Fixed point theorems for a family of hybrid pairs of mappings in metrically convex spaces
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The present paper establishes some coincidence and common fixed point theorems for a sequence of hybrid-type nonself-mappings defined on a closed subset of a metrically convex metric space. Our results generalize some earlier results due to Khan et al. (2000), Itoh (1977), Khan (1981), Ahmad and Imdad (1992 and 1998), and several others. Some related results are also discussed. 1. Introduction In recent years several fixed point theorems for hybrid pairs of mappings are proved and by now there exists considerable literature in this direction. To mention a few, one can cite Imdad and Ahmad [10], Pathak [19], Popa [20] and references cited therein. On the other hand Assad and Kirk [4] gave a sufficient condition enunciating fixed point of set-valued mappings enjoying specific boundary condition in metrically convex metric spaces. In the current years the work due to Assad and Kirk [4] has inspired extensive activities which includes Itoh [12], Khan [14], Ahmad and Imdad [1, 2], Imdad et al. [11] and some others. Most recently, Huang and Cho [9] and Dhage et al. [6] proved some fixed point theorems for a sequence of set-valued mappings which generalize several results due to Itoh [12], Khan [14], Ahmad and Khan [3] and others. The purpose of this paper is to prove some coincidence and common fixed point theorems for a sequence of hybrid type nonself mappings satisfying certain contraction type condition which is essentially patterned after Khan et al. [15]. Our results either partially or completely generalize earlier results due to Khan et al. [15], Itoh [12], Khan [14], Ahmad and Imdad [1, 2], Ahmad and Khan [3] and several others. 2. Preliminaries Before proving our results, we collect the relevant definitions and results for our future use. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 281–294 DOI: 10.1155/FPTA.2005.281
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Hybrid fixed point theorems in metrically convex spaces
Let (X,d) be a metric space. Then following Nadler [17], we recall (i) CB(X) = {A : A is nonempty closed and bounded subset of X }. (ii) C(X) = {A : A is nonempty compact subset of X }. (iii) For nonempty subsets A, B of X and x ∈ X,
d(x,A) = inf d(x,a) : a ∈ A , H(A,B) = max
supd(a,B) : a ∈ A , supd(A,b) : b ∈ B .
(2.1)
It is well known (cf. Kuratowski [16]) that CB(X) is a metric space with the distance H which is known as Hausdorff-Pompeiu metric on X. The following definitions and lemmas will be frequently used in the sequel. Definition 2.1. Let K be a nonempty subset of a metric space (X,d), T : K → X and F : K → CB(X). The pair (F,T) is said to be pointwise R-weakly commuting on K if for given x ∈ K and Tx ∈ K, there exists some R = R(x) > 0 such that d(T y,FTx) ≤ R · d(Tx,Fx) for each y ∈ K ∩ Fx.
(2.2)
Moreover, the pair (F,T) will be called R-weakly commuting on K if (2.2) holds for each x ∈ K, Tx ∈ K with some R > 0. If R = 1, we get the definition of weak commutativity of (F,T) on K due to Hadzic and Gajic [8]. For K = X Definition 2.1 reduces to “pointwi
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