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

*Boundary Value Problems*
**volume 2011**, Article number: 8 (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.

## 1 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 [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*.

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: (*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 .

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 [4–7] obtained in case of positive Green's function, vanishing Green's function and sign-changing 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.

## References

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*Nonlinear Problems in Abstract Cones.*Academic Press, New York; 1988.

## 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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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.

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Zhong, S., An, Y. Existence of positive solutions to periodic boundary value problems with sign-changing Green's function.
*Bound Value Probl* **2011**, 8 (2011). https://doi.org/10.1186/1687-2770-2011-8

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DOI: https://doi.org/10.1186/1687-2770-2011-8