Cycle Of The Second Order Differential Equation Solution

In this paper, we study the existence of periodic solutions of the second order Duffing equationx" + cx' + g(x) = e(t),where c is a constant. Using continuation theorem, we obtain the existence of periodic solutions under the condition that g(x) satisfiesandwhere M, d are two positive constants.We also study the the existence of periodic solutions of Lienard equation with asymmetricnonlinearitiesx" + f(x)x' + ax⁺ - bx^- + φ(x) + h(x) = p(t).Assume that (a, b) lies on the Fucik spectrum, f(x), φ(x) have finite limits, H(x)(= ∫₀^x h(s)ds) is sublinear and eitheroris of detinite sign. We prove that the given equation has at least one periodic solution.