Fixed point theorems for mixed monotone operators and applications to integral equations

doi:10.1016/j.na.2010.10.047

Nonlinear Analysis: Theory, Methods & Applications

Volume 74, Issue 5, 1 March 2011, Pages 1749-1760

https://doi.org/10.1016/j.na.2010.10.047 Get rights and content

Abstract

The purpose of this paper is to present some coupled fixed point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions. We also present an application to integral equations.

Section snippets

Introduction and background

Mixed monotone operators were introduce by Guo and Lakshmikantham in [1]. Their study has not only important theoretical meaning but also wide applications in engineering, nuclear physics, biological chemistry technology, etc. (see [1], [2], [3], [4], [5], [6]).

The purpose of this paper is to present some coupled fixed point theorems for a mixed monotone operator in the context of ordered metric spaces involving altering distance functions. These theorems are generalizations of the results of

Coupled fixed point theorems

Let $(X, \leq)$ be a partially ordered set and $d$ a metric on $X$ such that $(X, d)$ is a complete metric space. Further, we consider in the product space $X \times X$ the following partial order: $if (x, y), (u, v) \in X \times X, (x, y) \leq (u, v) ⟺ x \leq u and y \geq v .$ Now, we present the following theorem which is a version of Theorem 2.1 of [12] in the context of mappings with the mixed monotone property.

Theorem 2

Let $(X, \leq)$ be a partially ordered set and suppose that there exists a metric $d$ in $X$ such that $(X, d)$ is a complete metric space. Let $F : X \times X ⟶ X$

Application to integral equations

In this section we study the existence and uniqueness of solutions of a nonlinear integral equation using the results proved in Section 2.

Consider the following integral equation: $x (t) = \int_{0}^{1} (k_{1} (t, s) + k_{2} (t, s)) (f (s, x (s)) + g (s, x (s))) d s + a (t), t \in [0, 1] .$ We will analyze Eq. (28) under the following assumptions:

(i)
$k_{i} : [0, 1] \times [0, 1] ⟶ R$ $(i = 1, 2)$ are continuous and $k_{1} (t, s) \geq 0$ and $k_{2} (t, s) \leq 0$ .
(ii)
$a \in C [0, 1]$ .
(iii)
$f, g : [0, 1] \times R ⟶ R$ are continuous functions.
(iv)
There exist constants $λ, μ > 0$ such that for all $x, y \in R$ and $x \geq y$ $0 \leq f (t, x) - f (t, y) \leq λ \sqrt{ln [(y - x)}$

Acknowledgement

This research was partially supported by “Ministerio de Educación y Ciencia”, Project MTM 2007/65706.

References (25)

D. Guo et al.
Coupled fixed points of nonlinear operators with applications
Nonlinear Anal.
(1987)
Y. Sun
A fixed point theorem for mixed monotone operator with applications
J. Math. Anal. Appl.
(1991)
T.G. Bhaskar et al.
Fixed point theorems in partially ordered metric spaces and applications
Nonlinear Anal.
(2006)
J. Harjani et al.
Fixed point theorems for weakly contractive mappings in partially ordered sets
Nonlinear Anal.
(2009)
J. Harjani et al.
Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations
Nonlinear Anal.
(2010)
V. Lakshmikantham et al.
Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces
Nonlinear Anal.
(2009)
D. O’Regan et al.
Fixed point theorems for generalized contractions in ordered metric spaces
J. Math. Anal. Appl.
(2008)
A. Cabada et al.
Fixed points and approximate solutions for nonlinear operator equations
J. Comput. Appl. Math.
(2000)
D. Guo et al.
Nonlinear Problems in Abstract Cones
(1988)
D. Guo
Existence and uniqueness of positive fixed point for mixed monotone operators with applications
Appl. Anal.
(1992)

Z. Zhang

New fixed point theorems of mixed monotone operators and applications

J. Math. Anal. Appl.

(1996)

S.S. Zhang et al.

Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solution for a class of functional equations arising in dynamic programming

J. Math. Anal. Appl.

(1991)

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Fixed point theorems for mixed monotone operators and applications to integral equations

Abstract

Section snippets

Introduction and background

Coupled fixed point theorems

Application to integral equations

Acknowledgement

Nonlinear Anal.

J. Math. Anal. Appl.

Nonlinear Anal.

Nonlinear Anal.

Nonlinear Anal.

Nonlinear Anal.

J. Math. Anal. Appl.

J. Comput. Appl. Math.

Nonlinear Problems in Abstract Cones

Existence and uniqueness of positive fixed point for mixed monotone operators with applications

Appl. Anal.

New fixed point theorems of mixed monotone operators and applications

J. Math. Anal. Appl.

Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solution for a class of functional equations arising in dynamic programming

J. Math. Anal. Appl.