M. Frigon

Théorèmes d’existence de solutions d’inclusions différentielles

Dans ce texte, on presente quelques applications de methodes topologiques permettant d’obtenir l’existence de solutions d’inclusions differentielles ordinaires. Trois types de fonctions multivoques sont distingues et un principe general d’existence de solutions est etabli pour chacun d’eux. Des resultats sont obtenus pour des systemes d’inclusions differentielles du second ordre et pour des inclusions differentielles dans des espaces de Banach. Les principaux theoremes obtenus decoulent soit de theoremes de point fixe, soit de la theorie de la transversalite topologique pour des operateurs compacts ou con-tractants, univoques ou multivoques.

Fixed point results for generalized contractions in gauge spaces and applications

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 {ft : 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 ft0 imply the existence of a fixed point for every ft. Finally, we apply those results to infinite systems of first order nonlinear differential equations and to integral equations on the real line.

EXISTENCE RESULTS FOR SOME INITIAL AND BOUNDARY VALUE PROBLEMS WITHOUT GROWTH RESTRICTION

In this paper, using the Schauder Fixed Point Theorem, we establish some existence results or initial and boundary value problems for differential equations withouth growth restriction on the right member.

Fuzzy contractive maps and fuzzy fixed points

This paper presents a variety of fuzzy fixed point theorems for contractive type maps. Our theory can be derived directly from results in the literature related to multivalued contractive maps with closed values; this observation seems to have been overlooked in the literature.

Fixed Points of Cone-Compressing and Cone-Extending Operators in Fréchet Spaces

Ag eneralization of norm type cone-compression and -expansion results due to Krasnosel’ski˘ (see Dokl. Akad. Nauk SSSR NS 135 (1960) 527–530) is presented here for single-valued completely continuous maps defined on a Fr´ echet space. Applications to second-order differential equations on the half line are presented, and the existence of nontrivial solutions is established.

Fixed Point Results for Multivalued Maps in Metric Spaces with Generalized Inwardness Conditions

We establish fixed point theorems for multivalued mappings defined on a closed subset of a complete metric space. We generalize Lims result on weakly inward contractions in a Banach space. We also generalize recent results of Azé and Corvellec, Maciejewski, and Uderzo for contractions and directional contractions. Finally, we present local fixed point theorems and continuation principles for generalized inward contractions.

Nonlinear first-order initial and periodic problems in Banach spaces

Abstract In this paper, we establish some existence results for periodic and initial value problems for first-order ordinary differential equations in Banach space, where the right member f has a decomposition f = g + h with g and h satisfying, respectively, a compactness and Lipschitz assumptions. Our results extend results of [1].

Heat equations with discontinuous nonlinearities on convex and nonconvex constraints

IN THIS paper we use a variational method for solving some semilinear parabolic equations with a discontinuous nonlinearity, possibly on either some convex or nonconvex constraints. The approach is based on the fact that the solutions of the above-mentioned problems can be viewed as “steepest descent curves”, in a suitable sense to be specified, for some lower semicontinuous functionals, possibly restricted to suitable constraints. This abstract framework, which is presented in Section 1, seems interesting to us in that it provides a unifying tool for treating various kinds of constrained problems, including cases in which the constraint is not convex (see Section 4); moreover the existence theorem that we get holds under reasonably weak assumptions. All these ideas in great part originated from the paper [lo], where a general framework for variational evolution was proposed, and are also related to the theory of maximal monotone operators (see [4]) and some of its extensions (see [5,6,9, 11-15, 17, 19-241 for some applications), the main difference being in the fact that, using compactness, we find existence theorems without uniqueness. The applications presented can be described, roughly speaking, as follows: given an open set Sz C RN and g: Q x R + R, possibly discontinuous, we search for U: [0, T[ -+ t2(sZ) which solve Wf) E G(Q) ‘u’(t) = A%(t) v t in I and a.e. in I:

Applications of Multivalued Contractions on Graphs to Graph-Directed Iterated Function Systems

We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graph and a suitable -contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system.

Multiple solutions of boundary value problems with ϕ-Laplacian operators and under a Wintner-Nagumo growth condition

In this paper, we establish the existence of multiple solutions to second-order differential equations with ϕ-Laplacian satisfying periodic, Dirichlet or Neumann boundary conditions. The right-hand side is a Carathéodory function satisfying a growth condition of Wintner-Nagumo type. The existence of upper and lower solutions is assumed. The proofs rely on the fixed point index theory.MSC: 34B15, 34C25, 47H10, 37C25, 47H11.