Linear difference operator with multiple variable parameters and applications to second-order differential equations

• Main Authors: Feifan Li, Zhonghua Bi, Shaowen Yao, Yun Xin
• Format: Article
• Language: English
• Published: SpringerOpen 2020-01-01
• Series: Boundary Value Problems
• Subjects: Difference operator; Multiple variable parameters; Periodic solution; Second-order differential equation
• Online Access: https://doi.org/10.1186/s13661-019-01312-4

Abstract In this article, we first investigate the linear difference operator ( A x ) ( t ) : = x ( t ) − ∑ i = 1 n c i ( t ) x ( t − δ i ( t ) ) $(Ax)(t):=x(t)-\sum_{i=1}^{n}c_{i}(t)x(t- \delta _{i}(t))$ in a continuous periodic function space. The existence condition and some properties of the inverse of the operator A are explicitly pointed out. Afterwards, as applications of properties of the operator A, we study the existence of periodic solutions for two kinds of second-order functional differential equations with this operator. One is a kind of second-order functional differential equation, by applications of Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of positive periodic solutions are established. Another one is a kind of second-order quasi-linear differential equation, we establish the existence of periodic solutions of this equation by an extension of Mawhin’s continuous theorem.