Differential Equation Existence And Solvability

In this paper, we study the existence of periodic solutions of the second order Duffing equationx" + g(x) = e(t),where g,e : Râ†'R are continuous and e(t) is 2Ï€-periodic. Assume that the following conditions hold,(Ï„₀) There exist positive constantμ, positive intergers m, n and (m,n)=1, {a_k}, {b_k},a_kâ†'+∞,b_kâ†'+∞, (kâ†'+∞) such thatWe prove the existence and multiplicity of periodic solutions by using Poincar(?)-Birkhoff theorem and Poincar(?) - Bohl fixed point theorem.On the other hand, we also study the existence of periodic solutions of second order differential equations with singularity,where a, e∈L¹[0,2Ï€], f∈Car([0,2Ï€]×R⁺, R). Moreover, assume the following conditions hold,(H₀) Green funtion G(t,s) is non-negative for all(t, s)∈[0,2π×[0,2Ï€];(H₁) f(t, x) + e(t)≥0, for all (t, x)∈[0, 2Ï€]×(0, +∞);(H₂) there exist continuous non-negative functions g(x), h(x), k(t) such thatf(t,x)≤k(t)[g(x) + h(x)). for all (t,x)∈[0.2Ï€]×(0, +∞)and g(x) + h(x) is non-increasing;(H₃)there exists a constant R, such that R >Φ_* +γ_* > 0 and R≥K^*(h(r) +g(r)) +γ^*.We prove the exsistence of positive periodic solution of the given equation.