Fixed point and continuation results for contractions in metric and gauge spaces

Abstract

Abstract. We present an overview of generalizations of Banach's fixed point theorem and continuation results for contractions, i.e., results establishing that the existence of a fixed point is preserved by suitable homotopies. We will consider single-valued and multi-valued contractions in metric and in gauge spaces.0. Introduction. We present an overview of fixed point results for contractions in metric and in gauge spaces. The first result is the famous contraction principle due to Banach [4]. Weakening the contraction condition permitted many authors to generalize the Banach fixed point theorem, see [6,7,8,14,15,16,27,40,50,51,60]. Banach's fixed point theorem was also generalized to locally convex spaces by Cain and Nashed [9], and to uniform spaces by Knill [38]. See also [20,23,28,62] for results in Fréchet or gauge spaces.The question of the convergence of a sequence of fixed points of a converging sequence of contractions is then raised. An affirmative answer to this question was obtained by Bonsall [5] for a sequence of contractions {f n } converging pointwise to f 0 when the constants of contraction are the same for every f n . This result was extended by Reich [51] for more general contractions. Moreover, the constants of contraction may vary with n if stronger assumptions are imposed on the space or if the convergence of {f n } is uniform, see Nadler [42].One could also ask if one can replace the sequence of contractions by a family of contractions or homotopies of contractions. We present an overview of continuation results for homotopies of contractions on metric or gauge spaces h : X × [0, 1] → E with X a closed subset of E. More precisely, we give conditions which ensure that if h(·, 0) has a fixed point then h(·, t) has a fixed point for every t ∈ [0, 1]. Usually the space X is 2000 Mathematics Subject Classification: Primary 47H10; Secondary 47H04, 47H09.

Perov type results in gauge spaces and their applications to integral systems on semi-axis

“…We note that this notion was first introduced by Marinescu [10] in locally convex spaces assuming that ϕ 2 = ϕ and then by Colojoarȃ [3] in uniform spaces, under the same condition. The case ϕ = 1 Λ (identity) was considered by Tarafdar [16] (see also Frigon [6]). Also note that a somewhat different notion of contraction in a uniform space was defined by Knill [9] in terms of entourages.…”

Existence of Semi Linear Impulsive Neutral Evolution Inclusions with Infinite Delay in Frechet Spaces

“…Theorem 2.1 (Nonlinear Alternative of Frigon, [38,39]). Let X be a Frechet space and U an open neighborhood of the origin in X and let N :Ū → P (X) be an admissible multivalued contraction.…”

On the continuation method and the nonlinear alternative for Caristi-type non-self-mappings

“…More general, the existence of a fixed point of a given mapping can be deduced by continuation, if that mapping is embedded into one-parameter family of mappings, called a homotopy, by which it is continuously deformed to a simpler "robust" mapping [8]. Such alternative and continuation results are known for several classes of mappings including contractive mappings [3,5,6,10,12] and compact-type operators (see, e.g. [7]).…”