Fixed point and continuity of multivalued mappings in complete metric spaces (Q2731021)

Let \((X,d)\) be a metric space, \(\text{Cpt} (X)\) the set of all nonempty compact subsets of \(X\) and \(d(x,A)= \inf \{d(x,y): y\in A\}\), \(H(A,B)= \max \{\sup_{y\in B} d(y,A), \sup_{x\in A} d(x,B)\}\), \(\delta(A,B)= \sup \{d(a,b): a\in A\), \(b\in B\}\). In this paper four fixed point theorems for set-valued and single valued mappings satisfying some inequalities of the types \(H(Fx,Gy)\leq k[d(x,Fx) d(y,Gy) d(x,y)]^{1/3}\) (resp. \(\delta (Fx,Fy)\leq k[H(Fx,x) H(Gy,y) d(x,y)]^{1/3})\) for \(x,y\in X\) and \(k\in [0,1)\), where \(F,G:X\to \text{Cpt} (X)\) are proved and it has been shown that these mappings are continuous at the fixed points.