Extended Two-Parameter F-Controlled Asymptotically Contractive Self-Mappings in Metric Spaces

Certain extensions of F-controlled self-mappings in metric spaces to the, as called in this manuscript, Fτ′τ and modified ∗Fτ′τ controlled self-mappings, which are parameterized by two parameters, are addressed. Those parameters govern the properties of local expansivity, asymptotic nonexpansivity, and contractivity properties of the generated sequences. Also, further generalizations to parameterizations by two real sequences of parameters, which are referred to as Fτj′j=0∞τjj=0∞-controlled self-mappings, are studied. The main formulated results rely on the asymptotic contractivity and the asymptotic nonexpansivity in metric spaces and some of their relevant properties. In particular, the properties of boundedness of the sequences of distances, as well as those of boundedness of the elements of the sequences themselves, are investigated under asymptotic contractivity or nonexpansivity related to the various types of the above-mentioned F.-controlled self-mappings. Also, existence and uniqueness results of fixed points are proved if the metric space is complete, and the resulting Cauchyness properties of sequences and properties of the convergence of such sequences to fixed points are also proved. Finally, two illustrative examples are described if the F.-controlled self-mappings are of a cyclic nature when defined using the union of two nonempty closed subsets of the metric space, in the case that those sets intersect, and also in the case when they are disjointed.