Positive Periodic Solutions for Second-Order Ordinary Differential Equations with Derivative Terms and Singularity in Nonlinearities

The existence results of positive -periodic solutions are obtained for the second-order ordinary differential equation where, is a continuous function, which is -periodic in and may be singular at . The discussion is based on the fixed point index theory in cones.


Introduction
In this paper, we discuss the existence of positive ω-periodic solutions of the second-order ordinary differential equation with first-order derivative term in the nonlinearity u t f t, u t , u t , t ∈ R, 1.1 where the nonlinearity f : R × 0, ∞ × R → R is a continuous function, which is ω-periodic in t and f t, u, v may be singular at u 0. The existence problems of periodic solutions for nonlinear second-order ordinary differential equations have attracted many authors' attention and concern, and most works are on the special equation u t f t, u t , t ∈ R, 1.2 that 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 in the Banach space C 0, ω , where σ > 0 is a constant.Hence the fixed point theorems of cone mapping can be applied to the second-order periodic problems.For more precise results using the theory of the fixed point index in cones to discuss the existence of positive periodic solutions of second-order ordinary differential equation, see 19-22 .However, all of these works are on the special second-order equation 1.2 , and few people consider the existence of the positive periodic solutions for the general second-order equation 1.1 that explicitly contains the first order derivative term.The purpose of this paper is to extend the results of 16-22 to the general secondorder equation 1.1 .We will use the theory of the fixed point index in cones to discuss the existence of positive periodic solutions of 1.1 .For the periodic problem of 1.1 , since the corresponding integral operator has no definition on the cone K 0 in C 0, ω , the argument methods used in 16-22 are not applicable.We will use a completely different method to treat 1.1 .Our main results will be given in Section 3. Some preliminaries to discuss 1.1 are presented in Section 2.

Preliminaries
Let C ω R denote the Banach space of all continuous ω-periodic function u t with norm u C max 0≤t≤ω |u t |.Let C 1 ω R be the Banach space of all continuous differentiable ωperiodic function u t with the norm Generally, C n ω R denotes the nth-order continuous differentiable ω-periodic function space for n ∈ N. Let C ω R be the cone of all nonnegative functions in C ω R .
Let M ∈ 0, π 2 /ω 2 be a constant.For h ∈ C ω R , we consider the linear second-order differential equation

2.2
The ω-periodic solutions of 2.2 are closely related with the linear second-order boundary value problem Since U t > 0 for every t ∈ 0, ω , by 2.5 , if h ∈ C ω R and h t / ≡ 0, then the ωperiodic solution of 2.2 u t > 0 for every t ∈ R, and we term it the positive ω-periodic solution.Let

2.8
Define the cone We have the following Lemma.
Now we consider the nonlinear equation 1.1 .Hereafter, we assume that the nonlinearity f satisfies the following condition.F0 There exists M ∈ 0, π 2 /ω 2 such that f t, x, y Mx ≥ 0, x > 0, t, y ∈ R.

2.15
Let f 1 t, x, y f t, x, y Mx, then f 1 t, x, y ≥ 0 for x > 0, t, y ∈ R, and 1.1 is rewritten to

2.16
For u ∈ K, if u / 0, then u C > 0 and by the definition of K, u t ≥ σ u C > 0 for every t ∈ R. Hence is well defined, and we can define the integral operator A : By the definition of operator S, the positive ω-periodic solution of 1.1 is equivalent to the nontrivial fixed point of A. From Assumption F0 , Lemmas 2.1 and 2.2, we easily see the following Lemma.

Lemma 2.3. A K \ {0} ⊂ K, and
We will find the nonzero fixed point of A by using the fixed point index theory in cones.Since the singularity of f at x 0 implies that A has no definition at u 0, the fixed point index theory in the cone K cannot be directly applied to A. We need to make some Preliminaries.
We recall some concepts and conclusions on the fixed point index in 23, 24 .Let E be a Banach space and K ⊂ E a closed convex cone in E. Assume Ω is a bounded open subset of E with boundary ∂Ω, and K ∩ Ω / ∅.Let A : K ∩ Ω → K be a completely continuous mapping.If Au / u for any u ∈ K ∩ ∂Ω, then the fixed point index i A, K ∩ Ω, K has a definition.One important fact is that if i A, K ∩ Ω, K / 0, then A has a fixed point in K ∩ Ω.The following two lemmas are needed in our argument.

Lemma 2.4 see 24 . Let Ω be a bounded open subset of E with θ ∈ Ω and A
Lemma 2.5 see 24 .Let Ω be a bounded open subset of E and A : K ∩ Ω → K a completely continuous mapping.If there exists an e ∈ K \ {θ} such that u − Au / τe for every u ∈ K ∩ ∂Ω and τ ≥ 0, then i A, K ∩ Ω, K 0.
We use Lemmas 2.4 and 2.5 to show the following fixed-point theorem in cones which is applicable to the operator A defined by 2.18 .
Theorem 2.6.Let E be a Banach space and a completely continuous mapping.If A satisfies the following conditions: 2 there exists e ∈ K \ {θ} such that u − Au / τe for u ∈ K ∩ ∂Ω 2 , τ ≥ 0, or the following conditions: Proof.By Dugundji's extension theorem, the operator If A satisfies conditions 1 and 2 of Theorem 2.6, then A also satisfies them.By Lemmas 2.4 and 2.5, respectively, we have By the additivity of the fixed point index, we have

2.20
Hence If A satisfies conditions 3 and 4 of Theorem 2.6, with a similar count, we obtain that

2.21
This means that A has a fixed-point in Theorem 2.6 is an improvement of the fixed point theorem of Krasnoselskii's cone expansion or compression.We will use it to discuss the existence of positive ω-periodic solutions of 1.1 in the next section.

Main Results
We consider the the existence of positive ω-periodic solutions of 1.1 .Let f ∈ C R× 0, ∞ ×R satisfy Assumption F0 and f t, x, y be ω-periodic in t.Let C 0 be the constant defined by 2.8 and I 0, ω .To be convenient, we introduce the notations x, y x , x, y x .

3.1
Our main results are as follows.
Theorem 3.1.Let f : R × 0, ∞ × R → R be continuous and f t, x, y be ω-periodic in t.If f satisfies Assumption F0 and the condition then 1.1 has at least one positive ω-periodic solution.
Theorem 3.2.Let f : R × 0, ∞ × R → R be continuous and f t, x, y be ω-periodic in t.If f satisfies Assumption F0 and the conditions

has at least one positive ω-periodic solution.
Noting that 0 is an eigenvalue of the associated linear eigenvalue problems of 1.1 with periodic boundary condition, if one inequality concerning comparison with 0 in F1 or F2 of Theorem 3.1 or Theorem 3.2 is not true, the existence of periodic solution to 1.1 cannot be guaranteed.Hence, the 0 is the optimal value in condition F1 and F2 .
In Theorem 3.2, the condition F2 allows that f t, x, y has singularity at x 0. For example, satisfies F0 with M π 2 /2ω 2 , and F2 with f 0 ∞ and f ∞ −π 2 /2ω 2 .The existence of periodic solutions for singular ordinary differential equations has been studied by several authors, see 20, 25, 26 .But the equations considered by these authors do not contain derivative term u t .

Proof of Theorem 3.1. Choose the working space
ω R defined by 2.9 and A : K \ {0} → K the operator defined by 2.18 .Then the positive ω-periodic solution of 1.1 is equivalent to the nontrivial fixed point of A. Let 0 < r < R < ∞ and set We show that the operator A has a fixed point in K ∩ Ω 2 \ Ω 1 by Theorem 2.6 when r is small enough and R large enough.By f 0 < 0 and the definition of f 0 , there exist ε ∈ 0, M and δ > 0, such that f t, x, y ≤ −εx, t ∈ 0, ω , y ≤ C 0 , 0 < x ≤ δ.

3.8
By this, 3.6 , and the definition of f 1 we have Integrating both sides of this inequality from 0 to ω and using the periodicity of u 0 , we obtain that
On the other hand, since f ∞ > 0, by the definition of f ∞ , there exist ε 1 > 0 and H > 0 such that Define a function g : 0, ∞ → R by

3.13
This implies that

3.15
Combining this with 3.11 , it follows that Choose e t ≡ 1.Clearly, e ∈ K \ {θ}.We show that A satisfies the Condition 2 of Theorem 2.6 if R is large enough, namely, u − Au / τe for every u ∈ K ∩ ∂Ω 2 and τ ≥ 0. In fact, if there exist u 1 ∈ K ∩ ∂Ω 2 and τ 1 ≥ 0 such that u 1 − Au 1 τ 1 e, since u 1 − τ 1 e Au 1 , by definition of A and Lemma 2.1, u 1 ∈ C 2 ω R satisfies the differential equation 3.17 From 3.17 and 3.16 , it follows that

3.18
Integrating this inequality on 0, ω and using the periodicity of u 1 , we get that Since u 1 ∈ K ∩ ∂Ω 2 , by the definition of K, we have

3.20
By the first inequality of 3.20 , we have From this and 3.19 , it follows that By this and the second inequality of 3.20 , we have Therefore, choose R > max{R, δ}, then A satisfies the Condition 2 of Theorem 2.6.Now by the first part of Theorem 2.6, A has a fixed point in ω R be defined by 3.4 .We use Theorem 2.6 to prove that the operator A has a fixed point in K ∩ Ω 2 \ Ω 1 if r is small enough and R large enough.

3.26
By this, 3.25 , and the definition of f 1 , we have

3.27
Integrating this inequality on 0, ω and using the periodicity of u 0 t , we obtain that

3.28
Since ω 0 u 0 t dt ≥ ωσ u 0 C > 0, from this inequality it follows that M ≥ M ε, which is a contradiction.Hence A satisfies the Condition 3 of Theorem 2.6.
Since f ∞ < 0, by the definition of f ∞ , there exist ε 1 ∈ 0, M and H > 0 such that

3.30
Since u 1 ∈ K ∩ ∂Ω 2 , by the definition of K, u 1 satisfies 3.20 .By the second inequality of 3.20 , we have

3.32
By 3.32 and the first inequality of 3.20 , we have

3.33
From this, the second inequality of 3.20 and 3.29 , it follows that

3.34
By this and 3.30 , we have

3.35
Integrating this inequality on 0, ω and using the periodicity of u 1 t , we obtain that

Remarks
Our discussion on the existence of the positive ω-periodic solutions to 1.1 is applicable to the following ordinary differential equation: where the nonlinearity f : R × 0, ∞ × R → R is continuous and f t, x, y is ω-periodic in t.For 4.1 , we need the following assumption.

4.2
Similarly to Lemma 2.1, we have the following conclusion.
Now, using the similar arguments to Theorems 3.1 and 3.2, we can obtain the following results.
which is a contradiction.This means that A satisfies the Condition 4 of Theorem 2.6.By the second part of Theorem 2.6, A has a fixed point in K ∩ Ω 2 \ Ω 1 , which is a positive ω-periodic solution of 1.1 . 2 t x 2 a 3 xy 2 satisfies the conditions F0 and F1 .By Theorem 3.1, 3.37 has at least one positive ω-periodic solution./x 2 satisfies the conditions F0 and F2 .By Theorem 3.2, the 3.38 has a positive ω-periodic solution.
i t ∈ C ω R , i 1, 2, 3.If −π 2 / ω 2 < a 1 t < 0 and a 2 t , a 3 t > 0 for t ∈ 0, ω , then f t, x, y a 1 t x a ω R .If −π 2 /ω 2 < a t < 0 and b t , c t > 0 for t ∈ 0, ω , then f t, x, y a t x b t x c t y 2 Lemma 4.1.Let M > 0 be a constant.Then for every h ∈ C ω R , the linear second order differential equation