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Contractivity of leader type and fixed points in uniform spaces with generalized pseudodistances. (English) Zbl 1233.54019
S. Leader [“Equivalent Cauchy sequences and contractive fixed points in metric spaces”, Stud. Math. 76, 63–67 (1983; Zbl 0469.54028)] proved two fixed point theorems for contractive mappings $T$, with a complete graph, on a metric space. In the present paper, the authors consider the following question: If the spaces $X$ are uniform with $𝒯$-families of generalized pseudo-distances, under what conditions does the fixed point theorem of Leader type for self-maps $T$ on $X$ hold even in the case where the spaces are not sequentially complete and the maps $T$ do not have complete graphs? Answering this question affirmatively, the authors define the $𝒯$-family of generalized pseudo-distances, apply it to construct $𝒯$-contractions of Leader type on $X$, and provide conditions guaranteeing the existence and uniqueness of fixed points of these contractions and the convergence to these fixed points of all iterative sequences of these contractions. Examples are constructed to illustrate the results obtained.
##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54E15 Uniform structures and generalizations