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

Manuel De La Sen,Ravi P Agarwal,Asier Ibeas

doi:10.1186/1687-1812-2014-169

Abstract

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 T n : A n → B n for n ∈ Z 0 + and T : ⋃ j ∈ Z 0 + A 0 j → ⋃ j ∈ Z 0 + B 0 j , or T : A → ( ⋃ B n ) , subject to T ( A 0 n ) ⊆ B 0 n and T n ( A n ) ⊆ B n , such that T n converges uniformly to T, and the distances D n = d ( A n , B n ) are iteration-dependent, where A 0 n , A n , B 0 n and B n are non-empty subsets of X, for n ∈ Z 0 + , where ( X , d ) is a metric space, provided that the set-theoretic limit of the sequences of closed sets { A n } and { B n } exist as n → ∞ 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.

Highlights

1 Introduction The characterization and study of existence and uniqueness of best proximity points is an important tool in fixed point theory concerning cyclic nonexpansive mappings including the problems of contractions, asymptotic contractions, contractive and weak contractive mappings and in related problems of proximal contractions, weak proximal contractions and approximation results and methods [ – ]
The problem of proximal contractions associated with uniformly converging non-self-mappings {Tn} ⇒ {T} of the form Tn : An → Bn; ∀n ∈ Z +, where An and Bn are in general distinct, with a settheoretic limit of the form T : j∈Z + A j → j∈Z + B j, provided that the set-theoretic limits of the involved set exist and that the infinite unions of the involved closed sets are closed
Further related results are obtained for generalized weak proximal and proximal contractions in metric spaces [, ], which are subject to certain parametrical constraints on the contractive conditions

Summary

Introduction

The characterization and study of existence and uniqueness of best proximity points is an important tool in fixed point theory concerning cyclic nonexpansive mappings including the problems of (strict) contractions, asymptotic contractions, contractive and weak contractive mappings and in related problems of proximal contractions, weak proximal contractions and approximation results and methods [ – ]. ), assume that (X, d) is complete, that A and Bn; ∀n ∈ Z +, are non-empty subsets of X such that A is closed, A is non-empty, the set-theoretic limit B := limn→∞ Bn exists, is closed and approximatively compact with respect to A (or the weaker condition that A is closed) and T(A ) ⊆ B .

Results

Conclusion