Times Linear Li¨¦nard Equation And The Existence Of The Singularity Of The Duffing Equation Periodic Solution

In this paper, we deal with the existence of periodic solutions of sublinear Liénard differential equations and Duffing differential equations with singularity.Firstly, we study the existence of periodic solutions of Lienard differential equations x" + f(x)x' + g(x) = e(t), where f,g,e : R â†' R are continuous and e(t) is 2Ï€-periodic. In chapter two, using phase-plane analysis methods and continuation theroem based on coincidence degree, we obtain the existence of periodic solutions provided that F(x)(= ∫x0 f(s)ds) is sublinear when x tends to be positive infinity and g(x) satisfies a new conditionwhere M, d are two positive constants. At the end of chapter two, we construct an example to show the applications of Theorem 2.1.Secondly, we deal with the existence of periodic solutions of Duffing differential equations with singularity x"+g(x) = p(t), where g : R â†' R is continuous and has singularity at the origin, p(t) is continuous and 2Ï€-periodic. In chapter three, by combining phase-plane analysis methods and Poincare-Bohl theorem, we prove that the given Duffing differential equation has at least one periodic solution when g(x) satisfiesand G(x)(=∫x1 g(s)ds) satisfiesand G(x) â†' 0+ as x â†' +∞, where n ≥ 0 is an integer.