Open Access

Positive solutions for Sturm-Liouville BVPs on time scales via sub-supersolution and variational methods

Boundary Value Problems20132013:123

DOI: 10.1186/1687-2770-2013-123

Received: 20 January 2013

Accepted: 26 April 2013

Published: 13 May 2013

Abstract

This paper is concerned with the existence of one and two positive solutions for the following Sturm-Liouville boundary value problem on time scales

{ ( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = f ( t , u σ ( t ) ) , t [ 0 , T ] T , α 1 u ( 0 ) α 2 u Δ ( 0 ) = 0 , α 3 u ( σ 2 ( T ) ) + α 4 u Δ ( σ ( T ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equa_HTML.gif

Under a locally nonnegative assumption on the nonlinearity f and some other suitable hypotheses, positive solutions are sought by considering the corresponding truncated problem, constructing the variational framework and combining the sub-supersolution method with the mountain pass lemma.

MSC:34B10, 34B18.

Keywords

positive solution Sturm-Liouville time scales sub-supersolution variational methods

1 Introduction

The theory of dynamic equations on time scales has become a new important mathematical branch [1, 2] since it was initiated by Hilger in 1988 [3]. Since then, boundary value problems (BVPs) for dynamic equations on time scales have received considerable attention, various fixed point theorems, sub-supersolution method and Leray-Schauder degree theory have been applied to get many interesting results about the existence of solutions; see [1, 2, 412] and the references therein. Variational method is also an important method for dealing with the existence of BVPs. Recently, some authors have used the theory to study the existence of solutions of some BVPs on time scales [4, 5, 13, 14].

Especially, in [4, 5], Agarwal et al. studied the following dynamic equation on time scales:
u Δ Δ ( t ) = f ( t , u σ ( t ) ) , t ( a , b ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equb_HTML.gif
and
u Δ Δ ( t ) = f ( σ ( t ) , u σ ( t ) ) , t ( a , b ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equc_HTML.gif

with the Dirichlet boundary condition. They gave some sufficient conditions for the existence of single and multiple positive solutions by using the variational method and critical point theory.

In [14], we considered the problem
{ ( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = λ f ( t , u σ ( t ) ) , t [ 0 , T ] T , α 1 u ( 0 ) α 2 u Δ ( 0 ) = 0 , α 3 u ( σ 2 ( T ) ) + α 4 u Δ ( σ ( T ) ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ1_HTML.gif
(1.1)

and obtained the existence of many solutions depending on the value of the parameter λ which lie in some different intervals under some suitable hypotheses. The main approaches are also the variational method and some known critical point theorems and a three critical point theorem established in [15]. Erbe et al. [8] also established some existence criteria of positive solutions by a fixed point theorem in a cone with the globally nonnegative hypothesis of f.

Motivated by the papers [4, 5, 8, 14], in this paper, we continue to study the problem (1.1) in the case of λ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq1_HTML.gif, that is,
{ ( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = f ( t , u σ ( t ) ) , t [ 0 , T ] T , α 1 u ( 0 ) α 2 u Δ ( 0 ) = 0 , α 3 u ( σ 2 ( T ) ) + α 4 u Δ ( σ ( T ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ2_HTML.gif
(1.2)

Here T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq2_HTML.gif is a time scale and [ 0 , T ] T = { t T : 0 t T  and  T T κ 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq3_HTML.gif, p C 1 ( [ 0 , σ ( T ) ] T , ( 0 , + ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq4_HTML.gif, q C ( [ 0 , T ] T , [ 0 , + ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq5_HTML.gif, f C ( [ 0 , T ] T × R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq6_HTML.gif, α i 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq7_HTML.gif, i = 1 , 2 , 3 , 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq8_HTML.gif, α 1 + α 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq9_HTML.gif, α 3 + α 4 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq10_HTML.gif and α 1 + α 3 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq11_HTML.gif. The purpose of this paper is to discuss the existence and multiplicity of positive solutions to the problem (1.2) under the local non-negativity assumption of f and some other hypotheses. The main tools are the truncated method, the variational method, the sub-supersolution method and the mountain pass lemma. First, inspired by the method in [4], we convert the existence of a positive solution of (1.2) to the existence of a solution of an associated problem of (1.2). In contrast with the paper [4], the appearance of term p ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq12_HTML.gif, our problem is more complicated and the proof is also different from [4] (see Lemma 2.3 for details). Next, we construct a supersolution of (1.2) and give the existence of one positive solution. Finally, under our weaker assumption on f (see (H1) and (H2) for details), since we cannot verify that the corresponding functional for the associated problem satisfies the P.S. condition, we consider the corresponding truncated problem. To prove the existence of the second positive solution by the mountain pass lemma, we also give an estimate of a nonnegative solution of (1.2) and prove the solution of a truncated problem is also a solution of (1.2) for n large enough (see Theorem 3.3 for details). To the best of our knowledge, the results are new both in the continuous and in the discrete case.

The paper is organized as follows. In Section 2, we present some basic properties of some related Sobolev space on time scales, construct the variational framework, give some properties of this framework and some necessary lemmas. In Section 3, we firstly get the existence of a single positive solution of (1.2) by using the sub-supersolution method; then applying the truncated method, analytic technique and mountain pass lemma, we establish the existence of two positive solutions.

2 Preliminaries and variational formulation

In this section, we list the definition and properties of the Sobolev space on time scales [16], give some lemmas which we need for the proof of the main result and construct a variational framework.

For convenience, for f L Δ 1 ( [ a , b ) T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq13_HTML.gif [16, 17], we denote a b f ( s ) Δ s = [ a , b ) T f ( s ) Δ s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq14_HTML.gif. We let A C ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq15_HTML.gif [18] denote the class of absolutely continuous functions on [ 0 , σ 2 ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq16_HTML.gif and the Sobolev space is defined as
H Δ 1 ( [ 0 , σ 2 ( T ) ] T ) = { u | u A C [ 0 , σ 2 ( T ) ] T , u Δ L Δ 2 ( [ 0 , σ 2 ( T ) ) T ) } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equd_HTML.gif
with the norm
u = ( 0 σ 2 ( T ) ( | u ( t ) | 2 + | u Δ ( t ) | 2 ) Δ t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Eque_HTML.gif

We know the immersion H Δ 1 ( [ 0 , σ 2 ( T ) ] T ) C ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq17_HTML.gif is compact. Analogous to the proof in the real numbers situation, one can deduce the following result on time scales.

Lemma 2.1 If f C 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq18_HTML.gif, f L ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq19_HTML.gif, u H Δ 1 ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq20_HTML.gif, then
f ( u ) H Δ 1 ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equf_HTML.gif
and
( f ( u ( t ) ) ) Δ = u Δ ( t ) 0 1 f ( u ( t ) + ( σ ( t ) t ) s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equg_HTML.gif

Using Lemma 2.1, by a similar proof of T = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq21_HTML.gif, one can derive the following.

Lemma 2.2 If u H Δ 1 ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq20_HTML.gif, then u + , u H Δ 1 ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq22_HTML.gif and ( u + ) Δ ( u ) Δ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq23_HTML.gif, Δ-a.e. in [ 0 , σ 2 ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq16_HTML.gif, where u + = max { u , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq24_HTML.gif, u = ( u ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq25_HTML.gif.

For convenience, we denote
β 1 = { α 1 α 2 if  α 2 0 , 0 if  α 2 = 0 , β 2 = { α 3 α 4 if  α 4 0 , 0 if  α 4 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equh_HTML.gif
and for u , v H Δ 1 ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq26_HTML.gif, we set
( u , v ) 0 = 0 σ 2 ( T ) p ( t ) u Δ ( t ) v Δ ( t ) Δ t + 0 σ ( T ) q ( t ) u σ ( t ) v σ ( t ) Δ t + β 1 p ( 0 ) u ( 0 ) v ( 0 ) + β 2 p ( σ ( T ) ) u ( σ 2 ( T ) ) v ( σ 2 ( T ) ) , u 0 = ( u , u ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equi_HTML.gif
In order to discuss the existence of a positive solution of (1.2), we consider the following problem:
{ ( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = f ( t , ( u + ) σ ( t ) ) , t [ 0 , T ] T , α 1 u ( 0 ) α 2 u Δ ( 0 ) = 0 , α 3 u ( σ 2 ( T ) ) + α 4 u Δ ( σ ( T ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ3_HTML.gif
(2.1)

First, we give an important lemma.

Lemma 2.3 If f ( t , 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq27_HTML.gif for t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq28_HTML.gif, u is a solution of (2.1), then u is nonnegative in [ 0 , σ 2 ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq16_HTML.gif. Furthermore, if f ( t , 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq29_HTML.gif for t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq28_HTML.gif, and p ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq12_HTML.gif is nondecreasing in [ 0 , σ ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq30_HTML.gif, then u ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq31_HTML.gif, t ( 0 , σ 2 ( T ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq32_HTML.gif.

Proof Let u be a solution of (2.1). In view of Lemma 2.2, we know u + , u H Δ 1 ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq22_HTML.gif and ( u + ) Δ ( u ) Δ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq23_HTML.gif, Δ-a.e. in [ 0 , σ 2 ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq16_HTML.gif. Multiplying (2.1) by ( u ) σ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq33_HTML.gif, integrating over [ 0 , σ ( T ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq34_HTML.gif and employing the integration by parts formula for an absolutely continuous function on T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq2_HTML.gif, we find that
u 0 2 0 σ 2 ( T ) p ( t ) u Δ ( t ) ( u ) Δ ( t ) Δ t + 0 σ ( T ) q ( t ) u σ ( t ) ( u ) σ ( t ) Δ t + β 1 p ( 0 ) u ( 0 ) u ( 0 ) + β 2 p ( σ ( T ) ) u ( σ 2 ( T ) ) u ( σ 2 ( T ) ) = 0 σ ( T ) f ( t , ( u + ) σ ( t ) ) ( u ) σ ( t ) Δ t = 0 σ ( T ) f ( t , 0 ) ( u ) σ ( t ) Δ t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equj_HTML.gif

Therefore, u ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq35_HTML.gif for t [ 0 , σ 2 ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq36_HTML.gif.

Next, we show that if p ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq12_HTML.gif is nondecreasing in [ 0 , σ ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq30_HTML.gif, and f ( t , 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq29_HTML.gif for t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq28_HTML.gif, then u ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq31_HTML.gif, t ( 0 , σ 2 ( T ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq37_HTML.gif.

In fact, if the conclusion is false, we can suppose that there exists c ( 0 , σ 2 ( T ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq38_HTML.gif such that u ( c ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq39_HTML.gif and one of the following two cases holds:
  1. (i)

    u ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq31_HTML.gif for t ( c , σ 2 ( T ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq40_HTML.gif and σ ( c ) σ 2 ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq41_HTML.gif,

     
  2. (ii)

    u ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq42_HTML.gif for t ( c , σ 2 ( T ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq43_HTML.gif and there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq44_HTML.gif such that u ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq31_HTML.gif for t [ c δ , c ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq45_HTML.gif.

     
For the case (i), if ρ ( c ) = c < σ ( c ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq46_HTML.gif, then u Δ ( c ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq47_HTML.gif, there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq44_HTML.gif such that u Δ ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq48_HTML.gif for t ( c δ , c ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq49_HTML.gif. According to the nonnegativity of u on [ 0 , σ 2 ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq16_HTML.gif and u ( c ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq39_HTML.gif, it is easy to see that ρ ( c ) = c < σ ( c ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq46_HTML.gif is impossible. Thus we have
σ ( ρ ( c ) ) = c . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ4_HTML.gif
(2.2)
If ρ ( c ) < c σ ( c ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq50_HTML.gif, then we know u Δ ( c ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq51_HTML.gif, u Δ ( ρ ( c ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq52_HTML.gif and u Δ Δ ( ρ ( c ) ) = u Δ ( c ) u Δ ( ρ ( c ) ) c ρ ( c ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq53_HTML.gif. If ρ ( c ) = c = σ ( c ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq54_HTML.gif, then u Δ ( c ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq55_HTML.gif, u Δ Δ ( ρ ( c ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq56_HTML.gif. Hence, in this case, we always have
u Δ ( c ) 0 , u Δ Δ ( ρ ( c ) ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ5_HTML.gif
(2.3)
Therefore
( p ( ρ ( c ) ) u Δ ( ρ ( c ) ) ) Δ = p ( ρ ( c ) ) u Δ Δ ( ρ ( c ) ) p Δ ( ρ ( c ) ) u Δ ( c ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ6_HTML.gif
(2.4)
However, since u is a solution of (1.2), (2.2) and u ( c ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq39_HTML.gif, we know that
( p ( ρ ( c ) ) u Δ ( ρ ( c ) ) ) Δ = f ( ρ ( c ) , 0 ) > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ7_HTML.gif
(2.5)

which contradicts (2.4).

For the case (ii), it can be divided into two cases to consider.
  1. (1)

    If σ ( c ) < σ 2 ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq57_HTML.gif, then one can deduce that u Δ ( c ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq55_HTML.gif, u Δ Δ ( c ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq58_HTML.gif. From this and p ( σ ( c ) ) u Δ Δ ( c ) = f ( c , 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq59_HTML.gif, we have a contradiction.

     
  2. (2)

    If σ ( c ) = σ 2 ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq60_HTML.gif, then we always have σ ( c ) > c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq61_HTML.gif. If u ( σ 2 ( T ) ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq62_HTML.gif, then u Δ ( c ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq47_HTML.gif, u Δ Δ ( ρ ( c ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq56_HTML.gif. Similar to (i), we get (2.2) and (2.3). But this is impossible from (2.4) and (2.5).

     

If u ( σ 2 ( T ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq63_HTML.gif and c = σ ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq64_HTML.gif, then σ ( T ) > T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq65_HTML.gif, u Δ ( c ) = u Δ ( σ ( T ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq66_HTML.gif, u Δ Δ ( ρ ( c ) ) = u Δ Δ ( T ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq67_HTML.gif. So, we get a contradiction to (2.4) and (2.5).

If u ( σ 2 ( T ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq63_HTML.gif and ρ ( c ) < c < σ ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq68_HTML.gif, we have u Δ ( c ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq55_HTML.gif, u Δ Δ ( ρ ( c ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq56_HTML.gif. Hence, we get (2.2) and (2.3). But this contradicts (2.4) and (2.5).

If u ( σ 2 ( T ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq63_HTML.gif and ρ ( c ) = c < σ ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq69_HTML.gif, then u Δ ( c ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq55_HTML.gif. By assumption, u is a solution of (1.2), so we have lim t c u Δ Δ ( t ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq70_HTML.gif, which contradicts u Δ ( c ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq55_HTML.gif and u ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq31_HTML.gif for t [ c δ , c ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq71_HTML.gif. □

Remark 2.4 From the proof of Lemma 2.3, we can easily find that if T = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq21_HTML.gif, then the monotonicity assumption of p ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq12_HTML.gif can be omitted.

By Lemma 2.3, under the hypothesis

(H1) p ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq12_HTML.gif is nondecreasing in [ 0 , σ ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq30_HTML.gif, and f ( t , 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq29_HTML.gif for t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq28_HTML.gif,

in order to prove the existence of a positive solution of (1.2), it suffices to consider the existence of a solution of (2.1). Now we establish the corresponding variational formulations for (2.1). We set
E = { u H Δ 1 ( [ 0 , σ 2 ( T ) ] ) T | u ( 0 ) = 0  if  α 2 = 0 , u ( σ 2 ( T ) ) = 0  if  α 4 = 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equk_HTML.gif
then E is a Banach space with the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq72_HTML.gif, and we can find that 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq73_HTML.gif can be taken as an equivalent norm on E. Define the functional I on E as
I ( u ) = 1 2 0 σ 2 ( T ) p ( t ) | u Δ ( t ) | 2 Δ t + 1 2 0 σ ( T ) q ( t ) | u σ ( t ) | 2 Δ t 0 σ ( T ) F ( t , ( u + ) σ ( t ) ) Δ t 0 σ ( T ) f ( t , 0 ) ( u ) σ ( t ) Δ t + 1 2 β 1 p ( 0 ) u 2 ( 0 ) + 1 2 β 2 p ( σ ( T ) ) u 2 ( σ 2 ( T ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equl_HTML.gif

where F ( t , ξ ) = 0 ξ f ( t , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq74_HTML.gif.

Note that the appearance of the term 0 σ ( T ) f ( t , 0 ) ( u ) σ ( t ) Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq75_HTML.gif in the functional I guarantees that I is C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq76_HTML.gif, see next lemma. By the definition of Fréchet derivative and the fact that the immersion H Δ 1 ( [ 0 , σ 2 ( T ) ] T ) C ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq77_HTML.gif is compact, we have the following results.

Lemma 2.5 The following statements are valid.
  1. (i)
    I C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq78_HTML.gif, and for every u , v E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq79_HTML.gif,
    I ( u ) v = ( u , v ) 0 0 σ ( T ) f ( t , ( u + ) σ ( t ) ) v σ ( t ) Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equm_HTML.gif
     
  2. (ii)
    We define
    J ( u ) = 0 σ ( T ) F ( t , ( u + ) σ ( t ) ) Δ t + 0 σ ( T ) f ( t , 0 ) ( u ) σ ( t ) Δ t , u E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equn_HTML.gif
     
Then
J ( u ) v = 0 σ ( T ) f ( t , ( u + ) σ ( t ) ) v σ ( t ) Δ t , u , v E , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equo_HTML.gif
J is weakly continuous in E and J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq80_HTML.gif is compact.
  1. (iii)

    The solutions of (2.1) match up to the critical points of I in E.

     
For the eigenvalue problem
{ ( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = λ u σ ( t ) , t [ 0 , T ] T , α 1 u ( 0 ) α 2 u Δ ( 0 ) = 0 , α 3 u ( σ 2 ( T ) ) + α 4 u Δ ( σ ( T ) ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ8_HTML.gif
(2.6)

we have the following lemma.

Lemma 2.6 [[14], Lemma 3.1]

The eigenvalues of (2.6) may be arranged as 0 < λ 1 < λ 2 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq81_HTML.gif , and there exists a countable orthonormal basis of E consisting of eigenfunction associated eigenvalues of (2.6) and
λ 1 = inf u E , u 0 u 0 2 0 σ ( T ) | u σ ( t ) | 2 Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ9_HTML.gif
(2.7)
Remark 2.7 By (2.7) and Lemma 2.2, we know the eigenfunction φ 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq82_HTML.gif corresponding to the eigenvalue λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq83_HTML.gif satisfies φ 1 ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq84_HTML.gif for t ( 0 , σ 2 ( T ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq37_HTML.gif. Furthermore, by the Krein-Rutman theorem [[19], Theorem 7.C], we know φ 1 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq85_HTML.gif with
K = { u E | u ( t ) > 0  for  t ( 0 , σ 2 ( T ) ) T , u Δ ( 0 ) > 0  if  α 1 > 0 , u ( 0 ) > 0  if  α 1 = 0 , u Δ ( σ ( T ) ) < 0  if  α 3 > 0 , u ( σ 2 ( T ) ) > 0  if  α 3 = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equp_HTML.gif

Lemma 2.8 [[1], Theorem 4.73], [8]

The problem (1.2) has the Green function
G q ( t , s ) = { 1 w φ ( t ) ψ ( σ ( s ) ) , t s , 1 w φ ( σ ( s ) ) ψ ( t ) , t σ ( s ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equq_HTML.gif
where w = p ( t ) [ φ Δ ( t ) ψ ( t ) φ ( t ) ψ Δ ( t ) ] = const > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq86_HTML.gif, ψ, φ are solutions of
( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = 0 for t [ 0 , T ] T , u ( 0 ) = α 2 , u Δ ( 0 ) = α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equr_HTML.gif
and
( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = 0 for t [ 0 , T ] T , u ( σ 2 ( T ) ) = α 4 , u Δ ( σ ( T ) ) = α 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equs_HTML.gif

respectively, and satisfy φ ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq87_HTML.gif, t ( 0 , σ 2 ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq88_HTML.gif, φ Δ ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq89_HTML.gif, t [ 0 , σ ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq90_HTML.gif, ψ ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq91_HTML.gif, t [ 0 , σ 2 ( T ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq92_HTML.gif, ψ Δ ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq93_HTML.gif, t [ 0 , σ ( T ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq94_HTML.gif.

Lemma 2.9 The function defined by
Γ ( t , s ) = G q ( t , s ) φ 1 σ ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equt_HTML.gif

belongs to L ( [ 0 , σ ( T ) ] T × [ 0 , σ ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq95_HTML.gif, where φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq96_HTML.gif is given in Remark  2.7.

Proof Clearly, Γ ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq97_HTML.gif is well defined in [ 0 , σ ( T ) ] T × ( 0 , σ ( T ) ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq98_HTML.gif.

If α 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq99_HTML.gif, by φ 1 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq85_HTML.gif and α 1 φ 1 ( 0 ) α 2 φ 1 Δ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq100_HTML.gif, we have φ 1 ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq101_HTML.gif. Hence φ 1 ( σ ( 0 ) ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq102_HTML.gif.

If α 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq103_HTML.gif, then φ 1 ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq104_HTML.gif. By Remark 2.7, we know φ 1 Δ ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq105_HTML.gif. If σ ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq106_HTML.gif, then u σ ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq107_HTML.gif. If σ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq108_HTML.gif, then there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq44_HTML.gif such that φ 1 ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq84_HTML.gif, φ 1 Δ ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq109_HTML.gif for t ( 0 , δ ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq110_HTML.gif, then by L’Hôspital rule [[1], Theorem 1.119] and Lemma 2.8, we know
< lim δ 0 + inf s ( 0 , δ ) T φ Δ ( s ) φ 1 Δ ( s ) lim δ 0 + inf s ( 0 , δ ) T φ ( s ) φ 1 ( s ) lim δ 0 + sup s ( 0 , δ ) T φ ( s ) φ 1 ( s ) lim δ 0 + sup s ( 0 , δ ) T φ Δ ( s ) φ 1 Δ ( s ) < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equu_HTML.gif

Hence, Γ ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq97_HTML.gif is bounded for s close to 0.

Similarly, we can derive that Γ ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq97_HTML.gif is bounded for s close to σ ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq111_HTML.gif. □

In order to derive the main result, we list the following well-known mountain pass lemma.

Lemma 2.10 [[20], Theorem 6.1]

Suppose I C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq112_HTML.gif satisfies the P.S. condition. Suppose I ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq113_HTML.gif and
  1. (i)

    there exist ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq114_HTML.gif, α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq115_HTML.gif such that I ( u ) α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq116_HTML.gif for u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq117_HTML.gif with u = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq118_HTML.gif;

     
  2. (ii)

    there is u 1 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq119_HTML.gif such that u 1 > ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq120_HTML.gif and I ( u 1 ) < α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq121_HTML.gif.

     
Define
P = { p C 0 ( [ 0 , 1 ] ; E ) ; p ( 0 ) = 0 , p ( 1 ) = u 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equv_HTML.gif

Then β = inf p P sup u p I ( u ) α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq122_HTML.gif is a critical value.

3 Main results

In this section, we establish some existence criteria of a positive solution of (1.1) by employing the sub-supersolution method and critical point theory.

First, using a method analogous to that in [21], we construct a supersolution to employ the sub-supersolution method.

Theorem 3.1 Assume that (H1) holds and there are constants a 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq123_HTML.gif and a 2 a 1 a μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq124_HTML.gif such that
f ( t , ξ ) < μ ξ + a 1 , t [ 0 , T ] T , ξ [ 0 , a 2 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equw_HTML.gif
where μ < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq125_HTML.gif is fixed, a μ = u μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq126_HTML.gif, u μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq127_HTML.gif represents the unique positive solution of
{ ( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = μ u σ ( t ) + 1 , t [ 0 , T ] T , α 1 u ( 0 ) α 2 u Δ ( 0 ) = 0 , α 3 u ( σ 2 ( T ) ) + α 4 u Δ ( σ ( T ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equx_HTML.gif

Then the problem (1.2) has at least one positive solution.

Proof For fixed μ < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq125_HTML.gif, let v be the unique positive solution of
{ ( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = μ u σ ( t ) + a 1 , t [ 0 , T ] T , α 1 u ( 0 ) α 2 u Δ ( 0 ) = 0 , α 3 u ( σ 2 ( T ) ) + α 4 u Δ ( σ ( T ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ10_HTML.gif
(3.1)

Then, from the definition of u μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq127_HTML.gif, we have v a 1 a μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq128_HTML.gif. Then, by the assumptions, it is easy to see that v is a supersolution of (1.2). In addition, condition (H1) guarantees that the constant function 0 is a strict subsolution of (1.2). Therefore, the sub-supersolution method implies (1.2) has a positive solution u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif. □

Remark 3.2 Furthermore, by Lemma 2.8, we know
a μ 0 σ ( T ) max t [ 0 , T ] T | G q μ ( t , s ) | Δ s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equy_HTML.gif

Theorem 3.3 Under the hypothesis of Theorem  3.1 and suppose the condition

(H2) lim inf ξ + f ( t , ξ ) ξ > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq130_HTML.gif uniformly for t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq28_HTML.gif

holds, then the problem (1.2) has at least two positive solutions.

In order to prove this theorem, we first present some necessary lemmas.

Lemma 3.4 Let v, u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif be given in the proof of Theorem  3.1, then u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif is a local minimizer of I in E.

Proof Denote
W = { u C 1 ( [ 0 , σ ( T ) ] T , R ) | α 1 u ( 0 ) α 2 u Δ ( 0 ) = 0 , α 3 u ( σ 2 ( T ) ) + α 4 u Δ ( σ ( T ) ) = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equz_HTML.gif

By the assumptions and Lemma 2.8, we easily find v u 1 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq131_HTML.gif, u 1 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq132_HTML.gif. Hence u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif is a local minimizer of I in W.

Next, by a similar argument to that in [22], we assert that u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif is also a local minimizer of I in E.

In fact, if u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif is not a local minimizer of I in E, then for every ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq133_HTML.gif there is v ε E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq134_HTML.gif such that 0 < v ε ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq135_HTML.gif, I ( u 1 + v ε ) < I ( u 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq136_HTML.gif and I ( u 1 + v ε ) = inf v E , v ε I ( v + u 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq137_HTML.gif. By the Lagrange multiplier rule, we know there exists a constant μ ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq138_HTML.gif such that
I ( u 1 + v ε ) φ = μ ε ( v ε , φ ) 0 for every  φ E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ11_HTML.gif
(3.2)
Note that u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif is a solution of (1.2), so
I ( u 1 + v ε ) φ = ( u 1 + v ε , φ ) 0 0 σ ( T ) f ( t , u 1 σ + v ε σ ) φ σ Δ t = ( v ε , φ ) 0 0 σ ( T ) f ( t , u 1 σ + v ε σ ) φ σ Δ t + 0 σ ( T ) f ( t , u 1 σ ) φ σ Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equaa_HTML.gif
Thus, from (3.2), we have
( 1 μ ε ) ( v ε , φ ) 0 0 σ ( T ) f ( t , u 1 σ + v ε σ ) φ σ Δ t + 0 σ ( T ) f ( t , u 1 σ ) φ σ Δ t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equab_HTML.gif
Therefore, v ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq139_HTML.gif is a solution of
{ ( 1 μ ε ) ( p ( t ) u Δ ( t ) ) Δ + q ( t ) u σ ( t ) = f ( t , u σ ( t ) + u 1 σ ( t ) ) f ( t , u 1 σ ( t ) ) , t [ 0 , T ] T , α 1 u ( 0 ) α 2 u Δ ( 0 ) = 0 , α 3 u ( σ 2 ( T ) ) + α 4 u Δ ( σ ( T ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ12_HTML.gif
(3.3)

It is easy to show that v ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq140_HTML.gif as ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq141_HTML.gif in C 1 ( [ 0 , σ ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq142_HTML.gif. But this contradicts the fact that u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif is a local minimizer of I in W. □

Next, under hypothesis (H2), in order to show the existence of the second positive solution of (1.2) by employing the mountain pass lemma, we need to show that I satisfies the P.S. condition. However, by (H2), we cannot justify this; therefore, we consider the truncation function f n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq143_HTML.gif and the truncation functional I n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq144_HTML.gif defined as follows.

Let γ > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq145_HTML.gif and { ξ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq146_HTML.gif be an increasing positive sequence with ξ n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq147_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq148_HTML.gif. For n = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq149_HTML.gif , define
f n ( t , ξ ) = { f ( t , 0 ) , ξ < 0 , f ( t , ξ ) , 0 ξ ξ n , γ ( ξ ξ n ) + f ( t , ξ n ) , ξ > ξ n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equac_HTML.gif
and
I n ( u ) = 1 2 u 0 2 0 σ ( T ) F n ( t , ( u + ) σ ( t ) ) Δ t 0 σ ( T ) f ( t , 0 ) ( u ) σ ( t ) Δ t , u E , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equad_HTML.gif
where F n ( t , s ) = 0 s f n ( t , ξ ) d ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq150_HTML.gif. Then
I n ( u ) v = ( u , v ) 0 0 σ ( T ) f n ( t , ( u + ) σ ( t ) ) v σ ( t ) Δ t , u , v E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equae_HTML.gif

Lemma 3.5 Assume that (H1) and (H2) hold, then there exists n 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq151_HTML.gif such that the functional I n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq144_HTML.gif satisfies the P.S. condition in E for n > n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq152_HTML.gif.

Proof For given n, let { u m } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq153_HTML.gif be the P.S. sequence of I n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq144_HTML.gif, that is, { I n ( u m ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq154_HTML.gif is bounded and I n ( u m ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq155_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq156_HTML.gif. If { u m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq157_HTML.gif is bounded, one can deduce that I satisfies the P.S. condition by a similar proof to Proposition B.35 in [23].

Suppose that { u m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq157_HTML.gif is unbounded. Since
u m 0 2 0 σ ( T ) f ( t , 0 ) ( u m ) σ ( t ) Δ t + ( u m , u m ) 0 = I n ( u m ) u m I n ( u m ) u m 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equaf_HTML.gif
we have u m 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq158_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq156_HTML.gif. Denote v m = u m u m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq159_HTML.gif, then v m 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq160_HTML.gif. So, without loss of generality, we can assume that v m v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq161_HTML.gif in E, v m v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq162_HTML.gif in C ( [ 0 , σ 2 ( T ) ] T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq163_HTML.gif. Note that
I n ( u m ) ( v m v ) u m 0 = ( v m , v m v ) 0 1 u m 0 0 σ ( T ) f n ( t , ( u m + ) σ ( t ) ) ( v m v ) σ ( t ) Δ t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ13_HTML.gif
(3.4)

by the definition of f n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq143_HTML.gif and passing to limit in (3.4), one can derive that v 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq164_HTML.gif.

In view of (H2) and the definition of f n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq143_HTML.gif, for ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq133_HTML.gif small enough, there exist ξ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq165_HTML.gif independent of n, n 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq151_HTML.gif such that
f n ( t , ξ ) > ( λ 1 + ε ) ξ for  ξ > ξ n > n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ14_HTML.gif
(3.5)
Since ( u m σ , φ 1 ) 0 = λ 1 0 σ ( T ) u m σ ( t ) φ 1 σ ( t ) Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq166_HTML.gif, then
ε 0 σ ( T ) v m σ ( t ) φ 1 σ ( t ) Δ t = 1 u m 0 ( 0 σ ( T ) ( λ 1 + ε ) u m σ ( t ) φ 1 σ ( t ) Δ t ( u m , φ 1 ) 0 ) = 1 u m 0 ( 0 σ ( T ) f n ( t , ( u m + ) σ ( t ) ) φ 1 σ ( t ) + ( λ 1 + ε ) ( u m + ) σ ( t ) φ 1 σ ( t ) Δ t I n ( u m ) φ 1 + 0 σ ( T ) ( λ 1 + ε ) ( u m ) σ ( t ) φ 1 σ ( t ) Δ t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ15_HTML.gif
(3.6)

Passing to the limit in (3.6), we know ε 0 σ ( T ) v σ ( t ) φ 1 σ ( t ) Δ t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq167_HTML.gif. Hence v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq168_HTML.gif, which contradicts v 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq164_HTML.gif. Therefore { u m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq157_HTML.gif is bounded. So, I n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq144_HTML.gif satisfies the P.S. condition for n > n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq152_HTML.gif. □

By Lemmas 3.4 and 3.5, we deduce that for n large enough, I n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq144_HTML.gif has a nontrivial critical point w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq169_HTML.gif by using the mountain pass lemma and Theorem 1 in [24]. In order to obtain a solution of (1.2), we need to get an estimate of w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq169_HTML.gif. Therefore, we first give an estimate of a nonnegative solution of (1.2) employing a method similar to that in [25].

Lemma 3.6 Suppose (H2) holds, then there is M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq170_HTML.gif such that for any nonnegative solution u of (1.2), we have u M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq171_HTML.gif.

Proof If u is a nonnegative solution of (1.2), then by the definition of φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq96_HTML.gif, we have
λ 1 0 σ ( T ) u σ ( t ) φ 1 σ ( t ) Δ t = 0 σ 2 ( T ) p ( t ) u Δ ( t ) φ 1 Δ ( t ) Δ t + 0 σ ( T ) q ( t ) u σ ( t ) φ 1 σ ( t ) Δ t + 1 2 β 1 p ( 0 ) u ( 0 ) φ 1 ( 0 ) + 1 2 β 2 p ( σ ( T ) ) u ( σ 2 ( T ) ) φ 1 ( σ 2 ( T ) ) = 0 σ ( T ) f ( t , u σ ( t ) ) φ 1 σ ( t ) Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ16_HTML.gif
(3.7)
Condition (H2) implies there exist ρ > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq172_HTML.gif, C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq173_HTML.gif such that
f ( t , ξ ) ρ ξ C , t [ 0 , T ] , ξ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equ17_HTML.gif
(3.8)
Hence, from (3.7) and (3.8), we derive that
0 σ ( T ) u σ ( t ) φ 1 σ ( t ) Δ t C ρ λ 1 0 σ ( T ) φ 1 σ ( t ) Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equag_HTML.gif
So, using (3.7), we know
0 σ ( T ) f ( t , u σ ( t ) ) φ 1 σ ( t ) Δ t λ 1 C ρ λ 1 0 σ ( T ) φ 1 σ ( t ) Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equah_HTML.gif
Thus, by (H2), we know 0 σ ( T ) | f ( t , u σ ( t ) ) | φ 1 σ ( t ) Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq174_HTML.gif is uniformly bounded. Note that by Lemma 2.9, for t [ 0 , σ 2 ( T ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq175_HTML.gif,
| u ( t ) | = | 0 σ ( T ) G q ( t , s ) f ( s , u σ ( s ) ) Δ s | Γ 0 σ ( T ) | f ( s , u σ ( s ) ) | φ 1 σ ( s ) Δ s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_Equai_HTML.gif

Hence, there exists M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq170_HTML.gif such that u M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq171_HTML.gif. □

Remark 3.7 Note that we only need (H2) to derive (3.8). Hence, (3.5) implies Lemma 3.6 is also valid for the truncation problem.

Proof of Theorem 3.3 Since the positive solution u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif derived from Theorem 3.1 is a local minimizer of I and 0 < u 1 v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq176_HTML.gif, we can choose n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq177_HTML.gif large enough such that I ( u 1 ) = I n ( u 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq178_HTML.gif for every n > n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq152_HTML.gif. Hence, u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq129_HTML.gif is also a local minimizer of I n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq144_HTML.gif for n > n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq152_HTML.gif. Then, from the definition of f n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq143_HTML.gif and Lemma 3.5, we know the mountain pass lemma and Theorem 1 in [24] imply that I n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq144_HTML.gif has the second critical point u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq179_HTML.gif. Furthermore, by Remark 3.7, we know u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq179_HTML.gif is also a critical point of I. Thus the problem (1.2) has the second positive solution u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq179_HTML.gif. □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The research of HR Sun has been supported by the program for New Century Excellent Talents in University (NECT-12-0246) and FRFCU (lzujbky-2013-k02).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Lanzhou University
(2)
Center of Teaching Guidance, Gansu Radio and TV University

References

  1. Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Boston; 2001.View Article
  2. Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2003.View Article
  3. Hilger, S: Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. thesis, Universität Würzburg (1988) (in German)
  4. Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods. J. Math. Anal. Appl. 2007, 331: 1263–1274. 10.1016/j.jmaa.2006.09.051MathSciNetView Article
  5. Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions of singular Dirichlet problems on time scales via variational methods. Nonlinear Anal. 2007, 67: 368–381. 10.1016/j.na.2006.05.014MathSciNetView Article
  6. Anderson D: Solutions to second-order three-points problems on time scales. J. Differ. Equ. Appl. 2002, 8: 673–688. 10.1080/1023619021000000717View Article
  7. Atici EM, Guseinov GS: On Green’s functions and positive solutions for boundary value problems on time scales. J. Comput. Appl. Math. 2002, 141: 75–99. 10.1016/S0377-0427(01)00437-XMathSciNetView Article
  8. Erbe L, Peterson A, Mathsen R: Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain. J. Comput. Appl. Math. 2000, 113: 365–380. 10.1016/S0377-0427(99)00267-8MathSciNetView Article
  9. Rynne BP: L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq180_HTML.gif spaces and boundary value problems on time scales. J. Math. Anal. Appl. 2007, 328: 1217–1236. 10.1016/j.jmaa.2006.06.008MathSciNetView Article
  10. Sun HR: Triple positive solutions for p -Laplacian m -point boundary value problem on time scales. Comput. Math. Appl. 2009, 58: 1736–1741. 10.1016/j.camwa.2009.07.083MathSciNetView Article
  11. Sun HR, Li WT: Positive solutions for nonlinear three-point boundary value problems on time scales. J. Math. Anal. Appl. 2004, 299: 508–524. 10.1016/j.jmaa.2004.03.079MathSciNetView Article
  12. Sun HR, Li WT: Existence theory for positive solutions to one-dimensional p -Laplacian boundary value problems on time scales. J. Differ. Equ. 2007, 240: 217–248. 10.1016/j.jde.2007.06.004View Article
  13. Jiang L, Zhou Z: Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales. Nonlinear Anal. 2008, 69: 1376–1388. 10.1016/j.na.2007.06.034MathSciNetView Article
  14. Zhang QG, Sun HR: Variational approach for Sturm-Liouville boundary value problems on time scales. J. Appl. Math. Comput. 2011, 36: 219–232. 10.1007/s12190-010-0398-3MathSciNetView Article
  15. Bonanno G, Candito P: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ. 2008, 244: 3031–3059. 10.1016/j.jde.2008.02.025MathSciNetView Article
  16. Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Basic properties of Sobolev’s spaces on time scales. Adv. Differ. Equ. 2006., 2006: Article ID 38121
  17. Guseinov GS: Integration on time scales. J. Math. Anal. Appl. 2003, 285: 107–127. 10.1016/S0022-247X(03)00361-5MathSciNetView Article
  18. Cabada A, Vivero DR: Criterions for absolute continuity on time scales. J. Differ. Equ. Appl. 2005, 11: 1013–1028. 10.1080/10236190500272830MathSciNetView Article
  19. Zeidler E: Nonlinear Functional Analysis and Its Application I: Fixed-Point Theorems. Springer, New York; 1985.View Article
  20. Struwe M: Variational Methods. Springer, Berlin; 1990.View Article
  21. DE Figueiredo DG, Lions PL: On pairs of positive solutions for a class of semilinear elliptic problems. Indiana Univ. Math. J. 1985, 34: 591–606. 10.1512/iumj.1985.34.34031MathSciNetView Article
  22. Brezis H, Nirenberg L: H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq181_HTML.gif versus C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-123/MediaObjects/13661_2013_Article_372_IEq76_HTML.gif local minimizers. C. R. Acad. Sci., Sér. 1 Math. 1993, 317: 465–472.MathSciNet
  23. Rabinowitz PH CBMS Reg. Conf. Ser. Math. 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.
  24. Ghoussoub N, Preiss D: A general mountain pass principle for locating and classifying critical points. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1989, 6: 321–330.MathSciNet
  25. Mawhin J, Omana W: A priori bounds and existence of positive solutions for some Sturm-Liouville superlinear boundary value problems. Funkc. Ekvacioj 1992, 35: 333–342.MathSciNet

Copyright

© Zhang et al.; 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.