Existence of positive solutions to periodic boundary value problems with sign-changing Green's function

This paper deals with the periodic boundary value problems

where is a constant and in which case the associated Green's function may changes sign. The existence result of positive solutions is established by using the fixed point index theory of cone mapping.

The periodic boundary value problems

where f is a continuous or L¹-Caratheodory type function have been extensively studied. A very popular technique to obtain the existence and multiplicity of positive solutions to the problem is Krasnosel'skii's fixed point theorem of cone expansion/compression type, see for example [1-4], and the references contained therein. In those papers, the following condition is an essential assumptions:

(A) The Green function G(t, s) associated with problem (1) is positive for all (t, s) ∈ [0, T] × [0, T].

Under condition (A), Torres get in [4] some existence results for (1) with jumping nonlinearities as well as (1) with a repulsive or attractive singularity, and the authors in [3] obtained the multiplicity results to (1) when f(t, u) has a repulsive singularity near x = 0 and f(t, u) is super-linear near x = +∞. In [2], a special case, a(t) ≡ m² and , was considered, the multiplicity results to (1) are obtained when the nonlinear term f(t, u) is singular at u = 0 and is super-linear at u = ∞.

Recently, in [5], the hypothesis (A) is weakened as

(B) The Green function G(t, s) associated with problem (1) is nonnegative for all (t, s) ∈ [0, T] × [0, T] but vanish at some interior points.

By defining a new cone, in order to apply Krasnosel'skii's fixed point theorem, the authors get an existence result when and is sub-linear at u = 0 and u = ∞ or is super-linear at u = 0 and u = ∞ with is convex and nondecreasing.

In [6], the author improve the result of [5] and prove the existence results of at least two positive solutions under conditions weaker than sub- and super-linearity.

In [7], the author study (1) with f(t, u) = λb(t)f(u) under the following condition:

(C) The Green function G(t, s) associated with problem (1) changes sign and where G ^- is the negative part of G.

Inspired by those papers, here we study the problem:

where is a constant and the associated Green's function may changes sign. The aim is to prove the existence of positive solutions to the problem.

Consider the periodic boundary value problem

where and e(t) is a continuous function on [0, T]. It is well known that the solutions of (3) can be expressed in the following forms

where G(t, s) is Green's function associated to (3) and it can be explicitly expressed

By direct computation, we get

and

for when , and

where G⁺ and G^- are the positive and negative parts of G.

We denote

and

Let E denote the Banach space C[0, T] with the norm ||u|| = max_t∈[0,T] |u(t)|.

Define the cone K in E by

We know that and therefore K ≠ ∅. For r > 0, let K_r = {u ∈ K : ||u|| < r}, and ∂K_r = {u ∈ K : ||u|| = r}, which is the relative boundary of K_r in K.

To prove our result, we need the following fixed point index theorem of cone mapping.

Lemma 1 (Guo and Lakshmikantham [8]). Let E be a Banach space and let K ⊂ E be a closed convex cone in E. Let L : K → K be a completely continuous operator and let i(L, K_r, K) denote the fixed point index of operator L.

(i) If μLu ≠ u for any u ∈ ∂K_r and 0 < μ ≤ 1, then

(ii) If and μLu ≠ u for any u ∈ ∂K_r and μ ≥ 1, then

We make the following assumptions: (H1) f : [0, +∞) → [0, +∞) is continuous;

(H2) 0 ≤ m = inf _{u∈[0,+ ∞]} f (u) and M = sup_{u∈[0,+ ∞)} f (u) ≤ +∞;

(H3) , when m = 0 we define .

To be convenience, we introduce the notations:

and suppose that f₀, f_∞ ∈ [0, ∞].

Define a mapping L : K → E by

It can be easily verified that u ∈ K is a fixed point of L if and only if u is a positive solution of (2).

Lemma 2. Suppose that (H₁), (H₂) and (H₃) hold, then L : E → E is completely continuous and L(K) ⊆ K.

Proof Let u ∈ K, then in case of γ = +∞, since G(t, s) ≥ 0, we have Lu(t) ≥ 0 on [0, T]; in case of γ < +∞, we have

On the other hand,

and

for t ∈ [0, T]. Thus,

i.e., L(K) ⊆ K. A standard argument can be used to show that L : E → E is completely continuous.

Now we give and prove our existence theorem:

Theorem 3. Assume that (H₁), (H₂) and (H₃) hold. Furthermore, suppose that f₀ > ρ² and f_∞ < ρ² in case of γ = +∞. Then problem (2) has at least one positive solution.

Proof Since f₀ > ρ², there exist ε > 0 and ξ > 0 such that

Let r ∈ (0, ξ), then for every u ∈ ∂K_r, we have

Hence, . Next, we show that μLu ≠ u for any u ∈ ∂K_r and μ ≥ 1. In fact, if there exist u₀ ∈ ∂K_r and μ₀ ≥ 1 such that μ₀Lu₀ = u₀, then u₀(t) satisfies

Integrating the first equation in (5) from 0 to T and using the periodicity of u₀(t) and (4), we have

Since , we see that ρ² ≥ (ρ² + ε), which is a contradiction. Hence, by Lemma 1, we have

On the other hand, since f_∞ < ρ², there exist ε ∈ (0, ρ²) and ζ > 0 such that

Set C = max_0≤u≤ζ |f (u) - (ρ² - ε)u| + 1, it is clear that

If there exist u₀ ∈ K and 0 < μ₀ ≤ 1 such that μ₀Lu₀ = u₀, then (5) is valid.

Integrating again the first equation in (5) from 0 to T, and from (7), we have

Therefore, we obtain that

i.e.,

Let , then μLu ≠ u for any u ∈ ∂K_R and 0 < μ ≤ 1. Therefore, by Lemma 1, we get

From (6) and (9) it follows that

Hence, L has a fixed point in , which is the positive solution of (2).

Remark 4. Theorem 3 contains the partial results of [4-7] obtained in case of positive Green's function, vanishing Green's function and sign-changing Green's function, respectively.

Let 0 ≠ q < 1 be a constant, h be the function:

and let

By the direct calculation, we get m = 1 and M = γ, and f₀ = ∞ and f_∞ = 0 in case of γ = +∞. Consider the following problem

where is a constant. We know that the conditions of Theorem 3 hold for the problem (10) and therefore, (10) have at least one positive solution from Theorem 3.

The authors declare that they have no competing interests.

YA conceived of the study, and participated in its coordination. SZ drafted the manuscript. All authors read and approved the final manuscript.

The authors are very grateful to the anonymous referee whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript.

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