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Coincidence and fixed point for weakly reciprocally continuous single-valued and multi-valued maps

100%

Gairola A. , Singh J. , Joshi M. C.

Demonstratio Mathematica

2014

tom Vol. 47, nr 3

704--713

In the present paper, we extend the concept of Weak Reciprocal Continuity for a hybrid pair of single-valued and multi-valued maps and introduce (T, f)-completeness of the space. Further, we establish some results on the existence of coincidence and fixed points for the hybrid pair of maps. Our results generalize several well known results available in the literature.

Conditional reciprocal continuity and common fixed points

100%

Singh J. , Bisht R. K. , Joshi M. C.

Fasciculi Mathematici

2013

tom Nr 51

149--158

The aim of the present paper is to obtain common fixed point theorems by employing the recently introduced notion of conditional reciprocal continuity. We demonstrate that conditional reciprocal continuity ensures the existence of fixed points under contractive conditions which otherwise do not ensure the existence of fixed points. Our results generalize and extend several well-known fixed point theorems in the setting of metric spaces. We also provide more answers to the open problem posed by B. E. Rhoades [Contractive Definitions and Continuity, Contemporary Math. 72 (1988), 233-245] regarding existence of a contractive condition which is strong enough to generate a fixed point, but which does not force the map to be continuous at the fixed point.