Open Access

Solutions to a boundary value problem of a fourth-order impulsive differential equation

Boundary Value Problems20132013:154

DOI: 10.1186/1687-2770-2013-154

Received: 26 March 2013

Accepted: 15 June 2013

Published: 1 July 2013

Abstract

This paper is concerned with the existence of solutions to a boundary value problem of a fourth-order impulsive differential equation with a control parameter λ. By employing some existing critical point theorems, we find the range of the control parameter in which the boundary value problem admits at least one solution. It is also shown that under certain conditions there exists an interval of the control parameter in which the boundary value problem possesses infinitely many solutions. The main results are also demonstrated with examples.

MSC:34B15, 34B18, 34B37, 58E30.

Keywords

critical point theorem impulsive differential equations boundary value problem

1 Introduction

Fourth-order two-point boundary value problems of ordinary differential equations are widely employed by engineers to describe the beam deflection with two simply supported ends [13]. One example is the following fourth-order two-point boundary value problem:
{ u ( i v ) ( t ) + A u ( t ) + B u ( t ) = λ f ( t , u ( t ) ) , t [ 0 , 1 ] , u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ1_HTML.gif
(1.1)
where u ( i v ) ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq1_HTML.gif, u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq2_HTML.gif, u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq3_HTML.gif are the fourth, third, and second derivatives of u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq4_HTML.gif with respect to t, respectively, f C ( [ 0 , 1 ] × R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq5_HTML.gif, A and B are two real constants. System (1.1) has been studied in [47] and the references therein. For a beam, t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq6_HTML.gif and t = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq7_HTML.gif in (1.1) refer to the two ends of the beam. At other locations of the beam, t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq8_HTML.gif, there may be some sudden changes in loads placed on the beam, or some unexpected forces working on the beam. These sudden changes may result in impulsive effects for the governing differential equation. This motivates us to consider the following boundary value problem for a fourth-order impulsive differential equation:
{ u ( i v ) ( t ) + A u ( t ) + B u ( t ) = λ f ( t , u ( t ) ) , t t j , t [ 0 , 1 ] , Δ u ( t j ) = I 1 j ( u ( t j ) ) , Δ u ( t j ) = I 2 j ( u ( t j ) ) , j = 1 , 2 , , m , u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ2_HTML.gif
(1.2)

where I 1 j , I 2 j C ( R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq9_HTML.gif, 0 = t 0 < t 1 < t 2 < < t m < t m + 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq10_HTML.gif, and the operator Δ is defined as Δ U ( t j ) = U ( t j + ) U ( t j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq11_HTML.gif, where U ( t j + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq12_HTML.gif ( U ( t j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq13_HTML.gif) denotes the right-hand (left-hand) limit of U at t j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq14_HTML.gif and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq15_HTML.gif is referred to as a control parameter.

We are mainly concerned with the existence of solutions of system (1.2). A function u ( t ) C ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq16_HTML.gif is said to be a (classical) solution of (1.2) if u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq4_HTML.gif satisfies (1.2). In literature, tools employed to establish the existence of solutions of impulsive differential equations include fixed point theorems, the upper and lower solutions method, the degree theory, critical point theory and variational methods. See, for example, [820]. In this paper, we establish the existence of solutions of (1.2) by converting the problem to the existence of critical points of some variational structure. In this paper we regard λ as a parameter and find the ranges in which (1.2) admits at least one and infinitely many solutions, respectively. Note that when λ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq17_HTML.gif system (1.2) reduces to the one studied in [21]. Our results extend those ones in [21].

The rest of this paper is organized as follows. In Section 2 we present some preliminary results. Our main results and their proofs are given in Section 3.

2 Preliminaries

Throughout we assume that A and B satisfy
A 0 B . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ3_HTML.gif
(2.1)
Let
H 0 1 ( [ 0 , 1 ] ) = { u L 2 ( [ 0 , 1 ] ) : u L 2 ( [ 0 , 1 ] ) , u ( 0 ) = u ( 1 ) = 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equa_HTML.gif
and
H 2 ( [ 0 , 1 ] ) = { u L 2 ( [ 0 , 1 ] ) : u , u L 2 ( [ 0 , 1 ] ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equb_HTML.gif
Take X : = H 2 ( [ 0 , 1 ] ) H 0 1 ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq18_HTML.gif and define
u X = ( 0 1 ( | u ( t ) | 2 A | u | 2 + B | u | 2 ) d t ) 1 2 , u X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ4_HTML.gif
(2.2)
Since A and B satisfy (2.1), it is straightforward to verify that (2.2) defines a norm for the Sobolev space X and this norm is equivalent to the usual norm defined as follows:
u = ( 0 1 u ( t ) 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equc_HTML.gif
It follows from (2.1) that
u u X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equd_HTML.gif
For the norm in C 1 ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq19_HTML.gif,
u = max ( max t [ 0 , 1 ] | u ( t ) | , max t [ 0 , 1 ] | u ( t ) | ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Eque_HTML.gif

we have the following relation.

Lemma 2.1 Let M 1 = 1 + 1 π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq20_HTML.gif. Then u M 1 u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq21_HTML.gif, u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq22_HTML.gif.

Proof The proof follows easily from Wirtinger’s inequality [22], Lemma 2.3 of [23] and Hölder’s inequality. The detailed argument is similar to the proof of Lemma 2.2 in [21], and we thus omit it here. □

Define a functional φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq23_HTML.gif as
φ λ ( u ) = Φ ( u ) λ Ψ ( u ) , u X , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ5_HTML.gif
(2.3)
where
Φ ( u ) = 1 2 u X 2 + j = 1 m 0 u ( t j ) I 1 j ( s ) d s + j = 1 m 0 u ( t j ) I 2 j ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ6_HTML.gif
(2.4)
and
Ψ ( u ) = 0 1 F ( t , u ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ7_HTML.gif
(2.5)
with
F ( t , u ) = 0 u ( t ) f ( t , s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equf_HTML.gif
Note that φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq23_HTML.gif is Fréchet differentiable at any u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq24_HTML.gif, and for any v X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq25_HTML.gif we have
φ λ ( u ) ( v ) = lim h 0 φ λ ( u + h v ) φ λ ( u ) h = 0 1 ( u ( t ) v ( t ) A u ( t ) v ( t ) + B u ( t ) v ( t ) ) d t + j = 1 m I 2 j ( u ( t j ) ) v ( t j ) + j = 1 m I 1 j ( u ( t j ) ) v ( t j ) λ 0 1 f ( t , u ( t ) ) v ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ8_HTML.gif
(2.6)

Next we show that a critical point of the functional φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq23_HTML.gif is a solution of system (1.2).

Lemma 2.2 If u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq24_HTML.gif is a critical point of φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq23_HTML.gif, then u is a solution of system (1.2).

Proof Suppose that u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq24_HTML.gif is a critical point of φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq26_HTML.gif. Then for any v X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq25_HTML.gif one has
λ 0 1 f ( t , u ( t ) ) v ( t ) d t = 0 1 ( u ( t ) v ( t ) A u ( t ) v ( t ) + B u ( t ) v ( t ) ) d t + j = 1 m I 2 j ( u ( t j ) ) v ( t j ) + j = 1 m I 1 j ( u ( t j ) ) v ( t j ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ9_HTML.gif
(2.7)
For j { 1 , 2 , , m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq27_HTML.gif, choose v X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq25_HTML.gif such that v ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq28_HTML.gif for t [ 0 , t j ] [ t j + 1 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq29_HTML.gif, then we have
t j t j + 1 ( u ( i v ) + A u ( t ) + B u ( t ) ) v ( t ) d t = λ t j t j + 1 f ( t , u ( t ) ) v ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equg_HTML.gif
Thus
u ( i v ) + A u ( t ) + B u ( t ) = λ f ( t , u ( t ) ) a.e.  t ( t j , t j + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equh_HTML.gif
Therefore, by (2.7) we have
j = 1 m ( Δ u ( t j ) + I 2 j ( u ( t j ) ) ) v ( t j ) j = 1 m ( Δ u ( t j ) I 1 j ( u ( t j ) ) ) v ( t j ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equi_HTML.gif
Next we show that u satisfies
Δ u ( t j ) = I 2 j ( u ( t j ) ) , j = 1 , 2 , , m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equj_HTML.gif
Suppose on the contrary that there exists some j { 1 , 2 , , m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq30_HTML.gif such that
Δ u ( t j ) + I 2 j ( u ( t j ) ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equk_HTML.gif
Pick
v ( t ) = ( t 3 3 t j 2 t ) i = 0 , i j m + 1 ( 1 3 t 3 1 2 ( t j + t i ) t 2 + t j t i t + 1 6 t i 3 1 2 t j t i 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equl_HTML.gif
then
v ( t ) = 3 ( t 2 t j 2 ) i = 0 , i j m + 1 ( 1 3 t 3 1 2 ( t j + t i ) t 2 + t j t i t + 1 6 t i 3 1 2 t j t i 2 ) + ( t 3 3 t j t ) k = 0 , k j m + 1 { ( t 2 ( t k + t j ) t + t k t j ) i = 0 , i j , k m + 1 ( 1 3 t 3 1 2 ( t j + t i ) t 2 + t j t i t + 1 6 t i 3 1 2 t j t i 2 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equm_HTML.gif
Clearly, v X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq25_HTML.gif. Simple calculations show that v ( t i ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq31_HTML.gif, i = 1 , 2 , , j 1 , j + 1 , , m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq32_HTML.gif, v ( t j ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq33_HTML.gif and v ( t i ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq34_HTML.gif, i = 1 , 2 , , m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq35_HTML.gif. Thus
1 3 t j 3 ( Δ u ( t j ) + I 2 j ( u ( t j ) ) ) i = 0 , i j m + 1 ( t i t j ) 3 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equn_HTML.gif

which is a contradiction. Similarly, one can show that Δ u ( t j ) = I 1 j ( u ( t j ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq36_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq37_HTML.gif. Therefore, u is a solution of (1.2). □

For r 1 , r 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq38_HTML.gif with r 1 < r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq39_HTML.gif, we define
α ( r 1 , r 2 ) = sup v Φ 1 ( ( r 1 , r 2 ) ) Ψ ( v ) sup u Φ 1 ( ( , r 1 ) ) Ψ ( u ) Φ ( v ) r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ10_HTML.gif
(2.8)
and
β ( r 1 , r 2 ) = inf v Φ 1 ( ( r 1 , r 2 ) ) sup u Φ 1 ( ( r 1 , r 2 ) ) Ψ ( u ) Ψ ( v ) r 2 Φ ( v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ11_HTML.gif
(2.9)
For r R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq40_HTML.gif, we define
ρ 1 ( r ) = inf v Φ 1 ( ( , r ) ) sup u Φ 1 ( ( , r ) ) Ψ ( u ) Ψ ( v ) r Φ ( v ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ12_HTML.gif
(2.10)
ρ 2 ( r ) = sup v Φ 1 ( ( r , + ) ) Ψ ( v ) sup u Φ 1 ( ( , r ] ) Ψ ( u ) Φ ( v ) r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ13_HTML.gif
(2.11)

3 Main results

3.1 Existence of at least one solution

In this section we derive conditions under which system (1.2) admits at least one solution. For this purpose, we introduce the following assumption.

(H1) Assume that there exist two positive constants k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq41_HTML.gif and k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq42_HTML.gif such that for each u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq24_HTML.gif
0 j = 1 m 0 u ( t j ) I 1 j ( s ) d s k 1 max j { 1 , 2 , , m } | u ( t j ) | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ14_HTML.gif
(3.1)
and
0 j = 1 m 0 u ( t j ) I 2 j ( s ) d s k 2 max j { 1 , 2 , , m } | u ( t j ) | 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ15_HTML.gif
(3.2)
Let k 0 = 2 A 6 + B 60 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq43_HTML.gif and k 3 = k 0 + k 1 + 1 4 k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq44_HTML.gif with k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq41_HTML.gif given in (3.1) and k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq42_HTML.gif given in (3.2). For constants c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq45_HTML.gif, c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq46_HTML.gif, and c satisfying
c 1 < 2 k 0 M 1 c < 2 k 3 M 1 c < c 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ16_HTML.gif
(3.3)
we define
a ( c 2 , c ) = 0 1 max | u | c 2 F ( t , u ) d t 0 1 F ( t , u 1 ( t ) ) d t c 2 2 2 k 3 M 1 2 c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ17_HTML.gif
(3.4)
and
b ( c 1 , c ) = 0 1 F ( t , u 1 ( t ) ) d t 0 1 max | u | c 1 F ( t , u ) d t 2 k 3 M 1 2 c 2 c 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ18_HTML.gif
(3.5)
where
u 1 ( t ) = c t ( 1 t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ19_HTML.gif
(3.6)

Note that for every c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq47_HTML.gif and t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq48_HTML.gif we have | u 1 ( t ) | = c t ( 1 t ) c 4 < c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq49_HTML.gif. Since A 0 B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq50_HTML.gif, then k 0 > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq51_HTML.gif. Thus, if c and c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq46_HTML.gif satisfy (3.3), then c 2 > c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq52_HTML.gif and 0 1 max | u | c 2 F ( t , u ) d t 0 1 F ( t , u 1 ( t ) ) d t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq53_HTML.gif and hence a ( c 2 , c ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq54_HTML.gif.

Theorem 3.1 Assume that (H1) is satisfied. If there exist constants c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq45_HTML.gif, c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq46_HTML.gif, and c satisfying (3.3) and
0 < a ( c 2 , c ) < b ( c 1 , c ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ20_HTML.gif
(3.7)

then, for each λ ( λ 1 , λ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq55_HTML.gif, system (1.2) admits at least one solution u and u X < c 2 M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq56_HTML.gif, where λ 1 = 1 2 M 1 2 b ( c 1 , c ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq57_HTML.gif and λ 2 = 1 2 M 1 2 a ( c 2 , c ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq58_HTML.gif.

Proof By Lemma 2.2, it suffices to show the functional φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq23_HTML.gif defined in (2.3) has at least one critical point. We prove this by verifying the conditions given in [[10], Theorem 5.1]. Note that Φ defined in (2.4) is a nonnegative Gâteaux differentiable, coercive, and sequentially weakly lower semicontinuous functional, and its Gâteaux derivative admits a continuous inverse on X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq59_HTML.gif. Moreover, Ψ defined in (2.5) is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Set
r 1 = c 1 2 2 M 1 2 , r 2 = c 2 2 2 M 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equo_HTML.gif
Note that u 1 ( t ) = c t ( 1 t ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq60_HTML.gif. It then follows from (H1) that
Φ ( u 1 ) = 1 2 u 1 X 2 + j = 1 m 0 u 1 ( t j ) I 1 j ( s ) d s + j = 1 m 0 u 1 ( t j ) I 2 j ( s ) d s = ( 2 A 6 + B 60 ) c 2 + j = 1 m 0 c 2 c t j I 1 j ( s ) d s + j = 1 m 0 c t j ( 1 t j ) I 2 j ( s ) d s = k 0 c 2 + j = 1 m 0 c 2 c t j I 1 j ( s ) d s + j = 1 m 0 c t j ( 1 t j ) I 2 j ( s ) d s k 0 c 2 + k 1 max j | c 2 c t j | 2 + k 2 max j | c t j ( 1 t j ) | 2 k 0 c 2 + k 1 c 2 + 1 4 k 2 c 2 = k 3 c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ21_HTML.gif
(3.8)
and
Φ ( u 1 ) 1 2 u 1 X 2 = k 0 c 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ22_HTML.gif
(3.9)
By (3.3) we have
r 1 = c 1 2 2 M 1 2 < k 0 c 2 Φ ( u 1 ) k 3 c 2 < c 2 2 2 M 1 2 = r 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equp_HTML.gif
For u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq24_HTML.gif satisfying Φ ( u ) < r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq61_HTML.gif, by Lemma 2.1, one has
| u | 2 u 2 M 1 2 u X 2 2 M 1 2 Φ ( u ) < 2 M 1 2 r 2 = c 2 2 , t [ 0 , 1 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equq_HTML.gif
which implies that
Ψ ( u ) = 0 1 F ( t , u ( t ) ) d t 0 1 max | u | c 2 F ( t , u ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equr_HTML.gif
Hence
sup u Φ 1 ( ( r 1 , r 2 ) ) Ψ ( u ) sup u Φ 1 ( ( , r 2 ) ) Ψ ( u ) 0 1 max | u | c 2 F ( t , u ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ23_HTML.gif
(3.10)
For u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq24_HTML.gif with Φ ( u ) < r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq62_HTML.gif, one can similarly obtain
sup u Φ 1 ( ( , r 1 ) ) Ψ ( u ) 0 1 max | u | c 1 F ( t , u ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ24_HTML.gif
(3.11)
It follows from the definition of β ( r 1 , r 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq63_HTML.gif that
β ( r 1 , r 2 ) sup u Φ 1 ( ( , r 2 ) ) Ψ ( u ) Ψ ( u 1 ) r 2 Φ ( u 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ25_HTML.gif
(3.12)
Note that Φ ( u 1 ) < r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq64_HTML.gif. By (3.10) one has
β ( r 1 , r 2 ) 0 1 max | u | c 2 F ( t , u ) d t 0 1 F ( t , u 1 ( t ) ) d t r 2 Φ ( u 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equs_HTML.gif
Making use of Φ ( u 1 ) < r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq64_HTML.gif, 0 1 max | u | c 2 F ( t , u ) d t 0 1 F ( t , u 1 ( t ) ) d t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq65_HTML.gif, and (3.8), we obtain
β ( r 1 , r 2 ) 0 1 max | u | c 2 F ( t , u ) d t 0 1 F ( t , u 1 ( t ) ) d t c 2 2 2 M 1 2 k 3 c 2 = 2 M 1 2 0 1 max | u | c 2 F ( t , u ) d t 0 1 F ( t , u 1 ( t ) ) d t c 2 2 2 k 3 M 1 2 c 2 = 2 M 1 2 a ( c 2 , c ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equt_HTML.gif
By (2.8), and note that Φ ( u 1 ) > r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq66_HTML.gif, one has
α ( r 1 , r 2 ) Ψ ( u 1 ) sup u Φ 1 ( ( , r 1 ) ) Ψ ( u ) Φ ( u 1 ) r 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equu_HTML.gif
By (3.11) we have
α ( r 1 , r 2 ) 0 1 F ( t , u 1 ( t ) ) d t 0 1 max | u | c 1 F ( t , u ) d t Φ ( u 1 ) r 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equv_HTML.gif
Note that (3.7) implies that
0 1 F ( t , u 1 ( t ) ) d t 0 1 max | u | c 1 F ( t , u ) d t > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equw_HTML.gif
which, together with (3.8), gives
α ( r 1 , r 2 ) 0 1 F ( t , u 1 ( t ) ) d t 0 1 max | u | c 1 F ( t , u ) d t k 3 c 2 r 1 = 2 M 1 2 0 1 F ( t , u 1 ( t ) ) d t 0 1 max | u | c 1 F ( t , u ) d t 2 k 3 M 1 2 c 2 c 1 2 = 2 M 1 2 b ( c 1 , d ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equx_HTML.gif

Therefore, β ( r 1 , r 2 ) 2 M 1 2 a ( c 2 , c ) < 2 M 1 2 b ( c 1 , d ) α ( r 1 , r 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq67_HTML.gif. Thus all the conditions in [[10], Theorem 5.1] are verified, and hence for each λ ( λ 1 , λ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq68_HTML.gif the functional φ λ = Φ λ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq69_HTML.gif admits at least one critical point u such that r 1 < Φ ( u ) < r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq70_HTML.gif. Consequently, system (1.2) admits at least one solution u and u X < c 2 M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq56_HTML.gif. □

In particular, if we take c 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq71_HTML.gif, then (3.4) and (3.5) become
a ( c 2 , c ) = 0 1 max | u | c 2 F ( t , u ) d t 0 1 F ( t , u 1 ( t ) ) d t c 2 2 2 k 3 M 1 2 c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equy_HTML.gif
and
b ( c 1 , c ) = b ( 0 , c ) = 0 1 F ( t , u 1 ( t ) ) d t 2 k 3 M 1 2 c 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equz_HTML.gif
Correspondingly, conditions (3.3) and (3.7) reduce to
2 k 3 M 1 c < c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ26_HTML.gif
(3.13)
and
0 1 max | u | c 2 F ( t , u ) d t < c 2 2 2 k 3 M 1 2 c 2 0 1 F ( t , u 1 ( t ) ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ27_HTML.gif
(3.14)
If (3.13) and (3.14) hold, then
λ 1 = k 3 c 2 0 1 F ( t , u 1 ( t ) ) d t : = λ ˆ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equaa_HTML.gif
and
λ 2 = c 2 2 2 k 3 M 1 2 c 2 2 M 1 2 ( 0 1 max | u | c 2 F ( t , u ) d t 0 1 F ( t , u 1 ( t ) ) d t ) c 2 2 2 M 1 2 0 1 max | u | c 2 F ( t , u ) d t : = λ ˆ 2 > λ ˆ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equab_HTML.gif

As a consequence, we have the following result.

Corollary 3.2 Assume that (H1) is satisfied. If there exist two constants c and c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq46_HTML.gif satisfying (3.13) and (3.14), then for each λ ( λ ˆ 1 , λ ˆ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq72_HTML.gif system (1.2) admits at least one nontrivial solution u.

Example 3.1 Consider the boundary value problem
{ u ( i v ) ( t ) = λ t , t t 1 , t [ 0 , 1 ] , Δ u ( t 1 ) = 1 4 u ( t 1 ) , t 1 = 1 2 , Δ u ( t 1 ) = 1 5 u ( t 1 ) , t 1 = 1 2 , u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ28_HTML.gif
(3.15)

Here, f ( t , u ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq73_HTML.gif, I 11 ( s ) = 1 4 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq74_HTML.gif, I 21 ( s ) = 1 5 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq75_HTML.gif, A = B = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq76_HTML.gif and m = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq77_HTML.gif. It is easy to verify that (H1) is satisfied with k 1 = 1 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq78_HTML.gif and k 2 = 1 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq79_HTML.gif. Direct calculations give F ( t , u ) = t u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq80_HTML.gif, k 0 = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq81_HTML.gif, k 3 = 43 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq82_HTML.gif and M 1 = 1 + 1 π 1.318 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq83_HTML.gif. Let c 1 = 0.01 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq84_HTML.gif, c = 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq85_HTML.gif, c 2 = 1 , 500 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq86_HTML.gif, then c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq45_HTML.gif, c, c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq46_HTML.gif satisfy (3.3) and a ( c 2 , c ) 3.33 × 10 4 < b ( c 1 , c ) 9.25 × 10 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq87_HTML.gif. Thus, it follows from Theorem 3.1 that system (3.15) has at least one solution for λ ( λ 1 , λ 2 ) = ( 311.2 , 864.4 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq88_HTML.gif.

3.2 Existence of infinitely many solutions

In this section, we derive some conditions under which system (1.2) admits infinitely many distinct solutions. To this end, we need the following assumptions.

(H2) Assume that
{ t 1 , t 2 , , t m } [ 1 4 , 3 4 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equac_HTML.gif
(H3) Assume that
F ( t , u ) 0 , for  ( t , u ) ( [ 0 , 1 4 ] [ 3 4 , 1 ] ) × R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equad_HTML.gif
Let k 4 = 2 , 048 ( 3 8 9 10 4 4 A + 79 14 4 8 B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq89_HTML.gif and k 5 = k 4 + k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq90_HTML.gif with k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq42_HTML.gif given in (3.2). We define
γ 1 : = lim inf r + ρ 1 ( r ) , γ 2 : = lim inf r ( inf X Φ ) + ρ 1 ( r ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ29_HTML.gif
(3.16)
where ρ 1 ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq91_HTML.gif is given (2.10). Let
μ 1 = 2 M 1 2 lim inf ξ + 0 1 max | u | ξ F ( t , u ) d t ξ 2 , μ 2 = 1 k 5 lim sup ξ + 1 4 3 4 F ( t , ξ ) d t ξ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equae_HTML.gif
Theorem 3.3 Assume that (H1), (H2), and (H3) are satisfied. If
μ 1 < μ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ30_HTML.gif
(3.17)

holds, then for each λ ( 1 μ 2 , 1 μ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq92_HTML.gif system (1.2) has an unbounded sequence of solutions in X.

Proof We apply [[5], Theorem 2.1] to show that the functional φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq23_HTML.gif defined in (2.3) has an unbounded sequence of critical points.

We first show that γ 1 < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq93_HTML.gif. Let { ξ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq94_HTML.gif be a sequence of positive numbers such that ξ n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq95_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq96_HTML.gif and
lim n + 0 1 max | u | ξ n F ( t , u ) d t ξ n 2 = lim inf ξ + 0 1 max | u | ξ F ( t , ξ ) d t ξ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equaf_HTML.gif
For any positive integer n, we let r n = ξ n 2 2 M 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq97_HTML.gif. For u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq24_HTML.gif satisfying Φ ( u ) < r n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq98_HTML.gif, similar to the proof of Theorem 3.1, one can show that
u 2 2 M 1 2 Φ ( u ) < ξ n 2 , t [ 0 , 1 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equag_HTML.gif
which implies that
Ψ ( u ) = 0 1 F ( t , u ) d t 0 1 max | u | ξ n F ( t , u ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equah_HTML.gif
Note that Ψ ( 0 ) = Φ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq99_HTML.gif, thus we have
ρ 1 ( r n ) = inf v Φ 1 ( ( , r n ) ) sup u Φ 1 ( ( , r n ) ) Ψ ( u ) Ψ ( v ) r n Φ ( v ) sup u Φ 1 ( ( , r n ) ) Ψ ( u ) Ψ ( 0 ) r n Φ ( 0 ) = sup u Φ 1 ( ( , r n ) ) Ψ ( u ) r n 2 M 1 2 0 1 max | u | ξ n F ( t , u ) d t ξ n 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equai_HTML.gif
which, together with (3.16), gives us
γ 1 2 M 1 2 lim inf ξ + 0 1 max | u | ξ F ( t , u ) d t ξ 2 = μ 1 < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equaj_HTML.gif

This shows that ( 1 μ 2 , 1 μ 1 ) ( 0 , 1 γ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq100_HTML.gif. For any fixed λ ( 1 μ 2 , 1 μ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq101_HTML.gif, it follows from [[5], Theorem 2.1] that either φ λ = Φ λ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq69_HTML.gif has a global minimum or there is a sequence { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq102_HTML.gif of critical points (local minima) of φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq26_HTML.gif such that lim n + u n X = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq103_HTML.gif.

Next we show that the functional φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq23_HTML.gif has no global minimum for λ ( 1 μ 2 , 1 μ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq101_HTML.gif. Since λ > 1 μ 2 = k 5 / lim sup ξ + 1 4 3 4 F ( t , ξ ) d t ξ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq104_HTML.gif, we can choose a constant M such that, for each n N = { 1 , 2 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq105_HTML.gif,
sup ξ n 1 4 3 4 F ( t , ξ ) d t ξ 2 > M > k 5 λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equak_HTML.gif
Thus, there exists ξ n n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq106_HTML.gif such that
1 4 3 4 F ( t , ξ n ) d t ξ n 2 > M . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equal_HTML.gif
Define u n ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq107_HTML.gif as follows:
u n ( t ) = { 64 ξ n ( t 3 3 4 t 2 + 3 16 t ) , t [ 0 , 1 4 ) , ξ n , t [ 1 4 , 3 4 ] , 64 ξ n ( t 3 + 9 4 t 2 27 16 t + 7 16 ) , t ( 3 4 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equam_HTML.gif
This, together with (H2), yields
Φ ( u n ) = 2 , 048 ( 3 8 9 10 4 4 A + 79 14 4 8 B ) ξ n 2 + j = 1 m 0 ξ n I 2 j ( s ) d s k 4 ξ n 2 + k 2 ξ n 2 = k 5 ξ n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equan_HTML.gif
It then follows from (H3) that
φ λ ( u n ) = Φ ( u n ) λ Ψ ( u n ) k 5 ξ n 2 λ 1 4 3 4 F ( t , ξ n ) d t ξ n 2 ( k 5 λ M ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equao_HTML.gif

Note that k 5 λ M < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq108_HTML.gif. Thus the functional φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq23_HTML.gif is unbounded from below and hence it has no global minimum and the proof is complete. □

Corollary 3.4 Assume that (H1), (H2), and (H3) are satisfied. If
lim inf ξ + 0 1 max | u | ξ F ( t , u ) d t ξ 2 < 1 2 M 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equap_HTML.gif
and
lim sup ξ + 1 4 3 4 F ( t , ξ ) d t ξ 2 > k 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equaq_HTML.gif

hold, then (1.2) has an unbounded sequence of solutions in X.

Let
μ 3 = 2 M 1 2 lim inf ω 0 + 0 1 max | u | ω F ( t , u ) d t ω 2 , μ 4 = 1 k 5 lim sup ω 0 + 1 4 3 4 F ( t , ω ) d t ω 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equar_HTML.gif
Theorem 3.5 Assume that (H1), (H2), and (H3) are satisfied. If
μ 3 < μ 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ31_HTML.gif
(3.18)

holds, then for each λ ( 1 μ 4 , 1 μ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq109_HTML.gif system (1.2) has a sequence of non-zero solutions in X, which weakly converges to 0.

Proof The proof is similar to that of Theorem 3.3 by showing that γ 2 < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq110_HTML.gif and 0 is not a local minimum of the functional φ λ = Φ λ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq111_HTML.gif. □

Example 3.2 Consider
{ u ( i v ) ( t ) 2 u ( t ) + u = λ f ( t , u ) , t t 1 , t [ 0 , 1 ] , Δ u ( t 1 ) = 1 10 u ( t 1 ) , t 1 = 1 2 , Δ u ( t 1 ) = 1 20 u ( t 1 ) , t 1 = 1 2 , u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ32_HTML.gif
(3.19)

where f ( t , u ) = 4 t u ( 1 + sin u ) + 2 t u 2 cos u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq112_HTML.gif.

Here I 11 ( s ) = 1 10 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq113_HTML.gif, I 21 ( s ) = 1 20 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq114_HTML.gif, A = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq115_HTML.gif, B = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq116_HTML.gif and m = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq77_HTML.gif. Note that
0 u ( t 1 ) I 11 ( s ) d s = 0 u ( t 1 ) 1 10 s d s = 1 20 | u ( t 1 ) | 2 , 0 u ( t 1 ) I 21 ( s ) d s = 0 u ( t 1 ) 1 20 s d s = 1 40 | u ( t 1 ) | 2 , t 1 = 1 2 [ 1 4 , 3 4 ] , F ( t , u ) = 2 t ( 1 + sin u ) u 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equas_HTML.gif
so (H1), (H2), and (H3) are satisfied. Moreover, we have k 5 782.6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq117_HTML.gif, and
lim ξ + inf 0 1 max | u | ξ F ( t , u ) d t ξ 2 = 0 , lim ξ + sup 1 4 3 4 F ( t , ξ ) d t ξ 2 = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equat_HTML.gif

Therefore, condition (3.17) holds and Theorem 3.3 applies: For λ ( 782.6 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq118_HTML.gif, (3.19) admits an unbounded sequence of solutions in X.

Example 3.3 Consider the boundary value problem
{ u ( i v ) ( t ) 2 u ( t ) + u = λ f ( t , u ) , t t 1 , t [ 0 , 1 ] , Δ u ( t 1 ) = 1 5 u ( t 1 ) , t 1 = 1 2 , Δ u ( t 1 ) = 1 8 u ( t 1 ) , t 1 = 1 2 , u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equ33_HTML.gif
(3.20)
where
f ( t , u ( t ) ) = { 4 t u ( 0.5001 + 1 2 cos ( ln | u | ) 1 4 sin ( ln ( | u | ) ) ) if  u 0 , 0 if  u = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equau_HTML.gif

In this example, I 11 ( s ) = 1 5 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq119_HTML.gif, I 21 ( s ) = 1 8 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq120_HTML.gif, A = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq115_HTML.gif and B = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq116_HTML.gif. The assumptions (H1), (H2), and (H3) clearly hold.

Direct calculations give
F ( t , u ( t ) ) = { 2 t u 2 ( 0.5001 + 1 2 cos ( ln | u | ) ) if  u 0 , 0 if  u = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equav_HTML.gif
k 5 762.64 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq121_HTML.gif and
lim w 0 + inf 0 1 max | u | w F ( t , u ) d t w 2 = 0.0001 , lim w 0 + sup 1 4 3 4 F ( t , w ) d t w 2 = 0.50005 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_Equaw_HTML.gif

Hence (3.18) holds. Therefore it follows from Theorem 3.5 that (3.20) admits a sequence of distinct solutions in X provided that λ ( 1525.1 , 2877.0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-154/MediaObjects/13661_2013_Article_413_IEq122_HTML.gif.

Authors’ information

JX is with the College of Mathematics and Statistics, Jishou University, China and is a PhD candidate at the Department of Mathematics, Hunan Normal University, China. ZL is a professor at the Department of Mathematics, Hunan Normal University, China.

Declarations

Acknowledgements

The authors are very grateful to the referees for their valuable comments and suggestions, which greatly improved the presentation of this paper. The work is partially supported by Hunan Provincial Natural Science Foundation of China (No: 11JJ3012).

Authors’ Affiliations

(1)
College of Mathematics and Statistics, Jishou University
(2)
Department of Mathematics, Hunan Normal University

References

  1. Aftabizadeh AR: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 1986, 116: 415–426. 10.1016/S0022-247X(86)80006-3MathSciNetView Article
  2. Gupta CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. 1988, 26: 289–304. 10.1080/00036818808839715MathSciNetView Article
  3. Ma R, Zhan J, Fu S: The method of lower and upper solutions for fourth-order two-point boundary value problems. J. Math. Anal. Appl. 1997, 215: 415–422. 10.1006/jmaa.1997.5639MathSciNetView Article
  4. Bonanno G, Dibella B, Regan DO: Non-trivial solutions for nonlinear fourth-order elastic beam equations. Comput. Math. Appl. 2011, 62: 1862–1869. 10.1016/j.camwa.2011.06.029MathSciNetView Article
  5. Bonanno G, Molica Bisci G: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 2009, 113: 1–20.MathSciNet
  6. Han G, Xu Z: Multiple solutions of some nonlinear fourth-order beam equations. Nonlinear Anal. 2008, 68: 3646–3656. 10.1016/j.na.2007.04.007MathSciNetView Article
  7. Liu X, Li W: Existence and multiplicity of solutions for fourth-order boundary values problems with parameters. J. Math. Anal. Appl. 2007, 327: 362–375. 10.1016/j.jmaa.2006.04.021MathSciNetView Article
  8. Agarwal RP, Rgean DO: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem. Appl. Math. Comput. 2005, 161: 433–439. 10.1016/j.amc.2003.12.096MathSciNetView Article
  9. Agarwal RP, Franco D, Regan DO: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math. 2005, 69: 83–96. 10.1007/s00010-004-2735-9View Article
  10. Bonanno G: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. 2012, 75: 2992–3007. 10.1016/j.na.2011.12.003MathSciNetView Article
  11. Chen J, Nieto JJ: Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc. 2007, 40: 902–912.
  12. De Coster C, Habets P: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. 371. In Nonlinear Analysis and Boundary Value Problem for Ordinary Differential Equations CISM-ICMS. Edited by: Zanolin F. Springer, New York; 1996:1–78.View Article
  13. Marek G, Szymon G: On the discrete boundary value problem for anisotropic equation. J. Math. Anal. Appl. 2012, 386: 956–965. 10.1016/j.jmaa.2011.08.053MathSciNetView Article
  14. Nieto JJ, Regan DO: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680–690. 10.1016/j.nonrwa.2007.10.022MathSciNetView Article
  15. Qin D, Li X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl. 2005, 303: 288–303. 10.1016/j.jmaa.2004.08.034MathSciNetView Article
  16. Shen JH, Wang BW: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 2008, 69: 4055–4062. 10.1016/j.na.2007.10.036MathSciNetView Article
  17. Tian Y, Ge W: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 2008, 51: 509–527.MathSciNetView Article
  18. Zhang H, Li Z: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal., Real World Appl. 2010, 11: 67–78. 10.1016/j.nonrwa.2008.10.016MathSciNetView Article
  19. Zhang X, Tang X: Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems. Nonlinear Anal., Real World Appl. 2012, 13: 113–130. 10.1016/j.nonrwa.2011.07.013MathSciNetView Article
  20. Zhang Z, Yuan R: Applications of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 2010, 11: 155–162. 10.1016/j.nonrwa.2008.10.044MathSciNetView Article
  21. Sun J, Chen H, Yang L: Variational methods to fourth-order impulsive differential equations. J. Appl. Math. Comput. 2011, 35: 323–340. 10.1007/s12190-009-0359-xMathSciNetView Article
  22. Dym H, McKean H: Fourier Series and Integrals. Academic Press, New York; 1985.
  23. Peletier LA, Troy WC, van der Vorst RCAM: Stationary solutions of a fourth order nonlinear diffusion equation. Differ. Equ. 1995, 31: 301–314.MathSciNet

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© Xie and Luo; 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.