Abstract
In this paper, a new class of a pair of generalized nonlinear contractions on partially ordered partial metric spaces is introduced, and some coincidence and common fixed-point theorems for these contractions are proved. Presented theorems are twofold generalizations of very recent fixed-point theorems of Altun and Erduran (Fixed Point Theory Appl 2011(Article ID 508730):10, 2011), Altun et al. (Topol Appl 157(18):2778-2785, 2010), Matthews (Proceedings of the 8th summer conference on general topology and applications, New York Academy of Sciences, New York, pp. 183-197, 1994) and many other known corresponding theorems.
2000 Mathematics Subject Classifications: 54H25; 47H10.
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1 Introduction
It is well known that the Banach contraction principle is a very useful, simple and classical tool in nonlinear analysis. There exist a vast literature concerning its various generalizations and extensions (see [1–45]). In [22], Matthews extended the Banach contraction mapping theorem to the partial metric context for applications in program verification. After that, fixed-point results in partial metric spaces have been studied [4, 8, 28, 31, 34, 45]. The existence of several connections between partial metrics and topological aspects of domain theory has been pointed by many authors (see [8, 9, 16, 23, 31, 33, 36–38, 41, 42, 46, 47]).
First, we recall some definitions of partial metric spaces and some their properties.
Definition 1.1 A partial metric on a set X is a function p : X × X → ℝ+ such that for all x, y, z ∈ X:
(p1) x = y ⇔ p(x, x) = p(x, y) = p(y, y),
(p2) p(x, x) ≤ p(x, y),
(p3) p(x, y) = p(y, x),
(p4) p(x, y) ≤ p(x, z) + p(z, y) - p(z, z).
Note that the self-distance of any point need not be zero, hence the idea of generalizing metrics so that a metric on a non-empty set X is precisely a partial metric p on X such that for any x ∈ X, p(x, x) = 0.
Similar to the case of metric space, a partial metric space is a pair (X, p) consisting of a non-empty set X and a partial metric p on X.
Example 1.1 Let a function p : ℝ+ × ℝ+ → ℝ+ be defined by p(x, y) = max{x, y} for any x, y ∈ ℝ+. Then, (ℝ+ , p) is a partial metric space where the self-distance for any point x ∈ ℝ+ is its value itself.
Example 1.2 Consider a function p : ℝ- × ℝ- → ℝ+ defined by p(x, y) = - min(x, y) for any x, y ∈ ℝ-. The pair (ℝ-, p) is a partial metric space for which p is called the usual partial metric on ℝ- and where the self-distance for any point x ∈ ℝ- is its absolute value.
Example 1.3 If X: = {[a, b] | a, b ∈ ℝ, a ≤ b}, then p : X × X → ℝ+ defined by p([a, b], [c, d]) = max{b, d} - min{a, b} defines a partial metric on X.
Each partial metric p on X generates a T0 topology τ p on X, which has as a base the family of open p-balls {B p (x, ε), x ∈ X, ε > 0}, where
If p is a partial metric on X, then the function ps : X × X → ℝ+ defined by
is a metric on X.
Definition 1.2 Let (X, p) be a partial metric space and {x n } be a sequence in X. Then,
(i) {x n } converges to a point x ∈ X if and only if p(x, x) = limn→+∞p(x, x n ),
(ii) {x n } is a Cauchy sequence if there exists (and is finite) limn,m→+∞p(x n , x m ).
Definition 1.3 A partial metric space (X, p) is said to be complete if every Cauchy sequence {x n } in X converges, with respect to τ p , to a point x ∈ X, such that p(x, x) = limn,m→+∞p(x n , x m ).
Remark 1.1 It is easy to see that every closed subset of a complete partial metric space is complete.
Lemma 1.1 ([22, 28]) Let (X, p) be a partial metric space. Then
(a) {x n } is a Cauchy sequence in (X, P) if and only if it is a Cauchy sequence in the metric space (X, Ps),
(b) (X, p) is complete if and only if the metric space (X, ps) is complete. Furthermore, limn→+∞ps (x n , x) = 0 if and only if
Matthews [22] obtained the following Banach fixed-point theorem on complete partial metric spaces.
Theorem 1.1 (Matthews [22]) Let f be a mapping of a complete partial metric space (X, p) into itself such that there is a constant c ∈ [0,1) satisfying for all x, y ∈ X :
Then, f has a unique fixed point.
Recently, Altun et al. [4] obtained the following nice result, which generalizes Theorem 1.1 of Matthews.
Theorem 1.2 (Altun et al. [4]) Let (X, p) be a complete partial metric space and let T : X → X be a map such that
for all x, y ∈ X, where φ : [0, +∞) → [0, +∞) satisfies the following conditions:
(i) φ is continuous and non-decreasing,
(ii) is convergent for each t > 0.
Then, T has a unique fixed point.
On the other hand, existence of fixed points in partially ordered sets has been considered recently in [32], and some generalizations of the result of [32] are given in [1–3, 5–7, 11, 12, 14, 15, 17, 19, 24–27, 29, 30, 39, 40, 43] in partial ordered metric spaces. Also, in [32], some applications to matrix equations are presented, and in [15] and [26], some applications to ordinary differential equations are given. In [29], O'Regan and Petruşel established some fixed-point results for self-generalized contractions in ordered metric spaces. Jachymski [19] established a geometric lemma [19, Lemma 1], giving a list of equivalent conditions for some subsets of the plane. Using this lemma, he proved that some very recent fixed-point theorems for generalized contractions on ordered metric spaces obtained by Harjani and Sadarangani [15] and Amini-Harandi and Emami [5] do follow from an earlier result of O'Regan and Petruşel [29, Theorem 3.6].
Very recently, Altun and Erduran [3] generalized Theorem 1.2 to partially ordered complete partial metric spaces and established the following new fixed-point theorems, involving a function φ : [0, +∞) → [0, +∞) satisfying the conditions (i)-(ii) in Theorem 1.2.
Theorem 1.3 (Altun and Erduran [3]). Let (X, ≼) be a partially ordered set and suppose that there is a partial metric p on X such that (X, p) is a complete partial metric space. Suppose F : X → X is a continuous and non-decreasing mapping (with respect to ≼) such that
for all x, y ∈ X with y ≼ x, where φ : [0, +∞) → [0, +∞) satisfies conditions (i)-(ii) in Theorem 1.2. If there exists x0 ∈ X such that x0 ≼ Fx0, then there exists x ∈ X such that Fx = x. Moreover, p (x, x) = 0.
Theorem 1.4 (Altun and Erduran [3]) Let (X, ≼) be a partially ordered set and suppose that there is a partial metric p on X such that (X, p) is a complete partial metric space. Suppose F : X → X is a non-decreasing mapping such that
for all x, y ∈ X with y ≺ x (y ≼ x and y ≠ x), where φ : [0, +∞) → [0, +∞) satisfies conditions (i)-(ii) in Theorem 1.2. Suppose also that the condition
holds. If there exists x0 ∈ X such that x0 ≼ Fx0, then there exists x ∈ X such that Fx = x. Moreover, p(x, x) = 0.
Theorem 1.5 (Altun and Erduran [3]) Let (X, ≼) be a partially ordered set and suppose that there is a partial metric p on X such that (X, p) is a complete partial metric space. Suppose F : X → X is a continuous and non-decreasing mapping such that
for all x, y ∈ X with y ≼ x, where φ : [0, +∞) → [0, +∞) satisfies conditions (i)-(ii) in Theorem 1.2. If there exists x0 ∈ X such that x0 ≼ Fx0, then there exists x ∈ X such that Fx = x. Moreover, p(x, x) = 0. If we suppose that for all x, y ∈ X there exists z ∈ X, which is comparable to x and y, we obtain uniqueness of the fixed point of F.
Altun et al. [4], Altun and Erduran [3] and many authors have obtained fixed-point theorems for contractions under the assumption that a comparison function φ : [0, +∞) → [0, +∞) is non-decreasing and such that for each t > 0 (see, e.g., [13] and the references in [11, 18]-Added in proof). However, the latter condition is strong and rather hard to verify in practice, though some examples and general criteria for this convergence are known (see, e.g., [3, 44]). So a natural question arises whether this strong condition can be omitted in partial metric fixed-point theory.
The aims of this paper is to establish coincidence and common fixed-point theorems in ordered partial metric spaces with a function φ satisfying the condition φ(t) < t for all t > 0, which is weaker than the condition Presented theorems generalize and extend to a pair of mappings the results of Altun and Erduran [3], Altun et al. [4], Matthews [22] and many other known corresponding theorems.
2 Main results
We start this section by some preliminaries.
Definition 2.1 (Altun and Erduran [3]) Let (X, p) be a partial metric space, F : X → X be a given mapping. We say that F is continuous at x0 ∈ X, if for every ε > 0, there exists δ > 0 such that F(B p (x0, δ)) ⊆ B p (Fx0, ε).
The following result is easy to check.
Lemma 2.1 Let (X, p) be a partial metric space, F : X → X be a given mapping. Suppose that F is continuous at x0 ∈ X. Then, for all sequence {x n } ⊂ X, we have
Definition 2.2 (Ćirić et al. [11]) Let (X, ≼) be a partially ordered set and F, g : X → X are mappings of X into itself. One says F is g-non-decreasing if for x, y ∈ X, we have
We introduce the following definition.
Definition 2.3 Let (X, p) be a partial metric space and F, g: X → X are mappings of X into itself. We say that the pair {F, g} is partial compatible if the following conditions hold:
(b1) p(x, x) = 0 ⇒ p(gx, gx) = 0,
(b2) limn→+∞p(Fgx n , gFx n ) = 0, whenever {x n } is a sequence in X such that Fx n → t and gx n → t for some t ∈ X.
It is clear that Definition 2.3 extends and generalizes the notion of compatibility introduced by Jungck [21].
Define by ϕ the set of functions φ : [0, +∞) → [0, +∞) satisfying the following conditions:
(c1) φ is continuous and non-decreasing,
(c2) φ(t) < t for each t > 0.
Now, we are ready to state and prove our first result.
Theorem 2.1 Let (X, ≼) be a partially ordered set and suppose that there is a partial metric p on X such that (X, p) is a complete partial metric space. Let F, g : X → X be two continuous self-mappings of X such that FX ⊆ gX, F is a g-non-decreasing mapping, the pair {F, g} is partial compatible, and
for all x, y ∈ X for which gy ≼ gx, where a function φ ∈ ϕ. If there exists x0 ∈ X with gx0 ≼ Fx0, then F and g have a coincidence point, that is, there exists x ∈ X such that Fx = gx. Moreover, we have p(x, x) = p(Fx, Fx) = p(gx, gx) = 0.
Proof. Let x0 ∈ X such that gx0 ≼ Fx0. Since FX ⊆ gX, we can choose x1 ∈ X so that gx1 = Fx0. Again, from FX ⊆ gX, there exists x2 ∈ X such that gx2 = Fx1. Continuing this process, we can choose a sequence {x n } ⊂ X such that
Since gx0 ≼ Fx0 and Fx0 = gx1, then gx0 ≼ gx1. Since F is a g-non-decreasing mapping, we have Fx0 ≼ Fx1, that is, gx1 ≼ gx2. Again, using that F is a g-non-decreasing mapping, we have Fx1 ≼ Fx2, that is, gx2 ≼ gx3. Continuing this process, we get
Suppose that there exists n ∈ N such that p(Fx n , Fxn+1) = 0. This implies that Fx n = Fxn+1, that is, gxn+1= Fxn+1. Then, xn+1is a coincidence point of F and g, and so we have finished the proof. Thus, we can assume that
We will show that
Using (2) and applying the considered contraction (1) with x = x n and y = xn+1, we get
Hence, as
and φ is non-decreasing, we have
If we suppose that , then from (5),
Using (3) and the fact that φ(t) < t for all t > 0, we have
a contradiction. Therefore,
and so from (5),
Thus, we proved (4).
Since φ is non-decreasing, repeating the inequality (4) n times, we get
Letting n → +∞ in the inequality (6) and using the fact that φn (t) → 0 as n → +∞ for all t > 0, we obtain
On the other hand, we have
Letting n → +∞ in this inequality, by (7), we get
Now, we shall prove that {Fx n } is a Cauchy sequence in the metric space (X, ps). Suppose, to the contrary, that {Fx n } is not a Cauchy sequence in (X, ps). Then, there exists ε > 0 such that for each positive integer k, there exist two sequences of positive integers {m(k)} and {n(k)} such that
Since ps(x, y) ≤ 2p(x, y) for all x, y ∈ X, from (9), for all positive integer k, we have
Without loss of generality, we can suppose that also
From (10) and the triangular inequality (that holds for a partial metric), we have
Letting k → +∞ and using (7), we get
Again, using the triangular inequality, we obtain
Letting k → +∞ in this inequality, and using (11) and (7), we get
Hence,
On the other hand, we have
From (1) with x = x n and y = xn+1, we get
Therefore, from (13) and since φ is a non-decreasing function, we get
Letting k → +∞ in the above inequality, using (7), (11), (12) and the continuity of φ, we have
a contradiction. Thus, our supposition that {Fx n } is not a Cauchy sequence was wrong. Therefore, {Fx n } is a Cauchy sequence in the metric space (X, ps), and so we have
Now, since (X, p) is complete, then from Lemma 1.1, (X, ps) is a complete metric space. Therefore, the sequence {Fx n } converges to some x ∈ X, that is,
From the property (b) in Lemma 1.1, we have
On the other hand, from property (p2) of a partial metric, we have
Letting n → +∞ in the above inequality and using (7), we obtain
Therefore, from the definition of ps and using (14), we get limm,n→+∞p(Fx n , Fx m ) = 0. Thus, from (15), we have
Now, since F is continuous, from (16) and using Lemma 2.1, we get
Using the triangular inequality, we obtain
Letting n → +∞ in the above inequality, using (17), (15), (16), the partial compatibility of {F, g}, the continuity of g and Lemma 2.1, we have
Now, suppose that p(Fx, gx) > 0. Then, from (1) with x = y, we get
Therefore, from (19), we have
a contradiction. Thus, we have p(Fx, gx) = 0, which implies that Fx = gx, that is, x is a coincidence point of F and g. Moreover, from (16) and since the pair {F, g} is partial compatible, we have p(x, x) = 0 = p(gx, gx) = p(Fx, Fx). This completes the proof. ■
An immediate consequence of Theorem 2.1 is the following result.
Theorem 2.2 Let (X, ≼) be a partially ordered set and suppose that there is a partial metric p on X such that (X, p) is a complete partial metric space. Suppose F : X → X is a continuous and non-decreasing mapping (with respect to ≼) such that
for all x, y ∈ X with y ≼ x, where φ : [0, +∞) → [0, +∞) is continuous non-decreasing and φ(t) < t for all t > 0. If there exists x0 ∈ X such that x0 ≼ Fx0, then there exists x ∈ X such that Fx = x. Moreover, p(x, x) = 0.
Proof. Putting gx = Ix = x in Theorem 2.1, we obtain Theorem 2.2. ■
Now we shall present an example in which F: X → X and φ : [0, +∞) → [0, +∞) satisfy all hypotheses of our Theorem 2.2, but not the hypotheses of Theorems of Altun et al. [4], Altun and Erduran [3] with φ given in an illustrative example in [3], Matthews [22] and of many other known corresponding theorems.
Before giving our example, we need the following result.
Lemma 2.2 Consider X = [0, +∞) endowed with the partial metric p : X × X → [0, +∞) defined by p(x, y) = max{x, y} for all x, y ≥ 0. Let F : X → X be a non-decreasing function. If F is continuous with respect to the standard metric d(x, y) = |x - y| for all x, y ≥ 0, then F is continuous with respect to the partial metric p.
Proof. Let {x n } be a sequence in X such that limn→+∞p(x n , x) = p(x, x) for some x ∈ X, that is, limn→+∞max{x n , x} = x. Using Lemma 2.1, we have to prove that limn→+∞p(Fx n , Fx) = p(Fx, Fx), that is, limn→+∞max{Fx n , Fx} = Fx.
Since F is a non-decreasing mapping, we have
Now, using that F is continuous with respect to the standard metric, we have
Therefore, from (21), it follows that
This makes end to the proof. ■
Example 2.1 Let X = [0, +∞) and (X, p) be a complete partial metric space, where p : X × X → ℝ+ is defined by p(x, y) = max{x, y}. Let us define a partial order ≼ on X as follows:
Define F : X → X by
and let φ : [0, +∞) → [0, +∞) be defined by
Clearly the function φ ∈ ϕ, that is, φ is continuous non-decreasing and φ(t) < t for each t > 0. On the other hand, using Lemma 2.2, since F is non-decreasing (with respect to the usual order) and continuous in X with respect to the standard metric, then it is continuous with respect to the partial metric p. The function F is also non-decreasing with respect to the partial order ≼.
We now show that F satisfies the nonlinear contractive condition (20) for all x, y ∈ X with y ≼ x. By definition of F, we have
Thus,
Therefore, the contractive condition (20) is satisfied for all x, y ∈ X for which y ≼ x.
Also, for x0 = 0, we have x0 ≼ Fx0.
Therefore, all hypotheses of Theorem 2.2 are satisfied and F has a fixed point. Note that it is easy to see that the hypothesis (23) as well as all other hypotheses in Theorems 2.3 and 2.4 below is also satisfied.
Observe that in this example, φ does not satisfy the condition for each t> 0 of Theorems in [3, 4]. Indeed, let t0 ∈ (0, 1] be arbitrary. Then, it is easy to show by induction that φn(t0) = t0/(1 + nt0). Thus,
Note that F does not satisfy the contractive condition (20) in Theorem 2.2 with a function
This function is given by Altun and Erduran in their illustrative example in [3]. It is easy to show that for y ≼ x,
Now, we will prove the following result.
Theorem 2.3 Let (X, ≼) be a partially ordered set and suppose that there is a partial metric p on X such that (X, p) is a complete partial metric space. Let F,g : X → X be two self-mappings of X such that FX ⊆ gX, F is a g-non-decreasing mapping and,
for all x, y ∈ X for which gx ≻ gy, whereφ ∈ ϕ. Also suppose
holds. Also suppose gX is closed. If there exists x0 ∈ X with gx0 ≼ Fx0, then F and g have a coincidence point x ∈ X such that p(Fx, Fx) = p(gx, gx) = 0. Further, if F and g commute at their coincidence points, then F and g have a common fixed point.
Proof. Denote
for all x, y ∈ X.
As in the proof of Theorem 2.1, we can construct a sequence {x n } in X by gxn+1= Fx n for all n ≥ 0. Also, we can assume that Fx n ≠ Fxn+1for all n ≥ 0; otherwise, we are finished. Therefore, we have
Again, as in the proof of Theorem 2.1, we can show that {Fx n } is a Cauchy sequence in the complete metric space (X, ps), and therefore, there exists y ∈ X such that
Since {Fx n } ⊂ gX and gX is closed, there exists x ∈ X such that y = gx. From (24) and hypothesis (23), we have
Now, we will show that x is a coincidence point of F and g. Using the triangular inequality, we have
From (26), using the considered contraction, we have
Thus,
Now, we have
Since φ is a non-decreasing function, using (25), the above inequality and n → +∞ in (27), we get
If p(gx, Fx) > 0, we obtain p(gx, Fx) ≤ φ(p(gx, Fx)) < p(gx, Fx): a contradiction. We deduce that p(gx, Fx) = 0, which implies that gx = Fx, that is, x is a coincidence point of F and g.
Suppose now that F and g commute at x. Set w = Fx = gx. Then,
From the hypothesis (23), we have gx ≼ g(gx) = gw. If gx = gw, we get w = gw = Fw, and the proof is finished. Then, suppose that gx ≺ gw. Applying the considered contraction, we get
where
Suppose that p(Fw, Fx) > 0, From (29), we get
which is a contradiction. Thus, we have p(Fw, Fx) = 0, which implies that Fw = Fx = w. Therefore, from (28), we have w = Fw = gw, and w is a common fixed point of F and g. This completes the proof. ■
Remark 2.1 The result given by Theorem 2.3 is also valid if the contraction condition (22) is satisfied for all x, y ∈ X with gx ≽ gy and (23) is replaced by
An immediate consequence of Theorem 2.3 is the following.
Theorem 2.4 Let (X, ≼) be a partially ordered set and suppose that there is a partial metric p on X such that (X, p) is a complete partial metric space. Suppose F : X → X is a non-decreasing mapping such that
for all x, y ∈ X with y ≺ x, where φ : [0, +∞) → [0, +∞) is continuous non-decreasing and φ(t) < t for all t > 0. Suppose also that the condition
holds. If there exists x0 ∈ X such that x0 ≼ Fx0, then there exists x ∈ X such that Fx = x. Moreover, p(x, x) = 0.
Now, we give a simple example to show that our result given by Theorem 2.3 is more general than Theorem 3.6 of O'Regan and Petruşel [29].
Example 2.2 Let X = [0, +∞) endowed with the partial metric p(x, y) = max{x, y} for all x, y ∈ X. We endow X with the usual order ≤. Consider the mappings F, g : X → X and φ : [0, +∞) → [0, +∞) defined by
Let y ≤ x. We have
Then, (22) is satisfied. It is easy to show that all the other hypotheses of Theorem 2.3 are also satisfied. Since F and g commute, we deduce that F and g have a common fixed point z = 0, that is, 0 = F(0) = g(0).
On the other hand, if we endow X with the standard metric d(x, y) = |x - y| for all x, y ∈ X, we have
for x ≠ y and for any φ : [0, +∞) → [0, +∞) satisfying φ(t) < t for t > 0. Therefore, Theorem 3.6 of O'Regan and Petruşel [29] is not applicable.
Note that F also does not satisfy the contractive conditions in the rest theorems of O'Regan and Petruşel [29].
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This work was supported by the Ministry of Sciences and technology of Republic Serbia (PROJECT 174025).
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Samet, B., Rajović, M., Lazović, R. et al. Common fixed-point results for nonlinear contractions in ordered partial metric spaces. Fixed Point Theory Appl 2011, 71 (2011). https://doi.org/10.1186/1687-1812-2011-71
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DOI: https://doi.org/10.1186/1687-1812-2011-71