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

- Shengren Zhong
^{1}and - Yukun An
^{2}Email author

**2011**:8

**DOI: **10.1186/1687-2770-2011-8

© Zhong and An; licensee Springer. 2011

**Received: **27 January 2011

**Accepted: **27 July 2011

**Published: **27 July 2011

## 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 sign-changing Green's function cone fixed point theorem## 1 Introduction

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 [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*) ≡ *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 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*.

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

*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

*G*(

*t*,

*s*) is Green's function associated to (3) and it can be explicitly expressed

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

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

*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: (*H* 1) *f* : [0, +∞) → [0, +∞) is continuous;

(*H* 2) 0 ≤ *m* = inf _{u∈[0,+ ∞]}*f* (*u*) and *M* = sup_{u∈[0,+ ∞)}*f* (*u*) ≤ +∞;

(*H* 3)
, when *m* = 0 we define
.

and suppose that *f*_{0}, *f*_{∞} ∈ [0, ∞].

*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

*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

*r*∈ (0,

*ξ*), then for every

*u*∈ ∂

*K*

_{ r }, we have

*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

*T*and using the periodicity of

*u*

_{0}(

*t*) and (4), we have

*ρ*

^{2}≥ (

*ρ*

^{2}+

*ε*), which is a contradiction. Hence, by Lemma 1, we have

*f*

_{∞}

*< ρ*

^{2}, there exist

*ε*∈ (0,

*ρ*

^{2}) and

*ζ >*0 such that

*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.

*T*, and from (7), we have

*μLu*≠

*u*for any

*u*∈ ∂

*K*

_{ R }and 0

*< μ*≤ 1. Therefore, by Lemma 1, we get

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.

## 4 An example

*q <*1 be a constant,

*h*be the function:

*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.

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

## References

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.