Moderne Strategies for solving inverse problems

International audienceUsing classical fixed point theory one needs strong conditions to establish existence of a solution of a problem and therefore restrict the applicability to certain classes of Partial and Fractional Differential Equations and their systems.Theorems called continuation theorems, represent a powerful existence tool in studying these operator equations. The main idea of a continuation theorem is that one can obtain the solution of a given equation starting from one of the solutions of a simpler equation.The first abstract formulation of the continuation principle was given by Leray and Schauder in their famous paper (Topologie et équations fonctionnelles. Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78).In this talk we presente the Leray-Schauder continuation theorem in terms of topological degree theory and give many examples of applications. Part of these examples comes from the author’s own work