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Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 48 (2012)
Abstract
In this article, we establish some common fixed point theorems for a hybrid pair {g, T} of single valued and multi-valued maps satisfying a generalized contractive condition defined on G-metric spaces. Our results unify, generalize and complement various known comparable results from the current literature.
2000 MSC: 54H25; 47H10; 54E50.
1. Introduction and preliminaries
Nadler [1] initiated the study of fixed points for multi-valued contraction mappings and generalized the well known Banach fixed point theorem. Then after, many authors studied many fixed point results for multi-valued contraction mappings see [2–13].
Mustafa and Sims [14] introduced the G-metric spaces as a generalization of the notion of metric spaces. Mustafa et al. [15–19] obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades [20] initiated the study of common fixed point in G-metric spaces. While Saadati et al. [21] studied some fixed point theorems in generalized partially ordered G-metric spaces. Gajić and Crvenković [22, 23] proved some fixed point results for mappings with contractive iterate at a point in G-metric spaces. For other studies in G-metric spaces, we refer the reader to [24–38]. Consistent with Mustafa and Sims [14], the following definitions and results will be needed in the sequel.
Definition 1.1. (See [14]). Let X be a non-empty set, G : X × X × X → ℝ+ be a function satisfying the following properties
(G1) G(x, y, z) = 0 if x = y = z,
(G2) 0 < G(x, x, y) for all x, y ∈ X with x ≠ y,
(G3) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with y ≠ z,
(G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = ... (symmetry in all three variables),
(G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a∈X (rectangle inequality).
Then the function G is called a generalized metric, or, more specially, a G-metric on X, and the pair (X, G) is called a G-metric space.
Definition 1.2. (See [14]). Let (X, G) be a G-metric space, and let (x n ) be a sequence of points of X, therefore, we say that (x n ) is G-convergent to x ∈ X if , that is, for any ε > 0, there exists N ∈ ℕ such that G(x, x n , x m ) < ε, for all n, m ≥ N. We call x the limit of the sequence and write x n → x or
Proposition 1.1. (See [14]). Let (X, G) be a G-metric space. The following statements are equivalent:
-
(1)
(x n ) is G-convergent to x,
-
(2)
G(x n , x n , x) → 0 as n → +∞,
-
(3)
G(x n , x, x) → 0 as n → +∞,
-
(4)
G(x n , x m , x) → 0 as n, m → +∞.
Definition 1.3. (See [14]). Let (X, G) be a G-metric space. A sequence (x n ) is called a G-Cauchy sequence if for any ε > 0, there is N ∈ ℕ such that G(x n , x m , x l ) < ε for all m, n, l ≥ N, that is, G(x n , x m , x l ) → 0 as n, m, l → +∞.
Proposition 1.2. (See [14]). Let (X, G) be a G-metric space. Then the following statements are equivalent:
-
(1)
the sequence (x n ) is G-Cauchy,
-
(2)
for any ε > 0, there exists N ∈ ℕ such that G(x n , x m , x m ) < ε, for all m, n ≥ N.
Definition 1.4. (See [14]). A G-metric space (X, G) is called G-complete if every G-Cauchy sequence is G-convergent in (X,G).
Every G-metric on X defines a metric d G on X given by
Recently, Kaewcharoen and Kaewkhao [34] introduced the following concepts. Let X be a G-metric space. We shall denote CB(X) the family of all nonempty closed bounded subsets of X. Let H(.,.,.) be the Hausdorff G-distance on CB(X), i.e.,
where
Recall that G(x, y, C) = inf {G(x, y, z), z ∈ C}. A mapping T : X → 2X is called a multi-valued mapping. A point x ∈ X is called a fixed point of T if x ∈ Tx.
Definition 1.5. Let X be a given non empty set. Assume that g : X → X and T : X → 2X.
If w = gx ∈ Tx for some x ∈ X, then x is called a coincidence point of g and T and w is a point of coincidence of g and T.
Mappings g and T are called weakly compatible if gx ∈ Tx for some x ∈ X implies gT(x) ⊆ Tg(x).
Proposition 1.3. (see [34]). Let X be a given non empty set. Assume that g : X → X and T : X → 2X are weakly compatible mappings. If g and T have a unique point of coincidence w = gx∈ Tx, then w is the unique common fixed point of g and T.
In this article, we establish some common fixed point theorems for a hybrid pair {g,T} of single valued and multi-valued maps satisfying a generalized contractive condition defined on G-metric spaces. Also, an example is presented.
2. Main results
We start this section with the following lemma, which is the variant of the one given in Nadler [1] or Assad and Kirk [4]. Its proof is a simple consequence of the definition of the Hausdorff G-distance H G (A, B, B).
Lemma 2.1. If A, B ∈ CB(X) and a ∈ A, then for each ε > 0, there exists b ∈ B such that G(a,b,b) ≤ H G (A, B, B) + ε.
The main result of the article is the following.
Theorem 2.1. Let (X, G) be a G-metric space. Set g : X → X and T : X → CB(X). Assume that there exists a function α : [0,+∞) → [0,1) satisfying for every t ≥ 0 such that
for all x, y, z ∈ X. If for any x ∈ X, Tx ⊆ g(X) and g(X) is a G-complete subspace of X, then g and T have a point of coincidence in X. Furthermore, if we assume that gp ∈ Tp and gq ∈ Tq implies G(gq, gp, gp) ≤ H G (Tq, Tp, Tp), then
(i) g and T have a unique point of coincidence.
(ii) If in addition g and T are weakly compatible, then g and T have a unique common fixed point.
Proof. Let x0 be arbitrary in X. Since Tx0 ⊆ g(X), choose x1 ∈ X such that gx1 ∈ Tx0. If gx1= gx0, we finished. Assume that gx0 ≠ gx1, so G(gx0, gx1, gx1) > 0. We can choose a positive integer n1 such that
By Lemma 2.1 and the fact that Tx1 ⊆ g(X), there exists gx2 ∈ Tx1 such that
Using the two above inequalities and (2), it follows that
If gx1 = gx2, we finished. Assume that gx1 ≠ gx2. Now we choose a positive integer n2> n1 such that
Since Tx2 ∈ CB(X) and the fact that Tx2 ⊆ g(X), we may select gx3 ∈ Tx2 such that from Lemma 2.1
and then, similarly to the previous case, we have
By repeating this process, for each k ∈ ℕ*, we may choose a positive integer n k such that
Again, we may select gx k+1 ∈ Tx k such that
The last two inequalities together imply that
which shows that the sequence of nonnegative numbers {d k }, given by d k = G(gx k-1 , gx k , gx k ), k = 1, 2,. . ., is non-increasing. This means that there exists d ≥ 0 such that
Let now prove that the {gx k } is a G-Cauchy sequence.
Using the fact that, by hypothesis for t = d, , it results that there exists a rank k0 such that for k ≥ k0, we have α(d k ) < h, where
Now, by (3) we deduce that the sequence {d k } satisfies the following recurrence inequality
By induction, from (4), we get
which, by using the fact that α < 1, can be simplified to
Referring to the proof of Theorem 2.1 in [11] or Lemma 3.2 in [12], we may obtain
where c is a positive constant. We deduce that
Now for k ≥ k0 and m is a positive arbitrary integer, we have using the property (G4)
since 0 < h < 1. This shows that the sequence {gx n } is G-Cauchy in the complete subspace g(X). Thus, there exists q ∈ g(X) such that, from Proposition 1.1
Since q ∈ g(X), then there exists p ∈ X such that q = gp. From (5), we have
We claim that gp ∈ Tp. Indeed, from (2), we have
Letting n → +∞ in (7) and using (6), we get
that is, gp ∈ Tp. That is T and g have a point of coincidence. Now, assume that if gp ∈ Tp and gq ∈ Tq, then G(gq, gp, gp) ≤ H G (Tq, Tp, Tp). We will prove the uniqueness of a point of coincidence of g and T. Suppose that gp ∈ Tp and gq ∈ Tq. By (2) and this assumption, we have
and since α(G(gq, gp, gp)) < G(gq, gp, gp), so necessarily from (8), we have G(gq, gp, gp) = 0, i.e., gp = gq. In view of
we get Tq = Tp. Thus, T and g have a unique point of coincidence. Suppose that g and T are weakly compatible. By applying Proposition 1.3, we obtain that g and T have a unique common fixed point.
Corollary 2.1. Let (X,G) be a complete G-metric space. Assume that T : X → CB(X) satisfies the following condition
for all x, y, z ∈ X, where α : [0,+∞) → [0,1) satisfies for every t ≥ 0. Then T has a fixed point in X. Furthermore, if we assume that p ∈ Tp and q ∈ Tq implies G(q, p, p) ≤ H G (Tq, Tp, Tp), then T has a unique fixed point.
Proof. It follows by taking g the identity on X in Theorem 2.1.
Corollary 2.2. Let (X, G) be a G-metric space. Assume that g : X → X and T : X → CB(X) satisfy the following condition
for all x, y, z ∈ X, where k ∈ [0,1). If for any x ∈ X, Tx ⊆ g(X) and g(X) is a G-complete subspace of X, then g and T have a point of coincidence in X. Furthermore, if we assume that gp ∈ Tp and gq ∈ Tq implies G(gq, gp, gp) ≤ H G (Tq, Tp, Tp), then
(i) g and T have a unique point of coincidence.
(ii) If in addition g and T are weakly compatible, then g and T have a unique common fixed point.
Proof. It follows by taking α(t) = k, k ∈ [0,1), in Theorem 2.1.
In the case of single-valued mappings, that is, if T : X → X, (i.e., Tx = {Tx} for any x ∈ X), it is obviously that
Furthermore, if gp ∈ Tp (i.e., gp = Tp) and gq ∈ Tq (i.e., gq = Tq), then clearly,
that is, the assumption given in Theorem 2.1 is verified.
Also, the single-valued mappings T, g : X → X are said weakly compatible if Tgx = gTx whenever Tx = gx for some x ∈ X.
Now, we may state the following corollaries from Theorem 2.1 and the precedent corollaries:
Corollary 2.3. Let (X, G) be a complete G-metric space. Assume that T : X → X satisfies the following condition
for all x, y, z ∈ X, where α : [0, +∞) → [0, 1) satisfies for every t ≥ 0. Then, T has a unique fixed point.
Corollary 2.4. Let (X, G) be a G-metric space. Assume that g : X → X and T : X → X satisfy the following condition
for all x, y, z ∈ X, where α : [0, +∞) → [0, 1) satisfies for every t ≥ 0. If T(X) ⊆ g(X) and g(X) is a G-complete subspace of X, then
(i) g and T have a unique point of coincidence.
(ii) Furthermore, if g and T are weakly compatible, then g and T have a unique common fixed point.
Now, we introduce an example to support the useability of our results.
Example 2.1. Let X = [0, 1]. Define T : X → CB(X) by and define g : X → X by . Define a G-metric on X by G(x, y, z) = max{|x-y|, |x-z|, |y-z|}. Also, define α : [0, +∞) → [0, 1) by Then:
-
(1)
Tx ⊆ g(X) for all x ∈ X.
-
(2)
g(X) is a G-complete subspace of X.
-
(3)
g and T are weakly compatible.
-
(4)
H G (Tx, Ty, Tz) ≤ α(G(gx, gy, gz))G(gx, gy, gz) for all x, y, z ∈ X.
Proof. The proofs of (1), (2), and (3) are clear. By (1), we have
To prove (4), let x, y, z ∈ X. If x = y = z = 0, then
Thus, we may assume that x, y, and z are not all zero. With out loss of generality, we assume that x ≤ y ≤ z. Then
Since x ≤ y ≤ z, so This implies that
For each we have
Also, for each , we have
This yields that
Moreover, for each , we have
This yields that
We deduce that
On the other hand, it is obvious that all other hypotheses of Theorem 2.1 are satisfied and so g and T have a unique common fixed point, which is u = 0.
Remark 1 • Theorem 2.1 improves Kaewcharoen and Kaewkhao [[34], Theorem 3.3] (in case b = c = d = 0).
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Tahat, N., Aydi, H., Karapinar, E. et al. Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces. Fixed Point Theory Appl 2012, 48 (2012). https://doi.org/10.1186/1687-1812-2012-48
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DOI: https://doi.org/10.1186/1687-1812-2012-48