Fixed point results for generalized contractions in gauge spaces and applications

Abstract

Abstract. In this paper, we present fixed point results for generalized contractions defined on a complete gauge space E. Also, we consider families of generalized contractions {f t : X → E} t∈ [0,1] where X ⊂ E is closed and can have empty interior. We give conditions under which the existence of a fixed point for some f t 0 imply the existence of a fixed point for every f t . Finally, we apply those results to infinite systems of first order nonlinear differential equations and to integral equations on the real line.

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

“…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 .…”

“…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.…”

On Common Property (E.A) for Hybrid Mappings in Gauge Spaces

“…As an application of our result, we derive the following extension of the main result of Frigon [6] , which itself is a generalization of Nadler's contraction principle [7] if |x -y| α = 0, then |Fx -Fy| α = 0.…”

“…Very recently, Liu, Wu and Li [4] defined the common property (E.A) in metric spaces which contains the property (E.A) for a hybrid pair of single-valued and multivalued mappings. On the other hand, Frigon [5] established the Banach contraction principle in complete gauge spaces. Frigon [6] presented a fixed point theorem for multivalued contractions on complete gauge spaces which generalizes fixed point results of Nadler [7] and Cain and Nashed [8] .…”

“…On the other hand, Frigon [5] established the Banach contraction principle in complete gauge spaces. Frigon [6] presented a fixed point theorem for multivalued contractions on complete gauge spaces which generalizes fixed point results of Nadler [7] and Cain and Nashed [8] . The purpose of this paper to extend the common property (E.A) to gauge spaces and give some coincidence point results and common fixed point results for mappings satisfying this property.…”