Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

This paper is concerned with the existence of periodic solutions for the fully second-order ordinary differential equation u󸀠󸀠(t) = f(t, u(t), u󸀠(t)), t ∈ R, where the nonlinearity f : R → R is continuous and f(t, x, y) is 2π-periodic in t. Under certain inequality conditions that f(t, x, y) may be superlinear growth on (x, y), an existence result of odd 2π-periodic solutions is obtained via Leray-Schauder fixed point theorem.


Introduction and Main Result
In this paper, we discuss the existence of odd 2-periodic solutions for the fully second-order ordinary differential equation   () =  (,  () ,   ()) ,  ∈ R, where the nonlinearity  : R 3 → R is continuous and (, , ) is 2-periodic with respect to .The existence of periodic solutions for nonlinear secondorder ordinary differential equations is an important topic in ordinary differential equation qualitative analysis.It has attracted many authors' attention and concern, and the most works are on the special equation which does not contain explicitly first-order derivative term in nonlinearity.Many theorems and methods of nonlinear functional analysis have been applied to the periodic problems of (2), such as the upper and lower solutions method and monotone iterative technique [1][2][3][4], the continuation method of topological degree [5][6][7][8][9], variational method and critical point theory [10][11][12][13][14], method of phase-plane analysis [15][16][17][18][19], the Krasnoselskii's type fixed point theorem in cone [20][21][22][23], and the theory of fixed point index [24][25][26].
However, there are not so many existence results for the second-order periodic problem that nonlinearity is dependent on the derivative.See [27][28][29][30].In [27,28], the authors discussed some special cases that nonlinearity is with a separated derivative term (  ).By finding the fixed point of the Poincaré mapping, they obtained several existence results.In [29], Hakl et al. considered the second-order periodic problem with linear derivative term by the method of lower and upper solutions.In [30], Li and Jiang researched the existence of positive 2-periodic solution of the general second-order differential equation (1) by employing fixed point index theory in cones under the nonlinearity (, , ) satisfying a sign condition.Contrary to the results in [30], whether BVP (1) has a sign-changing periodic solution, especially an odd periodic solution, is an interesting problem.For the special second-order ordinary differential equation ( 2), the existence of odd periodic solutions has been discussed by the present author in [31].But no one has discussed the general second-order equation (1).The main purpose of this paper is to obtain the existence of odd 2-periodic solutions for the general second-order equation (1).Our main result is as follows.
In Theorem 1, condition (F1) means that (, , ) is an odd function on (, ).Condition (F2) allows that (, , ) may be superlinear growth on  and .Condition (F3) is a Nagumo type growth condition on , which requests the growth of  on  which cannot be hyperquadric.
The proof of Theorem 1 is based on the Leray-Schauder fixed point theorem, which will be given in the next section.Two examples to illustrate the applicability of our main result are presented at the end of Section 2. Our result and method are different from those in the references mentioned above.

Proof of the Main Result
Clearly,   is a closed subspace of   2 (R) and hence it is a Banach space by the norm ‖‖ ,2 of   2 (R),   is a closed subspace of   2 (R), and hence it is a Banach space by the norm ‖‖   of   2 (R).Given ℎ ∈ , we consider the existence of odd 2periodic solution for the linear second-order differential equation Lemma 2. For every ℎ ∈ , the linear equation ( 7) has a unique odd 2-periodic solution  fl ℎ ∈  2 , and it satisfies Moreover, when ℎ ∈  the solution  = ℎ ∈  2 and  :  →  1 is a completely continuous linear operator.
Proof.Let ℎ ∈ .Since ℎ ∈  2 2 (R) is an odd function, it can be expressed by the Fourier sine series expansion where   = (2/)∫  0 ℎ() sin  ,  = 1,2,..., and the Parseval equality holds.It is easy to verify that belongs to  2 and is a unique odd 2-periodic solution of the linear equation (7) in Carathéodory sense.If ℎ ∈ , the solution  ∈  2 is a classical solution.From (11) we can easily see that  :  →  2 is a linearly bounded operator.By the compactness of the Sobolev embedding  2 2 (R) →  1 2 (R), we see that the embedding  2 →  1 is compact.Hence, by the boundedness of the embedding  → ,  maps  into  1 and  :  →  1 is completely continuous.
On the other hand, since   ∈  2 2 (R) is an even function, it can be expressed by the cosine series expansion where Hence we obtain the cosine series expansion of   : By the expansions ( 11) and ( 14), using Parseval equality, we obtain that By the continuity of  and Assumption (F1),  :  1 →  is continuous and it maps every bounded set of  1 into a bounded set of .Define a mapping by By Lemma 2,  :  →  1 is a completely continuous linear operator.Hence the composite mapping  :  1 →  1 is completely continuous.By the definition of the operator , the odd 2-periodic solution of ( 1) is equivalent to the fixed point of .We will use the Leray-Schauder fixed point theorem [32] to show that  has a fixed point.To do this, we consider the homotopic family of the operator equations We need to prove that the set of the solutions of the equations ( 18) is bounded in the space  1 .Let  ∈  1 be a solution of an equation of ( 18) for  ∈ (0, 1).Then  = (()).Set ℎ = ().Since ℎ ∈ , by the definition of ,  = ℎ is the unique odd 2 periodic solution of the linear equation (7).Hence  ∈  2 satisfies the differential equation Multiplying this equation by () and using Assumption (F2), we have Integrating this inequality on [0, 2], using integration by parts for the left side and ( 8), we have From this inequality it follows that By this and ( 8) we obtain that Hence, by the continuity of the Sobolev embedding where  is the Sobolev embedding constant.
We only consider Case 1; the other cases can be dealt with by a similar method.Let Case 1 hold.Set Then  1 <  1 , and by the definition of supremum, Hence, for every  ∈ [ Integrating both sides of this inequality on [ 1 ,  1 ] and making the variable transformation  =   () for the left side, we have From this inequality and ( 26) it follows that   ( 1 ) ≤  1 .Hence, ‖  ‖  =   ( 1 ) ≤  1 ; namely, (27) holds.Now by ( 24) and ( 27), we have This means that the set of the solutions of the equations ( 18) is bounded in  1 .By the Leray-Schauder fixed point theorem,  has a fixed point in  1 , which is an odd 2-periodic solution of (1).The proof of Theorem 1 is completed.
In the above proof, applying the Leray-Schauder fixed point theorem and the technique of prior estimation we proved Theorem 1.Since the nonlinearity (, , ) may be superlinear growth on  and , Theorem 1 cannot be proved by the simple Schauder fixed point theorem.
Example 3 shows that Theorem 1 is applicable to the superlinear fully second-order ordinary differential equation, and Example 4 shows that Theorem 1 is also applicable to the sublinear fully second-order ordinary differential equation.It should be pointed out that, in Theorem 1 since the nonlinearities contain derivative terms and do not have monotonicity, the conclusions of Examples 3 and 4 cannot be obtained from the known results of . 3.