Generalization of Meir-Keeler type fixed point theorems

Abstract

The purpose of this paper is two fold. In the following pages we prove common fixed point theorems for four mappings $ A$, $B$, $S$ and $ T $ (say) under the Meir-Keeler type $ (\varepsilon, \delta) $ condition, however, without imposing any additional condition on $delta$ or using a $ \phi $-contractive condition together with. Simultaneously we also show that none of the $ A $, $ B $, $ S $ or $ T $ 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 [19]) 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.