A comparison between applications of the Lyapunov’s second (direct) method and fixed point theory

Abstract

In this article, we will discuss the application of the Lyapunov's second method and fixed point theories to certain differential equations of first and second order. First, we will introduce some basic information about these subjects, and later, we give their applications concerning some specific attitude of Solutions of Delay Differential Equations. We will also do a comparison between them. Include keywords, mathematical subject classification numbers as needed.

Asymptotic Behavior of Solutions in Nonlinear Neutral System with Two Volterra Terms

“…However, the expressions of Lyapunov functionals are very complicated and hard to construct in many problems, as is the case when the differential equation has unbounded terms or when it contains unbounded functional delays. To overcome the shortcomings of Lyapunov's direct method, in recent years, several researchers have investigated different aspects of the qualitative theory of differential equations using the fixed-point method, which was found to be significantly advantageous compared to Lyapunov's direct method; see instance [20][21][22].…”