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Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces

✍ Scribed by Kazimierz Włodarczyk; Robert Plebaniak; Cezary Obczyński

Publisher
Elsevier Science
Year
2010
Tongue
English
Weight
1004 KB
Volume
72
Category
Article
ISSN
0362-546X

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✦ Synopsis

In cone uniform spaces X , using the concept of the D-family of cone pseudodistances, the distance between two not necessarily convex or compact sets A and B in X is defined, the concepts of cyclic and noncyclic set-valued dynamic systems of D-relatively quasiasymptotic contractions T : A ∪ B → 2 A∪B are introduced and the best approximation and best proximity point theorems for such contractions are proved. Also conditions are given which guarantee that for each starting point each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and the limit is a best proximity point. Moreover, D-families are constructed, characterized and compared.

The results are new for set-valued and single-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces. Various examples illustrating ideas, methods, definitions and results are constructed.

📜 SIMILAR VOLUMES

Erratum to: “Best proximity points for c
Erratum to: “Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces” [Nonlinear Anal. (2008), doi: 10.1016/j.na.2008.04.037]
✍ Kazimierz Włodarczyk; Robert Plebaniak; Artur Banach 📂 Article 📅 2009 🏛 Elsevier Science 🌐 English ⚖ 210 KB

We regret making the following mistakes in the above mentioned paper [1] and using [2] we would like to correct them. 1. In condition (A2) of Definition 2.4, change ''≤'' to ''<''. 2. In condition (C2) of Definition 2.6, change ''≤'' to ''<''. 3. The Remark 2.1 should be rewritten as: \* Correspondi