## Boundary Value Problems

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# Bifurcation of positive periodic solutions of first-order impulsive differential equations

Boundary Value Problems20122012:83

DOI: 10.1186/1687-2770-2012-83

Accepted: 20 July 2012

Published: 1 August 2012

## Abstract

We give a global description of the branches of positive solutions of first-order impulsive boundary value problem:

which is not necessarily linearizable. Where is a parameter, are given impulsive points. Our approach is based on the Krein-Rutman theorem, topological degree, and global bifurcation techniques.

MSC:34B10, 34B15, 34K15, 34K10, 34C25, 92D25.

### Keywords

Krein-Rutman theorem topological degree bifurcation from interval impulsive boundary value problem existence and multiplicity

## 1 Introduction

Some evolution processes are distinguished by the circumstance that at certain instants their evolution is subjected to a rapid change, that is, a jump in their states. Mathematically, this leads to an impulsive dynamical system. Differential equations involving impulsive effects occur in many applications: physics, population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc. Therefore, the study of this class of impulsive differential equations has gained prominence and it is a rapidly growing field. See [19] and the references therein.

Let us consider the equation
(1.1)
subjected to the impulsive boundary condition
(1.2)

where is a real parameter, , are given impulsive points. We make the following assumptions:

(H1) is a 1-periodic function and ;

(H2) , , for , there exist positive constants such that
(H3) is 1-periodic function with respect to the first variable, and , exist, . Moreover, there exist functions with in any subinterval of such that
where with as uniformly for (), and

where with as uniformly for ();

(H4) , ;

(H5) there exists function and in any subinterval of such that

Some special cases of (1.1), (1.2) have been investigated. For example, Nieto [3] considered the (1.1), (1.2) with , . By using Schaeffer’s theorem, some sufficient conditions for existence of solutions of the IBVP (1.1), (1.2) with , were obtained.

Li, Nieto, and Shen [4] studied the existence of at least one positive periodic solutions of (1.1), (1.2) with , (m is a constant). By using Schaeffer’s fixed-point theorem, they got the solvability under f satisfied at most linear growth and is bounded or f is bounded and satisfied at most linear growth.

Liu [7] studied the existence and multiplicity of (1.1), (1.2) with , by using the fixed- point theorem in cones, and he proved the following:

Theorem A ([7], Theorem 3.1.1])

Let (H 1) hold. Assume that , , , and
(1.3)
and
(1.4)
Then the problem (1.1), (1.2) with has at least one positive solution where will be defined in (2.2) and
(1.5)

Theorem B ([7], Theorem 3.1.2])

Let (H 1) hold. Assume that , , and
(1.6)
and
(1.7)
Then the problem (1.1), (1.2) with has at least one positive solution where W, w defined as (1.5) and
(1.8)

It is worth remarking that the [3, 4, 7] only get the existence of solutions, and there is not any information of global structure of positive periodic solutions.

By using global bifurcation techniques, we obtain a complete description of the global structure of positive solutions for (1.1), (1.2) under weaker conditions. More precisely, our main result is the following theorem.

Theorem 1.1 Let (H 1), (H 2), and (H 3) hold. Suppose , , , . Then

(i) is a bifurcation interval of positive solutions from infinity for (1.1), (1.2), and there exists no bifurcation interval of positive solutions from infinity which is disjoint with . More precisely, there exists a component of positive solutions of (1.1), (1.2) which meets , where , will be defined in Section  2;

(ii) is a bifurcation interval of positive solutions from the trivial solutions for (1.1), (1.2), and there exists no bifurcation interval of positive solutions from the trivial solutions which is disjoint with . More precisely, there exists a component of positive solutions of (1.1), (1.2) which meets , where , will be defined in Section  4;

(iii) If (H 4) and (H 5) also hold, then there is a number such that problem (1.1), (1.2) admits no positive solution with . In this case, .

Remark 1.1 There is no paper except [9] studying impulsive differential equations using bifurcation ideas. However, in [9], they only dealt with the case that , i.e. , do exist. Where

From (H3), it is easy to see that the , may be not exist, the method used in [9] is not helpful any more in this case.

Remark 1.2 From (iii) of Theorem 1.1, we know that , are involved in . Moreover, is a unique bifurcation interval of positive solutions from infinity for (1.1), (1.2), and is a unique bifurcation interval of positive solutions from the trivial solutions for (1.1), (1.2). Therefore, must be intersected with .

Remark 1.3 Obviously, (H3) is more general than (1.5), (1.8). Moreover, if we let , , under conditions (1.3), (1.4), we get , , respectively. Hence, cross the hyperplane . Therefore, Theorem 3.1.1 of [7] is the corollary of Theorems 1.1 even in the special case.

Remark 1.4 Similar, if we let , , only under condition (1.6), we can obtain . From Proposition 3.1, we will know that is unbounded in λ direction, so, cross the hyperplane . Therefore, Theorem 3.1.2 of [7] is the corollary of Theorems 1.1 even in the special case and weaker condition.

Remark 1.5 There are many papers which get the positive solutions using bifurcation from the interval. For example, see [10, 11]. However, in those papers, the linear operator corresponding problem is self-adjoint. It is easy to see that the linear operator corresponding (1.1), (1.2) is not self-adjoint. So, the method used in [9, 10] is not helpful in this case.

Remark 1.6 Condition (H3) means that f is not necessarily linearizable near 0 and infinity. So, we will apply the following global bifurcation theorems for mappings which are not necessarily smooth to get a global description of the branches of positive solutions of (1.1), (1.2), and then, we obtain the existence and multiplicity of positive solutions of (1.1), (1.2).

Theorem C (K. Schmitt, R. C. Thompson [12])

Let V be a real reflexive Banach space. Let be completely continuous such that , . Let () be such that is an isolated solution of the equation
(1.9)
for and , where , are not bifurcation points of (1.9). Furthermore, assume that
where is an isolating neighborhood of the trivial solution. Let
Then there exists a connected component of containing in , and either
1. (i)

is unbounded in , or

2. (ii)

.

Theorem D (K. Schmitt [13])

Let V be a real reflexive Banach space. Let be completely continuous, and let () be such that the solution of (1.9) are, a priori, bounded in V for and , i.e., there exists an such that
for all u with . Furthermore, assume that
for large. Then there exists a closed connected set of solutions of (1.9) that is unbounded in , and either
1. (i)

is unbounded in λ direction, or

2. (ii)

there exist an interval such that , and bifurcates from infinity in .

The rest of the paper is organized as follows: In Section 2, we state some notations and preliminary results. Sections 3 and 4 are devoted to study the bifurcation from infinity and from the trivial solution for a nonlinear problem which are not necessarily linearizable, respectively. Finally, in Section 5, we consider the intertwining of the branches bifurcating from infinity and from the trivial solution.

## 2 Preliminaries

Let

Then is a Banach space with the norm .

By a positive solution of the problem (1.1), (1.2), we mean a pair , where and u is a solution of (1.1), (1.2) with . Let be the closure of the set of positive solutions of (1.1), (1.2), where .

Lemma 2.1 ([14], Theorem 6.26])

The spectrum of compact linear operator T has following properties:
1. (i)

is a countable set with no accumulation point which is different from zero;

2. (ii)

each nonzero is an eigenvalue of T with finite multiplicity, and is an eigenvalue of with the same multiplicity, where denote the conjugate of λ, denote the conjugate operator of T.

Let , with inner product and norm .

Let and in any subinterval of . Further define the linear operator ,
(2.1)
where as defined in (H2), is the Green’s function of
and
(2.2)

where , it is easy to see that (H1) implies that .

By virtue of Krein-Rutman theorems (see [15]), we have the following lemma.

Lemma 2.2 Suppose that (H 1) holds, then for the operator defined by (2.1), has a unique characteristic value , which is positive, real, simple, and the corresponding eigenfunction is of one sign, i.e., we have .

Proof It is a direct consequence of the Krein-Rutman theorem [15], Theorem 19.3]. □

Remark 2.1 Since is real number, so from Lemma 2.1, is also the characteristic value of , let denote the nonnegative eigenfunction of corresponding to , where denote the conjugate operator of . Therefore, we have
We extend the function f to function , defined on by

Then on .

For , the problem
(2.3)
is equivalent to the operator equation .

Remark 2.2 For , if u is a nontrivial solution of (2.3), from the positivity of and , we have that on , so u is a nontrivial solution of (1.1), (1.2). Therefore, the closure of the set of nontrivial solutions of (2.3) in is exactly Σ.

The problem (2.3) is now equivalent to the operator equation
(2.4)
In the following, we shall apply the Leray-Schauder degree theory, mainly to the mapping ,

For , let , let denote the degree of on with respect to 0.

## 3 Bifurcation from infinity

In this section, we are devoted to study the bifurcation from infinity.

Lemma 3.1 Let be a compact interval with . Then there exists a number such that
Proof Suppose on the contrary that there exists with (), such that . We may assume . By Remark 2.2, in . Set . Then

From (H2), (H3), we know that is bounded in , so is a relatively compact set in since is bounded and continuous and . Suppose in . Then and in .

Now, from condition (H2), we know that there exist , such that
From (H3), we have that
So,
and
accordingly, we have
(3.1)
and
(3.2)
Let and denote the nonnegative eigenfunctions of , corresponding to , and , respectively. Then we have from the (3.1) that
Letting , we have
we obtain that
and consequently
Similarly, we deduce from (3.2) that

Thus, . This contradicts . □

Corollary 3.1 For and . Then .

Proof Lemma 3.1, applied to the interval , guarantees the existence of such that for ,
Hence, for any ,

which implies the assertion. □

On the other hand, we have

Lemma 3.2 Suppose . Then there exists with the property that with , ,

where is the nonnegative eigenfunction of corresponding to .

Proof Let us assume that for some sequence in with and numbers , such that . Then
and we conclude from Remark 2.2 that in . So we have
Choose such that
(3.3)
By (H3), there exists , such that
From , then exists , such that
and consequently
(3.4)
Let , applying (3.4), it follows that
Thus,

Corollary 3.2 For and , .

Proof By Lemma 3.2, there exists such that
Then

for all . The assertion follows. □

We are now ready to prove

Proposition 3.1 is a bifurcation interval of positive solutions from infinity for the problem (2.4). There exists an unbounded component of positive solutions of (2.4) which meets , and is unbounded in λ direction. Moreover, there exists no bifurcation interval of positive solutions from infinity which is disjointed with .

Proof For fixed with , let us take that , and . It is easy to check that for , all of the conditions of Theorem D are satisfied. So, there exists a closed connected set of solutions of (2.4) that is unbounded in , and either
1. (i)

is unbounded in λ direction, or else

2. (ii)

such that and bifurcates from infinity in .

By Lemma 3.1, the case (ii) cannot occur. Thus, bifurcates from infinity in and is unbounded in λ direction. Furthermore, we have from Lemma 3.1 that for any closed interval , the set is bounded in . So, must be bifurcated from infinity in and is unbounded in λ direction. □

Assertion (i) of Theorem 1.1 follows directly.

## 4 Bifurcation from the trivial solutions

In this section, we shall study the bifurcation from the trivial solution for a nonlinear problem which is not necessarily linearizable near 0 and infinity.

As in Section 2, let and in any subinterval of . Further define the linear operator ,
(4.1)

where is defined in (H2), is defined in (2.2).

Similar as Lemma 2.2, we have the following lemma.

Lemma 4.1 Suppose that (H 1) holds, then the operator has a unique characteristic value , which is positive, real, simple, and the corresponding eigenfunction is of one sign, i.e., we have .

Remark 4.1 Since is real number, so from Lemma 2.1, is also the characteristic value of , where denote the conjugate operator of , let denote the nonnegative eigenfunction of corresponding to . Therefore, we have
Lemma 4.2 Let be a compact interval with . Then there exists a number such that
Proof Suppose on the contrary that there exists with (), such that . We may assume . By Remark 2.2, in . Set . Then

From (H2), (H3), we know that is bounded in , so we infer that is a relatively compact set in , hence (for a subsequence) with in , .

Now, from condition (H2), we know that there exist , such that
From (H3), we have that
So,
and
accordingly, we have
(4.2)
and
(4.3)
Let and denote the nonnegative eigenfunctions of , corresponding to , and , respectively. Then we have from the (4.2) that
Letting , we have
we obtain that
and consequently
Similarly, we deduce from (4.3) that

Thus, . This contradicts . □

Corollary 4.1 For and . Then .

On the other hand, we have

Lemma 4.3 Suppose . Then there exists with the property that with , ,

where is the nonnegative eigenfunction of the corresponding to .

Proof We assume again on the contrary that there exists and a sequence with and in , such that for all .

Then
and we conclude from Remark 2.2 that in . So, we have
Choose such that
(4.4)
By (H3), there exists , such that
From , then exists , such that
and consequently
(4.5)
Let , applying (4.5), it follows that
Thus,

Corollary 4.2 For and . Then .

Proof By Lemma 4.3, there exists such that
Then

for all . Then the assertion follows. □

Now, using Theorem C and the similar method to prove Proposition 3.1 with obvious changes, we may prove the following proposition.

Proposition 4.1 is a bifurcation interval of positive solutions from the trivial solution for the problem (2.4). There exists an unbounded component of positive solutions of (2.4) which meets . Moreover, there exists no bifurcation interval of positive solutions from the trivial solution which is disjointed with .

This is exactly the assertion (ii) of Theorem 1.1.

## 5 Global behavior of the component of positive solutions

In this section, we consider the intertwining of the branches bifurcating from infinity and from the trivial solution.

Let , for . From (H2), we have , .

Define the linear operator ,
(5.1)

where is defined in (H5), is defined in (2.2).

Similar as Lemma 2.2, we have the following lemma.

Lemma 5.1 The operator has a unique characteristic value , which is positive, real, simple, and the corresponding eigenfunction is of one sign, i.e., we have .

Remark 5.1 Since is real number, so from Lemma 2.1, is also the characteristic value of , where denote the conjugate operator of , let denote the nonnegative eigenfunction of corresponding to . Therefore, we have

Lemma 5.2 Let (H 1)-(H 5) hold. Then there exists a number such that there is no positive solution of with .

Proof Let be a positive solution of . Then
From (H5) and the definition of , we have
(5.2)
From (5.2), we have
Thus,

□

The assertion that in Theorem 1.1(iii) now easily follows. For, in the case, and are contained in . Moreover, there exists no bifurcation interval of positive solution from infinity which is disjointed with , there exists no bifurcation interval of positive solution from the trivial solution which is disjointed with . In Theorem 1.1(iii), the unbounded component has to meet .

## Declarations

### Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11061030), NSFC (No. 11126296), and the Fundamental Research Funds for the Gansu Universities.

## Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University

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