Periodic Boundary Value Problem Of Nonlinear Wave Equations With Variable Coefficients And Damped Beam Equations

By using topological degree theory, in this paper we prove the weakened Ambrosetti-Prodi type multiplicity results for weak doubly periodic solutions of semi-linear damped beam equations and the existence results of periodic solutions for the resonance problem of nonlinear wave equation with variable coefficients.For semi-linear damped beam equation, we first talk about the compactness of weak solution operator A? 1. By using some basic results of the operator A,Fourier method and a prior estimate, we prove that A? 1 is a compact operator in the functional space in this chapter so that the coincidence degree is well defined. According to the hypothesis of the nonlinear part g and the properties of topological degree, we prove some lemmas. Then we obtain that the equation has no or at least one or at least two double periodic solutions when s belongs to different range by using these lemmas.For the resonance problem of nonlinear wave equation with variable coefficients, we first study some properties of linear part operator L . According to these properties, we get a prior estimate by using the similar method of Iannacci and Nkashama. Then we prove that the problem has at least one periodic solution by using the homotopy invariance of topological degree,Mawhin's continuation theorems and Fatou's lemma.