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

Authors : S. BALAMURALITHARAN

Pages : 158-168

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Publication Date : 2016-10-01

Article Type : Research Paper

Abstract :In this paper the periodic solutions for third order delay differential equation of the form \begin{center} $x```insert ignore into journalissuearticles values(t);+finsert ignore into journalissuearticles values(t,x``insert ignore into journalissuearticles values(t););+ginsert ignore into journalissuearticles values(t,x`insert ignore into journalissuearticles values(t););+hinsert ignore into journalissuearticles values(xinsert ignore into journalissuearticles values(t-\tauinsert ignore into journalissuearticles values(t););=pinsert ignore into journalissuearticles values(t);,t\geq0,t\neq t_k,$ \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.
Keywords : Third order delay differential equations, Impulses, Periodic solutions, Mawhin`s continuation theorem, Fredholm operator of index zero