Abstract
In this article we obtain, in the setting of metric spaces or ordered metric spaces, coincidence point, and common fixed point theorems for self-mappings in a general class of contractions defined by an implicit relation. Our results unify, extend, generalize many related common fixed point theorems from the literature.
Mathematics Subject Classification (2000): 47H10, 54H25.
Similar content being viewed by others
Introduction and preliminaries
It is well known that the contraction mapping principle, formulated and proved in the Ph.D. dissertation of Banach in 1920, which was published in 1922 [1], is one of the most important theorems in classical functional analysis. The study of fixed and common fixed points of mappings satisfying a certain metrical contractive condition attracted many researchers, see for example [2, 3] and for existence results for fixed points of contractive non-self-mappings, see [4–6]. Among these (common) fixed point theorems, only a few give a constructive method for finding the fixed points or the common fixed points of the mappings involved. Berinde in [7–15] obtained (common) fixed point theorems, which were called constructive (common) fixed point theorems, see [12]. These results have been obtained by considering self-mappings that satisfy an explicit contractive-type condition. On the other hand, several classical fixed point theorems and common fixed point theorems have been recently unified by considering general contractive conditions expressed by an implicit relation, see Popa [16, 17] and Ali and Imdad [18]. Following Popa's approach, many results on fixed point, common fixed point and coincidence point has been obtained, in various ambient spaces, see [16–25] and references therein.
In [21], Berinde obtained some constructive fixed point theorems for almost contractions satisfying an implicit relation. These results unify, extend, generalize related results (see [2, 3, 7–16, 21, 25–38]).
In this article we obtain, in the setting of metric spaces or ordered metric spaces, coincidence point, and common fixed point results for self-mappings in a general class of contractions defined by an implicit relation. Our results unify, extend, generalize many of related common fixed point theorems from literature.
Let X be a non-empty set and f, T: X → X. A point x ∈ X is called a coincidence point of f and T if Tx = fx. The mappings f and T are said to be weakly compatible if they commute at their coincidence point (i.e., Tfx = fTx whenever Tx = fx). Suppose TX ⊂ fX. For every x0 ∈ X we consider the sequence {x n } ⊂ X defined by fx n = Txn-1for all n ∈ ℕ, we say that {Tx n } is a T -f -sequence with initial point x0.
Let X be a non-empty set. If (X, d) is a metric space and (X, ≼) is partially ordered, then (X, d, ≼) is called an ordered metric space. Then, x, y ∈ X are called comparable if x ≼ y or y ≼ x holds. Let f, T: X → X be two mappings, T is said to be f -non-decreasing if fx ≼ fy implies Tx ≼ Ty for all x, y ∈ X. If f is the identity mapping on X, then T is non-decreasing.
Throughout this article the letters ℝ+ and ℕ will denote the set of all non-negative real numbers and the set of all positive integer numbers.
Fixed point theorems for mappings satisfying an implicit relation
A simple and natural way to unify and prove in a simple manner several metrical fixed point theorems is to consider an implicit contraction type condition instead of the usual explicit contractive conditions. Popa [16, 17] initiated this direction of research which produced so far a consistent literature (that cannot be completely cited here) on fixed point, common fixed point, and coincidence point theorems, for both single-valued and multi-valued mappings, in various ambient spaces; see the recent nice paper [21] of Berinde, for a partial list of references.
In [21], Berinde considered the family of all continuous real functions and the following conditions:
(F1a) F is non-increasing in the fifth variable and F (u, v, v, u, u + v, 0) ≤ 0 for u, v ≥ 0 implies that there exists h ∈ [0, 1) such that u ≤ hv;
(F1b) F is non-increasing in the fourth variable and F (u, v, 0, u + v, u, v) ≤ 0 for u, v ≥ 0 implies that there exists h ∈ [0, 1) such that u ≤ hv;
(F1c) F is non-increasing in the third variable and F (u, v, u+v, 0, v, u) ≤ 0 for u, v ≥ 0 implies that there exists h ∈ [0, 1) such that u ≤ hv;
(F2) F (u, u, 0, 0, u, u) > 0, for all u > 0.
He gave many examples of functions corresponding to well-known fixed point theorems and satisfying most of the conditions (F1a)-(F2) above, see Examples 1-11 of [21].
Example 1. The following functions satisfy properties F2 and F1a-F1c(see Examples 1-6, 9, and 11 of [21]).
-
(i)
F (t1, t2, t3, t4, t5, t6) = t1 − at2, where a ∈ [0, 1);
-
(ii)
F (t1, t2, t3, t4, t5, t6) = t1 − b(t3 + t4), where b ∈ [0, 1/ 2);
-
(iii)
F (t1, t2, t3, t4, t5, t6) = t1 − c(t5 + t6), where c ∈ [0, 1/ 2);
-
(iv)
, where a ∈ [0, 1);
-
(v)
F (t1, t2, t3, t4, t5, t6) = t1 − at2 − b(t3 + t4) − c(t5 + t6), where a, b, c ∈ [0, 1) and a + 2b + 2c < 1;
-
(vi)
, where a ∈ [0, 1);
-
(vii)
F (t1, t2, t3, t4, t5, t6) = t1 − at2 − L min{t3, t4, t5, t6}, where a ∈ [0, 1);
-
(viii)
, where a ∈ [0, 1) and L ≥ 0.
Example 2. The function , given by
where a ∈ [0, 1/ 2) satisfies properties F2 and F1a-F1cwith .
Motivated by [21], the following theorem is one of the main results in this article.
Theorem 1. Let (X, d) be a metric space and T, f: X → X be self-mappings such that TX ⊆ fX. Assume that there exists , satisfying (F1a), such that for all x, y ∈ X
If fX is a complete subspace of X, then T and f have a coincidence point. Moreover, if T and f are weakly compatible and F satisfies also F2, then T and f have a unique common fixed point. Further, for any x0 ∈ X, the T-f-sequence {Tx n } with initial point x0 converges to the common fixed point.
Proof. Let x0 ∈ X be an arbitrary point. As TX ⊆ fX, one can choose a T-f-sequence {Tx n } with initial point x0. If we take x = x n and y = xn+1in (1) and denote with u = d(Tx n , Txn+1) and v = d(Txn-1, Tx n ) we get that
By triangle inequality, d(Txn-1, Txn+1) ≤ d(Txn-1, Tx n ) + d(Tx n , Txn+1) = u + v and, since F is non-increasing in the fifth variable, we have
and hence, in view of assumption (F1a), there exists h ∈ [0, 1) such that u ≤ hv, i.e.,
By (2), in a straightforward way, we deduce that {Tx n } is a Cauchy sequence. Since fX is complete, there exist z, w ∈ X such that z = fw and
By taking x = x n and y = w in (1), we obtain that
As F is continuous, using (3) and letting n → +∞ in (4), we get
which, by assumption (F1a), yields d(fw, Tw) ≤ 0, i.e., fw = Tw = z. Thus, we have proved that T and f have a coincidence point.
Now, we assume that T and f are weakly compatible, then fz = fTw = Tfw = Tz.
We show that Tz = z = Tw.
Suppose d(Tz, Tw) > 0, by taking x = z and y = w in (1), we get
i.e.,
which is a contradiction by assumption (F2). This implies that d(Tz, Tw) = 0 and hence fz = Tz = Tw = z. So T and f have a common fixed point.
The uniqueness of the common fixed point is a consequence of assumption (F2). Clearly, for any x0 ∈ X, the T-f-sequence {Tx n } with initial point x0 converges to the unique common fixed point. □
Remark 1. From (2) we deduce the unifying error estimate
From this we get both the a priori estimate
and the a posteriori estimate
which are extremely important in applications, especially when approximating the solutions of nonlinear equations.
If f = I X from Theorem 1, we deduce the following result of fixed point for one self-mapping, see [21].
Corollary 1. Let (X, d) be a complete metric space and T: X → X. Assume that there exists, satisfying (F1a), such that for all x, y ∈ X
Then T has a fixed point. Moreover, if F satisfies also F2, then T has a unique fixed point. Further, for any x0 ∈ X, the Picard sequence {Tnx0} with initial point x0 converges to the fixed point.
Common fixed point in ordered metric spaces
The existence of fixed points in ordered metric spaces was investigated by Turinici [39], Ran and Reurings [40], Nieto and Rodríguez-López [41]. See, also [42–45], and references therein. A common fixed point result in ordered metric spaces for mappings satisfying implicit contractive conditions is given by the next theorem.
Theorem 2. Let (X, d, ≼) be a complete ordered metric space and T, f: X → X be self-mappings such that TX ⊆ fX. Assume that there exists , satisfying (F1a), such that for all x, y ∈ X with fx ≼ fy
If the following conditions hold:
-
(i)
there exists x0 ∈ X such that fx0 ≼ Tx0;
-
(ii)
T is f-non-decreasing;
-
(iii)
for a non-decreasing sequence {fx n } ⊆ X converging to fw ∈ X, we have fx n ≼ fw for all n ∈ ℕ and fw ≼ f fw;
then T and f have a coincidence point in X. Moreover, if
-
(iv)
T and f are weakly compatible;
-
(v)
F satisfies also F2,
then T and f have a common fixed point. Further, for any x0 ∈ X, the T-f-sequence {Tx n } with initial point x0 converges to a common fixed point.
Proof. Let x0 ∈ X such that fx0 ≼ Tx0 and let {Tx n } be a T-f-sequence with initial point x0. Since fx0 ≼ Tx0 and Tx0 = fx1, we have fx0 ≼ fx1. As T is f-non-decreasing we get that Tx0 ≼ Tx1. Continuing this process we obtain
In what follows we will suppose that d(Tx n , Txn+1) > 0 for all n ∈ ℕ, since if Tx n = Txn+1for some n, then fxn+1= Tx n = Txn+1. This implies that xn+1is a coincidence point for T and f and the result is proved. As fx n ≼ fxn+1for all n ∈ ℕ, if we take x = x n and y = xn+1in (5) and denote u = d(Tx n , Txn+1) and v = d(Txn- 1, Tx n ) we get that
By triangle inequality, d(Txn-1, Txn+1) ≤ d(Txn-1, Tx n ) + d(Tx n , Txn+1) = u + v and, since F is non-increasing in the fifth variable, we have
and hence, in view of assumption (F1a), there exists h ∈ [0, 1) such that u ≤ hv, i.e.,
By (6), we deduce that {Tx n } is a Cauchy sequence. Since (X, d) is complete, there exist z, w ∈ X such that z = fw and
By condition (iii), fx n ≼ fw for all n ∈ ℕ, if we take x = x n and y = w in (5) we get
As F is continuous, using (7) and letting n → +∞ we obtain
which, by assumption (F1a), yields d(fw, Tw) ≤ 0, i.e., fw = Tw. Thus we have proved that T and f have a coincidence point.
If T and f are weakly compatible we show that z is a common fixed point for T and f . As fz = fTw = Tfw = Tz, by condition (iii), we have that fw ≼ f fw = fz.
Now, by taking x = w and y = z in (5) we get
Assumption (F2) implies d(Tz, Tw) = 0 and hence fz = Tz = Tw = z. So T and f have a common fixed point. From the proof it follows that, for any x0 ∈ X, the T -f -sequence {Tx n } with initial point x0 converges to a common fixed point. □
We shall give a sufficient condition for the uniqueness of the common fixed point in Theorem 2.
Theorem 3. Let all the conditions of Theorem 2 be satisfied. If the following conditions hold
-
(vi)
for all x, y ∈ fX there exists v0 ∈ X such that fv0 ≼ x, fv0 ≼ y;
-
(vii)
F satisfies F1c,
then T and f have a unique common fixed point.
Proof. Let z, w be two common fixed points of T and f with z ≠ w. If z and w are comparable, say z ≼ y. Then taking x = z and y = w in (5), we obtain
which is a contradiction by assumption (F2) and so z = w.
If z and w are not comparable, then there exists v0 ∈ X such that fv0 ≼ fz = z and fv0 ≼ fw = w.
As T is f -non-decreasing from fv0 ≼ fz we get that
Continuing we obtain
Then, taking x = v n and y = z in (5) we obtain
i.e.,
Denote u = d(Tv n , Tz) and v = d(Tvn-1, Tz). As F is non-increasing in the third variable, we get
By assumption F1c, there exists h ∈ [0, 1) such that u ≤ hv, i.e.,
This implies that d(Tv n , Tz) = d(Tv n , z) → 0 as n → +∞.
With similar arguments, we deduce that d(Tv n , w) → 0 as n → +∞. Hence
as n → +∞, which is a contradiction. Thus T and f have a unique common fixed point. □
If f = I X from Theorems 2 and 3, we deduce the following results of fixed point for one self-mapping.
Corollary 2. Let (X, d, ≼) be a complete ordered metric space and T: X → X. Assume that there exists , satisfying (F1a), such that for all x, y ∈ X with x ≼ y
If the following conditions hold:
-
(i)
there exists x0 ∈ X such that x0 ≼ Tx0;
-
(ii)
T is non-decreasing;
-
(iii)
for a non-decreasing sequence {x n } ⊆ X converging to w ∈ X, we have x n ≼ w for all n ∈ ℕ,
then T has a fixed point in X. Further, for any x0 ∈ X, the Picard sequence {Tnx0} with initial point x0 converges to a fixed point.
Corollary 3. Let all the conditions of Corollary 2 be satisfied. If the following conditions hold
-
(v)
F satisfies F 2 ;
-
(vi)
for all x, y ∈ X there exists v0 ∈ X such that v0 ≼ x, v0 ≼ y;
-
(vii)
F satisfies F1c,
then T has a unique fixed point.
If F is the function in Example 2, then by Theorem 3 we obtain a fixed point theorem that extends the result of Theorem 3 of [44].
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund Math 1922, 3: 133–181.
Rus IA: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca; 2001.
Rus IA, Petruşel A, Petruşel G: Fixed Point Theory. Cluj University Press, Cluj-Napoca; 2008.
Reem D, Reich S, Zaslavski AJ: Two results in metric fixed point theory. J Fixed Point Theory Appl 2007, 1: 149–157. 10.1007/s11784-006-0011-4
Reich S, Zaslavski AJ: A fixed point theorem for Matkowski contractions. Fixed Point Theory 2007, 8: 303–307.
Reich S, Zaslavski AJ: A note on Rakotch contraction. Fixed Point Theory 2008, 9: 267–273.
Berinde V: On the approximation of fixed points of weak contractive mappings. Carpathian J Math 2003, 19: 7–22.
Berinde V: Approximating fixed points of weak φ -contractions. Fixed Point Theory 2003, 4: 131–142.
Berinde V: Approximation fixed points of weak contractions using the Picard iteration. Nonlinear Anal Forum 2004, 9: 43–53.
Berinde V: Error estimates for approximating fixed points of quasi contractions. Gen Math 2005, 13: 23–34.
Berinde V: Approximation of Fixed Points. Springer, Berlin, Heidelberg, New York; 2007.
Berinde V: General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces. Carpathian J Math 2008, 24: 10–19.
Berinde V: Some remarks on a fixed point theorem for Ćirić-type almost contractions. Carpathian J Math 2009, 25: 157–162.
Berinde V: Approximating common fixed points of noncommuting almost contractions in metric spaces. Fixed Point Theory 2010, 11: 179–188.
Berinde V: Common fixed points of noncommuting almost contractions in cone metric spaces. Math Commun 2010, 15: 229–241.
Popa V: Fixed point theorems for implicit contractive mappings. Stud Cerc St Ser Mat Univ Bacău 1997, 7: 127–133.
Popa V: Some fixed point theorems for compatible mappings satisfying an implicit relation. Demonstratio Math 1999, 32: 157–163.
Ali J, Imdad M: Unifying a multitude of common fixed point theorems employing an implicit relation. Commun Korean Math Soc 2009, 24: 41–55. 10.4134/CKMS.2009.24.1.041
Aliouche A, Djoudi A: Common fixed point theorems for mappings satisfying an implicit relation without decreasing assumption. Hacet J Math Stat 2007, 36: 11–18.
Aliouche A, Popa V: General common fixed point theorems for occasionally weakly compatible hybrid mappings and applications. Novi Sad J Math 2009, 39: 89–109.
Berinde V: Approximating fixed points of implicit almost contractions. Hacet J Math Stat 2012, 40: 93–102.
Popa V: A general fixed point theorem for weakly compatible mappings in compact metric spaces. Turkish J Math 2001, 25: 465–474.
Popa V: Fixed points for non-surjective expansion mappings satisfying an implicit relation. Bul Ştiinţ Univ Baia Mare Ser B 2002, 18: 105–108.
Popa V: A general fixed point theorem for four weakly compatible mappings satisfying an implicit relation. Filomat 2005, 19: 45–51.
Popa V, Imdad M, Ali J: Using implicit relations to prove unified fixed point theorems in metric and 2-metric spaces. Bull Malays Math Sci Soc 2010, 33: 105–120.
Abbas M, Ilic D: Common fixed points of generalized almost nonexpansive mappings. Filomat 2010, 24(3):11–18. 10.2298/FIL1003011A
Babu GVR, Sandhy ML, Kameshwari MVR: A note on a fixed point theorem of Berinde on weak contractions. Carpathian J Math 2008, 24: 8–12.
Berinde V: Stability of Picard iteration for contractive mappings satisfying an implicit relation. Carpathian J Math 2011, 27: 13–23.
Chatterjea SK: Fixed-point theorems. C R Acad Bulgare Sci 1972, 25: 727–730.
Hardy GE, Rogers TD: A generalization of a fixed point theorem of Reich. Canad Math Bull 1973, 16: 201–206. 10.4153/CMB-1973-036-0
Kannan R: Some results on fixed points. Bull Calcutta Math Soc 1968, 10: 71–76.
Reich S: Fixed points of contractive functions. Boll Un Mat Ital 1972, 5: 26–42.
Reich S: Kannan's fixed point theorem. Boll Un Mat Ital 1971, 4: 1–11.
Reich S: Some remarks concerning contraction mappings. Can Math Bull 1971, 14: 121–124. 10.4153/CMB-1971-024-9
Rhoades BE: A comparison of various definitions of contractive mappings. Trans Am Math Soc 1977, 226: 257–290.
Rhoades BE: Contractive definitions revisited. Contemp Math 1983, 21: 189–205.
Rhoades BE: Contractive definitions and continuity. Contemp Math 1988, 72: 233–245.
Zamfirescu T: Fix point theorems in metric spaces. Arch Math (Basel) 1972, 23: 292–298. 10.1007/BF01304884
Turinici M: Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J Math Anal Appl 1986, 117: 100–127. 10.1016/0022-247X(86)90251-9
Ran ACM, Reurings MC: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc Am Math Soc 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4
Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5
Cherichi M, Samet B: Fixed point theorems on ordered gauge spaces with applications to non-linear integral equations. Fixed Point Theory Appl 2012, 2012: 13. 10.1186/1687-1812-2012-13
Ćirić L, Agarwal RP, Samet B: Mixed monotone-generalized contractions in partially ordered probabilistic metric spaces. Fixed Point Theory Appl 2011, 2011: 56. 10.1186/1687-1812-2011-56
Golubović Z, Kadelburg Z, Radenović S: Common fixed points of ordered g-quasicontractions and weak contractions in ordered metric spaces. Fixed Point Theory Appl 2012, 2012: 20. 10.1186/1687-1812-2012-20
Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026
Acknowledgements
The first author research's was supported by the Grant PN-II-RU-TE-2011-3-239 of the Romanian Ministry of Education and and Research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Authors' Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Berinde, V., Vetro, F. Common fixed points of mappings satisfying implicit contractive conditions. Fixed Point Theory Appl 2012, 105 (2012). https://doi.org/10.1186/1687-1812-2012-105
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2012-105