Fixed Point Theorems for Times Reasonable Expansive Mapping
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Research Article Fixed Point Theorems for n Times Reasonable Expansive Mapping Chunfang Chen and Chuanxi Zhu Institute of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China Correspondence should be addressed to Chuanxi Zhu, [email protected] Received 29 February 2008; Revised 3 May 2008; Accepted 16 August 2008 Recommended by Jerzy Jezierski Based on previous notions of expansive mapping, n times reasonable expansive mapping is defined. The existence of fixed point for n times reasonable expansive mapping is discussed and some new results are obtained. Copyright q 2008 C. Chen and C. Zhu. 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.
1. Introduction and preliminaries The research about fixed points of expansive mapping was initiated by Machuca see 1. Later, Jungck discussed fixed points for other forms of expansive mapping see 2. In 1982, Wang et al. see 3 published a paper in Advances in Mathematics about expansive mapping which draws great attention of other scholars. Also, Zhang has done considerable work in this field. In order to generalize the results about fixed point theory, Zhang see 4 published his work Fixed Point Theory and Its Applications, in which the fixed point problem for expansive mapping is systematically presented in a chapter. As applications, he also investigated the existence of solutions of equations for locally condensing mapping and locally accretive mapping. In 1991, based on the results obtained by others, the author defined several new kinds of expansive-type mappings in 5, which expanded the expansive-type mapping from 19 to 23, and gave some new applications. Recently, the study about fixed point theorem for expansive mapping and nonexpansive mapping is deeply explored and has extended too many other directions. Motivated and inspired by the works see 1–13, in this paper, we define n times reasonable expansive mapping and discuss the existence of fixed point for n times reasonable expansive mapping. For the sake of convenience, we briefly recall some definitions. Let X, d be a complete metric space and let T : X → X be a mapping. Throughout this paper, we use N to denote the set of natural numbers and x to denote the maximum integral value that is not larger than x.
2
Fixed Point Theory and Applications
T : X → X is called an expansive mapping if there exists a constant h > 1 such that dT x, T y ≥ hdx, y, for all x, y ∈ X. T : X → X is called a two times reasonable expansive mapping if there exists a constant h > 1 such that dx, T 2 x ≥ hdx, T x, for all x ∈ X. T : X → X is called a twenty-one type expansive mapping if there exists a constant h > 1 such that dT x, T y ≥ h min dx, y, dx, T x, dy, T y, dx, T y, dy, T x ,
∀x, y ∈ X.
1.1
T : X → X is called a twenty-three type expansive mapping if there exists a constant h > 1 such that d2 T x,
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