A Few Types Of Delay Differential Equations Stability And Boundedness Of The Research

This paper studies the asymptotic stability of the zero solution and the boundedness of all solutions of the two kinds of third-order and a class of fourth-order delay differential equations. By the second method of Lyapunov,some sufficient conditions for the asymptotic stability of the zero solution and the boundedness of all solutions of them are established. The main content is as follows:1. The background and significance of the stability of delay differential equations and the research status of third-order and fourth-order delay differential equations are introduced.2. The basic concepts, the basic theories and the method of Lyapunov functional of delay differential equations are introduced.3. The asymptotic stability of the zero solution and the boundedness of all solutions of two kinds of third-order delay differential equations are studied. x"’(t)+f1(x(t),x(t-r(t)),x’(t),x’(t-r(t)),x"(t),x"(t-r(t)))x"(t)(1)+f2(x(t),x(t-r(t)),x’(t),x’(t-r(t)),x"(t),x"(t-r(t)))(x"(t))2+g1(x’(t))+g2(x’(t-r(t)))=p(x(t),x(t-r(t)),x’(t),x’(t-r(t)),x"(t),x"(t-r(t))) This third-order delay differential equation is extended by the second-order delay differential equation studied in [1]. x"’(t)+g1(x(t),x’(t))x"(t)+g2(x(t),x’(t))x’(t)(2)+f(x(t-r(t)),x’(t-r(t)))+h(x(t-r(t)))=p(t,x(t),x’(t),x(t-r(t)),x’(t-r(t))x"(t)) On the basis of the equation studied in [3], this equation is studied which adds an item as g2(x(t),x’(t))x’(t).4. The asymptotic stability of the zero solution of a certain fourth-order delay differential equation is studied.x4(t)+φ(x(t))z(t)+h(x(t))x(t)+φ(x(t-r),x(t-r))+f(x(t-r))=0On the basis of the equation studied in [4], this equation is studied which replaces φ(x(t-r))with φ(x(t-r),x(t-r)).