Abstract
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.
Keywords:
periodic boundary value problem; positive solution; signchanging Green's function; cone; fixed point theorem1 Introduction
The periodic boundary value problems
where f is a continuous or L^{1}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 [14], 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 superlinear near x = +∞. In [2], a special case, a(t) ≡ m^{2 }and , was considered, the multiplicity results to (1) are obtained when the nonlinear term f(t, u) is singular at u = 0 and is superlinear 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 sublinear at u = 0 and u = ∞ or is superlinear 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 superlinearity.
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.
2 Preliminaries
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
3 Existence result
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 f_{0}, 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 (H_{1}), (H_{2}) and (H_{3}) 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 (H_{1}), (H_{2}) and (H_{3}) hold. Furthermore, suppose that f_{0 }> ρ^{2 }and f_{∞ }< ρ^{2 }in case of γ = +∞. Then problem (2) has at least one positive solution.
Proof Since f_{0 }> ρ^{2}, 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 u_{0 }∈ ∂K_{r }and μ_{0 }≥ 1 such that μ_{0}Lu_{0 }= u_{0}, then u_{0}(t) satisfies
Integrating the first equation in (5) from 0 to T and using the periodicity of u_{0}(t) and (4), we have
Since , we see that ρ^{2 }≥ (ρ^{2 }+ ε), which is a contradiction. Hence, by Lemma 1, we have
On the other hand, since f_{∞ }< ρ^{2}, there exist ε ∈ (0, ρ^{2}) and ζ > 0 such that
Set C = max_{0≤u≤ζ }f (u)  (ρ^{2 } ε)u + 1, it is clear that
If there exist u_{0 }∈ K and 0 < μ_{0 }≤ 1 such that μ_{0}Lu_{0 }= u_{0}, 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 [47] obtained in case of positive Green's function, vanishing Green's function and signchanging Green's function, respectively.
4 An example
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 f_{0 }= ∞ 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.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
YA conceived of the study, and participated in its coordination. SZ drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
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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