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

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.

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 [1-9] and the references therein.

Let us consider the equation

subjected to the impulsive boundary condition

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 (H1) hold. Assume that, , , and

and

Then the problem (1.1), (1.2) withhas at least one positive solution wherewill be defined in (2.2) and

Theorem B ([7], Theorem 3.1.2])

Let (H1) hold. Assume that, , and

and

Then the problem (1.1), (1.2) withhas at least one positive solution whereW, wdefined as (1.5) and

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, , , . 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 componentof 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 componentof positive solutions of (1.1), (1.2) which meets, where, will be defined in Section 4;

(iii) If (H4) and (H5) also hold, then there is a numbersuch 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])

LetVbe a real reflexive Banach space. Letbe completely continuous such that, . Let () be such thatis an isolated solution of the equation

forand, where, are not bifurcation points of (1.9). Furthermore, assume that

whereis an isolating neighborhood of the trivial solution. Let

Then there exists a connected componentofℓcontainingin, and either

(i) is unbounded in, or

(ii) .

Theorem D (K. Schmitt [13])

LetVbe a real reflexive Banach space. Letbe completely continuous, and let () be such that the solution of (1.9) are, a priori, bounded inVforand, i.e., there exists ansuch that

for alluwith. Furthermore, assume that

forlarge. Then there exists a closed connected setof solutions of (1.9) that is unbounded in, and either

(i) is unbounded inλdirection, or

(ii) there exist an intervalsuch that, andbifurcates 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.

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 spectrumof compact linear operatorThas following properties:

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

(ii) each nonzerois an eigenvalue ofTwith finite multiplicity, andis an eigenvalue ofwith the same multiplicity, wheredenote the conjugate ofλ, denote the conjugate operator ofT.

Let , with inner product and norm .

Let and in any subinterval of . Further define the linear operator ,

where as defined in (H2), is the Green’s function of

and

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

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

Lemma 2.2Suppose that (H1) holds, then for the operatordefined by (2.1), has a unique characteristic value, which is positive, real, simple, and the corresponding eigenfunctionis 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

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

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.

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

Lemma 3.1Letbe a compact interval with. Then there exists a numbersuch 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

and

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 3.1Forand. 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 3.2Suppose. Then there existswith the property thatwith, ,

whereis the nonnegative eigenfunction ofcorresponding 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

By (H3), there exists , such that

From , then exists , such that

and consequently

Let , applying (3.4), it follows that

Thus,

this contradicts (3.3). □

Corollary 3.2Forand, .

Proof By Lemma 3.2, there exists such that

Then

for all . The assertion follows. □

We are now ready to prove

Proposition 3.1is a bifurcation interval of positive solutions from infinity for the problem (2.4). There exists an unbounded componentof positive solutions of (2.4) which meets, and is unbounded inλdirection. 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

(i) is unbounded in λ direction, or else

(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.

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 ,

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

Similar as Lemma 2.2, we have the following lemma.

Lemma 4.1Suppose that (H1) holds, then the operatorhas a unique characteristic value, which is positive, real, simple, and the corresponding eigenfunctionis 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 4.2Letbe a compact interval with. Then there exists a numbersuch 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

and

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 4.1Forand. Then.

On the other hand, we have

Lemma 4.3Suppose. Then there existswith the property thatwith, ,

whereis the nonnegative eigenfunction of thecorresponding 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

By (H3), there exists , such that

From , then exists , such that

and consequently

Let , applying (4.5), it follows that

Thus,

this contradicts with (4.4). □

Corollary 4.2Forand. 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 4.1is a bifurcation interval of positive solutions from the trivial solution for the problem (2.4). There exists an unbounded componentof 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.

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 ,

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

Similar as Lemma 2.2, we have the following lemma.

Lemma 5.1The operatorhas a unique characteristic value, which is positive, real, simple, and the corresponding eigenfunctionis 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 5.2Let (H1)-(H5) hold. Then there exists a numbersuch that there is no positive solutionofwith.

Proof Let be a positive solution of . Then

From (H5) and the definition of , we have

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 .

The authors declare that they have no competing interests.

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.

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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