Abstract

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.

1. Introduction

Prescribed mean curvature equation arises from some problems associated with differential geometry and physics such as combustible gas dynamics [1–3]. In the past years, the one-dimensional mean curvature equation of autonomous type has been studied by many authors [4–14]. The interesting thing is that most of them focus on the case in which the nonlinearity is chosen to be various power growth functions. For example, Li and Liu in [4] studied the exact number of solutions for the boundary value problem Clearly, the powers of growth with respect to the variable of functions on the right side of above two equations are not greater than . Pan in [5] studied the exact multiple solutions of boundary value problem for a one-dimensional prescribed mean curvature equation with exponential nonlinearity Equation (3) can be viewed as a variant of the one-dimensional Liouville-Bratu-Gelfand problem. By using the theory of time map, some results on the existence of multiple solutions are obtained. At the same time, we notice that Pan and Xing in [6] further studied the exact number of solutions for the problem where , , and , respectively. For other recent developments and applications on the study of mean curvature equation, we refer the reader to [15–20], while the problem of periodic solution for prescribed mean curvature equation has been rarely studied [21–24]. Considering the delay phenomenon which exists generally in nature, Feng [22] studied the existence of periodic solutions for the one-dimensional mean curvature type equation in the following form: By imposing some conditions on functions and as follows.

(H) there are two constants and 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, . 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.

2. Preliminaries

In order to investigate the existence of periodic solutions for (8), we give some definitions and lemmas in this section.

In this paper, unless otherwise specified, we use the following notation. Let with the norm defined by ; and the norm defined by . Clearly, and are two Banach spaces. Furthermore, define for all , where is a constant.

Lemma 1 (see [25]). Suppose and . Then the function has its inverse satisfying with . Furthermore, if , then

Now, let us recall Mawhin’s continuation theorem. Let and be real Banach spaces and let be a Fredholm operator with index zero; here denotes the domain of . This means that is closed in and . If is a Fredholm operator with index zero, then there exist continuous projectors such that and is invertible. Denote by the inverse of .

Let be an open bounded subset of ; a continuous map is said to be -compact in if is bounded and the operator is relatively compact.

Lemma 2 (see [26]). Suppose that and are two Banach spaces, and is a Fredholm operator with index zero. Furthermore, is an open bounded set and is -compact on . If all the following conditions hold:(1), , ,(2), ,(3), where is an isomorphism,then equation has a solution on .

Throughout this paper, for each , besides , we suppose in addition with , .

Remark 3. From above assumption, one can find from Lemma 1 that, for each , 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.

(A1) The functions , , and satisfy for all and where and are all continuous functions determined by Remark 3.

(A2) The function satisfies for all , and there is a constant such that, for all ,

Remark 4. Since (8) does not contain the term , condition (7) in assumption (H) of [22] does not hold. Furthermore, in our paper, the function is not required to satisfy the global Lipschitz condition (6). So the conditions in our paper are all essentially different from corresponding ones of [22].
Since the differential term of is nonlinear with respect to , the differential operator associated with Mawhin’s continuation Theorem is not Fredholm type. So we need to convert (8) to the following two-dimensional system: Clearly, if is a -periodic solution to (15), then must be a -periodic solution to (8). From this, we see that, in order to investigate the existence of -periodic solution for (8), it suffices for us to prove that (15) has a -periodic solution.
For using Mawhin’s continuation theorem, let where , where is a constant and is determined in assumption (A1).

3. Main Results

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

Theorem 5. Suppose that assumptions (A1) and (A2) hold. Then (8) possesses at least one -periodic solution.

Proof. Suppose that is an arbitrary solution to the equation for each , where and are defined by (16), respectively. This implies From the first formula of (18), we see Substituting (18) into the second formula of (17), we have Integrating both sides of (19) on the interval , we obtain Since for all , by using Lemma 1, we see that the function has its inverse . So by applying (9) to (20), we have where is determined by Remark 3. Similarly, from (19), we have which together with the condition of for all in assumption (A2) yields that It follows from the condition of for all in assumption (A1) that Substituting (21) into the above formula, we obtain that
On the other hand, since , there must be a point such that ; that is, . So This together with (25) implies that From assumption (A1), we know ; it follows from (18) and (27) that Furthermore, from the fact of the function being strongly increasing for , it follows from (27) that that is, By using (28), we have which implies Furthermore, from (21) and the condition of for all in assumption (A1), we see that there must be a point such that that is, By using the conclusion of in Remark 3, we have So by using assumption (A2), we see which together with (32) yields that If set , then, from the above proof, we see that for all . This means that condition (1) of Lemma 2 holds.
Now, suppose ; then is a constant vector with or . So which together with assumption (A2) yields that From (30), (37), and (39), we see that conditions (1) and (2) in Lemma 2 hold for .
Below, we will show that condition (3) of Lemma 2 also holds. In fact, let where . Clearly, if is the solution of equation for some , then is a constant vector. So This together with assumption (A2) results in and . By (37), we see that , and then . From this, we conclude that for all , which together with the property of homotopy invariance for Brouwer’s topological degree gives that This proves that condition (3) of Lemma 2 holds. Thus, by using Lemma 2, we have that (15) possesses at least one -periodic solution . Clearly, must be a -periodic solution to (8).