Some Fixed Point Properties of Self-Maps Constructed by Switched Sets of Primary Self-Maps on Normed Linear Spaces
- PDF / 499,588 Bytes
- 26 Pages / 600.05 x 792 pts Page_size
- 4 Downloads / 154 Views
Research Article Some Fixed Point Properties of Self-Maps Constructed by Switched Sets of Primary Self-Maps on Normed Linear Spaces M. De la Sen Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia), Aptdo. 644 Bilbao, 48080 Bilbao, Spain Correspondence should be addressed to M. De la Sen, [email protected] Received 15 September 2009; Revised 3 March 2010; Accepted 25 March 2010 Academic Editor: L. Gorniewicz ´ Copyright q 2010 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is devoted to the investigation of the existence of fixed points in a normed linear space X endowed with a norm · for self-maps f from T × X to X which are constructed from a given class of so-called primary self- maps being also from T × X to X. The construction of the self-maps of interest is performed via a so-called switching rule which is a piecewise-constant map from a set T to some finite subset of the positive integers or a sequence map which domain in some discrete subset of T .
1. Introduction This paper is devoted to the investigation of the existence of fixed points in a normed linear space X with norm · for self-maps from T × X to X which are constructed from a given class of so-called primary self-maps from T × X to X. The construction of the maps f : T × X → X of interest is performed via a so-called switching rule σ : T → N ⊂ Z which is a piecewise-constant map from a set T to some finite subset of the positive integers. The potential discontinuity points of such a self-map in a discrete subset ST ⊂ S are the so-called switching points at which a new primary self-map in a class PM is activated to construct the self-map f : T × X → X of interest, each of those self-maps depends also on some given switching rule σ : ST : {ti } ⊂ T → N : {1, 2, . . . , N} ⊂ Z . In particular, ft fi t ≡ fσt t ∈ PM , for all t ∈ tj , tj1 where tj , tj1 > tj are two consecutive elements in the sequence ST generated by the switching rule σ : T → N such that σt i ∈ N, for all t ∈ tj , tj1 , for all tj , tj1 > tj ∈ ST provided that there is no ST t ∈ tj , tj1 .
2
Fixed Point Theory and Applications
The class of primary self-maps PM used to generate the self-map f from T × X to X might contain itself, in the most general case, a class of contractive primary self-maps from T ×X to X, a class of large contractive self-maps from T ×X to X, a class of nonexpansive selfmaps from T ×X to X, as well as a class of expansive self-maps from T ×X to X. The problem is easily extendable to the case when the switching rule is a discrete sequence of domain in a discrete set of T and of codomain again being the set of nonnegative integers. The study is of particular interest for its potential application to the study of eventual fixed points in the state-trajectory solution of ei
Data Loading...
