Common fixed point theorems for (sub) compatible and set-valued generalized nonexpansive mappings in complete convex metric spaces (Q684878)

Let \((X,d)\) be a complete metric space, \(B(X)\) be the set of all nonempty bounded subsets of \(X\) and \(\delta(A,B)=\sup[d(a,b):a\in A\), \(b\in B]\) for all \(A,B\) in \(B(X)\). The mappings \(f:X\to X\) and \(F:X\to B(X)\) are compatible if \(\delta(Ffx_ n,fFx_ n)\to 0\) (in the sense of Definition 2.1) whenever \(\{x_ n\}\) is a sequence in \(X\) such that \(fFx_ n\in B(X)\), \(Fx_ n\to\{t\}\) and \(fx_ n\to t\) for some point \(t\in X\). The author proves two common fixed point theorems, of which the second one runs as follows: let \(K\) be a nonempty closed subset of a complete convex metric space \(X\), \(F:K\to B(K)\) and \(f:K\to K\) be such that \(\delta(Fx,Fy)\leq a\cdot d(fx,fy)+(1- a)\cdot\max\{\delta(Fx,fx),\delta(Fy,fy)\}\) for all \(x,y\) in \(K\), where \(0<a<1\). If \(fK\) is a convex subset of \(K\) such that \(FK\subseteq fK\) and if one of the following conditions is satisfied: (i) \(F\) and \(f\) are compatible and \(F\) is continuous (in the sense of Definition 2.2), (ii) \(F\) and \(f\) are compatible and \(f\) is continuous, (iii) \(F\) and \(f\) commute on their coincidence points and \(f\) is surjective, then \(f\) and \(F\) have a unique common fixed point \(u\in K\) such that \(Fu=\{u\}\). Results of \textit{B. Fisher} and the reviewer [Int. J. Math. Math. Sci. 9, 23-28 (1986; Zbl 0597.47036)], \textit{R. N. Mukherjee} and \textit{V. Verma} [Math. Jap. 33, No. 5, 745-749 (1988; Zbl 0655.47047)] and of \textit{G. Jungck} [Int. J. Math. Math. Sci. 13, No. 3, 497-500 (1990; Zbl 0705.54034)] are thus generalized.