박위태 Wee Tae Park

DOI: JANT Vol.26(No.1) 41-45, 1987

In this paper, we give several fixed point theorems in a complete metric space for two multi-valued mappings commuting with two single-valued mappings. In fact, our main theorems show the existence of solutions of functional equations f(x)＝g(x)∈Sx∩Tx and x＝f(x)＝g(x)∈Sx∩Tx under certain conditions. We also answer an open question proposed by Rhoades-Singh-Kulsherestha. Throughout this paper, let (X, d) be a complete metric space. We shall follow the following natations : CL(X) = {A; A is a nonempty closed subset of X}, CB(X) = {A; A a nonempty clouted and bounded subset of X}, C(X) = {A; A is a nonempty compact subset of X}. For each A, B ∈ CL and ε ＞ 0, N(ε, A) = {x ∈ X ; d(x, a) ＞ ε for some a ∈ A}, E_(A,B) = {ε ＞ 0 ; A ⊂ N(ε, B) and B ⊂ N(ε, A)}, and H(A, B) = inf E_(A,B) if E_(A,B) ≠ Φ, + ∞ if E_(A,B) = Φ Then H is called the generalized Hausdorff distance function for CL(X) induced by a metric d and H defined CB(X) is said to be the Hausdorff metric induced by d. D(x, A) will denote the ordinary distance between x∈X and a nonempty subset A of X. Let IR^+ and II^+ denote the sets of nonnegative real numbers and positive integers. respectively, and G the family of functions Φ from (IR^+)^s into IR^+ satisfying the following conditions: (1) Φ is nondecreasing and upper semicontinuous in each coordinate wariable, and (2) for each t＞0, Φ(t) = max{Φ(t, 0, 0, t, t), Φ(t, t, t, 2t, 0), Φ(0, t, 0, 0, t)}＜t, where Φ : IR^+ → IR^+ is a nondecreasing and upper semicontinuous function from the right. Before stating and proving our main theorems, we give the following lemmas: Lemma 1. [5] Let Φ: IR^+ → IR^+ be a nondecreasing and upper semicontinuous function from the right. Then ◎ if and only if Φ(t) ＜t for all t＞0, where Φⁿ is the n-th iteration of Φ.