Abstract
We give a global description of the branches of positive solutions of firstorder impulsive boundary value problem:
which is not necessarily linearizable. Where
MSC: 34B10, 34B15, 34K15, 34K10, 34C25, 92D25.
Keywords:
KreinRutman theorem; topological degree; bifurcation from interval; impulsive boundary value problem; existence and multiplicity1 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
subjected to the impulsive boundary condition
where
(H1)
(H2)
(H3)
where
where
(H4)
(H5) there exists function
Some special cases of (1.1), (1.2) have been investigated. For example, Nieto [3] considered the (1.1), (1.2) with
Li, Nieto, and Shen [4] studied the existence of at least one positive periodic solutions of (1.1), (1.2)
with
Liu [7] studied the existence and multiplicity of (1.1), (1.2) with
Theorem A ([7], Theorem 3.1.1])
Let (H1) hold. Assume that
and
Then the problem (1.1), (1.2) with
Theorem B ([7], Theorem 3.1.2])
Let (H1) hold. Assume that
and
Then the problem (1.1), (1.2) with
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 1.1Let (H1), (H2), and (H3) hold. Suppose
(i)
(ii)
(iii) If (H4) and (H5) also hold, then there is a number
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
From (H3), it is easy to see that the
Remark 1.2 From (iii) of Theorem 1.1, we know that
Remark 1.3 Obviously, (H3) is more general than (1.5), (1.8). Moreover, if we let
Remark 1.4 Similar, if we let
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 selfadjoint. It is easy to see that the linear operator corresponding (1.1), (1.2) is not selfadjoint. 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])
LetVbe a real reflexive Banach space. Let
for
where
Then there exists a connected component
(i)
(ii)
Theorem D (K. Schmitt [13])
LetVbe a real reflexive Banach space. Let
for alluwith
for
(i)
(ii) there exist an interval
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
By a positive solution of the problem (1.1), (1.2), we mean a pair
Lemma 2.1 ([14], Theorem 6.26])
The spectrum
(i)
(ii) each nonzero
Let
Let
where
and
where
By virtue of KreinRutman theorems (see [15]), we have the following lemma.
Lemma 2.2Suppose that (H1) holds, then for the operator
Proof It is a direct consequence of the KreinRutman theorem [15], Theorem 19.3]. □
Remark 2.1 Since
We extend the function f to function
Then
For
is equivalent to the operator equation
Remark 2.2 For
The problem (2.3) is now equivalent to the operator equation
In the following, we shall apply the LeraySchauder degree theory, mainly to the
mapping
For
3 Bifurcation from infinity
In this section, we are devoted to study the bifurcation from infinity.
Lemma 3.1Let
Proof Suppose on the contrary that there exists
From (H2), (H3), we know that
Now, from condition (H2), we know that there exist
From (H3), we have that
So,
and
accordingly, we have
and
Let
Letting
we obtain that
and consequently
Similarly, we deduce from (3.2) that
Thus,
Corollary 3.1For
Proof Lemma 3.1, applied to the interval
Hence, for any
which implies the assertion. □
On the other hand, we have
Lemma 3.2Suppose
where
Proof Let us assume that for some sequence
and we conclude from Remark 2.2 that
Choose
By (H3), there exists
From
and consequently
Let
Thus,
this contradicts (3.3). □
Corollary 3.2For
Proof By Lemma 3.2, there exists
Then
for all
We are now ready to prove
Proposition 3.1
Proof For fixed
(i)
(ii)
By Lemma 3.1, the case (ii) cannot occur. Thus,
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
where
Similar as Lemma 2.2, we have the following lemma.
Lemma 4.1Suppose that (H1) holds, then the operator
Remark 4.1 Since
Lemma 4.2Let
Proof Suppose on the contrary that there exists
From (H2), (H3), we know that
Now, from condition (H2), we know that there exist
From (H3), we have that
So,
and
accordingly, we have
and
Let
Letting
we obtain that
and consequently
Similarly, we deduce from (4.3) that
Thus,
Corollary 4.1For
On the other hand, we have
Lemma 4.3Suppose
where
Proof We assume again on the contrary that there exists
Then
and we conclude from Remark 2.2 that
Choose
By (H3), there exists
From
and consequently
Let
Thus,
this contradicts with (4.4). □
Corollary 4.2For
Proof By Lemma 4.3, there exists
Then
for all
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
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
Define the linear operator
where
Similar as Lemma 2.2, we have the following lemma.
Lemma 5.1The operator
Remark 5.1 Since
Lemma 5.2Let (H1)(H5) hold. Then there exists a number
Proof Let
From (H5) and the definition of
From (5.2), we have
Thus,
□
The assertion that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
RM completed the main study and carried out the results of this article. BY drafted the manuscript. ZW checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
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.
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