Research Article | Open Access
Existence of Subharmonic Periodic Solutions to a Class of Second-Order Non-Autonomous Neutral Functional Differential Equations
By introducing subdifferentiability of lower semicontinuous convex function and its conjugate function, as well as critical point theory and operator equation theory, we obtain the existence of multiple subharmonic periodic solutions to the following second-order nonlinear nonautonomous neutral nonlinear functional differential equation ,
The existence of periodic solutions for differential system has received a great deal of attention in the last few decades. Different from ordinary differential equations and partial differential equations that do not contain delay variate, it is very difficult to study the existence of periodic solutions for functional differential equations. For this reason, many mathematicians developed different approaches such as the averaging method , the Massera-Yoshizawa theory [2, 3], the Kaplan-York  method of coupled systems, the Grafton cone mapping method , the Nussbaum method of fixed point theory  and Mawhin  coincidence degree theory.
However, the critical point theory was rarely used in the literature, and most of the existing results were established for autonomous functional differential equations while little was done for nonautonomous equations via critical point theory.
In this paper, by using critical point and operator equation theories, we study the existence of the following second-order nonlinear and nonautonomous mixed-type functional differential equation:
Our basic assumptions are the following and to be given in Section 4:, and ; there exists a continuously differentiable function such that where and denote and , respectively; for all .
2. Variational Structure
Fix , where is a positive integer and
It is obvious that is a Sobolev space by defining the inner product and the norm as follows:
Moreover, can be expressed as
Let us consider the functional defined on by
For all and , we know that
It is then easy to see that where denotes the Frechet differential of the function . By the periodicity of , , and , we have Similarly, we have Hence, Therefore, the Euler equation corresponding to the functional is
It is not difficult to see that (2.10) is equivalent to (1.1). Thus, system (1.1) is the Euler equation of the functional . It follows that it is possible to obtain -periodic solutions of system (1.1) by seeking critical points of the functional .
Since has neither a supremum nor an infimum, we do not seek critical points of the functional by the extremum method. But we may use operator equation theory. First via the dual variational principle, we obtain new operator equations (see (4.16)) related to (1.1). Then solutions to system (1.1) are obtained by seeking critical points of operator equation (4.16).
In this paper, our main tool is the following.
Lemma 2.1 (Maintain Pass Theorem). Let be a real Banach space. If satisfies the Palais-Smale condition as well as the following additional conditions:(1)there exist constants and such that , where ,(2) and there exists such that , then is a positive critical value of , where
The rest of this paper is organized as follows. Subdifferentiability of lower semicontinuous convex function and its conjugate function are introduced in Section 3. In Section 4, we first give the definition of the weak solution to (1.1), then we establish the new operator equation (4.16) related to (1.1) by the conjugate function of and show that we can obtain solution to (1.1) from the solution to operator equation (4.16). In Section 5, by seeking critical points of operator equation (4.16), we obtain the result that there exist multiple subharmonic periodic solutions to system (1.1). Finally in Section 6, an example and a remark are given to illustrate our result.
3. The Subdifferentiability and the Conjugate Function of the Lower Semicontinuous Convex Function
Let be a space of all given -periodic functions in and a Banach space, where is a positive integer. Denote . Let be a lower semicontinuous convex function. Generally, is not always differentiable in conventional sense, but we may generalize the definition of “derivative” as follows.
Definition 3.1. Let . We say that is a subgradient of at point if
For all , the set of all subgradients of at point will be called the sub-differential of at point and will be denoted by .
By the definition of Subdifferentiability of function , we may define its conjugate function by where denotes the duality relation of and . So it is not difficult to obtain the following propositions.
Proposition 3.2. is a lower semicontinuous convex function ( may have functional value , but not functional value ).
Proposition 3.3. If , then .
Proposition 3.4 (Yang inequality). One has
Proposition 3.5. One has
Proposition 3.6. does not always equal .
Proof. Let and such that and . We consider the two convex sets in defined by By Hahn-Banach Theorem, we know that there exists and such that So, we have Thus , and
Since is a lower semicontinuous convex function that does not always equal , we may define its conjugate function by where denotes the duality relation of and .
Theorem 3.7. Let be a lower semicontinuous convex function that does not always equal , then .
Proof. We divide our proof into two parts. First we show that holds when and then holds for all lower semicontinuous convex functions that do not always equal .
(i) The case when .
From the definition of and Yang inequality, it is obvious that holds.
Next, to prove holds, suppose to the contrary that there exist a point , such that holds.
Consider the two convex sets By the Hahn-Banach Theorem, we know that there exist and such that So, it follows that . Let . Using and (3.11), one obtains that Thus, we have Then, by the definition of , we obtain that That is to say, which is a contradiction to (3.12).
(ii) For all , by Proposition 3.6, we know that . Choose , and define function by Then is a lower semicontinuous convex function that is not always equal to and satisfies . By the result of (i), we have that . On the other hand, we have That is, .
Corollary 3.8. Let be a lower semicontinuous convex function that is not always equal to . Then if and only if
Proof. One has
4. Weak Solution to (1.1)
Definition 4.1. For , we say that is a weak solution to (1.1), if
for all , where
when , where .
Our objective is to define the conjugate function of by using the definition of Subdifferentiability of lower semicontinuous convex function and by making use of the dual variational structure. So we add three more conditions on function as follows: is a continuously differentiable and strictly convex function and satisfies for , there exist constants , such that when we have
The conjugate function of function is defined by for .
Then is a continuously differentiable and strictly convex function. By the duality principle (Corollary 3.8), we have that where and denote and , respectively.
Example 4.2. Let . Then
Proof. The above expression holds since
Let denote the value field of operator . Then is a closed set. Let be the orthogonal projection operator of and . Then it is not difficult to see that maps continuous continuation into a compact operator of .
Let where .
Remark 4.3. In fact, for all , or , can be expressed as
So, it follows that
Thus can also be expressed as
We want to satisfy
where , that is, .
If is a solution to (4.16) and if , , then by the duality principle and (4.16), we have Thus, is a weak solution to (1.1).
5. Existence of Solutions to Operator Equation (4.16)
In this section, we discuss the existence of solutions to operator equation (4.16) by using the critical point theory. Our main result is the following.
Theorem 5.1. Under assumptions , problem (1.1) has at least one nontrivial weak -periodic solution.
To prove this theorem, we state and prove the following lemmas first.
Let , and
It is not difficult to verify that , where , that is, the operator is a symmetric operator.
We may obtain solutions to (4.16) by seeking critical points of functional defined by
may be regarded as the restriction to of function defined on since both functions have identical component on . Moreover Since there exist and such that
Lemma 5.2. The following two conditions are equivalent:(1), when ;(2).
Proof. For all let .
. By and , it is easy to see that . That is, . By it follows that
Lemma 5.3. Let satisfy assumptions and . Then there exist constants and such that where .
where denotes the ball of radius centered at the origin. By , one knows . On the other hand, by Lemma 5.2 and , we have
By the convexity of function , one has
Let run all over the ball of radius centered at , and choose the maximum of . Then it is not difficult to see that By we have that
Lemma 5.4. is a strictly convex function and satisfies where are constants, and are constants, depending on , and .
Proof. By Corollary 3.8 and , we have , for all . Moreover, by the definition of , we know that .
Now we show that (5.17) holds. By one obtains . So, by Proposition 3.3 and Example 4.2, it is easy to see that where .
Similar arguments to the proof of Lemma 5.3 show that there exists a constant such that Therefore, it follows that
Next we show that (5.18) holds. Again as in the proof of Lemma 5.3, we can estimate by where . By Lemma 5.3 again and the duality principle when , we have Since there exists a constant such that when , we have Choose Then it is not difficult to see
Finally, we show that (5.19) holds. By , for all , there exists so that when , we have Now, for all , choose and let . Then whenever , we have That is,
Lemma 5.5. There exist constants and depending on , such that and when .
Proof. By (5.19), we have So, as , we obtain . That is, when