Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces

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.

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 → 2^X 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 → 2^X.

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 → 2^X 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.

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).

• Corollary 2.3 generalizes Mustafa [[15], Theorem 5.1.7] and Shatanawi [[35], Corollary 3.4].

The authors declare that they have no competing interests.

The authors have contributed in obtaining the new results presented in this article. All authors read and approved the final manuscript.

The authors thank the editor and the referees for their useful comments and suggestions.

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