Theory of fuzzy differential equations and inclusions

Abstract

This chapter is devoted to the basic theory of fuzzy differential equations (FDEs). We begin Section 3.2 with the existence and uniqueness result for the initial value problem (IVP) employing the contraction mapping principle. Here the idea of weighted metric is utilized with effectiveness. Since the local existence result analogous to Peano's theorem in ordinary differential equations for the IVP of FDEs is still open, we prove in Section 3.3, an existence result under the stronger assumption of boundedness of the nonlinear function involved, everywhere. We establish in Section 3.4, a variety of comparison results for the solutions of FDEs which forrn the essential tools for studying the fundamental theory of FDEs. The comparison discussed shows how with the minimum linear structure, one can develop the theory of differential inequalities that are important in comparison principles. Section 3.5 deals with the convergence of successive approximations of the IVP of FDEs under the general uniqueness assumption of Perron type utilizing the comparison functions that is rather instructive. Continuous dependence of solutions of FDEs relative to the initial data is considered in Section 3.6. Section 3.7 investigates the global existence of solutions of FDEs. In Section 3.8, we discuss approximate solutions and error estimates between the solutions and approximate solutions. Finally, in Section 3.9, we initiate the study of stability criteria in a simpler way, suitably defining the stability concepts in the present framework. Notes and comments are provided in Section 3.10.