PERIODIC SOLUTIONS FOR THIRD ORDER DELAY DIFFERENTIAL EQUATION IMPULSES WITH FREDHOLM OPERATOR OF INDEX ZERO

Abstract

In this paper the periodic solutions for third order delay differential equation of the form \begin{center} $x'''(t)+f(t,x''(t))+g(t,x'(t))+h(x(t-\tau(t))=p(t),t\geq0,t\neq t_k,$ \end{center} is investigated. We derive a third order delay differential equation with Fredholm operator of index zero and periodic solution. We obtain the existence of periodic solution and Mawhin's continuation theorem. The delay conditions for the Schwarz inequality of the periodic solutions are also obtained. An example is also furnished which demonstrates validity of main result. Some new positive periodic criteria are given. Therefore it has at least one $2\pi$-periodic solution.