COMMON COUPLED FIXED POINT THEOREM UNDER GENERALIZED MIZOGUCHI-TAKAHASHI CONTRACTION FOR HYBRID PAIR OF MAPPINGS GENERALIZED MIZOGUCHI-TAKAHASHI CONTRACTION

Abstract

We establish a common coupled fixed point theorem for hybrid pair of mappings under generalized Mizoguchi-Takahashi contraction on a noncomplete metric space, which is not partially ordered. It is to be noted that to find coupled oincidence point, we do not employ the condition of continuity of any mapping involved therein. An example is also given to validate our results. We improve, extend and generalize several known results.

1. INTRODUCTION

Let (X, d) be a metric space. We denote by 2X the class of all nonempty subsets of X, by CL(X) the class of all nonempty closed subsets of X, by CB(X) the class of all nonempty closed bounded subsets of X and by K(X) the class of all nonempty compact subsets of X. A functional H : CL(X) × CL(X) → ℝ+ ∪ {+∞} is said to be the Pompeiu-Hausdorff generalized metric induced by d is given by for all A, B ∈ CB(X), where D(x, A) = infa∈A d(x, a) denote the distance from x to A ⊂ X. For simplicity, if x ∈ X, we denote g(x) by gx.

Markin [23] initiated to study the existence of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff metric which was further studied by many authors under different contractive conditions. The theory of multivalued mappings has application in control theory, convex optimization, differential inclusions and economics.

Bhaskar and Lakshmikantham [6] established some coupled fixed point theorems and applied these results to study the existence and uniqueness of solution for periodic boundary value problems, which were later extended by Lakshmikantham and Ciric [19]. For more details, see [5, 7, 8, 9, 10, 16, 17, 21, 22, 27, 30].

Samet et al. [28] claimed that most of the coupled fixed point theorems for single valued mappings on ordered metric spaces are consequences of well-known fixed point theorems.

The concepts related to coupled fixed point theory for multivalued mappings were extended by Abbas et al. [2] and obtained coupled coincidence point and common coupled fixed point theorems involving hybrid pair of mappings satisfying generalized contractive conditions in complete metric spaces. Very few researcher gave attention to coupled fixed point problems for hybrid pair of mappings including [1, 2, 11, 12, 13, 14, 15, 20, 29].

In [2], Abbas et al. introduced the following for multivalued mappings:

Definition 1.1. Let X be a nonempty set, F : X × X → 2X (a collection of all nonempty subsets of X) and g be a self-mapping on X. An element (x, y) ∈ X × X is called

(1) a coupled fixed point of F if x ∈ F(x, y) and y ∈ F(y, x). (2) a coupled coincidence point of hybrid pair {F, g} if gx ∈ F(x, y) and gy ∈ F(y, x). (3) a common coupled fixed point of hybrid pair {F, g} if x = gx ∈ F(x, y) and y = gy ∈ F(y, x).

We denote the set of coupled coincidence points of mappings F and g by C(F, g). Note that if (x, y) ∈ C(F, g), then (y, x) is also in C(F, g).

Definition 1.2. Let F : X × X → 2 X be a multivalued mapping and g be a self-mapping on X. The hybrid pair {F, g} is called w−compatible if gF(x, y) ⊆ F(gx, gy) whenever (x, y) ∈ C(F, g).

Definition 1.3. Let F : X × X → 2X be a multivalued mapping and g be a self-mapping on X. The mapping g is called F−weakly commuting at some point (x, y) ∈ X × X if g2x ∈ F(gx, gy) and g2y ∈ F(gy, gx).

Lemma 1.4 ([26]). Let (X, d) be a metric space. Then, for each a ∈ X and B ∈ K(X), there is b0 ∈ B such that D(a, B) = d(a, b0), where D(a, B) = infb∈B d(a, b).

Nadler [25] extended the famous Banach Contraction Principle [4] from single-valued mapping to multivalued mapping. Mizoguchi and Takahashi [24] proved the following generalization of Nadler’s fixed point theorem for weak contraction:

Theorem 1.5. Let (X, d) be a complete metric space and T : X → CB(X) be a multivalued mapping. Assume that H(T x, T y) ≤ Ψ(d(x, y))d(x, y),

for all x, y ∈ X, where Ψ is a function from [0, ∞) into [0, 1) satisfying

for all t ≥ 0. Then T has a fixed point.

Suzuki [31] gave its very simple proof. Amini-Harandi and O’Regan [3] obtained a generalization of Mizoguchi and Takahashi’s fixed point theorem.

In [8], Ciric et al. proved coupled fixed point theorems for mixed monotone mappings satisfying a generalized Mizoguchi-Takahashi’s condition in the setting of ordered metric spaces. Main results of Ciric et al. [8] extended and generalized the results of Bhaskar and Lakshmikantham [6], Du [17] and Harjani et al. [18].

In this paper, we prove a common coupled fixed point theorem for hybrid pair of mappings under generalized Mizoguchi-Takahashi contraction on a noncomplete metric space, which is not partially ordered. It is to be noted that to find coupled coincidence point, we do not employ the condition of continuity of any mapping involved therein. We improve, extend and generalize the results of Amini-Harandi and O’Regan [3], Bhaskar and Lakshmikantham [6], Ciric et al. [8], Du [17], Harjani et al. [18] and Mizoguchi and Takahashi [24]. The effectiveness of our generalization is demonstrated with the help of an example.

 

2. MAIN RESULTS

Let Ф denote the set of all functions φ : [0, +∞) → [0, +∞) satisfying

Let Ψ denote the set of all functions Ψ : [0, +∞) → [0, 1) which satisfies limr→t+ Ψ(r) < 1 for all t ≥ 0. For example, if φ(t) = ln(t + 1) and . Obviously, then φ ∈ Φ and Ψ ∈ Ψ, because φ is non-decreasing, positive in (0, +∞), φ(0) = 0 and . Also,

Theorem 2.1. Let (X, d) be a metric space, F : X × X → K(X) and g : X → X be two mappings. Assume that there exist some φ ∈ Φ and Ψ ∈ Ψ such that

for all x, y, u, v ∈ X. Furthermore assume that F(X × X) ⊆ g(X) and g(X) is a complete subset of X. Then F and g have a coupled coincidence point. Moreover, F and g have a common coupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limn→∞ gnx = u and limn→∞ gny = v for some (x, y) ∈ C(F, g) and for some u, v ∈ X and g is continuous at u and v. (b) g is F−weakly commuting for some (x, y) ∈ C(F, g) and gx and gy are fixed points of g, that is, g2x = gx and g2y = gy. (c) g is continuous at x and y. limn→∞ gnu = x and limn→∞ gnv = y for some (x, y) ∈ C(F, g) and for some u, v ∈ X. (d) g(C(F, g)) is a singleton subset of C(F, g).

Proof. Let x0, y0 ∈ X be arbitrary. Then F(x0, y0) and F(y0, x0) are well defined. Choose gx1 ∈ F(x0, y0) and gy1 ∈ F(y0, x0), because F(X × X) ⊆ g(X). Since F : X × X → K(X), therefore by Lemma 1.4, there exist z1 ∈ F(x1, y1) and z2 ∈ F(y1, x1) such that

Since F(X × X) ⊆ g(X), there exist x2, y2 ∈ X such that z1 = gx2 and z2 = gy2.

Thus

Continuing this process, we obtain sequences {xn} and {yn} in X such that for all n ∈ ℕ, we have gxn+1 ∈ F(xn, yn) and gyn+1 ∈ F(yn, xn) such that

which, by (iφ) and (2.1), implies

which, by the fact that Ψ < 1, implies

Similarly

Combining (2.2) and (2.3), we get

Since φ is non-decreasing, it follows that

Now (2.4) shows that {φ(max{d(gxn, gxn+1), d(gyn, gyn+1)})} is a non-increasing sequence. Therefore, there exists some δ ≥ 0 such that

Since Ψ ∈ Ψ, we have limr→δ+ Ψ(r) < 1 and Ψ(δ) < 1. Then there exists α ∈ [0, 1) and ε > 0 such that Ψ(r) ≤ α for all r ∈ [δ, δ + ε). From (2.5), we can take n0 ≥ 0 such that δ ≤ φ(max{d(gxn, gxn+1), d(gyn, gyn+1)}) ≤ δ + ε for all n ≥ n0. Then from (2.1), for all n ≥ n0, we have

Thus, for all n ≥ n0, we have

Similarly, for all n ≥ n0, we have

Combining (2.6) and (2.7), for all n ≥ n0, we get

Since φ is non-decreasing, it follows that, for all n ≥ n0,

Letting n → ∞ in (2.8) and using (2.5), we obtain that δ ≤ αδ. Since α ∈ [0, 1), therefore δ = 0. Thus

Since {φ(max{d(gxn, gxn+1), d(gyn, gyn+1)})} is a non-increasing sequence and φ is non-decreasing, then {max{d(gxn, gxn+1), d(gyn, gyn+1)}} is also a non-increasing sequence of positive numbers. This implies that there exists θ ≥ 0 such that

Since φ is non-decreasing, we have

Letting n → ∞ in this inequality, by using (2.9), we get 0 ≥ φ(θ), which, by (iiφ), implies that θ = 0. Thus, by (2.10), we get

Suppose that max{d(gxn, gxn+1), d(gyn, gyn+1)} = 0, for some n ≥ 0. Then, we have d(gxn, gxn+1) = 0 and d(gyn, gyn+1) = 0 which implies that gxn = gxn+1 ∈ F(xn, yn) and gyn = gyn+1 ∈ F(yn, xn), that is, (xn, yn) is a coupled coincidence point of F and g. Now, suppose that max{d(gxn, gxn+1), d(gyn, gyn+1)} ≠ 0, for all n ≥ 0.

Denote

From (2.8), we have

Then, we have

On the other hand, by (iiiφ), we have

Thus, by (2.12) and (2.13), we have ∑max{d(gxn, gxn+1), d(gyn, gyn+1)} < ∞. It means that and are Cauchy sequences in g(X). Since g(X) is complete, therefore there exist x, y ∈ X such that

Now, since gxn+1 ∈ F(xn, yn) and gyn+1 ∈ F(yn, xn), therefore by using condition (2.1) and (iφ), we get

Since φ is non-decreasing, we have

Letting n → ∞ in (2.15), by using (2.14), we obtain

which implies that

that is, (x, y) is a coupled coincidence point of F and g. Hence C(F, g) is nonempty.

Suppose now that (a) holds. Assume that for some (x, y) ∈ C(F, g),

where u, v ∈ X. Since g is continuous at u and v. We have, by (2.16), that u and v are fixed points of g, that is,

As F and g are w−compatible, so

that is,

Now, by using (2.1) and (2.18), we obtain

Since φ is non-decreasing, we have

On taking limit as n → ∞ in the above inequality, by using (2.16) and (2.17), we get

which implies that

Now, from (2.17) and (2.19), we have

that is, (u, v) is a common coupled fixed point of F and g.

Suppose now that (b) holds. Assume that for some (x, y) ∈ C(F, g), g is F−weakly commuting, that is g2x ∈ F(gx, gy), g2y ∈ F(gy, gx) and g2x = gx, g2y = gy. Thus gx = g2x ∈ F(gx, gy) and gy = g2y ∈ F(gy, gx), that is, (gx, gy) is a common coupled fixed point of F and g.

Suppose now that (c) holds. Assume that for some (x, y) ∈ C(F, g) and for some u, v ∈ X,

Since g is continuous at x and y, then x and y are fixed points of g, that is,

Since (x, y) ∈ C(F, g), so we obtain

that is, (x, y) is a common coupled fixed point of F and g.

Finally, suppose that (d) holds. Let g(C(F, g)) = {(x, x)}. Then {x} = {gx} = F(x, x). Hence (x, x) is a common coupled fixed point of F and g. ⃞

Example. Suppose that X = [0, 1], equipped with the metric d : X ×X → [0, +∞) defined as d(x, y) = max{x, y} and d(x, x) = 0 for all x, y ∈ X. Let F : X × X → K(X) be defined as

and g : X → X be defined as

Define φ : [0, +∞) → [0, +∞) by

and Ψ : [0, +∞) → [0, 1) defined by

Now, for all x, y, u, v ∈ X with x, y, u, v ∈ [0, 1), we have

Case (a). If x = u, then

which implies that

Case (b). If x ≠ u with x < u, then

which implies that

Similarly, we obtain the same result for u < x. Thus the contractive condition (2.1) is satisfied for all x, y, u, v ∈ X with x, y, u, v ∈ [0, 1). Again, for all x, y, u, v ∈ X with x, y ∈ [0, 1) and u, v = 1, we have

which implies that

Thus the contractive condition (2.1) is satisfied for all x, y, u, v ∈ X with x, y ∈ [0, 1) and u, v = 1. Similarly, we can see that the contractive condition (2.1) is satisfied for all x, y, u, v ∈ X with x, y, u, v = 1. Hence, the hybrid pair {F, g} satisfies the contractive condition (2.1), for all x, y, u, v ∈ X. In addition, all the other conditions of Theorem 2.1 are satisfied and z = (0, 0) is a common coupled fixed point of hybrid pair {F, g}. The function F : X × X → K(X) involved in this example is not continuous at the point (1, 1) ∈ X × X.

Remark 2.2. We improve, extend and generalize the results of Ciric et al. [8] in the sense that

(i) We prove our result for hybrid pair of mappings. (ii) We prove our result in the framework of non-complete metric space (X, d) and the product set X × X is not empowered with any order. (iii) We prove our result without the assumption of continuity and mixed gmonotone property for mapping F : X × X → K(X). (iv) The functions φ : [0, +∞) → [0, +∞) and Ψ : [0, +∞) → [0, 1) involved in our theorem and example are discontinuous.

If we put g = I (the identity mapping) in Theorem 2.1, we get the following result:

Corollary 2.3. Let (X, d) be a complete metric space, F : X × X → K(X) be a mapping. Assume that there exist some φ ∈ Φ and Ψ ∈ Ψ such that

for all x, y, u, v ∈ X. Then F has a coupled fixed point.

If we put for all t ≥ 0 in Theorem 2.1, then we get the following result:

Corollary 2.4. Let (X, d) be a metric space, F : X × X → K(X) and g : X → X be two mappings. Assume that there exist some φ ∈ Φ and such that

for all x, y, u, v ∈ X. Furthermore assume that F(X × X) ⊆ g(X) and g(X) is a complete subset of X. Then F and g have a coupled coincidence point. Moreover, F and g have a common coupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limn→∞ gnx = u and limn→∞ gny = v for some (x, y) ∈ C(F, g) and for some u, v ∈ X and g is continuous at u and v. (b) g is F−weakly commuting for some (x, y) ∈ C(F, g) and gx and gy are fixed points of g, that is, g2x = gx and g2y = gy. (c) g is continuous at x and y. limn→∞ gnu = x and limn→∞ gnv = y for some (x, y) ∈ C(F, g) and for some u, v ∈ X. (d) g(C(F, g)) is a singleton subset of C(F, g).

If we put g = I (the identity mapping) in the Corollary 2.4, we get the following result:

Corollary 2.5. Let (X, d) be a complete metric space, F : X × X → K(X) be a mapping. Assume that there exist some φ ∈ Φ and such that

for all x, y, u, v ∈ X. Then F has a coupled fixed point.

If we put φ(t) = 2t for all t ≥ 0 in Theorem 2.1, then we get the following result:

Corollary 2.6. Let (X, d) be a metric space, F : X × X → K(X) and g : X → X be two mappings. Assume that there exists some Ψ ∈ Ψ such that

for all x, y, u, v ∈ X. Furthermore assume that F(X × X) ⊆ g(X) and g(X) is a complete subset of X. Then F and g have a coupled coincidence point. Moreover, F and g have a common coupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limn→∞ gnx = u and limn→∞ gny = v for some (x, y) ∈ C(F, g) and for some u, v ∈ X and g is continuous at u and v. (b) g is F−weakly commuting for some (x, y) ∈ C(F, g) and gx and gy are fixed points of g, that is, g2x = gx and g2y = gy. (c) g is continuous at x and y. limn→∞ gnu = x and limn→∞ gnv = y for some (x, y) ∈ C(F, g) and for some u, v ∈ X. (d) g(C(F, g)) is a singleton subset of C(F, g).

If we put g = I (the identity mapping) in the Corollary 2.6, we get the following result:

Corollary 2.7. Let (X, d) be a complete metric space, F : X × X → K(X) be a mapping. Assume that there exists some Ψ ∈ Ψ such that

H(F(x, y), F(u, v)) ≤ Ψ (2 max {d(x, u), d(y, v)}) max {d(x, u), d(y, v)} , for all x, y, u, v ∈ X. Then F has a coupled fixed point.

If we put Ψ(t) = k where 0 < k < 1, for all t ≥ 0 in Corollary 2.6, then we get the following result:

Corollary 2.8. Let (X, d) be a metric space. Assume F : X × X → K(X) and g : X → X be two mappings satisfying

for all x, y, u, v ∈ X, where 0 < k < 1. Furthermore assume that F(X × X) ⊆ g(X) and g(X) is a complete subset of X. Then F and g have a coupled coincidence point. Moreover, F and g have a common coupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limn→∞ gnx = u and limn→∞ gny = v for some (x, y) ∈ C(F, g) and for some u, v ∈ X and g is continuous at u and v. (b) g is F−weakly commuting for some (x, y) ∈ C(F, g) and gx and gy are fixed points of g, that is, g2x = gx and g2y = gy. (c) g is continuous at x and y. limn→∞ gnu = x and limn→∞ gnv = y for some (x, y) ∈ C(F, g) and for some u, v ∈ X. (d) g(C(F, g)) is a singleton subset of C(F, g).

If we put g = I (the identity mapping) in the Corollary 2.8, we get the following result:

Corollary 2.9. Let (X, d) be a complete metric space. Assume F : X ×X → K(X) be a mapping satisfying

for all x, y, u, v ∈ X, where 0 < k < 1. Then F has a coupled fixed point.