This paper is concerned with the existence of periodic solutions for the fully second-order ordinary differential equation u′′(t)=ft,ut,u′t, t∈R, where the nonlinearity f:R3→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.

National Natural Science Foundation of China1166107111261053113610551. Introduction and Main Result

In this paper, we discuss the existence of odd 2π-periodic solutions for the fully second-order ordinary differential equation (1)u′′t=ft,ut,u′t,t∈R,where the nonlinearity f:R3→R is continuous and f(t,x,y) is 2π-periodic with respect to t.

The existence of periodic solutions for nonlinear second-order 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 (2)u′′t=ft,ut,t∈Rwhich 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 [1–4], the continuation method of topological degree [5–9], variational method and critical point theory [10–14], method of phase-plane analysis [15–19], the Krasnoselskii’s type fixed point theorem in cone [20–23], and the theory of fixed point index [24–26].

However, there are not so many existence results for the second-order periodic problem that nonlinearity is dependent on the derivative. See [27–30]. In [27, 28], the authors discussed some special cases that nonlinearity is with a separated derivative term gu′. 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 (3)u′′t+futu′t+gut=ht,ut,t∈0,ω,u0=uω,u′0=u′ωby the method of lower and upper solutions. In [30], Li and Jiang researched the existence of positive 2π-periodic solution of the general second-order differential equation (1) by employing fixed point index theory in cones under the nonlinearity f(t,x,y) 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 2π-periodic solutions for the general second-order equation (1). Our main result is as follows.

Theorem 1.

Assume that f:R3→R is continuous; f(t,x,y) is 2π-periodic with respect to t and satisfies the following conditions:

f(-t,-x,y)=-f(t,x,y), ∀(t,x,y)∈R3.

there exist nonnegative constants a and b satisfying a+b<1 and a positive constant C0, such that (4)ft,x,yx≥-ax2-by2-C0,t,x,y∈R3.

For any given M>0, there is a positive continuous function gM(ρ) on R+ satisfying (5)∫0+∞ρdρgMρ+1=+∞,

such that (6)ft,x,y≤gMy,t,x,y∈0,2π×-M,M×R.

Then (1) has at least one odd 2π-periodic solution.

In Theorem 1, condition (F1) means that f(t,x,y) is an odd function on (t,x). Condition (F2) allows that f(t,x,y) may be superlinear growth on x and y. Condition (F3) is a Nagumo type growth condition on y, which requests the growth of f on y 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.

2. Proof of the Main Result

Let C2π(R) denote the Banach space of all continuous 2π-periodic function u(t) in R with norm u=max0≤t≤2πut and C2πn(R) the Banach space of all nth-order continuously differentiable 2π-periodic functions u(t) in R with the norm uCn=maxuC,u′C,…,unC, where n is a positive integer. Let L2π2(R) be the Hilbert space of locally square integrable 2π-periodic functions in R with the interior product (u,v)=∫02πutvtdt and the norm u2=(∫02πut2dt)1/2. Let H2πn(R) be the Sobolev space of 2π-periodic functions with the norm un,2=(∑i=0nu(i)22)1/2. u∈H2πn(R) means that u∈C2πn-1(R) and u(n-1)(t) is absolutely continuous on any finite interval of R and u(n)∈L2π2(R). Let V be the subspace of odd functions in L2π2(R). Denote Vn=H2πn(R)∩V, W=C2π(R)∩V, and Wn=C2πn(R)∩V. Clearly, Vn is a closed subspace of H2πn(R) and hence it is a Banach space by the norm un,2 of H2πn(R), Wn is a closed subspace of C2πn(R), and hence it is a Banach space by the norm uCn of C2πn(R).

Given h∈V, we consider the existence of odd 2π-periodic solution for the linear second-order differential equation (7)u′′t=ht,t∈R.

Lemma 2.

For every h∈V, the linear equation (7) has a unique odd 2π-periodic solution u≔Sh∈V2, and it satisfies (8)u2≤u′2.Moreover, when h∈W the solution u=Sh∈W2 and S:W→W1 is a completely continuous linear operator.

Proof.

Let h∈V. Since h∈L2π2(R) is an odd function, it can be expressed by the Fourier sine series expansion (9)ht=∑k=1∞bksinkt,where bk=(2/π)∫0πhssinksds, k=1,2,…, and the Parseval equality (10)h22=π2∑k=1∞bk2holds. It is easy to verify that (11)ut=-∑k=1∞bkk2sinkt≔Shtbelongs to V2 and is a unique odd 2π-periodic solution of the linear equation (7) in Carathéodory sense. If h∈W, the solution u∈W2 is a classical solution. From (11) we can easily see that S:V→V2 is a linearly bounded operator. By the compactness of the Sobolev embedding H2π2(R)↪C2π1(R), we see that the embedding V2↪W1 is compact. Hence, by the boundedness of the embedding W↪V, S maps W into W1 and S:W→W1 is completely continuous.

On the other hand, since u′∈L2π2R is an even function, it can be expressed by the cosine series expansion (12)u′t=a02+∑k=1∞akcoskt,where(13)a0=2π∫0πu′sds=uπ-u0=0,ak=2π∫0πu′scosksds=-2πk∫0πu′′ssinksds=-2πk∫0πhssinksds=-bkk,k=1,2,….Hence we obtain the cosine series expansion of u′: (14)u′t=-∑k=1∞bkkcoskt.By the expansions (11) and (14), using Parseval equality, we obtain that (15)u22=π2∑k=1∞bkk22≤π2∑k=1∞bkk2=u′22.Hence (8) holds.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

For every u∈W1, set (16)Fut≔ft,ut,u′t.By the continuity of f and Assumption (F1), F:W1→W is continuous and it maps every bounded set of W1 into a bounded set of W. Define a mapping by (17)A=S∘F.By Lemma 2, S:W→W1 is a completely continuous linear operator. Hence the composite mapping A:W1→W1 is completely continuous. By the definition of the operator S, the odd 2π-periodic solution of (1) is equivalent to the fixed point of A. We will use the Leray-Schauder fixed point theorem [32] to show that A has a fixed point. To do this, we consider the homotopic family of the operator equations (18)u=λAu,0<λ<1.We need to prove that the set of the solutions of the equations (18) is bounded in the space W1.

Let u∈W1 be a solution of an equation of (18) for λ∈(0,1). Then u=S(λF(u)). Set h=λF(u). Since h∈W, by the definition of S, u=Sh is the unique odd 2π periodic solution of the linear equation (7). Hence u∈W2 satisfies the differential equation (19)u′′t=λft,ut,u′t,t∈R.Multiplying this equation by u(t) and using Assumption (F2), we have(20)u′′tut=λft,ut,u′tut≥λ-aut2-bu′t2-C0=-aut2-bu′t2-C0,t∈R.Integrating this inequality on [0,2π], using integration by parts for the left side and (8), we have(21)-u′22≥-au22-bu′22-2πC0≥-a+bu′22-2πC0.From this inequality it follows that (22)u′22≤2πC01-a+b.By this and (8) we obtain that (23)u1,2≤u22+u′221/2≤2u′2≤2πC01-a+b1/2≔C1.Hence, by the continuity of the Sobolev embedding H2π1(R)↪C2π(R), we have (24)uC≤Cu1,2≤CC1≕M,where C is the Sobolev embedding constant.

For this M>0, by Assumption (F3), there is a positive continuous function gM(ρ) on R+ satisfying (5) such that (6) holds. Hence by (6) and (24), (25)ft,ut,u′t≤gMu′t,t∈R.By (5), there exists M1>M such that (26)∫0M1ρdρgMρ>2M.We use (25) and (26) to show that (27)u′C≤M1.

Let u′t≢0. By the periodicity of u, there exist t0∈(0,2π) and t1∈[0,2π], t1≠t0, such that (28)u′t0=0,u′C=u′t1>0.Clearly, one of the following cases holds.

Case 1. u′t1>0, t0<t1.

Case 2. u′t1>0, t0>t1.

Case 3. u′t1<0, t0<t1.

Case 4. u′t1<0, t0>t1.

We only consider Case 1; the other cases can be dealt with by a similar method. Let Case 1 hold. Set (29)s1=sups∈t0,t1:u′s=0.Then s1<t1, and by the definition of supremum, (30)u′t>0,t∈s1,t1;u′s1=0.Hence, for every t∈[s1,t1], by (19) and (25), we have(31)u′′t=λft,ut,u′t≤λft,ut,u′t≤ft,ut,u′t≤gMu′t,so we obtain that (32)u′′tu′tgMu′t≤u′t,t∈s1,t1.Integrating both sides of this inequality on s1,t1 and making the variable transformation ρ=u′t for the left side, we have (33)∫0u′t1ρdρgMρ≤ut1-us1≤2uC≤2M.From this inequality and (26) it follows that u′t1≤M1. Hence, u′C=u′(t1)≤M1; namely, (27) holds.

Now by (24) and (27), we have (34)uC1≤M1.This means that the set of the solutions of the equations (18) is bounded in W1. By the Leray-Schauder fixed point theorem, A has a fixed point in W1, which is an odd 2π-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 f(t,x,y) may be superlinear growth on x and y, Theorem 1 cannot be proved by the simple Schauder fixed point theorem.

Example 3.

Consider the superlinear second-order ordinary differential equation(35)u′′t=u3t+utu′t2+sint,t∈R.We verify that the corresponding nonlinearity (36)ft,x,y=x3+xy2+sintsatisfies the conditions of Theorem 1. Clearly, f(t,x,y) satisfies conditions (F1) and (F3). For any (t,x,y)∈R3, by definition (36) we have(37)ft,x,yx=x4+x2y2+xsint≥xsint≥-12x2-12sin2t≥-12x2-12;that is, f satisfies condition (F2) for a=1/2, b=0, and C0=1/2. Hence f satisfies the conditions of Theorem 1. By Theorem 1, (35) has at least one odd 2π-periodic solution.

Example 4.

Consider the sublinear second-order ordinary differential equation (38)u′′t=ut3cost+u′t3sint+sin2t,t∈R.Evidently, the corresponding nonlinearity (39)ft,x,y=x3cost+y3sint+sin2tsatisfies conditions (F1) and (F3). We show that it also satisfies condition (F2). For any (t,x,y)∈R3, by definition (39),(40)ft,x,yx≤x4/3+y1/3x+x≤16x2+163+13y+23x+x≤16x2+163+16y2+16x2+53x≤13x2+16y2+163+130x2+1256≤1130x2+16y2+1576,which is derived by using the Young inequality (41)AB≤1pAp+1qBq,A,B≥0,p,q>1,1p+1q=1to the terms of |x|4/3, |y|1/3, 1/3yx, and (5/3)|x|. From this inequality it follows that (42)ft,x,yx≥-ft,x,yx≥-1130x2-16y2-1576,t,x,y∈R3.Hence, f satisfies condition (F2). By Theorem 1, (38) has at least one odd 2π-periodic solution.

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 [1–31].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by NNSFs of China (11661071, 11261053, and 11361055).

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