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
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
References
 Cherichi M, Samet B: Fixed point theorems on ordered gauge spaces with applications to nonlinear integral equations. Fixed Point Theory Appl. 2012., 2012: Article ID 13Google Scholar
 Banach S: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundam. Math. 1922, 3: 133–181.MATHGoogle Scholar
 Hardy GC, Rogers T: A generalization of fixed point theorem of Reich. Can. Math. Bull. 1973, 16: 201–206. 10.4153/CMB19730360MathSciNetView ArticleMATHGoogle Scholar
 Kannan R: Some results on fixed points  II. Am. Math. Mon. 1969, 76: 71–76.View ArticleMathSciNetMATHGoogle Scholar
 Reich S: Some remarks concerning contraction mappings. Can. Math. Bull. 1971, 14: 121–124. 10.4153/CMB19710249View ArticleMathSciNetMATHGoogle Scholar
 Chu SC, Diaz JB: Remarks on a generalization of Banach’s mappings. J. Math. Anal. Appl. 1965, 11: 440–446.MathSciNetView ArticleMATHGoogle Scholar
 Bonsall FF: Lectures on Some Fixed Point Theorems of Functional Analysis. Tata Institute of Fundamental Research, Bombay; 1962.Google Scholar
 Smart DR: Fixed Point Theorems. Cambridge University Press, Cambridge; 1974.MATHGoogle Scholar
 Sehgal VM: A fixed point theorem for mappings with a contractive iterate. Proc. Am. Math. Soc. 1969, 23: 631–634. 10.1090/S0002993919690250292XMathSciNetView ArticleMATHGoogle Scholar
 Reich S: Kannan’s fixed point theorem. Boll. Unione Mat. Ital. 1971, 4: 1–11.MATHGoogle Scholar
 Edelstein M: On fixed points and periodic points under contraction mappings. J. Lond. Math. Soc. 1962, 37: 74–79. 10.1112/jlms/s137.1.74MathSciNetView ArticleMATHGoogle Scholar
 Holmes RD: On fixed and periodic points under sets of mappings. Can. Math. Bull. 1969, 12: 813–822. 10.4153/CMB19691061View ArticleMathSciNetMATHGoogle Scholar
 Nadler SB Jr.: Sequences of contractions and fixed points. Pac. J. Math. 1968, 27: 579–585. 10.2140/pjm.1968.27.579MathSciNetView ArticleMATHGoogle Scholar
 Wong CS: Common fixed points of two mappings. Pac. J. Math. 1973, 48: 299–312. 10.2140/pjm.1973.48.299View ArticleMathSciNetMATHGoogle Scholar
 Samet B: Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026MathSciNetView ArticleMATHGoogle Scholar
 Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002993903072204MathSciNetView ArticleMATHGoogle Scholar
 Nieto JJ, RodríguezLópez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22(3):223–239. 10.1007/s1108300590185MathSciNetView ArticleMATHGoogle Scholar
 Alber YI, GuerreDelabriere S: Principle of weakly contractive maps in Hilbert space. 98. In New Results in Operator Theory, Advances and Its Applications. Edited by: Gohberg I, Lyubich Y. Birkhäuser, Basel; 1997:7–22.View ArticleGoogle Scholar
 Boyd DW, Wong SW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S00029939196902395599MathSciNetView ArticleMATHGoogle Scholar
 Karapınar E: Fixed point theory for cyclic weak ϕ contraction. Appl. Math. Lett. 2011, 24: 822–825. 10.1016/j.aml.2010.12.016MathSciNetView ArticleMATHGoogle Scholar
 Kirk WA, Srinavasan PS, Veeramani P: Fixed points for mapping satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4: 79–89.MathSciNetMATHGoogle Scholar
 Petruşel G: Cyclic representations and periodic points. Stud. Univ. BabeşBolyai, Math. 2005, L(3):107–112.MathSciNetMATHGoogle Scholar
 Rus IA: Cyclic representations and fixed points. Ann. T. Popoviciu Semin. Funct. Equ. Approx. Convexity 2005, 3: 171–178.Google Scholar
 Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal. 2001, 47(4):2683–2693. 10.1016/S0362546X(01)003881MathSciNetView ArticleMATHGoogle Scholar
 Păcurar M, Rus IA: Fixed point theory for cyclic φ contractions. Nonlinear Anal. 2010, 72(3–4):1181–1187. 10.1016/j.na.2009.08.002MathSciNetView ArticleMATHGoogle Scholar
 Hussain N, Jungck G:Common fixed point and invariant approximation results for noncommuting generalized $(f,g)$nonexpansive maps. J. Math. Anal. Appl. 2006, 321: 851–861. 10.1016/j.jmaa.2005.08.045MathSciNetView ArticleMATHGoogle Scholar
 Song Y: Coincidence points for noncommuting f weakly contractive mappings. Int. J. Comput. Appl. Math. 2007, 2(1):51–57.MathSciNetMATHGoogle Scholar
 Song Y, Xu S: A note on common fixedpoints for Banach operator pairs. Int. J. Contemp. Math. Sci. 2007, 2: 1163–1166.MathSciNetMATHGoogle Scholar
 Zhang Q, Song Y: Fixed point theory for generalized φ weak contractions. Appl. Math. Lett. 2009, 22(1):75–78. 10.1016/j.aml.2008.02.007MathSciNetView ArticleMATHGoogle Scholar
 Abdeljawad T, Karapınar E: Quasicone metric spaces and generalizations of Caristi Kirk’s theorem. Fixed Point Theory Appl. 2009. doi:10.1155/2009/574387Google Scholar
 Berinde V 22. In Contractii generalizate si aplicatii. Editura Cub Press, Baia Mare; 2007.Google Scholar
 Berinde V: Approximating fixed point of almost contractions. Hacet. J. Math. Stat. 2012, 41(1):93–102.MathSciNetMATHGoogle Scholar
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