Open Access

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

Boundary Value Problems20122012:83

DOI: 10.1186/1687-2770-2012-83

Received: 18 May 2012

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:

{ 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equa_HTML.gif

which is not necessarily linearizable. Where λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq1_HTML.gif is a parameter, 0 < t 1 < t 2 < < t p < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq2_HTML.gif 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
u ( t ) + a ( t ) u ( t ) = λ f ( t , u ( t ) ) , t J , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ1_HTML.gif
(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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ2_HTML.gif
(1.2)

where λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq1_HTML.gif is a real parameter, J = [ 0 , 1 ] { t 1 , , t p } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq3_HTML.gif, 0 < t 1 < t 2 < < t p < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq2_HTML.gif are given impulsive points. We make the following assumptions:

(H1) a C ( [ 0 , 1 ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq4_HTML.gif is a 1-periodic function and 0 1 a ( t ) d t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq5_HTML.gif;

(H2) I k C ( [ 0 , ) , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq6_HTML.gif, k = 1 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq7_HTML.gif, I k ( u ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq8_HTML.gif for u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq9_HTML.gif, there exist positive constants I k ( 0 ) , I k ( ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq10_HTML.gif such that
I k ( 0 ) = lim u 0 + I k ( u ) u , I k ( ) = lim u + I k ( u ) u ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equb_HTML.gif
(H3) f C ( J × [ 0 , ) , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq11_HTML.gif is 1-periodic function with respect to the first variable, and f ( t k + , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq12_HTML.gif, f ( t k , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq13_HTML.gif exist, f ( t k , u ) = f ( t k , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq14_HTML.gif. Moreover, there exist functions a 0 , a 0 , b , b C ( [ 0 , 1 ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq15_HTML.gif with a 0 ( t ) , a 0 ( t ) , b ( t ) , b ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq16_HTML.gif in any subinterval of [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif such that
a 0 ( t ) u ξ 1 ( t , u ) f ( t , u ) a 0 ( t ) u + ξ 2 ( t , u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equc_HTML.gif
where ξ i C ( [ 0 , 1 ] × [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq18_HTML.gif with ξ i ( t , u ) = o ( | u | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq19_HTML.gif as | u | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq20_HTML.gif uniformly for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq21_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq22_HTML.gif), and
b ( t ) u ζ 1 ( t , u ) f ( t , u ) b ( t ) u + ζ 2 ( t , u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equd_HTML.gif

where ζ i C ( [ 0 , 1 ] × [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq23_HTML.gif with ζ i ( t , u ) = o ( | u | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq24_HTML.gif as | u | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq25_HTML.gif uniformly for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq21_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq26_HTML.gif);

(H4) f ( t , u ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq27_HTML.gif, ( t , u ) [ 0 , 1 ] × ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq28_HTML.gif;

(H5) there exists function c C ( [ 0 , 1 ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq29_HTML.gif and c ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq30_HTML.gif in any subinterval of [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif such that
f ( t , u ) c ( t ) u , ( t , u ) [ 0 , 1 ] × [ 0 , ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Eque_HTML.gif

Some special cases of (1.1), (1.2) have been investigated. For example, Nieto [3] considered the (1.1), (1.2) with λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq31_HTML.gif, a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq32_HTML.gif. By using Schaeffer’s theorem, some sufficient conditions for existence of solutions of the IBVP (1.1), (1.2) with λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq33_HTML.gif, a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq32_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq33_HTML.gif, a m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq34_HTML.gif (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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq35_HTML.gif is bounded or f is bounded and I k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq35_HTML.gif satisfied at most linear growth.

Liu [7] studied the existence and multiplicity of (1.1), (1.2) with λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq33_HTML.gif, 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 f ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq36_HTML.gif, I k ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq37_HTML.gif, u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq38_HTML.gif, and
max t [ 0 , 1 ] { M 0 1 G ( t , s ) d s + W k = 1 p G ( t , t k ) } < 1 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ3_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ4_HTML.gif
(1.4)
Then the problem (1.1), (1.2) with λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq33_HTML.gif has at least one positive solution where G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq39_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ5_HTML.gif
(1.5)

Theorem B ([7], Theorem 3.1.2])

Let (H 1) hold. Assume that f ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq36_HTML.gif, I k ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq37_HTML.gif, u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq38_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ6_HTML.gif
(1.6)
and
max t [ 0 , 1 ] { V 0 1 G ( t , s ) d s + w k = 1 p G ( t , t k ) } < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ7_HTML.gif
(1.7)
Then the problem (1.1), (1.2) with λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq33_HTML.gif has at least one positive solution where W, w defined 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ8_HTML.gif
(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 f ( t , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq40_HTML.gif, t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq41_HTML.gif, I k ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq42_HTML.gif, k = 1 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq7_HTML.gif. Then

(i) [ λ 1 ( b ) , λ 1 ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq43_HTML.gif 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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq43_HTML.gif. More precisely, there exists a component Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq44_HTML.gif of positive solutions of (1.1), (1.2) which meets [ λ 1 ( b ) , λ 1 ( b ) ] × { } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq45_HTML.gif, where λ 1 ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq46_HTML.gif, λ 1 ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq47_HTML.gif will be defined in Section  2;

(ii) [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq48_HTML.gif 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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq48_HTML.gif. More precisely, there exists a component Σ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq49_HTML.gif of positive solutions of (1.1), (1.2) which meets [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] × { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq50_HTML.gif, where λ ˜ 1 ( a 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq51_HTML.gif, λ ˜ 1 ( a 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq52_HTML.gif will be defined in Section  4;

(iii) If (H 4) and (H 5) also hold, then there is a number λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq53_HTML.gif such that problem (1.1), (1.2) admits no positive solution with λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq54_HTML.gif. In this case, Σ = Σ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq55_HTML.gif.

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 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq56_HTML.gif, i.e. f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq57_HTML.gif, f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq58_HTML.gif 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 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equf_HTML.gif

From (H3), it is easy to see that the f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq57_HTML.gif, f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq58_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq49_HTML.gif, Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq44_HTML.gif are involved in [ 0 , λ ] × PC [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq59_HTML.gif. Moreover, [ λ 1 ( b ) , λ 1 ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq43_HTML.gif is a unique bifurcation interval of positive solutions from infinity for (1.1), (1.2), and [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq48_HTML.gif is a unique bifurcation interval of positive solutions from the trivial solutions for (1.1), (1.2). Therefore, Σ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq49_HTML.gif must be intersected with [ λ 1 ( b ) , λ 1 ( b ) ] × { } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq45_HTML.gif.

Remark 1.3 Obviously, (H3) is more general than (1.5), (1.8). Moreover, if we let a 0 ( t ) : = v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq60_HTML.gif, b ( t ) : = M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq61_HTML.gif, under conditions (1.3), (1.4), we get λ 1 ( b ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq62_HTML.gif, λ ˜ 1 ( a 0 ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq63_HTML.gif, respectively. Hence, Σ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq49_HTML.gif cross the hyperplane { 1 } × PC [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq64_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq65_HTML.gif, b ( t ) : = m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq66_HTML.gif, only under condition (1.6), we can obtain λ 1 ( b ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq67_HTML.gif. From Proposition 3.1, we will know that Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq44_HTML.gif is unbounded in λ direction, so, Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq44_HTML.gif cross the hyperplane { 1 } × PC [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq68_HTML.gif. 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 F : R × V V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq69_HTML.gif be completely continuous such that F ( λ , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq70_HTML.gif, λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq71_HTML.gif. Let a , b R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq72_HTML.gif ( a < b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq73_HTML.gif) be such that u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq74_HTML.gif is an isolated solution of the equation
u F ( λ , u ) = 0 , u V , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ9_HTML.gif
(1.9)
for λ = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq75_HTML.gif and λ = b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq76_HTML.gif, where ( a , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq77_HTML.gif, ( b , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq78_HTML.gif 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equg_HTML.gif
where B r ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq79_HTML.gif 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 } ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equh_HTML.gif
Then there exists a connected component C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq80_HTML.gif of containing [ a , b ] × { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq81_HTML.gif in R × V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq82_HTML.gif, and either
  1. (i)

    C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq80_HTML.gif is unbounded in R × V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq82_HTML.gif, or

     
  2. (ii)

    C [ ( R [ a , b ] ) × { 0 } ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq83_HTML.gif.

     

Theorem D (K. Schmitt [13])

Let V be a real reflexive Banach space. Let F : R × V V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq69_HTML.gif be completely continuous, and let a , b R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq72_HTML.gif ( a < b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq73_HTML.gif) be such that the solution of (1.9) are, a priori, bounded in V for λ = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq75_HTML.gif and λ = b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq76_HTML.gif, i.e., there exists an R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq84_HTML.gif such that
F ( a , u ) u F ( b , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equi_HTML.gif
for all u with u R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq85_HTML.gif. Furthermore, assume that
deg ( I F ( a , ) , B R ( 0 ) , 0 ) deg ( I F ( b , ) , B R ( 0 ) , 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equj_HTML.gif
for R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq84_HTML.gif large. Then there exists a closed connected set C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq80_HTML.gif of solutions of (1.9) that is unbounded in [ a , b ] × V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq86_HTML.gif, and either
  1. (i)

    C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq80_HTML.gif is unbounded in λ direction, or

     
  2. (ii)

    there exist an interval [ c , d ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq87_HTML.gif such that ( a , b ) ( c , d ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq88_HTML.gif, and C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq80_HTML.gif bifurcates from infinity in [ c , d ] × V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq89_HTML.gif.

     

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 , . } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equk_HTML.gif

Then P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif is a Banach space with the norm u = sup t [ 0 , 1 ] | u ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq91_HTML.gif.

By a positive solution of the problem (1.1), (1.2), we mean a pair ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq92_HTML.gif, where λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq1_HTML.gif and u is a solution of (1.1), (1.2) with u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq9_HTML.gif. Let Σ R + × P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq93_HTML.gif be the closure of the set of positive solutions of (1.1), (1.2), where R + : = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq94_HTML.gif.

Lemma 2.1 ([14], Theorem 6.26])

The spectrum σ ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq95_HTML.gif of compact linear operator T has following properties:
  1. (i)

    σ ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq95_HTML.gif is a countable set with no accumulation point which is different from zero;

     
  2. (ii)

    each nonzero λ σ ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq96_HTML.gif is an eigenvalue of T with finite multiplicity, and λ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq97_HTML.gif is an eigenvalue of T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq98_HTML.gif with the same multiplicity, where λ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq97_HTML.gif denote the conjugate of λ, T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq98_HTML.gif denote the conjugate operator of T.

     

Let H : = L 2 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq99_HTML.gif, with inner product , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq100_HTML.gif and norm L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq101_HTML.gif.

Let Z ( ) C ( [ 0 , 1 ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq102_HTML.gif and Z ( ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq103_HTML.gif in any subinterval of [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif. Further define the linear operator L Z : P C [ 0 , 1 ] P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq104_HTML.gif,
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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ10_HTML.gif
(2.1)
where I k ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq105_HTML.gif as defined in (H2), G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq39_HTML.gif is the Green’s function of
{ u ( t ) + a ( t ) u ( t ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equl_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ11_HTML.gif
(2.2)

where A ( t ) = 0 t a ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq106_HTML.gif, it is easy to see that (H1) implies that G ( t , s ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq107_HTML.gif.

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 L Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq108_HTML.gif defined by (2.1), has a unique characteristic value λ 1 ( Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq109_HTML.gif, which is positive, real, simple, and the corresponding eigenfunction φ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq110_HTML.gif is of one sign, i.e., we have φ ( t ) = λ 1 ( Z ) L Z φ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq111_HTML.gif.

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

Remark 2.1 Since λ 1 ( Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq109_HTML.gif is real number, so from Lemma 2.1, λ 1 ( Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq109_HTML.gif is also the characteristic value of L Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq112_HTML.gif, let φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq113_HTML.gif denote the nonnegative eigenfunction of L Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq112_HTML.gif corresponding to λ 1 ( Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq109_HTML.gif, where L Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq112_HTML.gif denote the conjugate operator of L Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq108_HTML.gif. Therefore, we have
φ 1 ( t ) = λ 1 ( Z ) L Z φ 1 ( t ) , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equm_HTML.gif
We extend the function f to function f ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq114_HTML.gif, defined on [ 0 , 1 ] × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq115_HTML.gif by
f ¯ ( t , u ) = { f ( t , u ) , ( t , u ) [ 0 , 1 ] × [ 0 , ) , f ( t , 0 ) , ( t , u ) [ 0 , 1 ] × ( , 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equn_HTML.gif

Then f ¯ ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq116_HTML.gif on [ 0 , 1 ] × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq115_HTML.gif.

For λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq117_HTML.gif, 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ12_HTML.gif
(2.3)
is equivalent to the operator equation A λ : P C [ 0 , 1 ] P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq118_HTML.gif.
( 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 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equo_HTML.gif

Remark 2.2 For λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq1_HTML.gif, if u is a nontrivial solution of (2.3), from the positivity of G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq39_HTML.gif and f ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq114_HTML.gif, we have that u ( ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq119_HTML.gif on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif, so u is a nontrivial solution of (1.1), (1.2). Therefore, the closure of the set of nontrivial solutions ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq92_HTML.gif of (2.3) in R + × P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq120_HTML.gif is exactly Σ.

The problem (2.3) is now equivalent to the operator equation
u = A λ ( u ) , u P C [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ13_HTML.gif
(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 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq121_HTML.gif,
Φ λ ( u ) = u A λ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equp_HTML.gif

For R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq84_HTML.gif, let B R = { u P C [ 0 , 1 ] : u < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq122_HTML.gif, let deg ( Φ λ , B R , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq123_HTML.gif denote the degree of Φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq124_HTML.gif on B R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq125_HTML.gif with respect to 0.

3 Bifurcation from infinity

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

Lemma 3.1 Let Λ R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq126_HTML.gif be a compact interval with [ λ 1 ( b ) , λ 1 ( b ) ] Λ = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq127_HTML.gif. Then there exists a number R 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq128_HTML.gif such that
Φ λ ( u ) 0 , λ Λ , u P C [ 0 , 1 ] : u R 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equq_HTML.gif
Proof Suppose on the contrary that there exists { ( μ n , u n ) } Λ × P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq129_HTML.gif with u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq130_HTML.gif ( n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq131_HTML.gif), such that Φ μ n ( u n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq132_HTML.gif. We may assume μ n μ ¯ Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq133_HTML.gif. By Remark 2.2, u n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq134_HTML.gif in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif. Set v n = u n 1 u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq135_HTML.gif. Then
v n = A μ n ( u n ) u n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equr_HTML.gif

From (H2), (H3), we know that u n 1 A μ n ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq136_HTML.gif is bounded in P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif, so { v n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq137_HTML.gif is a relatively compact set in P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif since A μ n : P C [ 0 , 1 ] P C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq138_HTML.gif is bounded and continuous and P C 1 [ 0 , 1 ] P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq139_HTML.gif. Suppose v n v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq140_HTML.gif in P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif. Then v ¯ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq141_HTML.gif and v ¯ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq142_HTML.gif in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif.

Now, from condition (H2), we know that there exist ρ k C ( [ 0 , ) , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq143_HTML.gif, such that
I k ( u ) = I k ( ) u + ρ k ( ) ( u ) and lim | u | ρ k ( u ) u = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equs_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equt_HTML.gif
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 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equu_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equv_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ14_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ15_HTML.gif
(3.2)
Let φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq144_HTML.gif and φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq145_HTML.gif denote the nonnegative eigenfunctions of L b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq146_HTML.gif, L b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq147_HTML.gif corresponding to λ 1 ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq46_HTML.gif, and λ 1 ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq47_HTML.gif, respectively. Then we have from the (3.1) that
v n , φ μ n L b v n , φ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equw_HTML.gif
Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq131_HTML.gif, we have
v ¯ , φ μ ¯ L b v ¯ , φ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equx_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equy_HTML.gif
and consequently
μ ¯ λ 1 ( b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equz_HTML.gif
Similarly, we deduce from (3.2) that
μ ¯ λ 1 ( b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equaa_HTML.gif

Thus, λ 1 ( b ) μ ¯ λ 1 ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq148_HTML.gif. This contradicts μ ¯ Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq149_HTML.gif. □

Corollary 3.1 For μ ( 0 , λ 1 ( b ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq150_HTML.gif and R R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq151_HTML.gif. Then deg ( ϕ μ , B R , 0 ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq152_HTML.gif.

Proof Lemma 3.1, applied to the interval Λ = [ 0 , μ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq153_HTML.gif, guarantees the existence of R 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq128_HTML.gif such that for R R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq151_HTML.gif,
u τ A μ ( u ) 0 , u P C [ 0 , 1 ] : u R , τ [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equab_HTML.gif
Hence, for any R R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq151_HTML.gif,
deg ( ϕ μ , B R , 0 ) = deg ( I , B R , 0 ) = 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equac_HTML.gif

which implies the assertion. □

On the other hand, we have

Lemma 3.2 Suppose λ > λ 1 ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq154_HTML.gif. Then there exists R 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq155_HTML.gif with the property that u P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq156_HTML.gif with u R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq157_HTML.gif, τ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq158_HTML.gif,
Φ λ ( u ) τ φ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equad_HTML.gif

where φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq159_HTML.gif is the nonnegative eigenfunction of L b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq160_HTML.gif corresponding to λ 1 ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq47_HTML.gif.

Proof Let us assume that for some sequence { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq161_HTML.gif in P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif with u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq130_HTML.gif and numbers τ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq162_HTML.gif, such that Φ λ ( u n ) = τ n φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq163_HTML.gif. Then
u n = A λ ( u n ) + τ n φ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equae_HTML.gif
and we conclude from Remark 2.2 that u n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq134_HTML.gif in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif. So we have
u n , φ = A λ ( u n ) + τ n φ , φ = A λ ( u n ) , φ + τ n φ , φ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equaf_HTML.gif
Choose σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq164_HTML.gif such that
σ < λ λ 1 ( b ) λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ16_HTML.gif
(3.3)
By (H3), there exists M 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq165_HTML.gif, such that
f ( t , u ) ( 1 σ ) b ( t ) u , u > M 0 , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equag_HTML.gif
From u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq130_HTML.gif, then exists N > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq166_HTML.gif, such that
u n > M 0 , t [ 0 , 1 ] , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equah_HTML.gif
and consequently
f ( t , u n ) ( 1 σ ) b ( t ) u n , n N , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ17_HTML.gif
(3.4)
Let v n = u n u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq167_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equai_HTML.gif
Thus,
λ 1 ( b ) λ ( 1 σ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equaj_HTML.gif

this contradicts (3.3). □

Corollary 3.2 For λ > λ 1 ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq154_HTML.gif and R R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq168_HTML.gif, deg ( ϕ λ , B R , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq169_HTML.gif.

Proof By Lemma 3.2, there exists R 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq155_HTML.gif such that
Φ λ ( u ) τ φ , u P C [ 0 , 1 ] : u R 2 , τ [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equak_HTML.gif
Then
deg ( Φ λ , B R , 0 ) = deg ( Φ λ φ , B R , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equal_HTML.gif

for all R R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq168_HTML.gif. The assertion follows. □

We are now ready to prove

Proposition 3.1 [ λ 1 ( b ) , λ 1 ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq170_HTML.gif is a bifurcation interval of positive solutions from infinity for the problem (2.4). There exists an unbounded component Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq44_HTML.gif of positive solutions of (2.4) which meets [ λ 1 ( b ) , λ 1 ( b ) ] × { } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq171_HTML.gif, and is unbounded in λ direction. Moreover, there exists no bifurcation interval of positive solutions from infinity which is disjointed with [ λ 1 ( b ) , λ 1 ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq172_HTML.gif.

Proof For fixed n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq173_HTML.gif with λ 1 ( b ) 1 n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq174_HTML.gif, let us take that a n = λ 1 ( b ) 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq175_HTML.gif, b n = λ 1 ( b ) + 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq176_HTML.gif and R ˆ = max { R 1 , R 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq177_HTML.gif. It is easy to check that for R > R ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq178_HTML.gif, all of the conditions of Theorem D are satisfied. So, there exists a closed connected set C n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq179_HTML.gif of solutions of (2.4) that is unbounded in [ a n , b n ] × P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq180_HTML.gif, and either
  1. (i)

    C n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq179_HTML.gif is unbounded in λ direction, or else

     
  2. (ii)

    [ c , d ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq181_HTML.gif such that ( a n , b n ) ( c , d ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq182_HTML.gif and C n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq179_HTML.gif bifurcates from infinity in [ c , d ] × P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq183_HTML.gif.

     

By Lemma 3.1, the case (ii) cannot occur. Thus, C n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq179_HTML.gif bifurcates from infinity in [ a n , b n ] × P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq180_HTML.gif 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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq184_HTML.gif, the set { u P C [ 0 , 1 ] | ( λ , u ) Σ , λ I } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq185_HTML.gif is bounded in P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif. So, C n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq179_HTML.gif must be bifurcated from infinity in [ λ 1 ( b ) , λ 1 ( b ) ] × P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq186_HTML.gif 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 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq102_HTML.gif and Z ( ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq103_HTML.gif in any subinterval of [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif. Further define the linear operator L ˜ Z : P C [ 0 , 1 ] P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq187_HTML.gif,
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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ18_HTML.gif
(4.1)

where I k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq188_HTML.gif is defined in (H2), G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq39_HTML.gif 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 L ˜ Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq189_HTML.gif has a unique characteristic value λ ˜ 1 ( Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq190_HTML.gif, which is positive, real, simple, and the corresponding eigenfunction φ ˜ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq191_HTML.gif is of one sign, i.e., we have φ ˜ 1 ( t ) = λ ˜ 1 ( Z ) L ˜ Z φ ˜ 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq192_HTML.gif.

Remark 4.1 Since λ ˜ 1 ( Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq190_HTML.gif is real number, so from Lemma 2.1, λ ˜ 1 ( Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq190_HTML.gif is also the characteristic value of L ˜ Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq193_HTML.gif, where L ˜ Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq193_HTML.gif denote the conjugate operator of L ˜ Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq189_HTML.gif, let φ ˜ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq194_HTML.gif denote the nonnegative eigenfunction of L ˜ Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq193_HTML.gif corresponding to λ ˜ 1 ( Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq190_HTML.gif. Therefore, we have
φ ˜ 1 ( t ) = λ ˜ 1 ( Z ) L ˜ Z φ ˜ 1 ( t ) , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equam_HTML.gif
Lemma 4.2 Let Λ R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq126_HTML.gif be a compact interval with [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] Λ = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq195_HTML.gif. Then there exists a number δ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq196_HTML.gif such that
Φ λ ( u ) 0 , λ Λ , u P C [ 0 , 1 ] : 0 < u δ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equan_HTML.gif
Proof Suppose on the contrary that there exists { ( μ n , u n ) } Λ × P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq129_HTML.gif with u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq197_HTML.gif ( n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq131_HTML.gif), such that Φ μ n ( u n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq132_HTML.gif. We may assume μ n μ ¯ Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq133_HTML.gif. By Remark 2.2, u n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq134_HTML.gif in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif. Set v n = u n 1 u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq135_HTML.gif. Then
v n = A μ n ( u n ) u n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equao_HTML.gif

From (H2), (H3), we know that u n 1 A μ n ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq136_HTML.gif is bounded in P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif, so we infer that v n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq198_HTML.gif is a relatively compact set in P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif, hence (for a subsequence) v n v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq140_HTML.gif with v ¯ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq142_HTML.gif in P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif, v ¯ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq141_HTML.gif.

Now, from condition (H2), we know that there exist ρ k 0 C ( [ 0 , ) , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq199_HTML.gif, such that
I k ( u ) = I k ( 0 ) u + ρ k 0 ( u ) and lim u 0 + ρ k 0 ( u ) u = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equap_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equaq_HTML.gif
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 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equar_HTML.gif
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 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equas_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ19_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ20_HTML.gif
(4.3)
Let φ ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq200_HTML.gif and φ ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq201_HTML.gif denote the nonnegative eigenfunctions of L ˜ a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq202_HTML.gif, L ˜ a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq203_HTML.gif corresponding to λ ˜ 1 ( a 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq51_HTML.gif, and λ ˜ 1 ( a 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq52_HTML.gif, respectively. Then we have from the (4.2) that
v n , φ ˜ 0 μ n L ˜ a 0 v n , φ ˜ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equat_HTML.gif
Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq131_HTML.gif, we have
v ¯ , φ ˜ 0 μ ¯ L ˜ a 0 v ¯ , φ ˜ 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equau_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equav_HTML.gif
and consequently
μ ¯ λ ˜ 1 ( a 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equaw_HTML.gif
Similarly, we deduce from (4.3) that
μ ¯ λ ˜ 1 ( a 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equax_HTML.gif

Thus, λ ˜ 1 ( a 0 ) μ ¯ λ ˜ 1 ( a 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq204_HTML.gif. This contradicts μ ¯ Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq149_HTML.gif. □

Corollary 4.1 For μ ( 0 , λ ˜ 1 ( a 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq205_HTML.gif and δ ( 0 , δ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq206_HTML.gif. Then deg ( Φ μ , B δ , 0 ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq207_HTML.gif.

On the other hand, we have

Lemma 4.3 Suppose λ > λ ˜ 1 ( a 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq208_HTML.gif. Then there exists δ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq209_HTML.gif with the property that u P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq156_HTML.gif with 0 < u δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq210_HTML.gif, τ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq211_HTML.gif,
Φ λ ( u ) τ φ ˜ 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equay_HTML.gif

where φ ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq212_HTML.gif is the nonnegative eigenfunction of the L ˜ a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq213_HTML.gif corresponding to λ ˜ 1 ( a 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq52_HTML.gif.

Proof We assume again on the contrary that there exists τ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq162_HTML.gif and a sequence u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq214_HTML.gif with u n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq215_HTML.gif and u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq216_HTML.gif in P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq90_HTML.gif, such that Φ λ ( u n ) = τ n φ ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq217_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq173_HTML.gif.

Then
u n = A λ ( u n ) + τ n φ ˜ 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equaz_HTML.gif
and we conclude from Remark 2.2 that u n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq134_HTML.gif in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq17_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equba_HTML.gif
Choose σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq164_HTML.gif such that
σ < λ λ ˜ 1 ( a 0 ) λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ21_HTML.gif
(4.4)
By (H3), there exists r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq218_HTML.gif, such that
f ( t , u ) ( 1 σ ) a 0 ( t ) u , u [ 0 , r ] , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equbb_HTML.gif
From u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq197_HTML.gif, then exists N > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq166_HTML.gif, such that
0 u n r , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equbc_HTML.gif
and consequently
f ( t , u n ) ( 1 σ ) a 0 ( t ) u n , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ22_HTML.gif
(4.5)
Let v n = u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq219_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equbd_HTML.gif
Thus,
λ ˜ 1 ( a 0 ) λ ( 1 σ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Eqube_HTML.gif

this contradicts with (4.4). □

Corollary 4.2 For λ > λ ˜ 1 ( a 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq208_HTML.gif and δ ( 0 , δ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq220_HTML.gif. Then deg ( Φ λ , B δ , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq221_HTML.gif.

Proof By Lemma 4.3, there exists δ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq209_HTML.gif such that
Φ λ ( u ) τ φ ˜ 0 , u P C [ 0 , 1 ] : 0 < u δ 2 , τ [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equbf_HTML.gif
Then
deg ( Φ λ , B δ , 0 ) = deg ( Φ λ φ ˜ 0 , B δ , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equbg_HTML.gif

for all δ ( 0 , δ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq222_HTML.gif. 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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq223_HTML.gif is a bifurcation interval of positive solutions from the trivial solution for the problem (2.4). There exists an unbounded component Σ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq49_HTML.gif of positive solutions of (2.4) which meets [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] × { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq224_HTML.gif. 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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq225_HTML.gif.

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 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq226_HTML.gif, k = 1 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq7_HTML.gif for u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq227_HTML.gif. From (H2), we have m k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq228_HTML.gif, k = 1 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq7_HTML.gif.

Define the linear operator T c : P C [ 0 , 1 ] P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq229_HTML.gif,
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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ23_HTML.gif
(5.1)

where c ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq230_HTML.gif is defined in (H5), G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq39_HTML.gif is defined in (2.2).

Similar as Lemma 2.2, we have the following lemma.

Lemma 5.1 The operator T c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq231_HTML.gif has a unique characteristic value μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq232_HTML.gif, which is positive, real, simple, and the corresponding eigenfunction Φ c ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq233_HTML.gif is of one sign, i.e., we have Φ c ( t ) = μ 1 T c Φ c ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq234_HTML.gif.

Remark 5.1 Since μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq232_HTML.gif is real number, so from Lemma 2.1, μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq232_HTML.gif is also the characteristic value of T c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq235_HTML.gif, where T c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq235_HTML.gif denote the conjugate operator of T c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq231_HTML.gif, let Φ c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq236_HTML.gif denote the nonnegative eigenfunction of T c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq235_HTML.gif corresponding to μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq232_HTML.gif. Therefore, we have
Φ c ( t ) = μ 1 T c Φ c ( t ) , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equbh_HTML.gif

Lemma 5.2 Let (H 1)-(H 5) hold. Then there exists a number λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq53_HTML.gif such that there is no positive solution ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq92_HTML.gif of Φ λ ( u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq237_HTML.gif with λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq54_HTML.gif.

Proof Let ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq92_HTML.gif be a positive solution of Φ λ ( u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq237_HTML.gif. 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 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equbi_HTML.gif
From (H5) and the definition of m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq238_HTML.gif, 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 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equ24_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equbj_HTML.gif
Thus,
λ μ 1 : = λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_Equbk_HTML.gif

 □

The assertion that Σ 0 = Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq239_HTML.gif in Theorem 1.1(iii) now easily follows. For, in the case, Σ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq49_HTML.gif and Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq44_HTML.gif are contained in ( 0 , λ ] × P C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq240_HTML.gif. Moreover, there exists no bifurcation interval of positive solution from infinity which is disjointed with [ λ 1 ( b ) , λ 1 ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq241_HTML.gif, there exists no bifurcation interval of positive solution from the trivial solution which is disjointed with [ λ ˜ 1 ( a 0 ) , λ ˜ 1 ( a 0 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq242_HTML.gif. In Theorem 1.1(iii), the unbounded component Σ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq49_HTML.gif has to meet [ λ 1 ( b ) , λ 1 ( b ) ] × { } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-83/MediaObjects/13661_2012_Article_198_IEq243_HTML.gif.

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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