Abstract
In this article, we first investigate the linear difference operator (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.
Highlights
Difference operators play a very important role in solving functional differential equations, which derived from some practical problems, such as biology, economics and population models [11, 20, 25]
[26] in 1995 first introduced the properties of the linear autonomous difference operator (A1x)(t) := x(t) – cx(t – δ), where c, δ are constants, which became an effective tool for the research on differential equation, since it relieved the above stability restriction
A, we investigate the existence of periodic solutions for two kinds of second-order differential equations as follows
Summary
Difference operators play a very important role in solving functional differential equations, which derived from some practical problems, such as biology, economics and population models [11, 20, 25]. They obtained the existence of periodic solutions for the By applying Mawhin’s continuation theorem and the properties of A3, they obtained sufficient conditions for the existence of periodic solutions to a kind of Liénard differential equation.
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