Periodic Solutions for a Prescribed Mean Curvature Equation with Multiple Delays

We study the existence of periodic solutions for the one-dimensional prescribed mean curvature delay equation . By using Mawhin's continuation theorem, a new result is obtained. Furthermore, the nonexistence of periodic solution for the equation is investigated as well.

(H) there are two constants  > 0 and  > 0 such that      (, the author obtained that (3) has at least one periodic solution by using Mawhin's continuation theorem.From [22], we see that assumption (H) is crucial for estimating a priori bounds of all possible -periodic solutions.
In this paper, we consider the following prescribed mean curvature equation with multiple delays: where  ∈ (, ) and ,   , and   are all continuous -periodic functions,  = 1, 2, . . ., .By using Mawhin's continuation theorem, some new results are obtained; and the problem of nonexistence of periodic solution for (8) is investigated as well.The significance of this paper lies in the following two respects: firstly, we do not need assumption (7); secondly, the conditions imposed on function () and the methods to estimate a priori bounds of possible -periodic solutions for the equation  =  are all essentially different from corresponding ones of [22].For example, we do not require that the function  satisfies global Lipschitz condition (6).Especially, the function  is allowed to be exponential nonlinearity.

Preliminaries
In order to investigate the existence of periodic solutions for (8), we give some definitions and lemmas in this section.
Let Ω be an open bounded subset of ; a continuous map is bounded and the operator   ( − )(Ω) is relatively compact.

Remark 3.
From above assumption, one can find from Lemma 1 that, for each  ∈ {1, 2, . . ., }, the function  −   () has its inverse denoted by ().Define Since   ∈   , it follows from Lemma 1 again that For the sake of convenience, we list the following assumptions which will be used for us to study the existence of periodic solutions to (8) in Section 3.

Main Results
In this section, we will apply Lemma 2 to study the existence of periodic solutions for (8).