In this paper we give a new proof of a result by S. Reich and A.J. Zaslavski (S. Reich and A.J. Zaslavski, A fixed point theorem for Matkowski contractions, Fixed Point Theory, 8(2007), No. 2, 303-307). Some new fixed point theorems for nonself generalized contractions are also given.

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In this paper we present some basic problems of the metric fixed point theory (existence, uniqueness, settheoretic aspects (Bessaga, Janos, Rus, ...), order-theoretic aspects (Ekeland, Bronsted, Caristi, Kirk, Jachymski, ...), convergence of the succesive approximations, data dependence (general estimation, Ulam problem, dependence on the parameters, ...), well-posedness of the fixed point problem, limit shadowing property, stability, Gronwall lemmas, comparison lemmas, retractibility, ...). Following [I. A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory, 9 (2008), No. 2, 541–559] we define the relevance of a metrical fixed point theorem by the impact of the theorem on these basic problems. Some case studies are presented.

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In this paper we consider the following abstract Volterra equations:Using the weakly Picard operator technique we establish existence, data dependence and comparison results for the solution. The derivability of the solutions with respect to a parameter is also studied.