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 Open Access
On almost contractions in partially ordered metric spaces via implicit relations
 Uğur Gül^{1}Email author and
 Erdal Karapınar^{2}
https://doi.org/10.1186/1029242X2012217
© Gül and Karapınar; licensee Springer 2012
 Received: 28 June 2012
 Accepted: 19 September 2012
 Published: 2 October 2012
Abstract
In this paper, we prove general fixed point theorems for selfmaps of a partially ordered complete metric space which satisfy an implicit type relation. Our method relies on constructive arguments involving Picard type iteration processes and our uniqueness result uses comparability arguments. Our results generalize a multitude of fixed point theorems in the literature to the context of partially ordered metric spaces.
Keywords
 Point Theorem
 Fixed Point Theorem
 Triangle Inequality
 Cauchy Sequence
 Unique Fixed Point
1 Introduction
Fixed point theorems in nonlinear analysis have become indispensable tools in a vast area of the analysis ranging from proving the existence of solutions of certain partial differential equations to nonlinear optimization and related fields (see, for instance, [1]). Having their origin in the classical paper of Stefan Banach [2] as the ‘Banach Contraction Mapping Theorem’ (which is by now so classical that it appears in almost every book on Functional Analysis), fixed point theorems have attracted a lot of attention during the past five decades. This is mainly due to the fact that they have found many applications to the problems in applied mathematics such as boundary value problems in differential equations. The ‘Banach Contraction Mapping Theorem’ was generalized by many authors to mappings that satisfy much weaker conditions (see, for instance, [3–10]). Banach’s theorem was also extended to mappings which have an invariant subset that is finite, namely that have ‘periodic points’ [11, 12]. Another direction where the theorem was extended is for more than one mapping which have common fixed points [13–15]. In recent years, Banach’s theorem was extended in part to partially ordered metric spaces by Ran and Reuring [16] in order to obtain a solution of a matrix equation. Nieto and López [17] generalized the result of Ran and Reuring by removing the continuity condition of the mapping. They applied their result to get a solution of a boundary value problem. The efficiency of these kind of extensions of fixed point theorems in such kind of problems, as it is well known, is due to the fact that most real valued function spaces are partially ordered metric spaces.
We underline that for a lower semicontinuous mapping ϕ, the function $\mathrm{\Phi}(u)=u\varphi (u)$ coincides with Boyd and Wong types. These two notions, Φcontraction and weak ϕcontraction, have been studied heavily by many authors in fixed point theory (see, e.g., [20–30]).
Our aim in this paper is to obtain fixed point theorems for mappings acting on partially ordered complete metric spaces which satisfy certain implicit relations. There are indeed fixed point theorems for mappings satisfying such kind of relation in the literature; however, all of these fixed point theorems are on complete metric spaces that are not partially ordered spaces. The novelty of this work lies in generalizing these fixed point theorems to partially ordered spaces.
Throughout this paper, $(X,d,\u2aaf)$ denotes a partially ordered metric space where $(X,\u2aaf)$ is a partially ordered set and $(X,d)$ is a metric space for a given metric d on X. A partially ordered metric space $(X,d,\u2aaf)$ is called regular, if for each convergent sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}\subset X$, the following condition holds: either

if $\{{x}_{n}\}$ is a nonincreasing sequence in X such that ${x}_{n}\to {x}^{\ast}$ implies ${x}^{\ast}\u2aaf{x}_{n}$$\mathrm{\forall}n\in \mathbb{N}$,
or

if $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to {x}^{\ast}$ implies ${x}_{n}\u2aaf{x}^{\ast}$$\mathrm{\forall}n\in \mathbb{N}$.
Let Φ be the class of all strictly increasing lower semicontinuous functions $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $\varphi (0)=0$. Let $\mathbb{F}$ denote the class of all implicit continuous functions $F:{({\mathbb{R}}^{+})}^{6}\to \mathbb{R}$. We shall consider the following subclasses of $\mathbb{F}$:
(${\mathbb{F}}_{1}$) $F\in \mathbb{F}$ is nonincreasing in the fifth variable, and $F(u,v,v,u,u+v,0)\le 0$ for $u,v>0$ implies that there exists a function $\varphi \in \mathrm{\Phi}$ such that $u\le v\varphi (v)$.
(${\mathbb{F}}_{2}$) $F\in \mathbb{F}$ is nonincreasing in the fifth variable, and $F(u,v,v,u,u+v,0)\le 0$ for $u,v>0\u27f9\mathrm{\exists}k\in [0,1)$ such that $u\le kv$.
(${\mathbb{F}}_{3}$) $F\in \mathbb{F}$ is nonincreasing in the fourth variable, and $F(u,v,0,u+v,u,v)\le 0$ for $u,v>0\u27f9\mathrm{\exists}k\in [0,1)$ such that $u\le kv$.
(${\mathbb{F}}_{4}$) $F\in \mathbb{F}$ is nonincreasing in the third variable, and $F(u,v,u+v,0,v,u)\le 0$ for $u,v>0\u27f9\mathrm{\exists}k\in [0,1)$ such that $u\le kv$.
(${\mathbb{F}}_{5}$) $F\in \mathbb{F}$ such that $F(u,u,0,0,u,u)>0$ for all $u>0$.
See [31, 32] for examples of functions $F\in \mathbb{F}$ satisfying the above conditions ${\mathbb{F}}_{1}$${\mathbb{F}}_{5}$.
2 Main results
We start this section with the first main result.
where$F\in {\mathbb{F}}_{1}\cap {\mathbb{F}}_{5}$. Then T has a fixed point.
□
We remove the continuity condition of the mapping T in Theorem 1 by replacing the condition that X is regular.
where$F\in {\mathbb{F}}_{1}\cap {\mathbb{F}}_{5}$. Then T has a fixed point.
Proof Following the line in the proof of Theorem 1, we get a Cauchy sequence $\{{x}_{n}\}$ as it is defined above. Since X is a complete metric space, we have $lim{x}_{n}={x}^{\ast}\in X$. Since $(X,d)$ is regular, we have ${x}^{\ast}\u2aaf{x}_{n}$$\mathrm{\forall}n\in \mathbb{N}$.
Hence, by ${\mathbb{F}}_{5}$, we have $d({x}^{\ast},T{x}^{\ast})\le 0$, which implies that ${x}^{\ast}=T{x}^{\ast}$. □
We generalize the main result of Berinde [32] in the framework of partially ordered metric spaces.
where$F\in {\mathbb{F}}_{2}$. Then
(p1) $Fix(T)\ne \mathrm{\varnothing}$;
(p2) for any${x}_{0}\in X$, the Picard iteration${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$converges to a fixed point${x}^{\ast}\in X$of T;
and this proves (p1).
(p2): follows by the proof of (p1).
(p3): follows by equation (2.30).
Again, by assumption ${\mathbb{F}}_{3}$, this implies that $\mathrm{\exists}k\in [0,1)$ such that $d({x}_{n+1},{x}^{\ast})\le kd({x}_{n},{x}^{\ast})$. □
Remark 4 Let ${F}_{1}\in \mathbb{F}$ such that ${F}_{1}({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{1}k{t}_{2}$ where $k\in [0,1)$. If we take $F={F}_{1}$ in Theorem 3, then we get the main result of Ran and Reurings (Theorem 2.1 of [16]).
We get the same results by removing the continuity condition of the mapping T in Theorem 3 and by adding the condition that X is regular.
where$F\in {\mathbb{F}}_{2}$. Then
(p1) $Fix(T)\ne \mathrm{\varnothing}$;
(p2) for any${x}_{0}\in X$, the Picard iteration${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$converges to a fixed point${x}^{\ast}\in X$of T;
Hence, by ${\mathbb{F}}_{2}$, we have $d({x}^{\ast},T{x}^{\ast})\le 0$, which implies that ${x}^{\ast}=T{x}^{\ast}$ and this proves (p1).
The rest of the proof is the same as the proof of Theorem 3. □
Remark 6 Let ${F}_{1}\in \mathbb{F}$ such that ${F}_{1}({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{1}k{t}_{2}$ where $k\in [0,1)$. If we take $F={F}_{1}$ in Theorem 5, then we get the main result of Nieto and RodríguezLópez (Theorem 2.2 of [17]).
Remark 7 If we take $\varphi (t)=kt$ in Theorem 1 (respectively, Theorem 2) where $k\in [0,1)$ we get (p1) of Theorem 3 (respectively, Theorem 5).
3 Uniqueness of a fixed point
In this section, we investigate the uniqueness of fixed points in the theorems above. In order to assure the uniqueness of fixed points, we need the following notion on the partially ordered metric space $(X,\u2aaf)$ which is called the comparability condition:
(C) For every $x,y\in X$, there exists $z\in X$ such that either $x\u2aafz$ and $y\u2aafz$ or $z\u2aafx$ and $z\u2aafy$.
We also require the following condition:
(${\mathbb{F}}_{6}$) $F\in \mathbb{F}$ is nonincreasing in the fourth variable and such that $F(u,v,0,u+v,u,v)\ge 0$ for all $u,v>0$.
Adding condition (C) and ${\mathbb{F}}_{6}$ to the hypotheses of Theorem 1, we obtain the uniqueness of the fixed point:
where$F\in {\mathbb{F}}_{1}\cap {\mathbb{F}}_{5}\cap {\mathbb{F}}_{6}$. Then T has a unique fixed point.
Proof Due to Theorem 1, we guarantee that T has a fixed point. Suppose x and y are fixed points of T with $x\ne y$.
We need to examine two cases:
which contradicts ${\mathbb{F}}_{5}$. Hence $x=y$.
which contradicts ${\mathbb{F}}_{6}$. Hence, we have $x=y$. □
Adding condition (C) and ${\mathbb{F}}_{6}$ to the hypotheses of Theorem 2, we obtain the uniqueness of the fixed point.
where$F\in {\mathbb{F}}_{1}\cap {\mathbb{F}}_{5}\cap {\mathbb{F}}_{6}$. Then T has a unique fixed point.
Proof The proof is the same as the proof of Theorem 8. □
Adding condition (C) to the hypotheses of Theorem 3, we obtain the uniqueness of the fixed point:
where$F\in {\mathbb{F}}_{2}\cap {\mathbb{F}}_{6}$. Then
(p1) T has a unique fixed point;
(p2) for any${x}_{0}\in X$, the Picard iteration${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$converges to a fixed point${x}^{\ast}\in X$of T;
Proof The proof is the same as the proof of Theorem 8. □
Adding condition (C) and ${\mathbb{F}}_{6}$ to the hypotheses of Theorem 5, we obtain the uniqueness of a fixed point:
where$F\in {\mathbb{F}}_{2}\cap {\mathbb{F}}_{6}$. Then
(p1) T has a unique fixed point;
(p2) for any${x}_{0}\in X$, the Picard iteration${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$converges to a fixed point${x}^{\ast}\in X$of T;
Proof The proof is the same as the proof of Theorem 8. □
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable suggestions on improving the text.
Authors’ Affiliations
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