A Fixed Point Theory for Quasi-Contractive Mappings in Ordered Sets - Advanced Research on Computer Education, Simulation and Modeling - page 112

A Fixed Point Theory for Quasi-Contractive
Mappings in Ordered Sets
Jiawei Chen 1 , Zhongping Wan 1 , Zhanyi Ao 2 , and Liuyang Yuan 1
1 School of Mathematics and Statistics, Wuhan University, Wuhan. Hubei,
430072, People's Republic of China
[e-mail]
2 Department of Mathematics and Computer Science, Hebei Normal University of
Nationalities, Chengde. Hebei, 067000, People's Republic of China
{zpwan-whu,zyiao,lyyuan}@126.com
Abstract. In this paper, the author presents a fixed point theorem for quasi-
contractive maps in partially ordered sets, which extends corresponding results
of [Nieto, J. J. and Rodriguez-Lopez, R.: Contractive Mapping Theorems in
Partially Ordered Sets and Applicationsto Ordinary Differential Equations.
Order, 22(2005), 223-239; Nieto, J. J. and Rodriguez-Lopez, R.: Existence and
Uniqueness of Fixed Point in Partially Ordered Sets and Applications to
Ordinary Dierential Equations, Acta Mathematica Sinica, English Series, 23
(2006), 2205-2212].
Keywords: Fixed point, partially ordered set, quasi-contractive map.
1 Introduction
Fixed point theory is an important content of nonlinear analysis that widely applied to
optimization, computational algorithms, physics, building mathematical model,
economics, variational inequalities, complementary problems, equilibrium problems,
vector optimization problems, management science and so on (see, for example,
[1-9]). Very recently years, considerable interest has been shown in developing
various classes of fixed point under different maps, for instance, nonexpansive,
constractive, quasi-constractive and pseudo-constractive mappings and so on, both for
its own sake and for its applications.
In [1], some results on the existence of fixed points in partially ordered sets are
presented. In [2] some results on the existence of a unique fixed point for
nondecreasing mappings are applied to obtain a unique solution for a first order
ordinary differential equation with periodic boundary conditions. In [3], the authors
proved some fixed point theorems in partially ordered sets, providing an extension of
the Banach contractive mapping theorem.
Inspired and motivated by above works, the purpose of this paper is to investigate
that fixed point theorem for quasi-contractive maps is proved in partially ordered sets
under certain assumptions, which extends corresponding results of [2,3].