Some results on fixed point theorems for multivalued mappings in complete metric spaces (Q1608181)

Let \((X,d)\) be a metric space, then a function \(p:X\times X\to [0,\infty)\) is called a \(w\)-distance on \(X\) [\textit{O. Kada}, \textit{T. Suzuki} and \textit{W. Takahashi}, Math. Jap. 44, 381-391 (1996; Zbl 0897.54029)] if: (1) \(p(x,z)\leq p(x,y)+ p(y,z)\) for all \(x,y\), \(z\in X\); (2) for any \(x\in X\), \(p(x, .):X\to [0,\infty)\) is lower semicontinuous; (3) for any \(\varepsilon >0\), there exists \(\delta >0\) such that \(p(z,x)\leq \delta\) and \(p(z,y) \leq\delta\) imply \(d(x,y) \leq\varepsilon\). For any \(x\in X\) and \(A\subset X\) we denote by \(p(x,A)= \{\inf p(x,y): y\in A)\) and by \(p(A,x)= \inf\{p(y,x): y\in A\}\) and \(CB_p(X)= \{A \mid A\) is a nonempty closed susbset of \(X\) and \(\sup_{x,y\in A}p(x,y)\}< \infty\). Recently, \textit{J.-S. Ume} [J. Math. Anal. Appl. 225, 630-640 (1998; Zbl 0917.54047)] improved some fixed points theorem in a complete metric space using the concept of \(w\)-distance. In the paper under review the authors, using this concept, prove some common fixed point theorems for two multivalued mappings \(S\) and \(T\) in a complete metric space. The main result of this paper is the following theorem: Let \(X\) be a complete metric space with a metric \(d\) and let \(p\) be a \(w\)-distance on \(X\). Suppose that \(S\) and \(T\) are two mappings of \(X\) into \(CB_p(X)\) and \(f:X\times X\to [0,\infty)\) is a mapping such that \(\max\{p (u_1,u_2)\), \(p(v_1,v_2)\} \leq qf(x,y)\) for all nonempty subsets \(A,B\) of \(X\), \(u_1\in SA\), \(u_2\in S^2A\), \(v_1\in TB\), \(v_2\in T^2B\), \(x\in A\), \(y\in B\), and some \(q\in[0,1]\) with \(\sup\{\sup (f(x,y)/\min [p (x,SA)\), \(p(y,TB):x\in A\), \(y\in B):A, B<X\}<1/q\), \(\inf\{p(y,u)+ p(x,Sx)+p(y, Ty):x,y\in X\}>0\), for every \(u\in X\) with \(u\in Su\) or \(u\notin Tu\), where \(SA\) means \(\cup\{Sa:a\in A\}\). Then \(S\) and \(T\) have a common fixed point.