Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets

Abstract

This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms [InlineEquation not available: see fulltext.] for [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], or [InlineEquation not available: see fulltext.], subject to [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], such that [InlineEquation not available: see fulltext.] converges uniformly to T, and the distances [InlineEquation not available: see fulltext.] are iteration-dependent, where [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] are non-empty subsets of X, for [InlineEquation not available: see fulltext.], where [InlineEquation not available: see fulltext.] is a metric space, provided that the set-theoretic limit of the sequences of closed sets [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] exist as [InlineEquation not available: see fulltext.] and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems.