Like Functions

Abstract

In this chapter, we investigate essentially stability theory via Lyapunov-like functions. We shall also initiate development of fuzzy differential systems utilizing generalized metric spaces. In Section 4.2 we prove a comparison result in terms of Lyapunov-like functions which serves as a vehicle for the investigation of the stability theory of Lyapunov. Section 4.3 establishes results on stability parallel to the original theorems of Lyapunov in the present framework. We provide in Section 4.4 nonuniform stability crikria employing the method of perturbing Lyapunov functions, under much weaker assumptions. Section 4.5 considers the various boundedness notions parallel to those of stability and offer sufficient conditions for the concepts of boundedness to hold. In Section 4.6, we embark on initiating the study of fuzzy differential systems, the consideration of which leads to generalized metric spaces, in terms of which one proves comparison results utilizing the concept of vector Lyapunov functions. Section 4.7 discusses the method of vector Lyapunov functions and stability criteria. Since in this setup, one gets a comparison differential system, the study of which is sometimes difficult; we provide certain results to reduce the study of comparison systems to a single comparison equation. In Section 4.8, we consider the linear fuzzy differential systeni and its perturbation and develop the variation of parameters formula. We also discuss a simple periodic boundary value problem for nonhoniogeneous linear fuzzy differential systems.