Note on “ Common fixed point theorem for weakly compatible multi-valued mappings satisfying strongly tangential property ”

In our paper [1], the existence and uniqueness of a common fixed point for multi valued mappings was given, hence in this note we point out to no uniqueness of common fixed point in theorem 3.1, corollary 3.1 and corollary 3.2. We provide an example to demonstrate our purpose. 1. Main results Theorem 1. Let f, g : X → X, be single valued mappings and S, T : X → CB(X) multi-valued mappings of a metric space (X, d), such for all x, y in X we have: (1) ∫ H(Sx,Ty) 0 φ(t) ≤ φ (∫ d(fx,gy) 0 φ(t)dt, ∫ d(fx,Sx) 0 φ(t)dt, ∫ d(gy,Ty) 0 φ(t)dt, ∫ d(fx,Ty) 0 φ(t)dt, ∫ d(gy,Sx)


Main results
Theorem 1. Let f, g : X → X, be single valued mappings and S, T : X → CB(X) multi-valued mappings of a metric space (X, d), such for all x, y in X we have: where φ : R 5 + → R + is an upper semi continuous, non decreasing function satisfying: for any t > 0 and ϕ : R + → R + is a Lebesgue-integrable function which is summable on each compact subset of R + , non-negative, and such that for each ε > 0, ε 0 ϕ(t)dt > 0. Suppose that the two pairs (f, S), (g, T ) are weakly compatible, moreover if (f, g) is strongly tangential w.r.t (S, T ), then f, g, S and T have a common fixed point in X.

164
A common fixed point theorem Remark 1. In paper [1], the proof of uniqueness of the common fixed point is false, since the inequality d(a, b) ≤ H(A, B), for all a ∈ A and b ∈ B is note true in general. For example A = [0, 2], B = [3, 5] and H(A, B) = 3, but d(0, 5) = 5 (with euclidian metric).
If S = T and f = g, we obtain the following corollary: Corollary 1. Let f : X → X, and S : X → CB(X) be single and set valued mappings of metric space (X, d) such: where φ ∈ Φ and ϕ : R + → R + is a Lebesgue-integrable function which is summable on each compact subset of R + , non-negative, and such that for each ε > 0, ε 0 ϕ(t)dt > 0, if f is strongly tangential w.r.t S and (f, S) is weakly compatible, then f and S have a common fixed point.
Corollary 2. Let f, g : X → X, and S, T : X → CB(X) be single and set valued mappings of metric space (X, d) such: where φ ∈ Φ and ϕ : R + → R + is a Lebesgue-integrable function which is summable on each compact subset of R + , non-negative, and such that for each ε > 0, ε 0 ϕ(t)dt > 0. and a, b, c are nonnegative real numbers such a + b + b < 1 and p ∈ N , if {f, g} is strongly tangential w.r.t (S, T ) and the two pairs (f, g), (S, T ) are weakly compatible, then f, g, S and T have a common fixed point. Example 1. Let X = [0, 4] endowed with the euclidian metric. Define the mappings f, g, S and T as follows: then f is strongly tangential w.r.t S. We will utilize Corollary 1, with φ(t 1 , t 2 , t 3 , t 4 , t 5 ) = 2 5 max{t 1 , t 2 , t 3 , t 4 , t 5 } and ϕ(t) = 1.
Also 1 and 2 are two coincidence points satisfying f S1 = [0, 1] = Sf 1, f S2 = {2} = Sf 2, consequently all hypotheses of Corollary 1 are satisfied, then f and S have at least a common fixed point, therefore 1 and 2 are two common fixed point.