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

© 2014, Sen et al.; licensee Springer. 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 (Formula presented.) for (Formula presented.) and (Formula presented.), or (Formula presented.), subject to (Formula presented.) and (Formula presented.), such that (Formula presented.) converges uniformly to T, and the distances (Formula presented.) are iteration-dependent, where (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.) are non-empty subsets of X, for (Formula presented.), where (Formula presented.) is a metric space, provided that the set-theoretic limit of the sequences of closed sets (Formula presented.) and (Formula presented.) exist as (Formula presented.) 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.