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

Ruyun Ma*, Bianxia Yang and Zhenyan Wang

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China

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Boundary Value Problems 2012, 2012:83  doi:10.1186/1687-2770-2012-83


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/83


Received:18 May 2012
Accepted:20 July 2012
Published:1 August 2012

© 2012 Ma et al.; licensee Springer

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.

Abstract

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

{ u ( t ) + a ( t ) u ( t ) = λ f ( t , u ( t ) ) , t ( 0 , 1 ) , t t k , k = 1 , , p , u ( t k + ) = u ( t k ) + λ I k ( u ( t k ) ) , k = 1 , , p , u ( 0 ) = u ( 1 ) ,

which is not necessarily linearizable. Where λ > 0 is a parameter, 0 < t 1 < t 2 < < t p < 1 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 [1-9] and the references therein.

Let us consider the equation

u ( t ) + a ( t ) u ( t ) = λ f ( t , u ( t ) ) , t J , (1.1)

subjected to the impulsive boundary condition

u ( t k + ) = u ( t k ) + λ I k ( u ( t k ) ) , k = 1 , , p , u ( 0 ) = u ( 1 ) , (1.2)

where λ > 0 is a real parameter, J = [ 0 , 1 ] { t 1 , , t p } , 0 < t 1 < t 2 < < t p < 1 are given impulsive points. We make the following assumptions:

(H1) a C ( [ 0 , 1 ] , R ) is a 1-periodic function and 0 1 a ( t ) d t > 0 ;

(H2) I k C ( [ 0 , ) , [ 0 , ) ) , k = 1 , , p , I k ( u ) > 0 for u > 0 , there exist positive constants I k ( 0 ) , I k ( ) ( 0 , ) such that

I k ( 0 ) = lim u 0 + I k ( u ) u , I k ( ) = lim u + I k ( u ) u ;

(H3) f C ( J × [ 0 , ) , [ 0 , ) ) is 1-periodic function with respect to the first variable, and f ( t k + , u ) , f ( t k , u ) exist, f ( t k , u ) = f ( t k , u ) . Moreover, there exist functions a 0 , a 0 , b , b C ( [ 0 , 1 ] , [ 0 , ) ) with a 0 ( t ) , a 0 ( t ) , b ( t ) , b ( t ) 0 in any subinterval of [ 0 , 1 ] such that

a 0 ( t ) u ξ 1 ( t , u ) f ( t , u ) a 0 ( t ) u + ξ 2 ( t , u ) ,

where ξ i C ( [ 0 , 1 ] × [ 0 , ) ) with ξ i ( t , u ) = o ( | u | ) as | u | 0 uniformly for t [ 0 , 1 ] ( i = 1 , 2 ), and

b ( t ) u ζ 1 ( t , u ) f ( t , u ) b ( t ) u + ζ 2 ( t , u ) ,

where ζ i C ( [ 0 , 1 ] × [ 0 , ) ) with ζ i ( t , u ) = o ( | u | ) as | u | uniformly for t [ 0 , 1 ] ( i = 1 , 2 );

(H4) f ( t , u ) > 0 , ( t , u ) [ 0 , 1 ] × ( 0 , ) ;

(H5) there exists function c C ( [ 0 , 1 ] , [ 0 , ) ) and c ( t ) 0 in any subinterval of [ 0 , 1 ] such that

f ( t , u ) c ( t ) u , ( t , u ) [ 0 , 1 ] × [ 0 , ) .

Some special cases of (1.1), (1.2) have been investigated. For example, Nieto [3] considered the (1.1), (1.2) with λ 1 , a 0 . By using Schaeffer’s theorem, some sufficient conditions for existence of solutions of the IBVP (1.1), (1.2) with λ 1 , a 0 were obtained.

Li, Nieto, and Shen [4] studied the existence of at least one positive periodic solutions of (1.1), (1.2) with λ 1 , a m (m is a constant). By using Schaeffer’s fixed-point theorem, they got the solvability under f satisfied at most linear growth and I k is bounded or f is bounded and I k satisfied at most linear growth.

Liu [7] studied the existence and multiplicity of (1.1), (1.2) with λ 1 , 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 f ( t , u ) 0 , I k ( u ) 0 , u 0 , and

max t [ 0 , 1 ] { M 0 1 G ( t , s ) d s + W k = 1 p G ( t , t k ) } < 1 ; (1.3)

and

min t [ 0 , 1 ] { e 0 1 | a ( s ) | d s v 0 1 G ( t , s ) d s + e 0 1 | a ( s ) | d s w k = 1 p G ( t , t k ) } > 1 . (1.4)

Then the problem (1.1), (1.2) with λ 1 has at least one positive solution where G ( t , s ) will be defined in (2.2) and

M : = lim u + sup t [ 0 , 1 ] f ( t , u ) u , W k : = lim u + I k ( u ) u , v : = lim u 0 inf t [ 0 , 1 ] f ( t , u ) u , w k : = lim u 0 I k ( u ) u . (1.5)

Theorem B ([7], Theorem 3.1.2])

Let (H1) hold. Assume that f ( t , u ) 0 , I k ( u ) 0 , u 0 and

min t [ 0 , 1 ] { m 0 1 G ( t , s ) d s + W k = 1 p G ( t , t k ) } > e 0 1 | a ( t ) | d t , (1.6)

and

max t [ 0 , 1 ] { V 0 1 G ( t , s ) d s + w k = 1 p G ( t , t k ) } < 1 . (1.7)

Then the problem (1.1), (1.2) with λ 1 has at least one positive solution whereW, wdefined as (1.5) and

m : = lim u + inf t [ 0 , 1 ] f ( t , u ) u , V : = lim u 0 sup t [ 0 , 1 ] f ( t , u ) u . (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 1.1Let (H1), (H2), and (H3) hold. Suppose f ( t , 0 ) = 0 , t [ 0 , 1 ] , I k ( 0 ) = 0 , k = 1 , , p . Then

(i) [ λ 1 ( b ) , λ 1 ( b ) ] 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 [ λ 1 ( b ) , λ 1 ( b ) ] . More precisely, there exists a component Σ of positive solutions of (1.1), (1.2) which meets [ λ 1 ( b ) , λ 1 ( b ) ] × { } , where λ 1 ( b ) , λ 1 ( b ) will be defined in Section 2;

(ii) [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] 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 [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] . More precisely, there exists a component Σ 0 of positive solutions of (1.1), (1.2) which meets [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] × { 0 } , where λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) will be defined in Section 4;

(iii) If (H4) and (H5) also hold, then there is a number λ > 0 such that problem (1.1), (1.2) admits no positive solution with λ > λ . In this case, Σ = Σ 0 .

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 f 0 , f ( 0 , ) , i.e. f 0 , f do exist. Where

f 0 : = lim | u | 0 f ( t , u ) u and f : = lim | u | f ( t , u ) u both uniformly with respect to  t [ 0 , 1 ] .

From (H3), it is easy to see that the f 0 , f 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 Σ 0 , Σ are involved in [ 0 , λ ] × PC [ 0 , 1 ] . Moreover, [ λ 1 ( b ) , λ 1 ( b ) ] is a unique bifurcation interval of positive solutions from infinity for (1.1), (1.2), and [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] is a unique bifurcation interval of positive solutions from the trivial solutions for (1.1), (1.2). Therefore, Σ 0 must be intersected with [ λ 1 ( b ) , λ 1 ( b ) ] × { } .

Remark 1.3 Obviously, (H3) is more general than (1.5), (1.8). Moreover, if we let a 0 ( t ) : = v , b ( t ) : = M , under conditions (1.3), (1.4), we get λ 1 ( b ) > 1 , λ ˜ 1 ( a 0 ) < 1 , respectively. Hence, Σ 0 cross the hyperplane { 1 } × PC [ 0 , 1 ] . 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 a 0 ( t ) : = V , b ( t ) : = m , only under condition (1.6), we can obtain λ 1 ( b ) < 1 . From Proposition 3.1, we will know that Σ is unbounded in λ direction, so, Σ cross the hyperplane { 1 } × PC [ 0 , 1 ] . 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. Let F : R × V V be completely continuous such that F ( λ , 0 ) = 0 , λ R . Let a , b R ( a < b ) be such that u = 0 is an isolated solution of the equation

u F ( λ , u ) = 0 , u V , (1.9)

for λ = a and λ = b , where ( a , 0 ) , ( b , 0 ) are not bifurcation points of (1.9). Furthermore, assume that

deg ( I F ( a , ) , B r ( 0 ) , 0 ) deg ( I F ( b , ) , B r ( 0 ) , 0 ) ,

where B r ( 0 ) is an isolating neighborhood of the trivial solution. Let

= { ( λ , u ) : ( λ , u ) is a solution of (1.9) with  u 0 } ¯ ( [ a , b ] × { 0 } ) .

Then there exists a connected component C ofcontaining [ a , b ] × { 0 } in R × V , and either

(i) C is unbounded in R × V , or

(ii) C [ ( R [ a , b ] ) × { 0 } ] .

Theorem D (K. Schmitt [13])

LetVbe a real reflexive Banach space. Let F : R × V V be completely continuous, and let a , b R ( a < b ) be such that the solution of (1.9) are, a priori, bounded inVfor λ = a and λ = b , i.e., there exists an R > 0 such that

F ( a , u ) u F ( b , u )

for alluwith u R . Furthermore, assume that

deg ( I F ( a , ) , B R ( 0 ) , 0 ) deg ( I F ( b , ) , B R ( 0 ) , 0 ) ,

for R > 0 large. Then there exists a closed connected set C of solutions of (1.9) that is unbounded in [ a , b ] × V , and either

(i) C is unbounded inλdirection, or

(ii) there exist an interval [ c , d ] such that ( a , b ) ( c , d ) = , and C bifurcates from infinity in [ c , d ] × V .

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

P C [ 0 , 1 ] = { u | u : [ 0 , 1 ] R , u ( t )  is continuous at  t t k , left continuous at  t = t k ,  and the right limit  u ( t k + )  exists for  k = 1 , 2 , 3 , . }

Then P C [ 0 , 1 ] is a Banach space with the norm u = sup t [ 0 , 1 ] | u ( t ) | .

By a positive solution of the problem (1.1), (1.2), we mean a pair ( λ , u ) , where λ > 0 and u is a solution of (1.1), (1.2) with u > 0 . Let Σ R + × P C [ 0 , 1 ] be the closure of the set of positive solutions of (1.1), (1.2), where R + : = [ 0 , ) .

Lemma 2.1 ([14], Theorem 6.26])

The spectrum σ ( T ) of compact linear operatorThas following properties:

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

(ii) each nonzero λ σ ( T ) is an eigenvalue ofTwith finite multiplicity, and λ ¯ is an eigenvalue of T with the same multiplicity, where λ ¯ denote the conjugate ofλ, T denote the conjugate operator ofT.

Let H : = L 2 ( 0 , 1 ) , with inner product , and norm L 2 .

Let Z ( ) C ( [ 0 , 1 ] , [ 0 , ) ) and Z ( ) 0 in any subinterval of [ 0 , 1 ] . Further define the linear operator L Z : P C [ 0 , 1 ] P C [ 0 , 1 ] ,

L Z u = 0 1 G ( t , s ) Z ( s ) u ( s ) d s + k = 1 p G ( t , t k ) I k ( ) u ( t k ) , (2.1)

where I k ( ) as defined in (H2), G ( t , s ) is the Green’s function of

{ u ( t ) + a ( t ) u ( t ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 )

and

G ( t , s ) = { e [ A ( t ) A ( s ) ] 1 e A ( 1 ) , 0 s t 1 , e [ A ( 1 ) + A ( t ) A ( s ) ] 1 e A ( 1 ) , 0 t < s 1 , (2.2)

where A ( t ) = 0 t a ( s ) d s , it is easy to see that (H1) implies that G ( t , s ) > 0 .

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

Lemma 2.2Suppose that (H1) holds, then for the operator L Z defined by (2.1), has a unique characteristic value λ 1 ( Z ) , which is positive, real, simple, and the corresponding eigenfunction φ ( t ) is of one sign, i.e., we have φ ( t ) = λ 1 ( Z ) L Z φ ( t ) .

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

Remark 2.1 Since λ 1 ( Z ) is real number, so from Lemma 2.1, λ 1 ( Z ) is also the characteristic value of L Z , let φ 1 denote the nonnegative eigenfunction of L Z corresponding to λ 1 ( Z ) , where L Z denote the conjugate operator of L Z . Therefore, we have

φ 1 ( t ) = λ 1 ( Z ) L Z φ 1 ( t ) , t [ 0 , 1 ] .

We extend the function f to function f ¯ , defined on [ 0 , 1 ] × R by

f ¯ ( t , u ) = { f ( t , u ) , ( t , u ) [ 0 , 1 ] × [ 0 , ) , f ( t , 0 ) , ( t , u ) [ 0 , 1 ] × ( , 0 ) .

Then f ¯ ( t , u ) 0 on [ 0 , 1 ] × R .

For λ 0 , the problem

{ u ( t ) + a ( t ) u ( t ) = λ f ¯ ( t , u ( t ) ) , t J , u ( t k + ) = u ( t k ) + λ I k ( u ( t k ) ) , k = 1 , , p , u ( 0 ) = u ( 1 ) (2.3)

is equivalent to the operator equation A λ : P C [ 0 , 1 ] P C [ 0 , 1 ] .

( A λ u ) ( t ) = λ 0 1 G ( t , s ) f ¯ ( s , u ( s ) ) d s + λ k = 1 p G ( t , t k ) I k ( u ( t k ) ) .

Remark 2.2 For λ > 0 , if u is a nontrivial solution of (2.3), from the positivity of G ( t , s ) and f ¯ , we have that u ( ) > 0 on [ 0 , 1 ] , so u is a nontrivial solution of (1.1), (1.2). Therefore, the closure of the set of nontrivial solutions ( λ , u ) of (2.3) in R + × P C [ 0 , 1 ] is exactly Σ.

The problem (2.3) is now equivalent to the operator equation

u = A λ ( u ) , u P C [ 0 , 1 ] . (2.4)

In the following, we shall apply the Leray-Schauder degree theory, mainly to the mapping Φ λ : P C [ 0 , 1 ] P C [ 0 , 1 ] ,

Φ λ ( u ) = u A λ ( u ) .

For R > 0 , let B R = { u P C [ 0 , 1 ] : u < R } , let deg ( Φ λ , B R , 0 ) denote the degree of Φ λ on B R with respect to 0.

3 Bifurcation from infinity

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

Lemma 3.1Let Λ R + be a compact interval with [ λ 1 ( b ) , λ 1 ( b ) ] Λ = . Then there exists a number R 1 > 0 such that

Φ λ ( u ) 0 , λ Λ , u P C [ 0 , 1 ] : u R 1 .

Proof Suppose on the contrary that there exists { ( μ n , u n ) } Λ × P C [ 0 , 1 ] with u n ( n ), such that Φ μ n ( u n ) = 0 . We may assume μ n μ ¯ Λ . By Remark 2.2, u n > 0 in [ 0 , 1 ] . Set v n = u n 1 u n . Then

v n = A μ n ( u n ) u n .

From (H2), (H3), we know that u n 1 A μ n ( u n ) is bounded in P C [ 0 , 1 ] , so { v n } is a relatively compact set in P C [ 0 , 1 ] since A μ n : P C [ 0 , 1 ] P C 1 [ 0 , 1 ] is bounded and continuous and P C 1 [ 0 , 1 ] P C [ 0 , 1 ] . Suppose v n v ¯ in P C [ 0 , 1 ] . Then v ¯ = 1 and v ¯ 0 in [ 0 , 1 ] .

Now, from condition (H2), we know that there exist ρ k C ( [ 0 , ) , [ 0 , ) ) , such that

I k ( u ) = I k ( ) u + ρ k ( ) ( u ) and lim | u | ρ k ( u ) u = 0 .

From (H3), we have that

b ( t ) u n ξ 1 ( t , u n ) f ( t , u n ) b ( t ) u n + ξ 2 ( t , u n ) .

So,

u n μ n 0 1 G ( t , s ) b ( s ) u n ( s ) d s + μ n k = 1 p G ( t , t k ) I k ( ) u n ( t k ) + μ n 0 1 G ( t , s ) ξ 2 ( s , u n ( s ) ) d s + μ n k = 1 p G ( t , t k ) ρ k ( u n ( t k ) ) ,

and

μ n 0 1 G ( t , s ) b ( s ) u n ( s ) d s + μ n k = 1 p G ( t , t k ) I k ( ) u n ( t k ) μ n 0 1 G ( t , s ) ξ 1 ( s , u n ( s ) ) d s + μ n k = 1 p G ( t , t k ) ρ k ( u n ( t k ) ) u n ,

accordingly, we have

v n μ n 0 1 G ( t , s ) b ( s ) v n ( s ) d s + μ n k = 1 p G ( t , t k ) I k ( ) v n ( t k ) + μ n 0 1 G ( t , s ) ξ 2 ( s , u n ( s ) ) u n ( s ) v n ( s ) d s + μ n k = 1 p G ( t , t k ) ρ k ( u n ( t k ) ) u n (3.1)

and

v n μ n 0 1 G ( t , s ) b ( s ) v n ( s ) d s + μ n k = 1 p G ( t , t k ) I k ( ) v n ( t k ) m u n 0 1 G ( t , s ) ξ 1 ( s , u n ( s ) ) u n ( s ) v n ( s ) d s + μ n k = 1 p G ( t , t k ) ρ k ( u n ( t k ) ) u n . (3.2)

Let φ and φ denote the nonnegative eigenfunctions of L b , L b corresponding to λ 1 ( b ) , and λ 1 ( b ) , respectively. Then we have from the (3.1) that

v n , φ μ n L b v n , φ .

Letting n , we have

v ¯ , φ μ ¯ L b v ¯ , φ ,

we obtain that

v ¯ , φ μ ¯ L b v ¯ , φ = μ ¯ v ¯ , L b φ = μ ¯ v ¯ , 1 λ 1 ( b ) φ = μ ¯ 1 λ 1 ( b ) v ¯ , φ ,

and consequently

μ ¯ λ 1 ( b ) .

Similarly, we deduce from (3.2) that

μ ¯ λ 1 ( b ) .

Thus, λ 1 ( b ) μ ¯ λ 1 ( b ) . This contradicts μ ¯ Λ . □

Corollary 3.1For μ ( 0 , λ 1 ( b ) ) and R R 1 . Then deg ( ϕ μ , B R , 0 ) = 1 .

Proof Lemma 3.1, applied to the interval Λ = [ 0 , μ ] , guarantees the existence of R 1 > 0 such that for R R 1 ,

u τ A μ ( u ) 0 , u P C [ 0 , 1 ] : u R , τ [ 0 , 1 ] .

Hence, for any R R 1 ,

deg ( ϕ μ , B R , 0 ) = deg ( I , B R , 0 ) = 1 ,

which implies the assertion. □

On the other hand, we have

Lemma 3.2Suppose λ > λ 1 ( b ) . Then there exists R 2 > 0 with the property that u P C [ 0 , 1 ] with u R 2 , τ 0 ,

Φ λ ( u ) τ φ ,

where φ is the nonnegative eigenfunction of L b corresponding to λ 1 ( b ) .

Proof Let us assume that for some sequence { u n } in P C [ 0 , 1 ] with u n and numbers τ n 0 , such that Φ λ ( u n ) = τ n φ . Then

u n = A λ ( u n ) + τ n φ ,

and we conclude from Remark 2.2 that u n > 0 in [ 0 , 1 ] . So we have

u n , φ = A λ ( u n ) + τ n φ , φ = A λ ( u n ) , φ + τ n φ , φ .

Choose σ > 0 such that

σ < λ λ 1 ( b ) λ . (3.3)

By (H3), there exists M 0 > 0 , such that

f ( t , u ) ( 1 σ ) b ( t ) u , u > M 0 , t [ 0 , 1 ] .

From u n , then exists N > 0 , such that

u n > M 0 , t [ 0 , 1 ] , n N ,

and consequently

f ( t , u n ) ( 1 σ ) b ( t ) u n , n N , t [ 0 , 1 ] . (3.4)

Let v n = u n u n , applying (3.4), it follows that

v n , φ A λ ( u n ) u n , φ λ ( 1 σ ) L b v n , φ = λ ( 1 σ ) v n , L b φ = λ ( 1 σ ) v n , 1 λ 1 ( b ) φ .

Thus,

λ 1 ( b ) λ ( 1 σ ) ,

this contradicts (3.3). □

Corollary 3.2For λ > λ 1 ( b ) and R R 2 , deg ( ϕ λ , B R , 0 ) = 0 .

Proof By Lemma 3.2, there exists R 2 > 0 such that

Φ λ ( u ) τ φ , u P C [ 0 , 1 ] : u R 2 , τ [ 0 , 1 ] .

Then

deg ( Φ λ , B R , 0 ) = deg ( Φ λ φ , B R , 0 ) = 0

for all R R 2 . The assertion follows. □

We are now ready to prove

Proposition 3.1 [ λ 1 ( b ) , λ 1 ( b ) ] is a bifurcation interval of positive solutions from infinity for the problem (2.4). There exists an unbounded component Σ of positive solutions of (2.4) which meets [ λ 1 ( b ) , λ 1 ( b ) ] × { } , and is unbounded inλdirection. Moreover, there exists no bifurcation interval of positive solutions from infinity which is disjointed with [ λ 1 ( b ) , λ 1 ( b ) ] .

Proof For fixed n N with λ 1 ( b ) 1 n > 0 , let us take that a n = λ 1 ( b ) 1 n , b n = λ 1 ( b ) + 1 n and R ˆ = max { R 1 , R 2 } . It is easy to check that for R > R ˆ , all of the conditions of Theorem D are satisfied. So, there exists a closed connected set C n of solutions of (2.4) that is unbounded in [ a n , b n ] × P C [ 0 , 1 ] , and either

(i) C n is unbounded in λ direction, or else

(ii) [ c , d ] such that ( a n , b n ) ( c , d ) = and C n bifurcates from infinity in [ c , d ] × P C [ 0 , 1 ] .

By Lemma 3.1, the case (ii) cannot occur. Thus, C n bifurcates from infinity in [ a n , b n ] × P C [ 0 , 1 ] and is unbounded in λ direction. Furthermore, we have from Lemma 3.1 that for any closed interval I [ a n , b n ] [ λ 1 ( b ) , λ 1 ( b ) ] , the set { u P C [ 0 , 1 ] | ( λ , u ) Σ , λ I } is bounded in P C [ 0 , 1 ] . So, C n must be bifurcated from infinity in [ λ 1 ( b ) , λ 1 ( b ) ] × P C [ 0 , 1 ] 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 Z ( ) C ( [ 0 , 1 ] , [ 0 , ) ) and Z ( ) 0 in any subinterval of [ 0 , 1 ] . Further define the linear operator L ˜ Z : P C [ 0 , 1 ] P C [ 0 , 1 ] ,

L ˜ Z u = 0 1 G ( t , s ) Z ( s ) u ( s ) d s + k = 1 p G ( t , t k ) I k ( 0 ) u ( t k ) , (4.1)

where I k ( 0 ) is defined in (H2), G ( t , s ) is defined in (2.2).

Similar as Lemma 2.2, we have the following lemma.

Lemma 4.1Suppose that (H1) holds, then the operator L ˜ Z has a unique characteristic value λ ˜ 1 ( Z ) , which is positive, real, simple, and the corresponding eigenfunction φ ˜ ( t ) is of one sign, i.e., we have φ ˜ 1 ( t ) = λ ˜ 1 ( Z ) L ˜ Z φ ˜ 1 ( t ) .

Remark 4.1 Since λ ˜ 1 ( Z ) is real number, so from Lemma 2.1, λ ˜ 1 ( Z ) is also the characteristic value of L ˜ Z , where L ˜ Z denote the conjugate operator of L ˜ Z , let φ ˜ 1 denote the nonnegative eigenfunction of L ˜ Z corresponding to λ ˜ 1 ( Z ) . Therefore, we have

φ ˜ 1 ( t ) = λ ˜ 1 ( Z ) L ˜ Z φ ˜ 1 ( t ) , t [ 0 , 1 ] .

Lemma 4.2Let Λ R + be a compact interval with [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] Λ = . Then there exists a number δ 1 > 0 such that

Φ λ ( u ) 0 , λ Λ , u P C [ 0 , 1 ] : 0 < u δ 1 .

Proof Suppose on the contrary that there exists { ( μ n , u n ) } Λ × P C [ 0 , 1 ] with u n 0 ( n ), such that Φ μ n ( u n ) = 0 . We may assume μ n μ ¯ Λ . By Remark 2.2, u n > 0 in [ 0 , 1 ] . Set v n = u n 1 u n . Then

v n = A μ n ( u n ) u n .

From (H2), (H3), we know that u n 1 A μ n ( u n ) is bounded in P C [ 0 , 1 ] , so we infer that v n is a relatively compact set in P C [ 0 , 1 ] , hence (for a subsequence) v n v ¯ with v ¯ 0 in P C [ 0 , 1 ] , v ¯ = 1 .

Now, from condition (H2), we know that there exist ρ k 0 C ( [ 0 , ) , [ 0 , ) ) , such that

I k ( u ) = I k ( 0 ) u + ρ k 0 ( u ) and lim u 0 + ρ k 0 ( u ) u = 0 .

From (H3), we have that

a 0 ( t ) u n ζ 1 ( t , u n ) f ( t , u n ) a 0 ( t ) u n + ζ 2 ( t , u n ) .

So,

u n μ n 0 1 G ( t , s ) a 0 ( s ) u n ( s ) d s + μ n k = 1 p G ( t , t k ) I k ( 0 ) u n ( t k ) + μ n 0 1 G ( t , s ) ζ 2 ( s , u n ( s ) ) d s + μ n k = 1 p G ( t , t k ) ρ k 0 ( u n ( t k ) ) ,

and

u n μ n 0 1 G ( t , s ) a 0 ( s ) u n ( s ) d s + μ n k = 1 p G ( t , t k ) I k ( 0 ) u n ( t k ) μ n 0 1 G ( t , s ) ζ 1 ( s , u n ( s ) ) d s + μ n k = 1 p G ( t , t k ) ρ k 0 ( u n ( t k ) ) ,

accordingly, we have

v n μ n 0 1 G ( t , s ) a 0 ( s ) v n ( s ) d s + μ n k = 1 p G ( t , t k ) I k ( 0 ) v n ( t k ) , + μ n 0 1 G ( t , s ) ζ 2 ( s , u n ( s ) ) u n ( s ) v n ( s ) d s + μ n k = 1 p G ( t , t k ) ρ k 0 ( u n ( t k ) ) u n (4.2)

and

v n μ n 0 1 G ( t , s ) a 0 ( s ) v n ( s ) d s + μ n k = 1 p G ( t , t k ) I k ( 0 ) v n ( t k ) μ n 0 1 G ( t , s ) ζ 1 ( s , u n ( s ) ) u n ( s ) v n ( s ) d s + μ n k = 1 p G ( t , t k ) ρ k 0 ( u n ( t k ) ) u n . (4.3)

Let φ ˜ 0 and φ ˜ 0 denote the nonnegative eigenfunctions of L ˜ a 0 , L ˜ a 0 corresponding to λ ˜ 1 ( a 0 ) , and λ ˜ 1 ( a 0 ) , respectively. Then we have from the (4.2) that

v n , φ ˜ 0 μ n L ˜ a 0 v n , φ ˜ 0 .

Letting n , we have

v ¯ , φ ˜ 0 μ ¯ L ˜ a 0 v ¯ , φ ˜ 0 ,

we obtain that

v ¯ , φ ˜ 0 μ ¯ L ˜ a 0 v ¯ , φ ˜ 0 = μ ¯ v ¯ , L ˜ a 0 φ ˜ 0 = μ ¯ v ¯ , 1 λ ˜ 1 ( a 0 ) φ ˜ 0 = μ ¯ 1 λ ˜ 1 ( a 0 ) v ¯ , φ ˜ 0 ,

and consequently

μ ¯ λ ˜ 1 ( a 0 ) .

Similarly, we deduce from (4.3) that

μ ¯ λ ˜ 1 ( a 0 ) .

Thus, λ ˜ 1 ( a 0 ) μ ¯ λ ˜ 1 ( a 0 ) . This contradicts μ ¯ Λ . □

Corollary 4.1For μ ( 0 , λ ˜ 1 ( a 0 ) ) and δ ( 0 , δ 1 ) . Then deg ( Φ μ , B δ , 0 ) = 1 .

On the other hand, we have

Lemma 4.3Suppose λ > λ ˜ 1 ( a 0 ) . Then there exists δ 2 > 0 with the property that u P C [ 0 , 1 ] with 0 < u δ 2 , τ 0 ,

Φ λ ( u ) τ φ ˜ 0 ,

where φ ˜ 0 is the nonnegative eigenfunction of the L ˜ a 0 corresponding to λ ˜ 1 ( a 0 ) .

Proof We assume again on the contrary that there exists τ n 0 and a sequence u n with u n > 0 and u n 0 in P C [ 0 , 1 ] , such that Φ λ ( u n ) = τ n φ ˜ 0 for all n N .

Then

u n = A λ ( u n ) + τ n φ ˜ 0 ,

and we conclude from Remark 2.2 that u n > 0 in [ 0 , 1 ] . So, we have

u n , φ ˜ 0 = A λ ( u n ) + τ n φ ˜ 0 , φ ˜ 0 = A λ ( u n ) , φ ˜ 0 + τ n φ ˜ 0 , φ ˜ 0 .

Choose σ > 0 such that

σ < λ λ ˜ 1 ( a 0 ) λ . (4.4)

By (H3), there exists r > 0 , such that

f ( t , u ) ( 1 σ ) a 0 ( t ) u , u [ 0 , r ] , t [ 0 , 1 ] .

From u n 0 , then exists N > 0 , such that

0 u n r , n N ,

and consequently

f ( t , u n ) ( 1 σ ) a 0 ( t ) u n , n N . (4.5)

Let v n = u n u , applying (4.5), it follows that

v n , φ ˜ 0 A λ ( u n ) u n , φ ˜ 0 λ ( 1 σ ) L ˜ a 0 v n , φ ˜ 0 = λ ( 1 σ ) v n , L ˜ a 0 φ ˜ 0 = λ ( 1 σ ) v n , 1 λ ˜ 1 ( a 0 ) φ ˜ 0 .

Thus,

λ ˜ 1 ( a 0 ) λ ( 1 σ ) ,

this contradicts with (4.4). □

Corollary 4.2For λ > λ ˜ 1 ( a 0 ) and δ ( 0 , δ 2 ) . Then deg ( Φ λ , B δ , 0 ) = 0 .

Proof By Lemma 4.3, there exists δ 2 > 0 such that

Φ λ ( u ) τ φ ˜ 0 , u P C [ 0 , 1 ] : 0 < u δ 2 , τ [ 0 , 1 ] .

Then

deg ( Φ λ , B δ , 0 ) = deg ( Φ λ φ ˜ 0 , B δ , 0 ) = 0

for all δ ( 0 , δ 2 ) . 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 [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] is a bifurcation interval of positive solutions from the trivial solution for the problem (2.4). There exists an unbounded component Σ 0 of positive solutions of (2.4) which meets [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] × { 0 } . Moreover, there exists no bifurcation interval of positive solutions from the trivial solution which is disjointed with [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] .

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 m k : = min { I k ( u ) u } , k = 1 , , p for u 0 . From (H2), we have m k > 0 , k = 1 , , p .

Define the linear operator T c : P C [ 0 , 1 ] P C [ 0 , 1 ] ,

T c u = 0 1 G ( t , s ) c ( s ) u ( s ) d s + k = 1 p G ( t , t k ) m k u ( t k ) , (5.1)

where c ( ) is defined in (H5), G ( t , s ) is defined in (2.2).

Similar as Lemma 2.2, we have the following lemma.

Lemma 5.1The operator T c has a unique characteristic value μ 1 , which is positive, real, simple, and the corresponding eigenfunction Φ c ( t ) is of one sign, i.e., we have Φ c ( t ) = μ 1 T c Φ c ( t ) .

Remark 5.1 Since μ 1 is real number, so from Lemma 2.1, μ 1 is also the characteristic value of T c , where T c denote the conjugate operator of T c , let Φ c denote the nonnegative eigenfunction of T c corresponding to μ 1 . Therefore, we have

Φ c ( t ) = μ 1 T c Φ c ( t ) , t [ 0 , 1 ] .

Lemma 5.2Let (H1)-(H5) hold. Then there exists a number λ > 0 such that there is no positive solution ( λ , u ) of Φ λ ( u ) = 0 with λ > λ .

Proof Let ( λ , u ) be a positive solution of Φ λ ( u ) = 0 . Then

u = λ 0 1 G ( t , s ) f ( s , u ( s ) ) d s + λ k = 1 p I k ( u ( t k ) ) , u P C [ 0 , 1 ] .

From (H5) and the definition of m k , we have

u λ 0 1 G ( t , s ) c ( s ) u ( s ) d s + λ k = 1 p m k u ( t k ) , u P C [ 0 , 1 ] . (5.2)

From (5.2), we have

u , Φ c λ T c u , Φ c = λ u , T c Φ c = λ u , 1 μ 1 Φ c = λ 1 μ 1 u , Φ c .

Thus,

λ μ 1 : = λ .

 □

The assertion that Σ 0 = Σ in Theorem 1.1(iii) now easily follows. For, in the case, Σ 0 and Σ are contained in ( 0 , λ ] × P C [ 0 , 1 ] . Moreover, there exists no bifurcation interval of positive solution from infinity which is disjointed with [ λ 1 ( b ) , λ 1 ( b ) ] , there exists no bifurcation interval of positive solution from the trivial solution which is disjointed with [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] . In Theorem 1.1(iii), the unbounded component Σ 0 has to meet [ λ 1 ( b ) , λ 1 ( b ) ] × { } .

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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