Open Access

Structure of positive solution sets of differential boundary value problems

Boundary Value Problems20132013:100

DOI: 10.1186/1687-2770-2013-100

Received: 26 September 2012

Accepted: 8 March 2013

Published: 22 April 2013

Abstract

In this paper, we first obtain some results on the structure of positive solution sets of differential boundary value problems. Then by using the results, we obtain an existence result for differential boundary value problems. The method used to show the main result is the global bifurcation theory.

Keywords

structure of positive solution sets differential boundary value problems bifurcation theory

1 Introduction

This paper considers the differential boundary value problem
{ ( p ( t ) ϕ ( u ) ) + λ p ( t ) f ( t , u ) = 0 , a < t < b , u ( a ) = u ( b ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ1_HTML.gif
(1.1)

where f is ϕ-superlinear at ∞ and f ( t , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq1_HTML.gif maybe negative and p is a positive continuous function, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq2_HTML.gif is a parameter.

Equations of form (1.1) occur in the study for the p-Laplacian equation, non-Newtonial fluid theory and the turbulent flow of a gas in a porous medium. The case where
α ( | u | 2 ) | u | = | u | p 2 | u | , p > 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equa_HTML.gif

i.e., perturbations of the p-Laplacian, has received much attention in the recent literature. Also, problem (1.1) with f ( t , 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq3_HTML.gif has been studied by several authors in recent years (see [1] and the references therein). Here, we are interested in the case when f ( t , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq4_HTML.gif may be negative (the so-called semipositone case) (see [2] and its references for a review). As pointed out by Lions in [3], semi-positone problems are mathematically very challenging. During the last ten years, finding positive solutions to semi-positone problems has been actively pursued and significant progress on semi-positone problems has taken place; see [48] and the references therein. For instance, Hai et al. [9] considered the existence positive solution of (1.1). Under some super-linear conditions on the non-linear term f, they proved that there exists λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq5_HTML.gif such that (1.1) has one positive solution for 0 < λ < λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq6_HTML.gif. The main method in [9] used to show the main result are the fixed-point theorems.

The main purpose of this paper is going to study the structure of the positive set of (1.1). Rabinowitz [10] gave the first important results on the structure of the solution sets of non-linear equations and obtained by the degree theoretic method. Amamn [11] studied the structure of the positive solution set of non-linear equations; the reader is referred to [12, 13] for other results concerning the structure of solution sets of non-linear equations. In our paper, we will study the existence results for an unbounded connected component of a positive solution set for the differential boundary value problem of (1.1). This paper generalizes some results from the literature [9]. The paper is arranged as follows. In Section 2, we will give some preliminary lemmas. The main results will be given in Section 3.

2 Some lemmas

For convenience, we make the following assumptions:

(A1) ϕ is an odd, increasing homeomorphism on R with ϕ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq7_HTML.gif concave on R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq8_HTML.gif.

(A2) For each c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq9_HTML.gif, there exists A c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq10_HTML.gif such that ϕ 1 ( c u ) A c ϕ 1 ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq11_HTML.gif, u R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq12_HTML.gif and lim c A c = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq13_HTML.gif (note that (A2) implies the existence of B c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq14_HTML.gif such that ϕ 1 ( c u ) B c ϕ 1 ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq15_HTML.gif, u R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq12_HTML.gif and lim c B c = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq16_HTML.gif).

(A3) p : [ a , b ] ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq17_HTML.gif is continuous.

(A4) f : [ a , b ] × R 1 R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq18_HTML.gif is continuous and
lim u + f ( t , u ) ϕ ( u ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equb_HTML.gif

uniformly for t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq19_HTML.gif.

Let E = C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq20_HTML.gif, the usual real Banach space of continuous functions with the maximum norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq21_HTML.gif. Let e ( t ) = ( t a ) ( b t ) ( b a ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq22_HTML.gif for t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq19_HTML.gif and P = { x E : u ( t ) 0 , t [ a , b ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq23_HTML.gif. Then P is a cone of E. Define
f ( t , u ) = { f ( t , u ) , u 0 , f ( t , 0 ) , u < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equc_HTML.gif
Then f ( t , u ) m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq24_HTML.gif for u R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq25_HTML.gif (here m > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq26_HTML.gif is a constant). For u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq27_HTML.gif, λ R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq28_HTML.gif, define
A ( λ , u ) = a t ϕ 1 ( C p ( s ) λ p ( s ) a s p ( r ) f ( r , u ) d r ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equd_HTML.gif
here C is a constant such that
a b ϕ 1 ( C p ( s ) λ p ( s ) a s p ( r ) f ( r , u ) d r ) d s = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Eque_HTML.gif
We know that C exists and is unique for every u P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq29_HTML.gif (see [14]). Then u = A ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq30_HTML.gif if and only if u is a solution of
{ ( p ( t ) ϕ ( u ) ) + λ p ( t ) f ( t , u ) = 0 , a < t < b , u ( a ) = u ( b ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equf_HTML.gif

From [9], we have the following Lemmas 2.1 and 2.2.

Lemma 2.1 Let ϕ satisfy (A1) and (A2). Then for each d > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq31_HTML.gif, there exist constants K d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq32_HTML.gif and L d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq33_HTML.gif such that
K d ϕ ( x ) ϕ ( d x ) L d ϕ ( x ) , x 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equg_HTML.gif

Further, L d 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq34_HTML.gif as d 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq35_HTML.gif and K d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq36_HTML.gif as d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq37_HTML.gif.

Lemma 2.2 Let ϕ be as in Lemma  2.1. Then there exist 0 < α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq38_HTML.gif and β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq39_HTML.gif such that ϕ 1 ( x y ) α ϕ 1 ( x ) β ϕ 1 ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq40_HTML.gif for x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq41_HTML.gif, y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq42_HTML.gif.

Lemma 2.3 Let p 0 , p 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq43_HTML.gif such that p 0 p ( t ) p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq44_HTML.gif for t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq19_HTML.gif. Let λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq45_HTML.gif and ω be a solution of
{ ( p ( t ) ϕ ( ω ) ) + λ p ( t ) h ( t ) = 0 , t [ a , b ] , ω ( a ) = ω ( b ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ2_HTML.gif
(2.1)
here h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq46_HTML.gif is continuous function with h ( t ) m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq47_HTML.gif (here m > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq26_HTML.gif is a constant) for t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq19_HTML.gif, if ω β ( b a ) α ϕ 1 ( λ m δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq48_HTML.gif, then
ω ( t ) ( α ω β ϕ 1 ( λ m δ ) ( b a ) ) e ( t ) for t [ a , b ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equh_HTML.gif

where δ = p 1 ( b a ) p 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq49_HTML.gif.

Proof By integrating, it follows that (3.1) has the unique solution given by
ω ( t ) = a t ϕ 1 { 1 p ( s ) ( C λ a s p ( r ) h ( r ) d r ) } d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equi_HTML.gif
where C is such that ω ( b ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq50_HTML.gif. Let ω = | ω ( t 0 ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq51_HTML.gif for some t 0 ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq52_HTML.gif. Then
ω ( t ) = a t ϕ 1 { λ p ( s ) ( s t 0 p ( r ) h ¯ ( r ) d r λ m p ( s ) s t 0 p ( r ) d r ) } d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equj_HTML.gif
where h ¯ ( t ) = h ( t ) + m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq53_HTML.gif. By Lemma 2.2, we get
ω ( t ) α a t ϕ 1 [ λ p ( s ) s t 0 p ( r ) h ¯ ( r ) d r ] d s β a t ϕ 1 ( λ m p ( s ) s t 0 p ( r ) d r ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equk_HTML.gif
Now
a t ϕ 1 ( λ m p ( s ) s t 0 p ( r ) d r ) d s a t ϕ 1 ( λ m p 0 s t 0 p 1 d r ) d s a t ϕ 1 ( λ m p 1 p 0 ( b a ) ) d s = ϕ 1 ( λ m δ ) ( t a ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equl_HTML.gif
And so
ω ( t ) α ω ¯ ( t ) β ϕ 1 ( λ m δ ) ( t a ) β ϕ 1 ( λ m δ ) ( t a ) for  t [ a , t 0 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ3_HTML.gif
(2.2)
here
ω ¯ ( t ) = a t ϕ 1 [ λ p ( s ) s t 0 p ( r ) h ¯ ( r ) d r ] d s for  t [ a , t 0 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equm_HTML.gif
Note that ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq54_HTML.gif satisfies
{ ( p ( t ) ϕ ( ω ¯ ) ) + λ p ( t ) h ¯ ( t ) = 0 , t [ a , t 0 ] , ω ¯ ( a ) = 0 , ω ¯ ( t 0 ) | ω ( t 0 ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equn_HTML.gif
In fact, ω ¯ ( t ) ω ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq55_HTML.gif for t [ a , t 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq56_HTML.gif. We next prove that ω ¯ ( t ) ν ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq57_HTML.gif for t [ a , t 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq56_HTML.gif, here ν satisfies
{ ( p ( t ) ϕ ( ν ) ) = 0 , t [ a , t 0 ] , ν ( a ) = 0 , ν ( t 0 ) = ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equo_HTML.gif
Suppose it is not true, then ω ¯ ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq58_HTML.gif has a negative absolute minimum at τ ( a , t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq59_HTML.gif. Since
ω ¯ ( a ) ν ( a ) = 0 , ω ¯ ( t 0 ) ν ( t 0 ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equp_HTML.gif
there exist τ 0 , τ 1 [ a , t 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq60_HTML.gif such that
ω ¯ ( τ 0 ) ν ( τ 0 ) = ω ¯ ( τ 1 ) ν ( τ 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equq_HTML.gif
and
ω ¯ ( t ) ν ( t ) < 0 , t ( τ 0 , τ 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equr_HTML.gif
Then
( p ( t ) ϕ ( ω ( t ) ) ) ( p ( t ) ϕ ( ν ( t ) ) ) = λ p ( t ) h ¯ ( t ) 0 for  t ( τ 0 , τ 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equs_HTML.gif
Let u ( t ) = ω ¯ ( t ) ν ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq61_HTML.gif, t ( τ 0 , τ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq62_HTML.gif, then
τ 0 τ 1 ( ( p ( t ) ϕ ( ω ¯ ( t ) ) ) ( p ( t ) ϕ ( ν ( t ) ) ) ) u ( t ) d t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equt_HTML.gif
On the other hand, using the inequality
( ϕ ( b ) ϕ ( a ) ) ( b a ) 0 , a , b R 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equu_HTML.gif
and the fact that there exists τ [ τ 0 , τ 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq63_HTML.gif such that ω ¯ ( τ ) ν ( τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq64_HTML.gif, we have
τ 0 τ 1 ( ( p ( t ) ϕ ( ω ¯ ( t ) ) ) ( p ( t ) ϕ ( ν ( t ) ) ) ) u ( t ) d t = τ 0 τ 1 ( ( p ( t ) ϕ ( ω ¯ ( t ) ) ) ( p ( t ) ϕ ( ν ( t ) ) ) ) ( ω ¯ ν ) d t < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equv_HTML.gif
which is a contradiction. So, ω ¯ ( t ) ν ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq57_HTML.gif for t [ a , t 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq56_HTML.gif. Obviously, ν ( t ) = ω t 0 a ( t a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq65_HTML.gif, t [ a , t 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq56_HTML.gif, since ω ¯ θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq66_HTML.gif for each t [ a , t 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq56_HTML.gif. From (2.2), we have
ω ( t ) ( α ω β ϕ 1 ( λ m δ ) ( b a ) ) t a t 0 a , t [ a , t 0 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equw_HTML.gif
Similarly,
ω ( t ) ( α ω β ϕ 1 ( λ m δ ) ( b a ) ) b t b t 0 , t [ t 0 , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equx_HTML.gif
If ω β ( b a ) α ϕ 1 ( λ m δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq67_HTML.gif, then
ω ( t ) ( α ω β ϕ 1 ( λ m δ ) ( b a ) ) e ( t ) for  t [ a , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equy_HTML.gif

The proof is complete. □

Let Q λ = { u E u ( α u β ϕ 1 ( λ m δ ) ( b a ) ) e ( t ) , t [ a , b ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq68_HTML.gif for each λ R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq28_HTML.gif, where α , β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq69_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq70_HTML.gif are defined as that in Lemma 2.2 and Lemma 2.3, respectively. Then Q λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq71_HTML.gif is also a cone of E. From Lemma 2.3, we know that A : P Q λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq72_HTML.gif is completely continuous.

Let
L ( P ) = { ( λ , u ) | λ R + , u P { θ }  is a solution of  (1.1) } ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equz_HTML.gif

From [[15], Lemma 29.1], we have Lemma 2.4.

Lemma 2.4 Let X be a compact metric space. Assume that A and B are two disjoint closed subsets of X. Then either there exist a connected component of X meeting both A and B or X = Ω A Ω B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq73_HTML.gif where Ω A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq74_HTML.gif, Ω B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq75_HTML.gif are disjoint compact subsets of X containing A and B, respectively.

Let U be an open and bounded subset of the metric space [ a , b ] × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq76_HTML.gif. We set U ( λ ) = { x E : ( λ , x ) U } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq77_HTML.gif, whose boundary is denoted by U ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq78_HTML.gif. Consider a map h ( λ , x ) = x k ( λ , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq79_HTML.gif, such that k ( λ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq80_HTML.gif is compact and θ h ( U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq81_HTML.gif. Such a map h will also be called an admissible homotopy on U. If h is an admissible homotopy, for every λ [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq82_HTML.gif and every x U ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq83_HTML.gif, one has that h λ ( x ) : = h ( λ , x ) θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq84_HTML.gif and it makes sense to evaluate deg ( h λ , U ( λ ) , θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq85_HTML.gif.

Lemma 2.5 If h is an admissible homotopy on U [ a , b ] × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq86_HTML.gif, the deg ( h λ , U ( λ ) , θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq87_HTML.gif is constant for all λ [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq82_HTML.gif.

Lemma 2.6 Let h E { θ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq88_HTML.gif such that h α h e ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq89_HTML.gif. Then for arbitrary λ ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq90_HTML.gif, there exists R λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq91_HTML.gif such that for each λ [ λ , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq92_HTML.gif, R R λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq93_HTML.gif and μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq94_HTML.gif,
u A ( λ , u ) + μ h , u B ( θ , R ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equaa_HTML.gif

where B ( θ , R ) = { u : u < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq95_HTML.gif.

Proof From (A4), for m 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq96_HTML.gif, such that
α ( b a ) 16 ϕ 1 ( m 1 λ 0 p 0 ( b a ) 16 p 1 ) α ( b a ) 8 > 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equab_HTML.gif
there exists m 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq97_HTML.gif such that f ( t , u ) m 1 ϕ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq98_HTML.gif for u m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq99_HTML.gif. Let
R λ = max { 1 α [ m 0 min t [ b + 3 a 4 , 3 b + a 4 ] e ( t ) + β ϕ 1 ( λ 0 m δ ( b a ) ) ] , β ( b a ) α ϕ 1 ( λ 0 m δ ) } + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equac_HTML.gif
Assume by contradiction that
u 0 = A ( λ 0 , u 0 ) + μ 0 h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ4_HTML.gif
(2.3)
for some R R λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq93_HTML.gif, u 0 B ( θ , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq100_HTML.gif, λ 0 [ λ , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq101_HTML.gif and μ 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq102_HTML.gif. Let
y 0 ( t ) = A ( λ 0 , u 0 ) = a t ϕ 1 ( C p ( s ) λ 0 p ( s ) a s p ( r ) f ( r , u 0 ) d r ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equad_HTML.gif
From Lemma 2.3, we know that y 0 ( t ) Q λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq103_HTML.gif. Namely,
y 0 ( t ) ( α y 0 β ϕ 1 ( λ 0 m δ ) ( b a ) ) e ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ5_HTML.gif
(2.4)
From (2.3) and (2.4), we have
u 0 = y 0 ( t ) + μ 0 h ( α y 0 β ϕ 1 ( λ 0 m δ ( b a ) ) ) e ( t ) + μ 0 α h e ( t ) ( α y 0 + μ 0 h β ϕ 1 ( λ 0 m δ ( b a ) ) e ( t ) = ( α u 0 β ϕ 1 ( λ 0 m δ ) ( b a ) ) e ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equae_HTML.gif
so u 0 Q λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq104_HTML.gif. For λ 0 [ λ , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq101_HTML.gif, we have
u 0 ( t ) ( α u 0 β ϕ 1 ( λ 0 m δ ( b a ) ) ) e ( t ) m 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equaf_HTML.gif
for t ( 3 a + b 4 , a + b 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq105_HTML.gif. Therefore, let u 0 = u ( t u 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq106_HTML.gif, t u 0 ( 3 a + b 4 , a + b 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq107_HTML.gif, assume that t u 0 5 a + 3 b 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq108_HTML.gif, then
u 0 = u ( t u 0 ) = 3 a + b 4 t u 0 ϕ 1 ( λ 0 p ( s ) s t u 0 p ( τ ) f ( t , u 0 ) d τ ) d s + μ 0 h α 3 a + b 4 t u 0 ϕ 1 ( λ 0 p 0 p 1 s t u 0 p ( τ ) ( f ( t 0 , u 0 ) + m ) d τ ) d s β 3 a + b 4 t u 0 ϕ 1 ( λ 0 p 1 m p 0 ( t u 0 s ) ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equag_HTML.gif
where
3 a + b 4 t u 0 ϕ 1 ( λ 0 p 0 p 1 s t u 0 ( f ( τ , u 0 ) + m ) d τ ) d s 3 a + b 4 11 a + 5 b 16 ϕ 1 ( λ 0 p 0 p 1 s t u 0 ( f ( τ , u 0 ) + m ) d τ ) d s 3 a + b 4 11 a + 5 b 16 ϕ 1 ( λ 0 p 0 p 1 11 a + 5 b 16 5 a + 3 b 8 m 1 ϕ ( u 0 ) d τ ) d s = 3 a + b 4 11 a + 5 b 16 ϕ 1 ( m 1 λ 0 p 0 ( b a ) 4 p 1 ) u 0 d s = b a 16 ϕ 1 ( m 1 λ 0 p 0 ( b a ) 16 p 1 ) u 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equah_HTML.gif
and
3 a + b 4 t u 0 ϕ 1 ( λ 0 p 1 m p 0 ( t u 0 s ) ) d s 3 a + b 4 t u 0 ϕ 1 ( λ 0 p 1 m p 0 ( b a ) ) d s = ϕ 1 ( λ 0 p 1 m p 0 ( b a ) ) ( t u 0 3 a + b 4 ) ( b a ) 8 ϕ 1 ( λ 0 m δ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equai_HTML.gif
Thus,
u 0 α ( b a ) 16 ϕ 1 ( m 1 λ 0 p 0 ( b a ) 16 p 1 ) u 0 β ( b a ) 8 ϕ 1 ( λ 0 m δ ) α ( b a ) 16 ϕ 1 ( m 1 λ 0 p 0 ( b a ) 16 p 1 ) u 0 α ( b a ) 8 u 0 = ( α ( b a ) 16 ϕ 1 ( m 1 λ 0 p 0 ( b a ) 16 p 1 ) α ( b a ) 8 ) u 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equaj_HTML.gif

so α ( b a ) 16 ϕ 1 ( m 1 λ 0 p 0 ( b a ) 16 p 1 ) α ( b a ) 8 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq109_HTML.gif, which is a contradiction. Then (2.3) holds. The proof is complete. □

3 Main results

For convenience, let us introduce the following symbols. For any r,
M ( r ) = { ( λ ˜ , u ) [ 0 , 1 ] × E : u = r } , M [ r , + ) = { ( λ ˜ , u ) [ 0 , 1 ] × E : r u < + } , M [ 0 , r ) = { ( λ ˜ , u ) [ 0 , 1 ] × E : 0 u < r } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equak_HTML.gif

Now we give our main results of this paper.

Theorem 3.1 Suppose (A1) to (A4) hold. Then L ( P ) ( [ 0 , 1 ] × P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq110_HTML.gif possesses an unbounded connected component D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq111_HTML.gif such that Proj λ D ( 0 , λ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq112_HTML.gif for some λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq113_HTML.gif and
lim λ 0 + , ( λ , u ) D u = + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equal_HTML.gif

where Proj λ D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq114_HTML.gif denotes the projection of D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq111_HTML.gif onto the λ-axis.

Proof We divide our proof into four steps.

Step 1. Let
T ( λ , u ) = { A ( λ , u ) , ( [ 0 , 1 ] × E ) M [ 2 , + ) , θ , ( [ 0 , 1 ] × B ( θ , 1 ) ¯ ) ( { 0 } × E ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ6_HTML.gif
(3.1)

where B ( θ , 1 ) ¯ = { u E : u 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq115_HTML.gif. Obviously, T ( , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq116_HTML.gif is a completely continuous operator.

Note that T ( λ , u ) = A ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq117_HTML.gif for all λ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq118_HTML.gif and u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq27_HTML.gif with u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq119_HTML.gif. From Lemma 2.5, there exists R 1 > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq120_HTML.gif large enough such that Fix T ( 1 , ) B ( θ , R 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq121_HTML.gif and
deg ( I T ( 1 , ) , B ( θ , R 1 ) , θ ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ7_HTML.gif
(3.2)
Obviously,
deg ( I T ( 1 , ) , B ( θ , 1 ) , θ ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ8_HTML.gif
(3.3)
Therefore,
deg ( I T ( 1 , ) , B ( θ , R 1 ) B ( θ , 1 ) ¯ , θ ) = deg ( I T ( 1 , ) , B ( θ , R 1 ) , θ ) deg ( I T ( 1 , ) , B ( θ , 1 ) , θ ) = 0 1 = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ9_HTML.gif
(3.4)

So, Fix T ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq122_HTML.gif and Fix T ( 1 , ) U : = B ( θ , R 1 ) B ( θ , 1 ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq123_HTML.gif.

Step 2. Let
S = { ( λ , u ) R + × E : u = T ( λ , u ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equam_HTML.gif
From Lemma 2.6, we have
L ( P ) [ 0 , 1 ] × { u E : 0 < u 1 } = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equan_HTML.gif
This implies that T has no bifurcation point on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq124_HTML.gif. From step 1,we have L ( P ) ( { 1 } × E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq125_HTML.gif, then for each ( 1 , u ) L ( P ) ( { 1 } × E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq126_HTML.gif, denote by D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq127_HTML.gif the connected component of the metric space L ( P ) ( { 1 } × E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq128_HTML.gif emitting from ( 1 , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq129_HTML.gif. Now we will show that, there must exist ( 1 , u 0 ) L ( P ) ( { 1 } × E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq130_HTML.gif such that D u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq131_HTML.gif is unbounded. Assume on the contrary that D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq127_HTML.gif is bounded for each ( 1 , u ) L ( P ) ( { 1 } × E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq132_HTML.gif. Take a bounded open neighborhood U u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq133_HTML.gif in L ( P ) ( { 1 } × E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq128_HTML.gif for each D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq127_HTML.gif such that
C l ( { 1 } × E ) U u 1 ( [ 0 , 1 ] × { θ } ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ10_HTML.gif
(3.5)
and U u 1 ( { 0 } × E ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq134_HTML.gif, where C l ( { 1 } × E ) U u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq135_HTML.gif denotes the closure of U u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq133_HTML.gif in the metric space [ 0 , 1 ] × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq136_HTML.gif. Let [ 0 , 1 ] × E U u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq137_HTML.gif denote the boundary of U u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq133_HTML.gif in the metric space [ 0 , 1 ] × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq136_HTML.gif. Obviously, [ 0 , 1 ] × E U u 1 L ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq138_HTML.gif is a compact subset. Assume that [ 0 , 1 ] × E U u 1 L ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq139_HTML.gif. From the maximal connectedness of D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq127_HTML.gif, there is no connected subset of [ 0 , 1 ] × E U u 1 L ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq138_HTML.gif meeting both [ 0 , 1 ] × E U u 1 L ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq140_HTML.gif and D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq127_HTML.gif. From Lemma 2.4, there exist compact disjoint subsets Ω u ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq141_HTML.gif and Ω u ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq142_HTML.gif of C l [ 0 , 1 ] × E Ω u ( 1 ) L ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq143_HTML.gif such that D u Ω u ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq144_HTML.gif and [ 0 , 1 ] × E U u ( 1 ) L ( P ) Ω u ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq145_HTML.gif, and C l [ 0 , 1 ] × E U u ( 1 ) L ( P ) = Ω u ( 1 ) Ω u ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq146_HTML.gif. Let d = d ( Ω u ( 1 ) , Ω u ( 2 ) ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq147_HTML.gif and U u ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq148_HTML.gif be the d 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq149_HTML.gif-neighborhood of Ω u ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq141_HTML.gif in the metric space [ 0 , 1 ] × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq136_HTML.gif. Set
U u = { U u ( 1 ) U u ( 2 ) , when  [ 0 , 1 ] × E U u ( 1 ) L ( P ) , U u ( 1 ) , when  [ 0 , 1 ] × E U u ( 1 ) L ( P ) = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ11_HTML.gif
(3.6)
Then we have D u U u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq150_HTML.gif and
[ 0 , 1 ] × E U u L ( P ) = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ12_HTML.gif
(3.7)
Obviously, the collection of the subsets
{ U u { 1 } × E : ( 1 , u ) L ( P ) ( { 1 } × E ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equao_HTML.gif
is an open cover of L ( P ) ( { 1 } × E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq128_HTML.gif. Since L ( P ) ( { 1 } × E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq151_HTML.gif is compact, then there exist finite points, namely
( 1 , u 1 ) , ( 1 , u 2 ) , , ( 1 , u n ) ( { 1 } × E ) L ( P ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equap_HTML.gif
such that
( { 1 } × E ) L ( P ) i = 1 n ( U u i ( { 1 } × E ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equaq_HTML.gif
Let U = i = 1 n U u i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq152_HTML.gif. Then U is a bounded open subset of [ 0 , 1 ] × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq136_HTML.gif. From (3.7), we have
[ 0 , 1 ] × E U L ( P ) i = i n ( ( [ 0 , 1 ] × E U u i ) L ( P ) ) = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ13_HTML.gif
(3.8)
Thus,
[ 0 , 1 ] × E U L ( P ) = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equar_HTML.gif
From (3.5) and (3.8), we have
[ 0 , 1 ] × E U S = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equas_HTML.gif
Then from Lemma 2.5, we have
deg ( I T ( λ , ) , U ( 1 ) , θ ) = deg ( I T ( λ , ) , U ( 0 ) , θ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ14_HTML.gif
(3.9)
Since U ( 0 ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq153_HTML.gif, then
deg ( I T ( λ , ) , U ( 0 ) , θ ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ15_HTML.gif
(3.10)
Since Fix T B ( θ , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq154_HTML.gif, then from (3.4) we have
deg ( I T ( λ , ) , U ( 1 ) , θ ) = 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ16_HTML.gif
(3.11)

which contradicts to (3.10) and (3.11). Therefore, there must exist ( 1 , u 0 ) ( { 1 } × E ) L ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq155_HTML.gif such that D u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq131_HTML.gif is bounded.

Step 3. Obviously, the projection of D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq127_HTML.gif is a interval, denote it by [ 0 , λ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq156_HTML.gif, then λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq157_HTML.gif. Then D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq127_HTML.gif is a bounded connected component of ( [ 0 , λ ] × E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq158_HTML.gif. Take r 0 = β ( b a ) α ϕ ( λ m δ ) + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq159_HTML.gif, let
Y 1 = ( { 1 } × E ) M [ r 0 , + ) , Y 2 = ( [ 0 , 1 ] × E ) M ( r 0 ) , Y = ( [ 0 , 1 ] × E ) M [ r 0 , + ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equat_HTML.gif
Obviously, D u ( Y 1 Y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq160_HTML.gif. For each p D u ( Y 1 Y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq161_HTML.gif, denote by E ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq162_HTML.gif the connected component of the metric space D u Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq163_HTML.gif, which passes the point p. Now we shall prove that there must exist a p 0 D u ( Y 1 Y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq164_HTML.gif such that E ( p 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq165_HTML.gif is an unbounded connected component of the metric space D u Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq163_HTML.gif. On the contrary, assume that E ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq162_HTML.gif is bounded for each p D u ( Y 1 Y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq161_HTML.gif. Then, for each p D u ( Y 1 Y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq161_HTML.gif, in the same way as in the construction of U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq166_HTML.gif in (3.6) we can show that there exists a neighborhood U ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq167_HTML.gif of E ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq162_HTML.gif in Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq168_HTML.gif such that
Y U ( p ) D u = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ17_HTML.gif
(3.12)
Obviously, the set of { U ( p ) ( Y 1 Y 2 ) | p D u ( Y 1 Y 2 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq169_HTML.gif is an open cover of the set D u ( Y 1 Y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq170_HTML.gif and D u ( Y 1 Y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq170_HTML.gif is a compact set. Thus, there exist finite subsets of { U ( p ) ( Y 1 Y 2 ) | p D u ( Y 1 Y 2 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq171_HTML.gif, say
U ( p 1 ) ( Y 1 Y 2 ) , U ( p 2 ) ( Y 1 Y 2 ) , , U ( p n ) ( Y 1 Y 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equau_HTML.gif
which is also an open cover of D u ( Y 1 Y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq170_HTML.gif, that is,
i = 1 n ( U ( p i ) ( Y 1 Y 2 ) ) D u ( Y 1 Y 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ18_HTML.gif
(3.13)
Let U 0 = i = 1 n U ( p i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq172_HTML.gif, then U 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq173_HTML.gif is a bounded open subset of Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq168_HTML.gif. Since
Y U 0 i = 1 n Y U ( p i ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equav_HTML.gif
then by (3.12) we have
Y U 0 D u = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equ19_HTML.gif
(3.14)
Let
W 1 = ( ( [ 0 , 1 ] × E ) M [ 0 , r 0 ) ) U 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equaw_HTML.gif
and W 2 = ( [ 0 , 1 ] × E ) C l [ 0 , 1 ] × E W 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq174_HTML.gif. It is easy to see that
[ 0 , 1 ] × E W 1 Y U 0 [ ( M ( r 0 ) ( [ 0 , 1 ] × E ) ) ( U 0 ( Y 1 Y 2 ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equax_HTML.gif

From (3.12) and (3.13), we see that [ 0 , 1 ] × E W 1 D u = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq175_HTML.gif. Obviously, W 1 D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq176_HTML.gif and W 1 W 2 = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq177_HTML.gif. Note the unboundedness of D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq127_HTML.gif, then W 2 D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq178_HTML.gif. Now we have D u = ( W 1 D u ) ( W 2 D u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq179_HTML.gif, which is a contradiction of the connectedness of D u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq127_HTML.gif. Therefore, there must exist p 0 D u ( Y 1 Y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq164_HTML.gif such that E ( p 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq165_HTML.gif is an unbounded connected component of D u Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq163_HTML.gif.

Step 4. Since u = T ( λ , u ) Q λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq180_HTML.gif for each ( λ , u ) E ( p 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq181_HTML.gif, we have
u ( α u β ϕ 1 ( λ m δ ) ) e ( t ) > θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equay_HTML.gif
So, u = A ( λ , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq182_HTML.gif for each ( λ , u ) E ( p 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq183_HTML.gif. This implies E ( p 0 ) L ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq184_HTML.gif is an unbounded subset. Let D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq111_HTML.gif be the connected component of L ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq185_HTML.gif containing E ( p 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq165_HTML.gif. Obviously, there exists λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq113_HTML.gif such that Proj D ( 0 , λ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq186_HTML.gif. As D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq111_HTML.gif is unbounded, we easily see that
lim λ 0 + , ( λ , u ) D u = + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_Equaz_HTML.gif

The proof is complete. □

Corollary 3.1 Let (A1) to (A4) hold. Then there exists λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq187_HTML.gif such that problem (1.1) has a positive solution u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq188_HTML.gif for 0 < λ < λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq189_HTML.gif with u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq190_HTML.gif as λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq191_HTML.gif.

Declarations

Acknowledgements

This paper is supported by Innovation Project of Jiangsu Province postgraduate training project (CXLX12_0979).

Authors’ Affiliations

(1)
Department of Mathematics, Jiangsu Normal University

References

  1. Manasevich R, Schmitt K: Boundary value problems for quasilinear second order differential equations. CISM Lecture Notes. In Nonlinear Analysis and Boundary Value Problems. Springer, Berlin; 1996:79-119.
  2. Castro A, Shivaji R: Semipositone Problems, Semigroups of Linear and Nonlinear Operators and Applications. Kluwer Academic, Dordrecht; 1993:109-119.View Article
  3. Lions PL: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 1982, 24: 441-467. 10.1137/1024101MathSciNetView Article
  4. Xian X, Jingxian S: Unbounded connected component of the positive solutions set of some semi-position problems. Topol. Methods Nonlinear Anal. 2011, 37: 283-302.MathSciNet
  5. Jingxian S, Xian X: Positive fixed points of semi-positone nonlinear operator and its applications. Acta Math. Sin. 2012, 55: 55-64. in Chinese
  6. Anuradha V, Hai DD, Shivaji R: Existence results for superlinear semipositone BVPs. Proc. Am. Math. Soc. 1996, 124: 757-763. 10.1090/S0002-9939-96-03256-XMathSciNetView Article
  7. Xian X, O’Regan D, Yanfeng C: Structure of positive solution sets of simi-positon singular boundary value problems. Nonlinear Anal. 2010, 72: 3535-3550. 10.1016/j.na.2009.12.035MathSciNetView Article
  8. Haisen LU: Existence and multiplicity of positive solutions for semiposition p -Laplacian equations. Acta Math. Appl. Sin. 2006, 19(3):546-553.
  9. Hai DD, Schmitt K, Shivaji R: Positive solutions of quasilinear boundary value problems. J. Math. Anal. Appl. 1998, 217: 672-686. 10.1006/jmaa.1997.5762MathSciNetView Article
  10. Rabinowitz PH: Some global results for nonlinear eigenvalues. J. Funct. Anal. 1971, 7: 487-513. 10.1016/0022-1236(71)90030-9MathSciNetView Article
  11. Amann H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 1976, 18: 620-709. 10.1137/1018114MathSciNetView Article
  12. Dancer EN: Solution branches for mappings in cones and applications. Bull. Aust. Math. Soc. 1974, 11: 133-145.MathSciNetView Article
  13. Rabinowitz PH: On bifurcation from infinity. J. Differ. Equ. 1973, 14: 462-475. 10.1016/0022-0396(73)90061-2MathSciNetView Article
  14. Xiyu L:Some existence principles for the singular second order differential equation ( ϕ ( x ) ) + f ( t , x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-100/MediaObjects/13661_2012_Article_346_IEq192_HTML.gif. Acta Math. Sin. 1996, 39(3):366-375.
  15. Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985.View Article

Copyright

© Xian and Xunxia; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. 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.