A Prešić type contractive condition and its applications

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Abstract

We study discrete dynamic systems in a complete metric space (M, d) defined by mappings which satisfy Prešić type contractive conditions. Their counterparts in an ordered Banach space are investigated and applied to solve the global asymptotic stability of the equilibriums of some nonlinear difference equations.

Introduction

The global asymptotic stability of an equilibrium of a difference equation is one of the actively discussed topics in the literature. Recently, Kruse and Nesemunn [1] introduced the Thompson’s metric (or part metric) to the study of the stability of an equilibrium of a Putnam difference equation and utilized the results of discrete dynamical systems in finite dimensional complete metric spaces. Many authors (e.g., Chen [2], Li et al. [3], [4], Yang et al. [5], [6] and the references therein.) continued their work and used various part-metric-related inequalities to investigate various types of difference equations. This approach is proved to be very fruitful. We present some recent results in this direction in the current paper. The detailed proofs will appear elsewhere.

Let (M,d) be a complete metric space, and T:MkM satisfy a Prešić type contractive condition (cf. Ćirić and Prešić [7]) with respect to a fixed point x, i.e., d(T(xn+k1,,xn),x)<max0jk1d(xn+j,x) for all (xn+k1,,xn)(x,,x), where (xk1,,x0)Mk and xn+k=T(xn+k1,,xn) for n=0,1,2,. In Section 2, the results on the global asymptotic stability of the equilibrium point of T in (M,d) and their counterparts in ordered Banach spaces will be presented. The applications to nonlinear difference equations are given in Section 3.

Section snippets

Main results

We first give two theorems in a complete metric space (M, d).

Theorem 2.1

Let (M , d ) be a complete metric space, and T:MkM be continuous, (xk1,,x0)Mk and define xn+k=T(xn+k1,,xn) for n=0,1,2, . Suppose there exists xM and an integer l , 0lk1 , such thatd(T(xn+k1,,xn),x)d(xn+l,x)with equality iff xn+l=x . If {xn:n=0,1,2,} is contained in a compact subset of M , then {xn} will be a sequence which is convergent to x , andlimnT(xn+k,,xn+1)=x.

Idea of Proof

{xn} is essentially consists of the

Applications to difference equations

As applications to our results, we will study the nonlinear difference equation of kth order of the form xn+k=f(xn+k1,,xn),n=0,1,2,, where f:R+kR+, R+=(0,), is a continuous function with an equilibrium xR+. Noting that M(x,y)=xy for x,yR+ and that a bounded sequence is contained in a compact subset of R+, the following theorems are the immediate consequences of Theorem 2.3, Theorem 2.4 respectively.

Theorem 3.1

Let f:R+kR+ be continuous, (xk1,,x0)R+k and define xn+k=f(xn+k1,,xn) for n=0,1,2,

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