] Pϕ n = ψn 0 s=nτ j n n 0 1 τ j n 0 [1 Qs] a j g j sϕs a j g j sψs Qs Qk s1 u=sτ j s a j g j uϕu. 2.6 Clearly, Pϕ is continuous. We first show that P maps from S to S. By 2.6 Pϕ { N n Mδ Mαδ ɛ. [1 Qs] M Mαδ αɛ s=nτ j n Qk a j g j s s1 u=sτ j s } a j g j uϕu ϕ Thus P maps from S into itself. We next show that P is a contraction. Let ζ, η S. Then P ζt { N P ηt α ζ η s=nτ j n [1 Qs] a j g j s Qk s1 u=sτ j s } a j g j u ζ η This shows that P is a contraction. Thus, by the contraction mapping principle, P has a unique fixed point in S which solves 1.2 and for any ϕ S, Pϕ ɛ. This proves that the zero solution of 1.2 is stable. EJQTDE, 2009 No. 8, p. 5

6 Theorem 2.2 Assume that the hypotheses of Theorem 2.1 hold. Also assume that k=n 0 Qk 0 as n. 2.7 Then the zero solution of 1.2 is asymptotically stable. Proof. We have already proved that the zero solution of 1.2 is stable. Let ψ Dn 0 such that ψn δ and define S = {ϕ B : ϕn = ψn if n [mn 0, n 0 ], ϕ ɛ and ϕn 0, as n }. Define P : S S by 2.6. From the proof of Theorem 2.2, the map P is a contraction and for every ϕ S, Pϕ ɛ. We next show that Pϕn 0 as n. The first term on the right side of 2.6 goes to zero because of condition 2.7. It is clear from 2.5 and the fact that ϕn 0 as n that N s=nτ j n a jg j s ϕs 0 as n. Now we show that the last term on the right side of 2.6 goes to zero as n. Since ϕn 0 and n τ j n as n, for each ɛ 1 > 0, there exists a N 1 > n 0 such that s N 1 implies ϕs τ j s < ɛ 1 for j = 1, 2, 3,..., N. Thus for n N 1, the last term, I 3 in 2.6 satisfies I 3 = 1 1 [1 Qs] [1 Qs] s=n 1 max ϕσ σ mn 0 ɛ 1 [1 Qs] s=n [1 Qs] Qk Qk s1 u=sτ j s Qk [1 Qs] s1 u=sτ j s u=sτ j s Qk a j g j uϕu s1 Qk a j g j u ϕu s1 a j g j u ϕu u=sτ j s By 2.7, there exists N 2 > N 1 such that n N 2 implies N11 max ϕσ [1 Qs] σ mn 0 Qk s1 u=sτ j s s1 u=sτ j s a j g j u a j g j u a j g j u < ɛ 1. EJQTDE, 2009 No. 8, p. 6

7 Apply 2.5 to obtain I 3 ɛ 1 ɛ 1 α < 2ɛ 1. Thus, I 3 0 as n. Hence Pϕn 0 as n, and so Pϕ S. By the contraction mapping principle, P has a unique fixed point that solves 1.2 and goes to zero as n goes to infinity. Therefore, the zero solution of 1.2 is asymptotically stable. References [1] T. Burton and T. Furumochi, Fixed points and problems in stability theory, Dynam. Systems Appl , [2] S. Elaydi, An Introduction to Difference Equations, Springer, New York, [3] S. Elaydi, Periodicity and stability of linear Volterra difference systems, J. Math. Anal. Appl , [4] S. Elaydi and S. Murakami, Uniform asymptotic stability in linear Volterra difference equations, J. Differerence Equ. Appl , [5] P. Eloe, M. Islam and Y. Raffoul, Uniform asymptotic stability in nonlinear Volterra discrete systems, Special Issue on Advances in Difference Equations IV, Computers Math. Appl , [6] M. Islam and Y. Raffoul, Exponential stability in nonlinear difference equations, J. Differerence Equ. Appl , [7] M. Islam and E. Yankson, Boundedness and Stability in nonlinear delay difference equations employing fixed point theory, Electron. J. Qual. Theory Differ. Equ [8] W. Kelley and A. Peterson, Difference Equations: An Introduction with Applications, Harcourt Academic Press, San Diego, [9] Y. Raffoul, Stability in neutral nonlinear differential equations with functional delays using fixed point theory, Mathematical and Computer Modelling , [10] Y. Raffoul, General theorems for stability and boundedness for nonlinear functional discrete systems, J. Math. Anal. Appl , [11] Y. N. Raffoul, Stability and periodicity in discrete delay equations, J. Math. Anal. Appl [12] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Analysis , Received November 14, 2008 EJQTDE, 2009 No. 8, p. 7