ON CONTRACTIVE MAPPINGS AND DISCONTINUITY AT FIXED POINTS

Abstract

This paper deals with an interesting open problem of B.E. Rhoades (Contemporary Math. (Amer. Math. Soc.) 72(1988), 233-245) on the existence of general contractive conditions which have fixed points, but are not necessarily continuous at the fixed points. We propose some more solutions to this problem by introducing two new types of contractive mappings, that is, A-contractive and A'-contractive, which are, in some sense, more appropriate than those of the important previous attempts. We establish some new fixed point results involving these two contractive mappings in compact metric spaces and also in complete metric spaces and show that these contractive mappings are not necessarily continuous at their fixed points. Finally, we suggest an applicable area, where our main results may be employed.