In this paper, using a combination of methods used in [5], [14], [15] and [17] the results of [1, Theorems 2.2 and 2.3] and [15, Theorem 3] are improved by removing the assumptions of continuity and reciprocally continuity, relaxing compatibility and compatibility of type (A) to weakly compatibility and replacing the completeness of the space with a set of four alternative conditions for four mappings satisfying an implicit relation.