Periodic Solutions For Second-order Differential Equations With Delays In Banach Spaces

In this paper, by using the fixed point theorem of condensing mapping and the monotone iterative technique in presence of upper and lower solutions and the fixed point index theory in cones with Kuratowski measure of noncompactness and the par-tial order theory, we discuss the existence and uniqueness of w-periodic solutions for the second-order differential equation with multiple delays in Banach spaces E-u"{t)+a(t)u(t)= f(t, u(t-Ï„1),…, u(t-Ï„n)), t ∈R where a(t) E Cω(R) is positive w-periodic functions and f:R× Enâ†' E is a continuous function, f(t, v) is ω-periodic in t for v ∈ En and Ï„1, …, Ï„n∈ [0,+∞) are constants.The main results of this paper are as follows:1. With the existence of solutions for corresponding second-order linear differential equation in R, we get the existence and uniqueness of periodic solutions for the second-order linear differential equation and the property of the solution operator in Banach spaces E.2. Under the conditions of noncompactness, we obtain the existence and unique-ness of ω-periodic solutions for the second-order nonlinear differential equation with multiple delays in Banach spaces by using fixed point theorem of condensing mapping.3. With the monotone iterative technique in presence of upper and lower solutions, we obtain the results of the existence and uniqueness of periodic solutions for the second-order nonlinear differential equation with multiple delays in Banach spaces.4. Under the measure of noncompactness, by constructing a special cone and applying the fixed-point index theory of condensing mapping in cones, the existence of positive periodic solutions for the second-order nonlinear differential equation with multiple delays are obtained in order Banach spaces.