Electronic Journal of Qualitative Theory of Differential Equations

We study the existence and multiplicity of positive periodic solutions for second order differential equations with vanishing Green’s functions. The proof relies on a fixed point theorem in cones. Some recent results in the literature are generalized.


Introduction
During the last two decades, the existence of periodic solutions for second order differential equation has been extensively studied in the literature for the regular cases as well as the singular cases, where a ∈ X = C(R/TZ, R) and the nonlinearity f ∈ C((R/TZ) × (0, ∞), R).See, for example, [4][5][6]15,17,18].Some classical tools have been used in the study of periodic solutions of equation (1.1), including the method of upper and lower solutions [15], some fixed point theorems in cones for completely continuous operators [4,17], Schauder's fixed point theorem [5,18] and a nonlinear Leray-Schauder alternative principle [2,6].
In the above mentioned works, when one tried to apply some fixed point theorems in cones, or the nonlinear alternative principle of Leray-Schauder, to study the existence of periodic solutions of equation (1.1), one major assumption is that the corresponding Green's function G(t, s) for the linear differential equation x + a(t)x = 0 (1.2) Email: liao@shnu.edu.cn F.-F. Liao is positive, which is equivalent to the strict anti-maximum principle for equation (1.2).Such an assumption plays an important role in constructing the following cone When the Green's function vanishes, we know that m = 0 and K 1 becomes the cone of nonnegative functions, which is not effective in obtaining the desired estimates.
For example, when a(t) = k 2 with k > 0 and k = 2nπ T (n ∈ Z + ), the Green's function is given as Therefore the positiveness of the Green's function is equivalent to k < π/T.For the critical case k = π/T, the Green's function vanishes on the line t = s, and therefore the results in [2,4,6,17] cannot deal with such a critical case.In this paper, we focus on the case k ≤ π/T because we assume that the following condition holds In Section 2, we will make a brief comment on condition (A).We observe that even when the Green's function vanishes, the following fact also holds Based on this fact, Graef, Kong and Wang in [8] introduced the following cone Using the above cone, it was proved in [8] that equation is continuous with min t g(t) > 0 and one of the following two conditions holds: , f is convex and nondecreasing.
As an example, it was shown that equation has at least one nontrivial T-periodic solution if 0 < k ≤ π/T and α ∈ (0, 1) ∪ (1, ∞).Such a result was generalized in [19] using fixed point index theory and used conditions related to the principal eigenvalue of the corresponding linear problem.Theorem 2.2 of [19] also has a result for existence of two positive solutions in a sublinear case.The result of [8] was extended in [12] to systems.For the superlinear case, the result in [8] was improved in [9], in which the convexity assumption was removed.A modification of the cone K was used in [14,19] and some sharp existence conditions were given for (1.4) by assuming that the non-negative function g satisfies a weaker condition T 0 g(t)dt > 0. Such a cone was also used to deal with some singular case in [1].We remark that existence results for (1.4) were proved in [13] using the Schauder fixed point theorem even when the Green's function is sign-changing.
The aim of this paper is to use the cone defined in (1.3), together with fixed point theorems in cones, to establish the existence of at least one or at least two positive T-periodic solutions for equation (1.1).Our main motivation is to obtain new existence results for the following differential equation where a, p, q, e ∈ X, 0 < α < 1, β > α and µ > 0 is a parameter.Our new results generalize some recent results contained in [4,8,9,17], because not only we can deal with the critical case, but also we can obtain the multiplicity result for the case e 0, here the notation e 0 means that e(t) ≥ 0 for all t ∈ [0, T] and ē = 1

Preliminaries
First we make a brief comment on condition (A).When For a non-constant function a(t), there is an L p -criterion proved in [17], which is given in the following lemma for the sake of completeness.Let K(q) denote the best Sobolev constant in the following inequality: The explicit formula for K(q) is where Γ is the Gamma function.

Lemma 2.1 ([17]
). Assume that a(t) 0 and a We can obtain the first positive T-periodic solution of (1.1) as a consequence of [11,Lemma 2.8] or [20,Lemma 5.3].The second positive T-periodic solution will be found based on the following well-known fixed point theorem in cones.Recall that a completely continuous operator means a continuous operator which transforms every bounded set into a relatively compact set.If D is a subset X, we write Lemma 2.2.[7] Let X be a Banach space and K (⊂ X) be a cone.Assume that be a continuous and completely continuous operator such that one of the following conditions is satisfied Then A has at least one fixed point in

Main results
In this section, we state and prove the main results of this paper.We will use the notations Recall that we suppose that ν = min t ω(t) > 0.
(H 2 ) There exists a positive constant r such that Then equation (1.1) has at least one T-periodic solution x with 0 < x < r.
Proof.Let K be the cone in X defined by (1.3).Define the operator A : X → X as Since f is continuous and non-negative in (t, x) ∈ [0, T] × [0, ∞), using the similar proof in [8], we can obtain that A maps the set {x ∈ X : x(t) ≥ 0} into K.Moreover, T-periodic solutions of (1.1) are fixed points of the operator A.
Since (H 2 ) holds, as a consequence of [11,Lemma 2.8] or [20, Lemma 5.3], we can obtain that equation (1.1) has a non-negative T-periodic solution x with x < r.
By the condition (H 1 ), we obtain that which implies that x is a positive T-periodic solution of (1.1).
(H 4 ) There exists a constant R > r such that Then equation (1.1) has at least one T-periodic solution x with r ≤ x ≤ R.
Proof.Let K be the cone in X defined by (1.3).Define the open sets and define the operator A : Next we prove that Ax ≥ Since g is convex, using Jensen's inequality [16,Theorem 3.3], we have Proof.We will apply Theorem 3.3.To this end, we take As in Example 3.2, we know that (H 1 ) and (H 2 ) are satisfied for all µ < μ.Moreover, since β > 1, it is easy to see that (H 3 ) is satisfied and condition (H 4 ) becomes for some R > 0. Since β > 1, the right-hand side of (3.4) goes to 0 as R → +∞.Thus, for any given 0 < µ < μ, it is always possible to find R r such that (3.3) is satisfied.Now all conditions of Theorem 3.3 are satisfied.Thus, equation (3.2) has a positive T-periodic solution x.Remark 3.5.In Lemma 3.1, condition (H 1 ) guarantees that the periodic solution obtained is nontrivial, while (H 1 ) is not required in Theorem 3.3.For the equation (3.2), we require that the function e 0 in Example 3.2, while e is only required to be nonnegative in Example 3.4.
The following multiplicity result is a direct consequence of Lemma 3.1 and Theorem 3.3.Theorem 3.6.Suppose that a(t) satisfies (A) and f (t, x) satisfies (H 1 )-(H 2 )-(H 3 )-(H 4 ).Then equation (1.1) has at least two positive T-periodic solutions x and x with 0 < x < r ≤ x ≤ R. Remark 3.9.We generalize the results in [9] because we can obtain two T-periodic solutions in Example 3.7.In [14], based on bifurcation techniques, the existence of one or two positive solutions was proved.However, our method is different from [14].Remark 3.10.Similar hypotheses to those in Theorem 3.3 have been used in [2,3,10] to study the existence and multiplicity of periodic solutions of singular differential equations.

Now Lemma 2 . 2 2 K \ Ω 1 KExample 3 . 4 .
guarantees that A has at least one fixed point x ∈ Ω with r ≤ x ≤ R. Clearly, x is a T-periodic solution of (1.1).Let us consider the differential equation (3.2) again, where a(t) satisfies (A), 0 < α < 1 < β, µ > 0 is a positive parameter and e ∈ X is nonnegative.Then equation (3.2) has at least one positive T-periodic solutions for each 0 < µ < μ, where μ is the constant given as (3.3) in Example 3.2.

Example 3 . 7 .Remark 3 . 8 .
Let us assume that a(t) satisfy (A), 0 < α < 1 < β and e 0. Then equation (3.2) has at least two positive T-periodic solutions for each 0 < µ < μ, where μ is the constant given as(3.3).It is easy to obtain results analogous to equation (3.2) for the general equation (1.5) with p, q being positive T-periodic continuous functions, but the notation becomes cumbersome.Here we consider only (3.2) for simplicity.