Nonlinear Analysis: Theory, Methods & Applications
A Prešić type contractive condition and its applications
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 be a complete metric space, and satisfy a Prešić type contractive condition (cf. Ćirić and Prešić [7]) with respect to a fixed point , i.e., for all , where and for . In Section 2, the results on the global asymptotic stability of the equilibrium point of in 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 (, ).
Theorem 2.1 Let ( , ) be a complete metric space, and be continuous, and define for . Suppose there exists and an integer , , such thatwith equality iff . If is contained in a compact subset of , then will be a sequence which is convergent to , and
Idea of Proof is essentially consists of the
Applications to difference equations
As applications to our results, we will study the nonlinear difference equation of th order of the form where , , is a continuous function with an equilibrium . Noting that for and that a bounded sequence is contained in a compact subset of , the following theorems are the immediate consequences of Theorem 2.3, Theorem 2.4 respectively.
Theorem 3.1 Let be continuous, and define for
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