A Meir-Keeler type fixed point theorem

The aim of the present paper is to obtain a fixed point theorem for a sequence of mappings satisfying a Meir-Keeler type (ε, δ) contractive condition. As compared to the analogous results, the present theorem has been obtained without assuming additional conditions on the function δ and, threfore, generalizes a number of Meir-Keeler type theorems. As a corollary of the main theorem we proved an answer to a question, which remained an open problem for more than a decade, on the existence of a contractive definition which generates a fixed point but does not force the map to be continuous at the fixed point.