Reciprocal Continuity and Common Fixed Points

Abstract

While studying the fixed point theorems for four mappings A, B, S and T (say) a Meir-Keeler type (ε,δ) contractive condition alone is not sufficient unless some additional condition is imposed on δ or a ∅-contractive condition is also used together with it. In the following pages we prove two common fixed theorems assuming a Meir-Keeler type (ε,δ) contractive condition together with a plane contractive condition (Theorem 1) or Lipschitz type analogue of a plane contractive condition (Theorem 2); however, without imposing any additional restriction on δ or having a ∅-contractive condition used together with it. Simultaneously, we also show that none of the mappings involved in the following theorems is continuous at their common fixed point Thus we not only generalize the Meir-Keeler type and Boyd-Wong type fixed point theorems, but also provide one more answer to the problem (see Rhoades [22]) on the existence of a contractive definition, which is strong enough to generate a fixed point but does not force the map to be continuous at the fixed point. Our result extends the result of Pant [15].