Open Access

Contractive-Like Mapping Principles in Ordered Metric Spaces and Application to Ordinary Differential Equations

Fixed Point Theory and Applications20102010:916064

DOI: 10.1155/2010/916064

Received: 25 November 2009

Accepted: 30 March 2010

Published: 10 May 2010

Abstract

The purpose of this paper is to present a fixed point theorem for generalized contractions in partially ordered complete metric spaces. We also present an application to first-order ordinary differential equations.

1. Introduction

Existence of fixed point in partially ordered sets has been considered recently in [117]. Tarski's theorem is used in [9] to show the existence of solutions for fuzzy equations and in [11] to prove existence theorems for fuzzy differential equations. In [2, 6, 7, 10, 13] some applications to ordinary differential equations and to matrix equations are presented. In [35, 17] some fixed point theorems are proved for a mixed monotone mapping in a metric space endowed with partial order and the authors apply their results to problems of existence and uniqueness of solutions for some boundary value problems.

In the context of ordered metric spaces, the usual contraction is weakened but at the expense that the operator is monotone. The main tool in the proof of the results in this context combines the ideas in the contraction principle with those in the monotone iterative technique [18].

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq1_HTML.gif denote the class of the class of the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq2_HTML.gif which satisfies the condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ1_HTML.gif
(1.1)

In [19] the following generalization of Banach's contraction principle appears.

Theorem 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq3_HTML.gif be a complete metric space and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq4_HTML.gif be a mapping satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq5_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq6_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq8_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq9_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq10_HTML.gif .

Recently, in [2] the authors prove a version of Theorem 1.1 in the context of ordered complete metric spaces. More precisely, they prove the following result.

Theorem 1.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq11_HTML.gif be a partially ordered set and suppose that there exists a metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq12_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq13_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq14_HTML.gif is a complete metric space. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq15_HTML.gif be a nondecreasing mapping such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ3_HTML.gif
(1.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq16_HTML.gif . Assume that either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq17_HTML.gif is continuous or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq18_HTML.gif satisfies the following condition:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ4_HTML.gif
(1.4)

Besides, suppose that for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq19_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq20_HTML.gif which is comparable to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq22_HTML.gif . If there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq23_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq24_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq25_HTML.gif has a unique fixed point.

The purpose of this paper is to generalize Theorem 1.2 with the help of the altering functions.

We recall the definition of such functions.

Definition 1.3.

An altering function is a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq26_HTML.gif which satisfies the following.

(a) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq27_HTML.gif is continuous and nondecreasing.

(b) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq28_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq29_HTML.gif .

Altering functions have been used in metric fixed point theory in recent papers [2022].

In [7] the authors use these functions and they prove some fixed point theorems in ordered metric spaces.

2. Fixed Point Theorems

Definition 2.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq30_HTML.gif is a partially ordered set and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq31_HTML.gif , we say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq32_HTML.gif is monotone nondecreasing if for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq33_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ5_HTML.gif
(2.1)

This definition coincides with the notion of a nondecreasing function in the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq35_HTML.gif represents the usual total order in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq36_HTML.gif .

In the sequel, we prove the main result of the paper.

Theorem 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq37_HTML.gif be a partially ordered set and suppose that there exists a metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq38_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq39_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq40_HTML.gif is a complete metric space. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq41_HTML.gif be a continuous and nondecreasing mapping such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ6_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq42_HTML.gif is an altering function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq43_HTML.gif .

If there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq44_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq45_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq46_HTML.gif has a fixed point.

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq47_HTML.gif , then the proof is finished. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq48_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq50_HTML.gif is a nondecreasing mapping, we obtain by induction that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ7_HTML.gif
(2.3)
Put https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq51_HTML.gif . Taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq52_HTML.gif and since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq53_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq54_HTML.gif then, by (2.2), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ8_HTML.gif
(2.4)
Using the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq55_HTML.gif is nondecreasing, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ9_HTML.gif
(2.5)
If there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq56_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq57_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq58_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq59_HTML.gif is a fixed point and the proof is finished. In another case, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq60_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq61_HTML.gif . Then, taking into account (2.5), the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq62_HTML.gif is decreasing and bounded below, so
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ10_HTML.gif
(2.6)

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq63_HTML.gif .

Then, from (2.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ11_HTML.gif
(2.7)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq64_HTML.gif in the last inequality and by the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq65_HTML.gif is an altering function, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ12_HTML.gif
(2.8)
and, consequently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq66_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq67_HTML.gif this implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq68_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq69_HTML.gif and this contradicts our assumption that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq70_HTML.gif Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ13_HTML.gif
(2.9)

In what follows, we will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq71_HTML.gif is a Cauchy sequence.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq72_HTML.gif is not a Cauchy sequence. Then, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq73_HTML.gif for which we can find subsequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq75_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq76_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq77_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ14_HTML.gif
(2.10)
Further, corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq78_HTML.gif , we can choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq79_HTML.gif in such a way that it is the smallest integer with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq80_HTML.gif and satisfying (2.10), then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ15_HTML.gif
(2.11)
Using (2.10), (2.11), and the triangular inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ16_HTML.gif
(2.12)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq81_HTML.gif and using (2.9), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ17_HTML.gif
(2.13)
Again, the triangular inequality gives us
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ18_HTML.gif
(2.14)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq82_HTML.gif in the above two inequalities and using (2.9) and (2.13), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ19_HTML.gif
(2.15)
As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq84_HTML.gif , by (2.2), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ20_HTML.gif
(2.16)
Taking into account (2.13) and (2.15) and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq85_HTML.gif is continuous and letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq86_HTML.gif in (2.16), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ21_HTML.gif
(2.17)
As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq87_HTML.gif is an altering function, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq88_HTML.gif , the last inequality gives us
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ22_HTML.gif
(2.18)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq89_HTML.gif , this means that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ23_HTML.gif
(2.19)

This fact and (2.15) give us https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq90_HTML.gif which is a contradiction.

This shows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq91_HTML.gif is a Cauchy sequence.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq92_HTML.gif is a complete metric space, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq93_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq94_HTML.gif Moreover, the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq95_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ24_HTML.gif
(2.20)

and this proves that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq96_HTML.gif is a fixed point.

In what follows, we prove that Theorem 2.2 is still valid for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq97_HTML.gif not necessarily continuous, assuming the following hypothesis in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq98_HTML.gif (which appears in [10, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq99_HTML.gif ]):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ25_HTML.gif
(2.21)

Theorem 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq100_HTML.gif be a partially ordered set and suppose that there exists a metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq101_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq102_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq103_HTML.gif is a complete metric space. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq104_HTML.gif satisfies (2.21). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq105_HTML.gif be a nondecreasing mapping such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ26_HTML.gif
(2.22)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq106_HTML.gif is an altering function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq107_HTML.gif . If there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq108_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq109_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq110_HTML.gif has a fixed point.

Proof.

Following the proof of Theorem 2.2, we only have to check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq111_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq112_HTML.gif is a nondecreasing sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq113_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq114_HTML.gif then, by (2.21), we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq115_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq116_HTML.gif , and, consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ27_HTML.gif
(2.23)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq117_HTML.gif and using the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq118_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ28_HTML.gif
(2.24)
or, equivalently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ29_HTML.gif
(2.25)

As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq119_HTML.gif is an altering function, this gives us https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq120_HTML.gif and, thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq121_HTML.gif

Now, we present an example where it can be appreciated that the hypotheses in Theorems 2.2 and 2.3 do not guarantee uniqueness of the fixed point. This example appears in [10].

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq122_HTML.gif and consider the usual order
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ30_HTML.gif
(2.26)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq123_HTML.gif is a partially ordered set whose different elements are not comparable. Besides, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq124_HTML.gif is a complete metric space considering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq125_HTML.gif as the Euclidean distance. The identity map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq126_HTML.gif is trivially continuous and nondecreasing and condition (2.2) of Theorem 2.2 is satisfied since elements in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq127_HTML.gif are only comparable to themselves. Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq129_HTML.gif has two fixed points in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq130_HTML.gif .

In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 2.2 and 2.3. This condition appears in [16] and says that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ31_HTML.gif
(2.27)
In [10] it is proved that condition (2.27) is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ32_HTML.gif
(2.28)

Theorem 2.4.

Adding condition (2.28) to the hypotheses of Theorem 2.2 (resp., Theorem 2.3), we obtain uniqueness of the fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq131_HTML.gif .

Proof.

Suppose that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq132_HTML.gif which are fixed points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq134_HTML.gif . We distinguish two cases.

Case 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq136_HTML.gif are comparable, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq137_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq138_HTML.gif are comparable for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq139_HTML.gif Using the contractive condition appearing in Theorem 2.2 (or Theorem 2.3) and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq140_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ33_HTML.gif
(2.29)

which is a contradiction.

Case 2.

Using condition (2.28), there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq141_HTML.gif comparable to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq142_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq143_HTML.gif . Monotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq144_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq145_HTML.gif is comparable to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq146_HTML.gif and to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq147_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq148_HTML.gif Moreover, as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq149_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ34_HTML.gif
(2.30)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq150_HTML.gif is nondecreasing the above inequality gives us
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ35_HTML.gif
(2.31)

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq151_HTML.gif

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq152_HTML.gif .

Taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq153_HTML.gif is an altering function and letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq154_HTML.gif in (2.30), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ36_HTML.gif
(2.32)

and this implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq155_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq156_HTML.gif then we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ37_HTML.gif
(2.33)

and, consequently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq157_HTML.gif , which is a contradiction.

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq158_HTML.gif .

Analogously, it can be proved that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ38_HTML.gif
(2.34)
Finally, as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ39_HTML.gif
(2.35)

and taking limit, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq159_HTML.gif .

This finishes the proof.

Remark 2.5.

Under the assumptions of Theorem 2.4, it can be proved that for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq160_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq161_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq162_HTML.gif is the fixed point (i.e., the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq163_HTML.gif is Picard).

In fact, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq164_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq165_HTML.gif comparable to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq166_HTML.gif then using the same argument that is in Case 1 of Theorem 2.4 can prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq167_HTML.gif and, hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq168_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq169_HTML.gif is not comparable to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq170_HTML.gif , we take that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq171_HTML.gif is comparable to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq173_HTML.gif . Using a similar argument that is in Case 2 of Theorem 2.4, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ40_HTML.gif
(2.36)
Finally,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ41_HTML.gif
(2.37)

and taking limit as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq174_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq175_HTML.gif or, equivalently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq176_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq177_HTML.gif .

Remark 2.6.

Notice that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq178_HTML.gif is totally ordered, condition (2.28) is obviously satisfied.

Remark 2.7.

Considering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq179_HTML.gif the identity mapping in Theorem 2.4, we obtain Theorem 1.2, being the main result of [2].

3. Application to Ordinary Differential Equations

In this section we present an example where our results can be applied.

This example is inspired by [10].

We study the existence of solution for the following first-order periodic problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ42_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq180_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq181_HTML.gif is a continuous function.

Previously, we considered the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq182_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq183_HTML.gif ) of continuous functions defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq184_HTML.gif . Obviously, this space with the metric given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ43_HTML.gif
(3.2)
is a complete metric space. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq185_HTML.gif can also be equipped with a partial order given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ44_HTML.gif
(3.3)

Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq186_HTML.gif satisfies condition (2.28), since for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq187_HTML.gif the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq188_HTML.gif .

Moreover, in [10] it is proved that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq189_HTML.gif with the above-mentioned metric satisfies condition (2.21).

Now, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq190_HTML.gif denote the class of functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq191_HTML.gif satisfying the following.

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq192_HTML.gif is nondecreasing.

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq194_HTML.gif .

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq195_HTML.gif ,

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq196_HTML.gif is the class of functions defined in Section 1.

Examples of such functions are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq197_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq199_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq200_HTML.gif .

Recall now the following definition

Definition 3.1.

A lower solution for (3.1) is a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq201_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ45_HTML.gif
(3.4)

Now, we present the following theorem about the existence of solution for problem (3.1) in presence of a lower solution.

Theorem 3.2.

Consider problem (3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq202_HTML.gif continuous and suppose that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq203_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ46_HTML.gif
(3.5)
such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq204_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq205_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ47_HTML.gif
(3.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq206_HTML.gif . Then the existence of a lower solution for (3.1) provides the existence of a unique solution of (3.1).

Proof.

Problem (3.1) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ48_HTML.gif
(3.7)
This problem is equivalent to the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ49_HTML.gif
(3.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq207_HTML.gif is the Green function given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ50_HTML.gif
(3.9)
Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq208_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ51_HTML.gif
(3.10)

Notice that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq209_HTML.gif is a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq210_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq211_HTML.gif is a solution of (3.1).

In the sequel, we check that hypotheses in Theorem 2.4 are satisfied.

The mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq212_HTML.gif is nondecreasing since, by hypothesis, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq213_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ52_HTML.gif
(3.11)
and this implies, taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq214_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq215_HTML.gif , that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ53_HTML.gif
(3.12)
Besides, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq216_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ54_HTML.gif
(3.13)
Using the Cauchy-Schwarz inequality in the last integral, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ55_HTML.gif
(3.14)
The first integral gives us
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ56_HTML.gif
(3.15)
As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq217_HTML.gif is nondecreasing, the second integral in (3.14) can be estimated by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ57_HTML.gif
(3.16)
Taking into account (3.14), (3.15), and (3.16), from (3.13) we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ58_HTML.gif
(3.17)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq218_HTML.gif , the last inequality gives us
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ59_HTML.gif
(3.18)
or, equivalently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ60_HTML.gif
(3.19)
This implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ61_HTML.gif
(3.20)
Putting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq219_HTML.gif , which is an altering function, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq220_HTML.gif because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq221_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ62_HTML.gif
(3.21)

This proves that the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq222_HTML.gif satisfies condition (2.2) of Theorem 2.2.

Finally, letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq223_HTML.gif be a lower solution for (3.1), we claim that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq224_HTML.gif

In fact,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ63_HTML.gif
(3.22)
Multiplying by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq225_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ64_HTML.gif
(3.23)
and this gives us
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ65_HTML.gif
(3.24)
As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq226_HTML.gif , the last inequality implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ66_HTML.gif
(3.25)
and so
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ67_HTML.gif
(3.26)
This and (3.24) give us
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ68_HTML.gif
(3.27)
and, consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ69_HTML.gif
(3.28)

Finally, Theorem 2.4 gives that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq227_HTML.gif has a unique fixed point.

Remark 3.3.

Notice that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq228_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq229_HTML.gif . In fact, as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq230_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq231_HTML.gif is nondecreasing and, consequently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq232_HTML.gif is also nondecreasing.

Moreover, as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq233_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq234_HTML.gif , and, thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq235_HTML.gif .

Finally, as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq236_HTML.gif , and as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq237_HTML.gif , then it is easily seen that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq238_HTML.gif .

Example 3.4.

Consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq239_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ70_HTML.gif
(3.29)

It is easily seen that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq240_HTML.gif . Taking into account Remark 3.3, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq241_HTML.gif .

Now, we consider problem (3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq242_HTML.gif continuous and suppose that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq243_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ71_HTML.gif
(3.30)
such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq244_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq245_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_Equ72_HTML.gif
(3.31)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq246_HTML.gif is the function above mentioned.

This example can be treated by our Theorem 3.2 but it cannot be covered by the results of [6] because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq247_HTML.gif is not increasing.

Declarations

Acknowledgments

This research was partially supported by "Ministerio de Educación y Ciencia", Project MTM 2007/65706. This work is dedicated to Professor W. Takahashi on the occasion of his retirement.

Authors’ Affiliations

(1)
Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria

References

  1. Agarwal RP, El-Gebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Applicable Analysis 2008,87(1):109–116. 10.1080/00036810701556151MathSciNetView ArticleMATHGoogle Scholar
  2. Amini-Harandi A, Emami H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications 2010,72(5):2238–2242. 10.1016/j.na.2009.10.023MathSciNetView ArticleMATHGoogle Scholar
  3. Burgić D, Kalabušić S, Kulenović MRS: Global attractivity results for mixed-monotone mappings in partially ordered complete metric spaces. Fixed Point Theory and Applications 2009, 2009:-17.Google Scholar
  4. Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications 2008, 2008:-11.Google Scholar
  5. Gnana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis: Theory, Methods & Applications 2006,65(7):1379–1393. 10.1016/j.na.2005.10.017MathSciNetView ArticleMATHGoogle Scholar
  6. Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):3403–3410. 10.1016/j.na.2009.01.240MathSciNetView ArticleMATHGoogle Scholar
  7. Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):3403–3410. 10.1016/j.na.2009.01.240MathSciNetView ArticleMATHGoogle Scholar
  8. Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(12):4341–4349. 10.1016/j.na.2008.09.020MathSciNetView ArticleMATHGoogle Scholar
  9. Nieto JJ, Rodríguez-López R: Existence of extremal solutions for quadratic fuzzy equations. Fixed Point Theory and Applications 2005,2005(3):321–342. 10.1155/FPTA.2005.321View ArticleMathSciNetMATHGoogle Scholar
  10. Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005,22(3):223–239. 10.1007/s11083-005-9018-5MathSciNetView ArticleMATHGoogle Scholar
  11. Nieto JJ, Rodríguez-López R: Applications of contractive-like mapping principles to fuzzy equations. Revista Matemática Complutense 2006,19(2):361–383.View ArticleMathSciNetMATHGoogle Scholar
  12. Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces. Proceedings of the American Mathematical Society 2007,135(8):2505–2517. 10.1090/S0002-9939-07-08729-1MathSciNetView ArticleMATHGoogle Scholar
  13. Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Mathematica Sinica 2007,23(12):2205–2212. 10.1007/s10114-005-0769-0MathSciNetView ArticleMATHGoogle Scholar
  14. O'Regan D, Petruşel A: Fixed point theorems for generalized contractions in ordered metric spaces. Journal of Mathematical Analysis and Applications 2008,341(2):1241–1252. 10.1016/j.jmaa.2007.11.026MathSciNetView ArticleMATHGoogle Scholar
  15. Petruşel A, Rus IA: Fixed point theorems in ordered https://static-content.springer.com/image/art%3A10.1155%2F2010%2F916064/MediaObjects/13663_2009_Article_1365_IEq248_HTML.gif -spaces. Proceedings of the American Mathematical Society 2006,134(2):411–418.MathSciNetView ArticleMATHGoogle Scholar
  16. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society 2004,132(5):1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetView ArticleMATHGoogle Scholar
  17. Wu Y: New fixed point theorems and applications of mixed monotone operator. Journal of Mathematical Analysis and Applications 2008,341(2):883–893. 10.1016/j.jmaa.2007.10.063MathSciNetView ArticleMATHGoogle Scholar
  18. Cabada A, Nieto JJ: Fixed points and approximate solutions for nonlinear operator equations. Journal of Computational and Applied Mathematics 2000,113(1–2):17–25. 10.1016/S0377-0427(99)00240-XMathSciNetView ArticleMATHGoogle Scholar
  19. Geraghty MA: On contractive mappings. Proceedings of the American Mathematical Society 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5MathSciNetView ArticleMATHGoogle Scholar
  20. Babu GVR, Lalitha B, Sandhya ML: Common fixed point theorems involving two generalized altering distance functions in four variables. Proceedings of the Jangjeon Mathematical Society 2007,10(1):83–93.MathSciNetMATHGoogle Scholar
  21. Naidu SVR: Some fixed point theorems in metric spaces by altering distances. Czechoslovak Mathematical Journal 2003,53(1):205–212. 10.1023/A:1022991929004MathSciNetView ArticleMATHGoogle Scholar
  22. Sastry KPR, Babu GVR: Some fixed point theorems by altering distances between the points. Indian Journal of Pure and Applied Mathematics 1999,30(6):641–647.MathSciNetMATHGoogle Scholar

Copyright

© J. Caballero et al. 2010

This article is published under license to BioMed Central Ltd. 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.