TRIPLED FIXED POINT THEOREM FOR HYBRID PAIR OF MAPPINGS UNDER GENERALIZED NONLINEAR CONTRACTION

Abstract

In this paper, we introduce the concept of w¡compatibility and weakly commutativity for hybrid pair of mappings $F:X{\times}X{\times}X{\rightarrow}2^X$ and $g:X{\rightarrow}X$ and establish a common tripled fixed point theorem under generalized nonlinear contraction. An example is also given to validate our result. We improve, extend and generalize various known results.

1. INTRODUCTION AND PRELIMINARIES

Let (X, d) be a metric space and CB(X) be the set of all nonempty closed bounded subsets of X. Let D(x, A) denote the distance from x to A ⊂ X and H denote the Hausdorff metric induced by d, that is, and for all A, B ∈ CB(X).Markin [23] initiated the study of fixed points for multivalued contractions and non-expansive maps using the Hausdorff metric. Fixed points existence for various multivalued contractive mappings has been studied by several authors under different conditions. For details, we refer the reader to [1, 2, 12, 13, 14, 15, 16, 18, 19, 20, 21, 25, 26, 27] and the reference therein. Multivalued maps theory has application in control theory, convex optimization, differential equations and economics.

Bhaskar and Lakshmikantham [10], established some coupled fixed point theorems and apply these to study the existence and uniqueness of solution for periodic boundary value problems. Lakshmikantham and Ciric [22] proved coupled coinci-dence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces and extended the results of Bhaskar and Lakshmikantham [10].

Berinde and Borcut [8] introduced the concept of tripled fixed point for single valued mappings in partially ordered metric spaces. In [8], Berinde and Borcut established the existence of tripled fixed point of single-valued mappings in partially ordered metric spaces. For more details on tripled fixed point theory, we also refer the reader to [3, 4, 5, 6, 7, 9, 11]: Samet and Vetro [24] introduced the notion of fixed point of N order in case of single-valued mappings. In particular for N=3 (tripled case), we have the following definition:

Definition 1.1 ([24]). Let X be a non-empty set and F : X × X × X → X be a given mapping. An element (x, y, z) ∈ X × X × X is called a tripled fixed pointof the mapping F if F(x, y, z) = x, F(y, z, x) = y and F(z, x, y) = z. In this paper, we prove a common tripled fixed point for hybrid pair of mappings under generalized nonlinear contraction. We improve, extend and generalize the results of Ding, Li and Radenovic [17] and Abbas, Ciric, Damjanovic and Khan [2]. The effectiveness of the present work is validated with the help of suitable example.

 

2. MAIN RESULTS

First we introduce the following:

Definition 2.1. Let X be a nonempty set, F : X × X × X → 2X(a collection of all nonempty subsets of X) and g be a self-map on X. An element (x, y, z)∈ X × X × X is called (1) a tripled fixed point of F if x∈ F(x, y, z), y ∈ F(y, z, x) and z ∈ F(z, x, y). (2) a tripled coincidence point of hybrid pair {F, g} if g(x) ∈ F(x, y, z), g(y) ∈ F(y, z, x) and g(z) ∈ F(z, x, y). (3) a common tripled fixed point of hybrid pair {F, g} if x = g(x) ∈ F(x, y, z), y = g(y) ∈ F(y, z, x) and z = g(z) ∈ F(z, x, y).

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

Deginition 2.2. Let F : X × X × X→ 2X be a multivalued mapping and g be a self-map on X. The hybrid pair {F, g} is called w-compatible if g(F(x, y, z)) ⊆ F(gx, gy, gz) whenever (x, y, z) ∈C(F, g).

Deginition 2.3. Let F : X × X × X→ 2X be a multivalued mapping and g be a self-map on X. The mapping g is called F-weakly commuting at some point (x, y, z) ∈ X3 if g2x ∈ F(gx, gy, gz), g2y ∈ F(gy, gz, gx) and g2z ∈ F(gz, gx, gy).

Lemma 2.1. Let (X, d) be a metric space. Then, for each a ∈ X and B ∈ CB(X), there is b0 ∈ B such that D(a, B) = d(a, b0), where D(a, B) = infb∈Bd(a, b).

Proof. Let a ∈ X and B ∈ CB(X). Since the function d is continuous. Thus, by the closedness of B, there exists b0 ∈ B such that infb ∈B d(a, b) = d(a, b0), that is, D(a,B) = d(a, b0).

Let Φ denote the set of all functions φ : [0, ∞) → [0, ∞) satisfying(iφ) φ is non-decreasing, (iiφ) limn→ ∞φn(t) = 0 for all t > 0.

It is clear that φ(t) < t for each t > 0: In fact, if φ(t0) ≥ t0 for some t0 > 0. then, since φ is non-decreasing, φn(t0) ≥ t0 for all n ∈ , which contradicts with limn→∞ φn(t0) = 0. In addition, it is easy to see that φ(0) = 0. □

Theorem 2.1. Let (X, d) be a metric space. Assume F : X × X × X→ CB(X) and g : X → X be two mappings satisfying

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

(a) F and g are w-compatible.limn→∞ gnx = u, limn→∞ gny = v and limn→∞gnz = w for some (x, y, z) ∈ C(F, g) and for some u, v, w ∈ X and g is continuous at u, v and w. (b) g is F-weakly commuting for some (x, y, z) ∈ C(F, g) and gx, gy and gz are fixed points of g, that is, g2x = gx, g2y = gy and g2z = gz. (c) g is continuous at x, y and z. limn→∞ gnu = x, limn→∞ gnv = y and limn→∞ gnw = z for some (x, y, z) ∈ C(F, g) and for some u, v, w ∈ X: (d) g(C(g, F)) is singleton subset of C(g, F):

Proof. Let x0, y0, z0 ∈ X be arbitrary. Then F(x0, y0, z0), F(y0, z0, x0) and F(z0, x0, y0) are well defined. Choose gx1 ∈ F(x0, y0, z0), gy1 ∈ F(y0, z0, x0) and gz1 ∈ F(z0, x0, y0), because F(X×X×X) ⊆ g(X). Since F . X×X×X → CB(X), therefore by Lemma 2.1, there exist u1∈ F(x1, y1, z1), u2 ∈ F(y1, z1, x1) and u3 ∈ F(z1, x1, y1) such that

d(gx1, u1) ≤ H(F(x0, y0, z0), F(x1, y1, z1)), d(gy1, u2) ≤ H(F(y0, z0, x0), F(y1, z1, x1)), d(gz1, u3) ≤ H(F(z0, x0, y0), F(z1, x1, y1)).

Since F(X×X×X) ⊆ g(X), there exist x2, y2, z2 ∈ X such that u1 = gx2, u2 = gy2 and u3 = gz2, Thus

d(gx1, gx2) ≤ H(F(x0, y0, z0), F(x1, y1, z1)), d(gy1, gy2) ≤ H(F(y0, z0, x0), F(y1, z1, x1)), d(gz1, gz2) ≤ H(F(z0, x0, y0), F(z1, x1, y1)).

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

Thus,

Similarly

Combining (2,2), (2,3) and (2,4), we get

Thus

If we suppose that

then by (2, 5), (iφ) and (iiφ), we have

which is a contradiction. Thus, we must have

Hence by (2,5), we have for all n ∈ ,

Thus

where δ= max {d(gx0, gx1), d(gy0, gy1), d(gz0, gz1)} .

Without loss of generality, one can assume that max max {d(gx0, gx1), d(gy0, gy1), d(gz0, gz1)} ≠ In fact, if this is not true, then gx0 = gx1 ∈ F(x0, y0, z0), gy0 = gy1 ∈ F(y0, z0, x0) and gz0 = gz1 ∈ F(z0, x0, y0) that (x0, y0, z0) s a tripled coincidence point of F and g.

Thus, for m, n ∈ with m > n, by triangle inequality and (2,6), we get

which implies, by (iiφ), that {gxn} is a Cauchy sequence in g(X). Similarly we obtain that {gyn} and {gzn} are Cauchy sequences in g(X). Since g(X) is complete, there exist x, y, z ∈ X such that

Now, since gxn+1 ∈ F(xn, yn, zn), gyn+1 ∈ F(yn, zn, xn) and gzn+1 ∈ F(zn, xn, yn), therefore by using condition (2,1), we get

where

Since limn→∞ gxn = gx, limn→∞ gyn= gy and limn→∞ gzn= gzthere exists n0 ∈ such that for all n > n0,

Combining this with (2,8), (2,9) and (2,10), we get for all n > n0,

Now, we claim that

If this is not true, then

Thus, by (2,11), (iφ) and (iiφ), we get for all n > n0,

Thus

Letting n → ∞ in (2:13), by using (2,7), we obtain

which is a contradiction. So (2, 12) holds. Thus, it follows that gx∈ F(x, y, z), gy∈ F(y, z, x) and gz∈ F(z, x, y), that is, (x, y, z) is a tripled coincidence point of F and g. Hence C(F, g) is nonempty. Suppose now that (a) holds. Assume that for some (x, y, z) ∈ C(F, g),

where u, v, w ∈ X. Since g is continuous at u, v and w. We have, by (2,14), that u,v and w are fixed points of g, that is,

As F and g are w-compatible, so for all n ≥ 1,

Now, by using (2,1) and (2,16), we obtain

where

By (2,14) and (2,15), there exists n0 ∈ such that for all n > n0,

Combining this with (2,17), we get for all n > n0,

Now, we claim that

If this is not true, then

Thus, by (2,18), (iφ) and (iiφ), we get for all n > n0,

On taking limit as n → ∞ in (2, 20), by using (2, 14) and (2, 15), we get

which is a contradiction. So (2, 19) holds. Thus, it follows that

Now, from (2, 15) and (2, 21), we have that is, (u, v, w) is a common tripled fixed point of F and g.

Suppose now that (b) holds. Assume that for some (x, y, z) ∈ C(F, g), g is F- weakly commuting, that is, g2x ∈ F(gx, gy, gz), g2y ∈ F(gy, gz, gx), g2z ∈ F(gz, gx, gy), and g2x = gx, g2y = gy, g2z = gz. Thus gx = g2x ∈ F(gx, gy, gz), gy = g2y ∈ F(gy, gz, gx) and gz = g2z ∈ F(gz, gx, gy), that is (gx, gy, gz) is a common tripled fixed point of F and g.

Suppose now that (c) holds. Assume that for some (x, y, z) ∈ C(F, g) and for some u, v, w ∈ X, limn→∞ gnu = x, limn→∞ gnv = y and limn→∞ gnw = z. Since g is continuous at x, y and z. We have that x, y and z are fixed point of g, that is, gx = x, gy = y and gz = z. Since (x, y, z) ∈ C(F, g), therefore, we obtain

x = gx ∈ F(x, y, z), y = gy ∈ F(y, z, x)

and

z = gz ∈ F(z, x, y),

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

Example 2.1. Suppose that X = [0, 1], equipped with the metric d : X ×X →[0, +∞) defined by d(x, y) = max{x, y} and d(x, x) = 0 for all x, y ∈ X. Let F : X×X×X → CB(X) be defined as and g : X → X be defined as

g(x) = x2, for all x ∈ X.

Define φ : [0, ∞ ] → [0, ∞ ] by

Now, for all x, y, z, u, v, w ∈ X with x, y, z, u, v, w ∈ [0, 1), we have Case (a) If x2 + y2 + z2 = u2 + v2 + w2, then

Case (b) If x2 + y2 + z2 ≠ u2 + v2 + w2 with x2 + y2 + z2 < u2 + v2 + w2, then

Similarly, we obtain the same result for u2 + v2 + w2 < x2 + y2 + z2. Thus the contractive condition (2,1) is satisfied for all x, y, z, u, v, w ∈ X with x, y, z, u, v, w ∈ [0, 1). Again for all x, y, z, u, v, w ∈X with x, y, z, ∈ [0, 1) and u, v, w = 1, we have

Thus the contractive condition (2:1) is satisfied for all x, y, z, u, v, w ∈ X with x, y z ∈ [0, 1) and u, v, w = 1, Similarly, we can see that the contractive condition (2,1) is satisfied for all x, y, z, u, v, w ∈ X with x, y, z, u, v, w = 1. Hence, the hybrid pair {F, g} satisfies the contractive condition (2,1), for all x, y, z, u, v, w ∈ X. In addition, all the other conditions of Theorem 2.1 are satisfied and z = (0, 0, 0) is a common tripled fixed point of hybrid pair {F, g} . The function F : X×X×X → CB(X) involved in this example is not continuous at the point (1, 1, 1) ∈ X×X×X.

Remark 2.1. We improve, extend and generalize the result of Ding, Li and Radenovic [17] in the following sense:

(i) We prove our result in the settings of multivalued mapping and for hybrid pair of mappings while Ding, Li and Radenovic [17] proved result for single valued mappings. (ii) We prove tripled coincidence and common tripled fixed point theorem while Ding, Li and Radenovic [17] proved coupled coincidence and common coupled fixed point theorems. (iii) To prove the result we consider non complete metric space and the space is also not partially ordered. (iv) The mapping F : X ×X ×X → CB(X) is discontinuous and not satisfying mixed g-monotone property. (v) The function φ : [0, ∞) → [0, ∞) involved in our theorem and example is discontinuous. (vi) Our proof is simple and different from the other results in the existing literature.

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

Corollary 2.2. Let (X, d) be a complete metric space, F : X × X× X → CB(X) be a mapping satisfying for all x, y, z, u, v, w ∈ X, where φ ∈ Φ. Then F has a tripled fixed point.

If we put φ(t) = kt where 0 < k < 1 in Theorem 2.1, then we have the following result:

Corollary 2.3. Let (X, d) be a metric space. Assume F : X × X× X → CB(X) and g : X → X be two mapping satisfying

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

(a) F and g are w-compatible. limn→∞ gnx = u, limn→∞ gny = v and limn→∞ gnz = w for some (x, y, z) ∈C(F, g) and for some u, v, w ∈ X and g is con-tinuous at u, v and w. (b) g is F-weakly commuting for some (x, y, z) ∈ C(F, g) and gx, gy and gz are fixed points of g, that is, g2x = gx, g2y = gy and g2z = gz. (c) g is continuous at x, y and z. limn→∞ gnu = x, limn→∞ gnv = y and limn→∞ gnw = z for som (x, y, z ) ∈ C(F, g) and for some u, v, w ∈ X. (d) g(C(g, F)) is singleton subset of C(g, F).

If we put g = I (I is the identity mapping) in Corollary 2.3, then we have the following result:

Corollary 2.4. Let (X, d) be a complete metric space, F : X × X × X → CB(X) be a mapping satisfying for all x, y, z, u, v, w ∈ X . Then F has a tripled fixed point.