We set up the basic theory of existence and uniqueness of solutions for systems of differential equations with usual derivatives replaced by Stieltjes derivatives. This type of equations contains as particular cases dynamic equations on time scales and impulsive ordinary differential equations.

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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.

In this paper, we introduce a new notion of linking which includes in particular the notions of homotopical linking and local linking. Critical point theorems for continuous functionals on metric spaces are presented. Finally, an application to nonlinear elliptic problems at resonance is given. Academic Press

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.