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Common fixed points of mappings satisfying implicit contractive conditions
Fixed Point Theory and Applications volume 2012, Article number: 105 (2012)
Abstract
In this article we obtain, in the setting of metric spaces or ordered metric spaces, coincidence point, and common fixed point theorems for selfmappings 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.
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 nonselfmappings, 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 selfmappings that satisfy an explicit contractivetype 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 selfmappings 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 nonempty 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 x_{0} ∈ X we consider the sequence {x_{ n }} ⊂ X defined by fx_{ n } = Tx_{n1}for all n ∈ ℕ, we say that {Tx_{ n }} is a T f sequence with initial point x_{0}.
Let X be a nonempty 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 nondecreasing if fx ≼ fy implies Tx ≼ Ty for all x, y ∈ X. If f is the identity mapping on X, then T is nondecreasing.
Throughout this article the letters ℝ_{+} and ℕ will denote the set of all nonnegative 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 singlevalued and multivalued 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 F:{\mathbb{R}}_{+}^{6}\to {\mathbb{R}}_{+} and the following conditions:
(F_{1a}) F is nonincreasing 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;
(F_{1b}) F is nonincreasing 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;
(F_{1c}) F is nonincreasing 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;
(F_{2}) F (u, u, 0, 0, u, u) > 0, for all u > 0.
He gave many examples of functions corresponding to wellknown fixed point theorems and satisfying most of the conditions (F_{1a})(F_{2}) above, see Examples 111 of [21].
Example 1. The following functions F\in \mathcal{F} satisfy properties F_{2} and F_{1a}F_{1c}(see Examples 16, 9, and 11 of [21]).

(i)
F (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) = t_{1} − at_{2}, where a ∈ [0, 1);

(ii)
F (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) = t_{1} − b(t_{3} + t_{4}), where b ∈ [0, 1/ 2);

(iii)
F (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) = t_{1} − c(t_{5} + t_{6}), where c ∈ [0, 1/ 2);

(iv)
F\left({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right)={t}_{1}a\mathsf{\text{max}}\left\{{t}_{2},\frac{{t}_{3}+{t}_{4}}{2},\frac{{t}_{5}+{t}_{6}}{2}\right\}, where a ∈ [0, 1);

(v)
F (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) = t_{1} − at_{2} − b(t_{3} + t_{4}) − c(t_{5} + t_{6}), where a, b, c ∈ [0, 1) and a + 2b + 2c < 1;

(vi)
F\left({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right)={t}_{1}a\mathsf{\text{max}}\left\{{t}_{2},\frac{{t}_{3}+{t}_{4}}{2},{t}_{5},{t}_{6}\right\}, where a ∈ [0, 1);

(vii)
F (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) = t_{1} − at_{2} − L min{t_{3}, t_{4}, t_{5}, t_{6}}, where a ∈ [0, 1);

(viii)
F\left({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right)={t}_{1}a\mathsf{\text{max}}\left\{{t}_{2},{t}_{3},{t}_{4},\frac{{t}_{5}+{t}_{6}}{2}\phantom{\rule{2.77695pt}{0ex}}\right\}L\mathsf{\text{min}}\left\{{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right\}, where a ∈ [0, 1) and L ≥ 0.
Example 2. The function F\in \mathcal{F}, given by
where a ∈ [0, 1/ 2) satisfies properties F_{2} and F_{1a}F_{1c}with h=\frac{a}{1a}<1.
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 selfmappings such that TX ⊆ fX. Assume that there exists F\in \mathcal{F}, satisfying (F_{1a}), 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 F_{2}, then T and f have a unique common fixed point. Further, for any x_{0} ∈ X, the Tfsequence {Tx_{ n }} with initial point x_{0} converges to the common fixed point.
Proof. Let x_{0} ∈ X be an arbitrary point. As TX ⊆ fX, one can choose a Tfsequence {Tx_{ n }} with initial point x_{0}. If we take x = x_{ n } and y = x_{n+1}in (1) and denote with u = d(Tx_{ n }, Tx_{n+1}) and v = d(Tx_{n1}, Tx_{ n }) we get that
By triangle inequality, d(Tx_{n1}, Tx_{n+1}) ≤ d(Tx_{n1}, Tx_{ n }) + d(Tx_{ n }, Tx_{n+1}) = u + v and, since F is nonincreasing in the fifth variable, we have
and hence, in view of assumption (F_{1a}), 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 (F_{1a}), 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 (F_{2}). 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 (F_{2}). Clearly, for any x_{0} ∈ X, the Tfsequence {Tx_{ n }} with initial point x_{0} 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 selfmapping, see [21].
Corollary 1. Let (X, d) be a complete metric space and T: X → X. Assume that there exists, F\in \mathcal{F}satisfying (F_{1a}), such that for all x, y ∈ X
Then T has a fixed point. Moreover, if F satisfies also F_{2}, then T has a unique fixed point. Further, for any x_{0} ∈ X, the Picard sequence {T^{n}x_{0}} with initial point x_{0} 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 RodríguezLópez [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 selfmappings such that TX ⊆ fX. Assume that there exists F\in \mathcal{F}, satisfying (F_{1a}), such that for all x, y ∈ X with fx ≼ fy
If the following conditions hold:

(i)
there exists x_{0} ∈ X such that fx_{0} ≼ Tx_{0};

(ii)
T is fnondecreasing;

(iii)
for a nondecreasing 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 F_{2},
then T and f have a common fixed point. Further, for any x_{0} ∈ X, the Tfsequence {Tx_{ n }} with initial point x_{0} converges to a common fixed point.
Proof. Let x_{0} ∈ X such that fx_{0} ≼ Tx_{0} and let {Tx_{ n }} be a Tfsequence with initial point x_{0}. Since fx_{0} ≼ Tx_{0} and Tx_{0} = fx_{1}, we have fx_{0} ≼ fx_{1}. As T is fnondecreasing we get that Tx_{0} ≼ Tx_{1}. Continuing this process we obtain
In what follows we will suppose that d(Tx_{ n }, Tx_{n+1}) > 0 for all n ∈ ℕ, since if Tx_{ n } = Tx_{n+1}for some n, then fx_{n+1}= Tx_{ n } = Tx_{n+1}. This implies that x_{n+1}is a coincidence point for T and f and the result is proved. As fx_{ n } ≼ fx_{n+1}for all n ∈ ℕ, if we take x = x_{ n } and y = x_{n+1}in (5) and denote u = d(Tx_{ n }, Tx_{n+1}) and v = d(Tx_{n 1}, Tx_{ n }) we get that
By triangle inequality, d(Tx_{n1}, Tx_{n+1}) ≤ d(Tx_{n1}, Tx_{ n }) + d(Tx_{ n }, Tx_{n+1}) = u + v and, since F is nonincreasing in the fifth variable, we have
and hence, in view of assumption (F_{1a}), 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 (F_{1a}), 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 (F_{2}) 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 x_{0} ∈ X, the T f sequence {Tx_{ n }} with initial point x_{0} 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 v_{0} ∈ X such that fv_{0} ≼ x, fv_{0} ≼ y;

(vii)
F satisfies F_{1c},
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 (F_{2}) and so z = w.
If z and w are not comparable, then there exists v_{0} ∈ X such that fv_{0} ≼ fz = z and fv_{0} ≼ fw = w.
As T is f nondecreasing from fv_{0} ≼ 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(Tv_{n1}, Tz). As F is nonincreasing in the third variable, we get
By assumption F_{1c}, 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 selfmapping.
Corollary 2. Let (X, d, ≼) be a complete ordered metric space and T: X → X. Assume that there exists F\in \mathcal{F}, satisfying (F_{1a}), such that for all x, y ∈ X with x ≼ y
If the following conditions hold:

(i)
there exists x_{0} ∈ X such that x_{0} ≼ Tx_{0};

(ii)
T is nondecreasing;

(iii)
for a nondecreasing 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 x_{0} ∈ X, the Picard sequence {T^{n}x_{0}} with initial point x_{0} 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 v_{0} ∈ X such that v_{0} ≼ x, v_{0} ≼ y;

(vii)
F satisfies F_{1c},
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].
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The first author research's was supported by the Grant PNIIRUTE20113239 of the Romanian Ministry of Education and and Research.
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Berinde, V., Vetro, F. Common fixed points of mappings satisfying implicit contractive conditions. Fixed Point Theory Appl 2012, 105 (2012). https://doi.org/10.1186/168718122012105
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DOI: https://doi.org/10.1186/168718122012105