COMMON COINCIDENCE POINTS OF R-WEAKLY COMMUTING MAPS

A common coincidence point theorem for R-weakly commuting mappings is obtained. Our result extend several ones existing in literature. 2000 Mathematics Subject Classification. 54H25.

1. Introduction.Throughout this paper, X denotes a metric space with metric d.For x ∈ X and A ⊆ X, d(x, A) = inf{d(x, y) : y ∈ A}.We denote by CB(X) the class of all nonempty bounded closed subsets of X.Let H be the Hausdorff metric with respect to d, that is, for every A, B ∈ CB(X).The mappings T : X → CB(X), f : X → X are said to be commuting if, f T X ⊆ T f X.A point p ∈ X is said to be a fixed point of T : X → CB(X) if p ∈ T p.The point p is called a coincidence point of f and T if f p ∈ T p.The mappings f : X → X and T : X → CB(X) are called weakly commuting if, for all x ∈ X, f T x ∈ CB(X) and Recently Daffer and Kaneko [2] reaffirmed the positive answer [5] to the conjecture of Reich [8] by giving an alternative proof to Theorem 5 of Mizoguchi and Takahashi [5].We state Theorem 2.1 of Daffer and Kaneko [2] for convenience.
Theorem 1.1.Let X be a complete metric space and T : X → CB(X).If α is a function of (0, ∞) to (0, 1] such that lim sup r →t + α(r ) < 1 for each t ∈ [0, ∞) and if for each x, y ∈ X, then T has a fixed point in X.
The purpose of this paper is to obtain a coincidence point theorem for R-weakly commuting multivalued mappings analogous to Theorem 1.1.We follow the same technique used in [2].The notion of R-weak commutativity for single-valued mappings was defined by Pant [7] to generalize the concept of commuting and weakly commuting mappings [9].Recently, Shahzad and Kamran [10] extended this concept to the setting of single and multivalued mappings, and studied the structure of common fixed points.Definition 1.2 (see [10]).The mappings f : X → X and T : X → CB(X) are called R-weakly commuting if for all x ∈ X, f T x ∈ CB(X) and there exists a positive real number R such that (1.3) 2. Main result.Before giving our main result, we state the following lemmas which are noted in Nadler [6], and Assad and Kirk [1].
Now, we prove our main result.
Theorem 2.3.Let X be a complete metric space, f ,g : X → X and S, T : X → CB(X) are continuous mappings such that SX ⊆ gX and T X ⊆ f X.Let α : (0, ∞) → (0, 1] be such that lim sup r →t + α(r ) < 1 for each t ∈ [0, ∞) and for each x, y ∈ X.If the pairs (g, T ) and (f , S) are R-weakly commuting, then g, S and f ,T have a common coincidence point.
Proof.Our method is constructive.We construct sequences {x n }, {y n }, and {A n } in X and CB(X), respectively as follows.Let x 0 be an arbitrary point of X and y 0 = f x 0 .Since Sx 0 ⊆ gX, there exists a point x 1 ∈ X such that y 1 = gx 1 ∈ Sx 0 = A 0 .Choose a positive integer n 1 such that Now Lemma 2.1 and the fact T X ⊆ f X guarantee that there is a point The above inequality in view of (2.2) and (2.3) implies that d(y 2 ,y 1 ) < d(y 0 ,y 1 ).Now choose a positive integer n 2 > n 1 such that where (2.12) Now it follows from the continuity of f , g, T , and S that gz ∈ T z and f z ∈ Sz.
If we put T = S and f = g in Theorem 2.3, we get the following corollary.
Corollary 2.4.Let X be a complete metric space, and let f : X → X be a continuous mapping and T : X → CB(X) be a mapping such that T X ⊆ f X.Let α : (0, ∞) → (0, 1] be such that lim sup r →t + α(r ) < 1 for each t ∈ [0, ∞) and

H(T x, T y) ≤ α d(f x, f y) d(f x, f y)
(2.13) for each x, y ∈ X.If the mappings f and T are R-weakly commuting, then f and T have coincidence point.
(2) In Corollary 2.4, T is not assumed to be continuous.In fact the continuity of T follows from the continuity of f .
(3) If we put f = I, the identity map, in Corollary 2.4, we obtain Theorem 1.1.
[2]or each k.So we have d(y 2k+1 ,y 2k ) < d(y 2k ,y 2k−1 ).Therefore, the sequence {d(y 2k+1 ,y 2k )} is monotone nonincreasing.Then, as in the proof of Theorem 2.1 in[2], {y n } is a Cauchy sequence in X.Further, equation (2.2) ensures that {A n } is a Cauchy sequence in CB(X).It is well known that if X is complete, then so is CB(X).Therefore, there exist z ∈ X and A ∈ CB(X) such that y n → z and A n → A. Moreover, gx 2k+1 → z and f x 2k → z.Since 2k+1 ,Sf x 2k ≤ H f Sx 2k ,Sf x 2k ≤ R d f x 2k ,Sx 2k .