Coupled common fixed point theorems for mixed weakly monotone mappings in partially ordered metric spaces
- Madjid Eshaghi Gordji1Email author,
- Esmat Akbartabar1,
- Yeol Je Cho2Email author and
- Maryam Ramezani1
https://doi.org/10.1186/1687-1812-2012-95
© Gordji et al; licensee Springer. 2012
Received: 26 September 2011
Accepted: 12 June 2012
Published: 12 June 2012
Abstract
In this paper, we introduce the concept of a mixed weakly monotone pair of mappings and prove some coupled common fixed point theorems for a contractive-type mappings with the mixed weakly monotone property in partially ordered metric spaces. Our results are generalizations of the main results of Bhaskar and Lakshmikantham and Kadelburg et al.
Mathematics Subject Classification 2000: 54H25.
Keywords
1. Introduction
In 1922, Banach gave a theorem, which is well-known as Banach's Fixed Point Theorem (or Banach's Contractive Principle) to establish the existence of solutions for nonlinear operator equations and integral equations. Since then, because of their simplicity and usefulness, it has become a very popular tools in solving the existence problems in many branches of mathematical analysis. Since then, many authors have extended, improved and generalized Banach's theorem in several ways [1–11].
Recently, the existence of coupled fixed points for some kinds of contractive-type mappings in partially ordered metric spaces, (ordered) cone metric spaces, fuzzy metric spaces and other spaces with applications has been investigated by some authors, for example, Bhaskar and Lakshmikantham [5], Cho et al. [12–14], Dhage et al. [15], Gordji et al. [16, 17], Kadelburg et al. [18], Nieto and Lopez [10], Ran and Rarings [11], Sintunavarat et al. [19, 20], Yang et al. [21] and others.
Especially, in [5], Bhaskar and Lakshmikantham introduced the notions of a mixed monotone mapping and a coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings and discussed the existence and uniqueness of solution for periodic boundary value problems.
Definition 1.2. [5] An element (x, y) ∈ X × X is called a coupled fixed point of a mapping F: X × X → X if x = F (x, y) and y = F (y, x).
- (1)
f is continuous or
- (2)
X has the following properties:
- (a)
if {x n } is an increasing sequence with x n → x, then x n ≤ x for all n ≥ 1;
- (b)
if {y n } is a decreasing sequence y n → y, then y n ≥ y for all n ≥ 1.
If there exist x0, y0 ∈ X such that x0 ≤ f(x0, y0) and y0 ≥ f(y0, x0), then f has a coupled fixed point in X.
Very recently, Kadelburg et al. [18] proved the following theorem on cone metric spaces.
- (1)there exist p, q, r, s, t ≥ 0 satisfying p + q + r + s + t < 1 and q = r or s = t such that
- (2)
f or g is continuous or, if a nondecreasing {x n } converges to a point x ∈ X, then x n ≤ x for all n ≥ 1.
Then f and g have a common fixed point in X.
Note that a pair (f, g) of self-mappings on partially ordered set (X, ≤) is said to be weakly increasing if fx ≤ gfx and gx ≤ fgx for all x ∈ X.
Now, we introduce the following concept of the mixed weakly increasing property of mappings.
Then a pair (f, g) has the mixed weakly monotone property.
The purpose of this paper is to present some coupled common fixed point theorems for a pair of mappings with the mixed weakly monotone property in a partially ordered metric space. Our results generalize the main results of Bhaskar and Lakshmikantham [5], Kadelburg et al. [18] and others.
2. Coupled common fixed point theorems
for all (x, y), (u, v) ∈ X × X.
for all x, y, u, v ∈ X with x ≤ u and y ≥ v. Let x0, y0 ∈ X be such that x0 ≤ f(x0, y0), y0 ≥ f(y0, x0) or x0 ≤ g(x0, y0), y0 ≥ g(y0, x0). If f or g is continuous, then f and g have a coupled common fixed point in X.
as n → ∞, which implies that d(x n , x m ) → 0 and d(y n , y m ) → 0 as m, n → ∞. Therefore, the sequences {x n } and {y n } are Cauchy sequences in X. Since (X, d) is a complete metric space, then there exist x, y ∈ X such that x n → x and y n → y as n → ∞.
Hence (x, y) is a coupled common fixed point of f and g.
Similarly, we can prove that (x, y) is a coupled common fixed point of f and g when g is a continuous mapping. This completes the proof. □
- (1)
if {x n } is an increasing sequence with x n → x, then x n ≤ x for all n ≥ 1;
- (2)
if {y n } is a decreasing sequence with y n → y, then y n ≥ y for all n ≥ 1.
for all x, y, u, v ∈ X with x ≤ u and y ≥ v. If there exist x0, y0 ∈ X such that x0 ≤ f(x0, y0), y0 ≥ f(y0, x0) or x0 ≤ g(x0, y0), y0 ≥ g(y0, x0), then f and g have a coupled common fixed point in X.
and so f(x, y) = x and f(y, x) = y. Similarly, we can show that g(x, y) = x and g(y, x) = y. Therefore, (x, y) is a coupled common fixed point of f and g. This completes the proof. □
Now, we give an example to illustrate Theorem 2.1 as follows:
By putting and q = r = s = 0 in (2.1), we see that (1, 1) is a unique coupled common fixed point of f and g.
Corollary 2.4. In Theorems 2.1 and 2.2, if X is a total ordered set, then a coupled common fixed point of f and g is unique and x = y.
Since q + 2s < 1, we have d(x, x*) + d(y, y*) = 0, which implies that x = x* and y = y*.
Since p + 2s < 1, we have d(x, y) = 0 and x = y. This completes the proof. □
for all x, y ∈ X and n ≥ 1.
which implies that the pair (f n , f n ) has the mixed weakly monotone property on X.
- (1)
f is continuous or
- (2)
X has the following properties:
- (a)
if {x n } is an increasing sequence with x n → x, then x n ≤ x for all n ≥ 1;
- (b)
if {y n } is a decreasing sequence with y n → y, then y n ≥ y for all n ≥ 1.
If there exist x0, y0 ∈ X such that x0 ≤ f(x0, y0) and y0 ≥ f(y0, x0), then f has a coupled fixed point in X.
Proof. Taking f = g in Theorems 2.1, 2.2 and using Remark 2.5, we can get the conclusion. □
- (1)
f is continuous or
- (2)
X has the following properties:
- (a)
if {x n } is an increasing sequence with x n → x, then x n ≤ x for all n ≥ 1;
- (b)
if {y n } is a decreasing sequence with y n → y, then y n ≥ y for all n ≥ 1.
If there exist x0, y0 ∈ X such that x0 ≤ f(x0, y0) and y0 ≥ f(y0, x0), then f has a coupled fixed point in X.
Proof. Taking f = g, p = k and q = r = s = 0 in Theorems 2.1, 2.2 and using Remark 2.5, we can get the conclusion. □
Declarations
Acknowledgements
The authors would like to thank anonymous referees for their valuable comments and suggestions. This research was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2011-0021821).
Authors’ Affiliations
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