Open Access

Coupled Fixed Point Theorems for Nonlinear Contractions Satisfied Mizoguchi-Takahashi's Condition in Quasiordered Metric Spaces

Fixed Point Theory and Applications20102010:876372

DOI: 10.1155/2010/876372

Received: 10 November 2009

Accepted: 14 March 2010

Published: 30 March 2010

Abstract

The main aim of this paper is to study and establish some new coupled fixed point theorems for nonlinear contractive maps that satisfied Mizoguchi-Takahashi's condition in the setting of quasiordered metric spaces or usual metric spaces.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq1_HTML.gif be a metric space. For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq3_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq4_HTML.gif . Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq5_HTML.gif the class of all nonempty subsets of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq7_HTML.gif the family of all nonempty closed and bounded subsets of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq8_HTML.gif . A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq9_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ1_HTML.gif
(11)

is said to be the Hausdorff metric on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq10_HTML.gif induced by the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq11_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq12_HTML.gif . A point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq13_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq14_HTML.gif is a fixed point of a map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq15_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq16_HTML.gif (when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq17_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq18_HTML.gif is a single-valued map) or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq19_HTML.gif (when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq20_HTML.gif is a multivalued map). Throughout this paper we denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq22_HTML.gif the set of positive integers and real numbers, respectively.

The existence of fixed point in partially ordered sets has been investigated recently in [111] and references therein. In [6, 8], Nieto and Rodríguez-López used Tarski's theorem to show the existence of solutions for fuzzy equations and fuzzy differential equations, respectively. The existence of solutions for matrix equations or ordinary differential equations by applying fixed point theorems is presented in [2, 4, 7, 9, 10]. The authors in [3, 11] proved some fixed point theorems for a mixed monotone mapping in a metric space endowed with partial order and applied their results to problems of existence and uniqueness of solutions for some boundary value problems.

The various contractive conditions are important to find the existence of fixed point. There is a trend to weaken the requirement on the contraction. In 1989, Mizoguchi and Takahashi [12] proved the following interesting fixed point theorem for a weak contraction which is a partial answer of Problem 9 in Reich [13] (see also [1416] and references therein).

Theorem MT. (Mizoguchi and Takahashi [12]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq23_HTML.gif be a complete metric space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq24_HTML.gif a map from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq25_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq26_HTML.gif . Assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ2_HTML.gif
(12)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq27_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq28_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq29_HTML.gif is a function from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq30_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq31_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ3_HTML.gif
(13)

Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq32_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq33_HTML.gif .

In fact, Mizoguchi-Takahashi's fixed point theorem is a generalization of Nadler's fixed point theorem [17, 18] which extended the Banach contraction principle (see, e.g., [18]) to multivalued maps, but its primitive proof is different. Recently, Suzuki [19] gave a very simple proof of Theorem MT.

The purpose of this paper is to present some new coupled fixed point theorems for weakly contractive maps that satisfied Mizoguchi-Takahashi's condition (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq34_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq35_HTML.gif ) in the setting of quasiordered metric spaces or usual metric spaces. Our results generalize and improve some results in [2, 7, 9] and references therein.

2. Generalized Bhaskar-Lakshmikantham's Coupled Fixed Point Theorems and Others

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq36_HTML.gif be a nonempty set and " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq37_HTML.gif " a quasiorder (preorder or pseudo-order, i.e., a reflexive and transitive relation) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq38_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq39_HTML.gif is called a quasiordered set. A sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq40_HTML.gif is called https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq41_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq42_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq43_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq44_HTML.gif ) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq45_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq46_HTML.gif ) for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq47_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq48_HTML.gif be a metric space with a quasi-order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq49_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq50_HTML.gif for short). We endow the product space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq51_HTML.gif with the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq52_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ4_HTML.gif
(21)

A map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq53_HTML.gif is said to be continuous at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq54_HTML.gif if any sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq55_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq56_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq57_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq58_HTML.gif is said to be continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq59_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq60_HTML.gif is continuous at every point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq61_HTML.gif .

In this paper, we also endow the product space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq62_HTML.gif with the following quasi-order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq63_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ5_HTML.gif
(22)

Definition 2.1 . (see [2]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq64_HTML.gif be a quasiordered set and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq65_HTML.gif a map. We say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq66_HTML.gif has the mixed monotone property on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq67_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq68_HTML.gif is monotone nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq69_HTML.gif and is monotone nonincreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq70_HTML.gif that is, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq71_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq72_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ6_HTML.gif
(23)

It is quite obvious that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq73_HTML.gif has the mixed monotone property on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq74_HTML.gif , then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq75_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq76_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq77_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq79_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq80_HTML.gif .

Definition 2.2 . (see [2]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq81_HTML.gif be a nonempty set and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq82_HTML.gif a map. We call an element https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq83_HTML.gif a coupled fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq84_HTML.gif if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ7_HTML.gif
(24)

Definition 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq85_HTML.gif be a metric space with a quasi-order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq86_HTML.gif A nonempty subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq87_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq88_HTML.gif is said to be

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq89_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq90_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq91_HTML.gif if every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq92_HTML.gif -nondecreasing Cauchy sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq93_HTML.gif converges;

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq94_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq95_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq96_HTML.gif if every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq97_HTML.gif -nonincreasing Cauchy sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq98_HTML.gif converges;

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq99_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq100_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq101_HTML.gif if it is both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq102_HTML.gif -complete and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq103_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq104_HTML.gif .

Definition 2.4 . (see [20]).

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq105_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq106_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq107_HTML.gif is said to be a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq108_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq109_HTML.gif if it satisfies Mizoguchi-Takahashi's condition (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq110_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq111_HTML.gif ).

Remark 2.5.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq112_HTML.gif Obviously, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq113_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq114_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq115_HTML.gif is defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq116_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq117_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq118_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq119_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq120_HTML.gif -function.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq121_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq122_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq123_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq124_HTML.gif is a nondecreasing function, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq125_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq126_HTML.gif -function.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq127_HTML.gif Notice that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq128_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq129_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq130_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq131_HTML.gif -function if and only if for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq132_HTML.gif there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq134_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq135_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq136_HTML.gif Indeed, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq137_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq138_HTML.gif -function, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq139_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq140_HTML.gif So for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq141_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq142_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq143_HTML.gif . Therefore we can find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq144_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq145_HTML.gif , and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq146_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq147_HTML.gif . The converse part is obvious.

The following lemmas are crucial to our proofs.

Lemma 2.6 . (see [20]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq148_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq149_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq150_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq151_HTML.gif -function. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq152_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq153_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq154_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq155_HTML.gif is also a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq156_HTML.gif -function.

Proof.

Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq158_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq159_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq160_HTML.gif be fixed. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq161_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq162_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq163_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq164_HTML.gif -function, there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq166_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq167_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq168_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq169_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq170_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq171_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq172_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq173_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq174_HTML.gif -function.

Lemma 2.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq175_HTML.gif be a quasiordered set and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq176_HTML.gif a map having the mixed monotone property on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq177_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq179_HTML.gif . Define two sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq180_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq181_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ8_HTML.gif
(25)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq182_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq184_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq185_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq186_HTML.gif -nondecreasing and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq187_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq188_HTML.gif -nonincreasing.

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq189_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq190_HTML.gif , by (2.5), and the mixed monotone property of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq191_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ9_HTML.gif
(26)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq192_HTML.gif and assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq193_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq194_HTML.gif is already known. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ10_HTML.gif
(27)

Hence, by induction, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq195_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq196_HTML.gif -nondecreasing and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq197_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq198_HTML.gif -nonincreasing.

Theorem 2.8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq199_HTML.gif be a sequentially https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq200_HTML.gif -complete metric space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq201_HTML.gif a continuous map having the mixed monotone property on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq202_HTML.gif . Assume that there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq203_HTML.gif -function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq204_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq205_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq206_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq207_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq208_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ11_HTML.gif
(28)

If there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq209_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq210_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq211_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq212_HTML.gif , then there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq213_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq214_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq216_HTML.gif

Proof.

By Lemma 2.6, we can define a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq217_HTML.gif -function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq218_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq219_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq220_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq221_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq222_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq223_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq224_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq225_HTML.gif be defined as in Lemma 2.7. Then, by Lemma 2.7, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq226_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq227_HTML.gif -nondecreasing and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq228_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq229_HTML.gif -nonincreasing. So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq230_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq231_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq232_HTML.gif . By (2.8), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ12_HTML.gif
(29)
It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ13_HTML.gif
(210)
For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq233_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq234_HTML.gif Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq235_HTML.gif By induction, we can obtain the following: for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq236_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ14_HTML.gif
(211)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ15_HTML.gif
(212)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ16_HTML.gif
(213)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq237_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq238_HTML.gif , the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq239_HTML.gif is strictly decreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq240_HTML.gif from (2.13). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq241_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq242_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq243_HTML.gif -function, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq244_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq245_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq246_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq247_HTML.gif . Also, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq248_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ17_HTML.gif
(214)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq249_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq250_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq251_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq252_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq253_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq254_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq255_HTML.gif . We claim that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq256_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq257_HTML.gif -nondecreasing Cauchy sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq258_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq259_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq260_HTML.gif -nonincreasing Cauchy sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq261_HTML.gif . Indeed, from our hypothesis, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq262_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ18_HTML.gif
(215)
Similarly,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ19_HTML.gif
(216)
Hence we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ20_HTML.gif
(217)
So it follows from (2.17) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ21_HTML.gif
(218)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq263_HTML.gif For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq264_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq265_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq266_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ22_HTML.gif
(219)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq267_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq268_HTML.gif and hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ23_HTML.gif
(220)

So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq269_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq270_HTML.gif -nondecreasing Cauchy sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq271_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq272_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq273_HTML.gif -nonincreasing Cauchy sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq274_HTML.gif . By the sequentially https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq275_HTML.gif -completeness of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq276_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq277_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq278_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq279_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq280_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq281_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq282_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq283_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq284_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq285_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq286_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq287_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq288_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq289_HTML.gif be given. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq290_HTML.gif is continuous at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq291_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq292_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ24_HTML.gif
(221)
whenever https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq293_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq294_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq295_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq296_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq297_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq298_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq299_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq300_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq301_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ25_HTML.gif
(222)
So, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq302_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq303_HTML.gif , by (2.22),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ26_HTML.gif
(223)
and hence we have from (2.21) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ27_HTML.gif
(224)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ28_HTML.gif
(225)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq304_HTML.gif is arbitrary, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq305_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq306_HTML.gif . Similarly, we can also prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq307_HTML.gif . The proof is completed.

Remark 2.9.

Theorem 2.8 generalizes and improves Bhaskar-Lakshmikantham's coupled fixed points theorem [2, Theorem  2.1] and some results in [7, 9].

Following a similar argument as in the proof of [2, Theorem  2.2] and applying Theorem 2.8, one can verify the following result where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq308_HTML.gif is not necessarily continuous.

Theorem 2.10.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq309_HTML.gif be a sequentially https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq310_HTML.gif -complete metric space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq311_HTML.gif a map having the mixed monotone property on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq312_HTML.gif . Assume that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq313_HTML.gif any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq314_HTML.gif -nondecreasing sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq315_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq316_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq317_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq318_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq319_HTML.gif any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq320_HTML.gif -nonincreasing sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq321_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq322_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq323_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq324_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq325_HTML.gif there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq326_HTML.gif -function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq327_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq328_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq329_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq330_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq331_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq332_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ29_HTML.gif
(226)

If there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq333_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq334_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq335_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq336_HTML.gif , then there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq337_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq338_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq339_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq340_HTML.gif

Remark 2.11.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq341_HTML.gif [2, Theorem  2.2] is a special case of Theorem 2.10.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq342_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq343_HTML.gif Similarly, we can obtain the generalizations of Theorems 2.4–2.6 in [2] for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq344_HTML.gif -functions.

Finally, we discuss the following coupled fixed point theorem in (usual) complete metric spaces.

Theorem 2.12.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq345_HTML.gif be a complete metric space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq346_HTML.gif a map. Assume that there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq347_HTML.gif -function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq348_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq349_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq350_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq351_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq352_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ30_HTML.gif
(227)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq353_HTML.gif has a unique coupled fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq354_HTML.gif ; that is, there exists unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq355_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq356_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq357_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq358_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq359_HTML.gif be given. For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq360_HTML.gif define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq361_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq362_HTML.gif . By our hypothesis, we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq363_HTML.gif is continuous. Following the same argument as in the proof of Theorem 2.8, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq364_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq365_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq366_HTML.gif . We prove the uniqueness of the coupled fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq367_HTML.gif . On the contrary, suppose that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq368_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq369_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_IEq370_HTML.gif . Then we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ31_HTML.gif
(228)
It follows from (2.28) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F876372/MediaObjects/13663_2009_Article_1359_Equ32_HTML.gif
(229)

a contradiction. The proof is completed.

Declarations

Acknowledgment

This paper is dedicated to Professor Wataru Takahashi in celebration of his retirement. This research was supported by the National Science Council of the Republic of China.

Authors’ Affiliations

(1)
Department of Mathematics, National Kaohsiung Normal University

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© Wei-Shih Du. 2010

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