Continuation theory for general contractions in gauge spaces
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A continuation principle of Leray-Schauder type is presented for contractions with respect to a gauge structure depending on the homotopy parameter. The result involves the most general notion of a contractive map on a gauge space and in particular yields homotopy invariance results for several types of generalized contractions. 1. Introduction One of the most useful results in nonlinear functional analysis, the Banach contraction principle, states that every contraction on a complete metric space into itself has a unique fixed point which can be obtained by successive approximations starting from any element of the space. Further extensions have tried to relax the metrical structure of the space, its completeness, or the contraction condition itself. Thus, there are known versions of the Banach fixed point theorem for contractions defined on subsets of locally convex spaces: Marinescu [18, page 181], in gauge spaces (spaces endowed with a family of pseudometrics): Colojoar˘a [5] and Gheorghiu [11], in uniform spaces: Knill [16], and in syntopogenous spaces: Precup [21]. As concerns the completeness of the space, there are known results for a space endowed with two metrics (or, more generally, with two families of pseudometrics). The space is assumed to be complete with respect to one of them, while the contraction condition is expressed in terms of the second one. The first result in this direction is due to Maia [17]. The extensions of Maia’s result to gauge spaces with two families of pseudometrics and to spaces with two syntopogenous structures were given by Gheorghiu [12] and Precup [22], respectively. As regards the contraction condition, several results have been established for various types of generalized contractions on metric spaces. We only refer to the earlier pa´ c [4], and to the survey article of pers of Kannan [15], Reich [27], Rus [29], and Ciri´ Rhoades [28]. Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:3 (2004) 173–185 2000 Mathematics Subject Classification: 47H10, 54H25 URL: http://dx.doi.org/10.1155/S1687182004403027
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Continuation theory
We may say that almost every fixed point theorem for self-maps can be accompanied by a continuation result of Leray-Schauder type (or a homotopy invariance result). An elementary proof of the continuation principle for contractions on closed subsets of a Banach space (another proof is based on the degree theory) is due to Gatica and Kirk [10]. The homotopy invariance principle for contractions on complete metric spaces was ´ established by Granas [14] (see also Frigon and Granas [8] and Andres and Gorniewicz [2]), extended to spaces endowed with two metrics or two vector-valued metrics, and completed by an iterative procedure of discrete continuation along the fixed points curve by Precup [23, 24] (see also O’Regan and Precup [19] and Precup [26]). Continuation results for contractions on complete gauge spaces were given by Frigon [7] and for gener´ c, by Agarwal and O’Regan alized contractions in the sense of
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