Open Access

Existence and multiplicity of solutions for some second-order systems on time scales with impulsive effects

Boundary Value Problems20122012:148

DOI: 10.1186/1687-2770-2012-148

Received: 17 September 2012

Accepted: 6 December 2012

Published: 21 December 2012

Abstract

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effects

{ u Δ 2 ( t ) + A ( σ ( t ) ) u ( σ ( t ) ) + F ( σ ( t ) , u ( σ ( t ) ) ) = 0 , Δ -a.e.  t [ 0 , T ] T κ ; u ( 0 ) u ( T ) = u Δ ( 0 ) u Δ ( T ) = 0 , ( u i ) Δ ( t j + ) ( u i ) Δ ( t j ) = I i j ( u i ( t j ) ) , i = 1 , 2 , , N , j = 1 , 2 , , p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equa_HTML.gif

where t 0 = 0 < t 1 < t 2 < < t p < t p + 1 = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq1_HTML.gif, t j [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq2_HTML.gif ( j = 0 , 1 , 2 , , p + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq3_HTML.gif), u ( t ) = ( u 1 ( t ) , u 2 ( t ) , , u N ( t ) ) R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq4_HTML.gif, A ( t ) = [ d l m ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq5_HTML.gif is a symmetric N × N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq6_HTML.gif matrix-valued function defined on [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq7_HTML.gif with d l m L ( [ 0 , T ] T , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq8_HTML.gif for all l , m = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq9_HTML.gif, I i j : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq10_HTML.gif ( i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq11_HTML.gif, j = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq12_HTML.gif) are continuous and F : [ 0 , T ] T × R N R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq13_HTML.gif. Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.

MSC:34B37, 34N05.

Keywords

nonautonomous second-order systems time scales impulse variational approach

1 Introduction

Consider the nonautonomous second-order system on time scales with impulsive effects
{ u Δ 2 ( t ) + A ( σ ( t ) ) u ( σ ( t ) ) + F ( σ ( t ) , u ( σ ( t ) ) ) = 0 , Δ -a.e.  t [ 0 , T ] T κ ; u ( 0 ) u ( T ) = u Δ ( 0 ) u Δ ( T ) = 0 , ( u i ) Δ ( t j + ) ( u i ) Δ ( t j ) = I i j ( u i ( t j ) ) , i = 1 , 2 , , N , j = 1 , 2 , , p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ1_HTML.gif
(1.1)
where t 0 = 0 < t 1 < t 2 < < t p < t p + 1 = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq1_HTML.gif, t j [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq2_HTML.gif ( j = 0 , 1 , 2 , , p + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq3_HTML.gif),
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equb_HTML.gif
u ( t ) = ( u 1 ( t ) , u 2 ( t ) , , u N ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq14_HTML.gif, A ( t ) = [ d l m ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq5_HTML.gif is a symmetric N × N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq6_HTML.gif matrix-valued function defined on [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq7_HTML.gif with d l m L ( [ 0 , T ] T , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq8_HTML.gif for all l , m = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq9_HTML.gif, I i j : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq10_HTML.gif ( i = 1 , 2 , , N , j = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq15_HTML.gif) are continuous and F : [ 0 , T ] T × R N R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq13_HTML.gif satisfies the following assumption:
  1. (A)
    F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq16_HTML.gif is Δ-measurable in t for every x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq17_HTML.gif and continuously differentiable in x for Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq18_HTML.gif, and there exist a C ( R + , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq19_HTML.gif, b σ L 1 ( 0 , T ; R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq20_HTML.gif such that
    | F ( t , x ) | a ( | x | ) b ( t ) , | F ( t , x ) | a ( | x | ) b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equc_HTML.gif
     

for all x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq21_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq22_HTML.gif, where F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq23_HTML.gif denotes the gradient of F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq16_HTML.gif in x.

For the sake of convenience, in the sequel, we denote Γ = { 1 , 2 , 3 , , N } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq24_HTML.gif, Λ = { 1 , 2 , 3 , , p } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq25_HTML.gif.

When I i j 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq26_HTML.gif, i A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq27_HTML.gif, j B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq28_HTML.gif and A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq29_HTML.gif is a zero matrix, (1.1) is the Hamiltonian system on time scales
{ u Δ 2 ( t ) + F ( σ ( t ) , u ( σ ( t ) ) ) = 0 , a.e.  t [ 0 , T ] , u ( 0 ) u ( T ) = u Δ ( 0 ) u Δ ( T ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ2_HTML.gif
(1.2)

In [1], the authors study the Sobolev’s spaces on time scales and their properties. As applications, they present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for (1.2).

When I i j ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq30_HTML.gif, i A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq27_HTML.gif, j B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq28_HTML.gif and A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq29_HTML.gif is not a zero matrix, until now the variational structure of (1.1) has not been studied.

Problem (1.1) covers the second-order Hamiltonian system with impulsive effects (when T = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq31_HTML.gif)
{ u ¨ ( t ) + A ( t ) u ( t ) + F ( t , u ( t ) ) = 0 , a.e.  t [ 0 , T ] ; u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) = 0 , Δ u ˙ i ( t j ) = u ˙ i ( t j + ) u ˙ i ( t j ) = I i j ( u i ( t j ) ) , i Γ , j Λ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ3_HTML.gif
(1.3)
as well as the second-order discrete Hamiltonian system (when T = Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq32_HTML.gif, T N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq33_HTML.gif, T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq34_HTML.gif)
{ u ( t + 2 ) 2 u ( t + 1 ) + u ( t ) + A ( t + 1 ) u ( t + 1 ) + F ( t + 1 , u ( t + 1 ) ) = 0 , t [ 1 , T 1 ] Z , u ( 0 ) u ( T ) = 0 , u ( T ) u ( 0 ) = u ( T + 1 ) u ( 1 ) , u i ( t j + 2 ) u i ( t j + 1 ) u i ( t j ) + u i ( t j 1 ) = I i j ( u i ( t j ) ) , i Γ , j Λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equd_HTML.gif

In [2], the authors establish some sufficient conditions on the existence of solutions of (1.3) by means of some critical point theorems when A ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq35_HTML.gif. When T R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq36_HTML.gif, until now, it is unknown whether problem (1.1) has a variational structure or not.

Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians (see [35]). Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory and so on (see [69]).

For a second-order differential equation u = f ( t , u , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq37_HTML.gif, one usually considers impulses in the position u and the velocity u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq38_HTML.gif. However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position (see [10]). The impulses only on the velocity occur also in impulsive mechanics (see [11]). An impulsive problem with impulses in the derivative only is considered in [12].

The study of dynamical systems on time scales is now an active area of research. One of the reasons for this is the fact that the study on time scales unifies the study of both discrete and continuous processes, besides many others. The pioneering works in this direction are Refs. [1317]. The theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis in 1988, providing a rich theory that unifies and extends discrete and continuous analysis [18, 19]. The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks and social sciences (see [16]). For example, it can model insect populations that are continuous while in season (and may follow a difference scheme with variable step-size), die out in winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population.

There have been many approaches to study solutions of differential equations on time scales, such as the method of lower and upper solutions, fixed-point theory, coincidence degree theory and so on (see [1, 2029]). In [24], authors used the fixed point theorem of strict-set-contraction to study the existence of positive periodic solutions for functional differential equations with impulse effects on time scales. However, the study of the existence and multiplicity of solutions for differential equations on time scales using the variational method has received considerably less attention (see, for example, [1, 29]). The variational method is, to the best of our knowledge, novel and it may open a new approach to deal with nonlinear problems, with some type of discontinuities such as impulses.

Motivated by the above, we research the existence of variational construction for problem (1.1) in an appropriate space of functions and study the existence of solutions for (1.1) by some critical point theorems in this paper. All these results are new.

2 Preliminaries and statements

In this section, we present some fundamental definitions and results from the calculus on time scales and Sobolev’s spaces on time scales that will be required below. These are a generalization to R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq39_HTML.gif of definitions and results found in [17].

Definition 2.1 ([[17], Definition 1.1])

Let T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq40_HTML.gif be a time scale. For t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq41_HTML.gif, the forward jump operator σ : T T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq42_HTML.gif is defined by
σ ( t ) = inf { s T , s > t } for all  t T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Eque_HTML.gif
while the backward jump operator ρ : T T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq43_HTML.gif is defined by
ρ ( t ) = sup { s T , s < t } for all  t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equf_HTML.gif

(supplemented by inf = sup T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq44_HTML.gif and sup = inf T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq45_HTML.gif, where denotes the empty set). A point t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq41_HTML.gif is called right-scattered, left-scattered, if σ ( t ) > t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq46_HTML.gif, ρ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq47_HTML.gif hold, respectively. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if t < sup T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq48_HTML.gif and σ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq49_HTML.gif, then t is called right-dense, and if t > inf T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq50_HTML.gif and ρ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq51_HTML.gif, then t is called left-dense. Points that are right-dense and left-dense at the same time are called dense. The set T κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq52_HTML.gif which is derived from the time scale T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq40_HTML.gif as follows. If T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq40_HTML.gif has a left-scattered maximum m, then T κ = T { m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq53_HTML.gif; otherwise, T κ = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq54_HTML.gif.

When a , b T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq55_HTML.gif, a < b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq56_HTML.gif, we denote the intervals [ a , b ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq57_HTML.gif, [ a , b ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq58_HTML.gif and ( a , b ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq59_HTML.gif in T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq40_HTML.gif by
[ a , b ] T = [ a , b ] T , [ a , b ) T = [ a , b ) T , ( a , b ] T = ( a , b ] T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equg_HTML.gif

respectively. Note that [ a , b ] T κ = [ a , b ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq60_HTML.gif if b is left-dense and [ a , b ] T κ = [ a , b ) T = [ a , ρ ( b ) ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq61_HTML.gif if b is left-scattered. We denote [ a , b ] T κ 2 = ( [ a , b ] T κ ) κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq62_HTML.gif, therefore [ a , b ] T κ 2 = [ a , b ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq63_HTML.gif if b is left-dense and [ a , b ] T κ 2 = [ a , ρ ( b ) ] T κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq64_HTML.gif if b is left-scattered.

Definition 2.2 ([[17], Definition 1.10])

Assume that f : T R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq65_HTML.gif is a function and let t T κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq66_HTML.gif. Then we define f Δ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq67_HTML.gif to be the number (provided it exists) with the property that given any ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq68_HTML.gif, there is a neighborhood U of t (i.e., U = ( t δ , t + δ ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq69_HTML.gif for some δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq70_HTML.gif) such that
| [ f ( σ ( t ) ) f ( s ) ] f Δ ( t ) [ σ ( t ) s ] | ϵ | σ ( t ) s | for all  s U . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equh_HTML.gif

We call f Δ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq67_HTML.gif the delta (or Hilger) derivative of f at t. The function f is delta (or Hilger) differentiable on T κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq52_HTML.gif provided f Δ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq67_HTML.gif exists for all t T κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq66_HTML.gif. The function f Δ : T κ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq71_HTML.gif is then called the delta derivative of f on T κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq52_HTML.gif.

Definition 2.3 ([[1], Definition 2.3])

Assume that f : T R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq72_HTML.gif is a function,
f ( t ) = ( f 1 ( t ) , f 2 ( t ) , , f N ( t ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equi_HTML.gif

and let t T κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq66_HTML.gif. Then we define f Δ ( t ) = ( f 1 Δ ( t ) , f 2 Δ ( t ) , , f N Δ ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq73_HTML.gif (provided it exists). We call f Δ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq67_HTML.gif the delta (or Hilger) derivative of f at t. The function f is delta (or Hilger) differentiable provided f Δ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq67_HTML.gif exists for all t T κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq74_HTML.gif. The function f Δ : T κ R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq75_HTML.gif is then called the delta derivative of f on T κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq52_HTML.gif.

Definition 2.4 ([[17], Definition 2.7])

For a function f : T R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq65_HTML.gif, we will talk about the second derivative f Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq76_HTML.gif provided f Δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq77_HTML.gif is differentiable on T κ 2 = ( T κ ) κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq78_HTML.gif with derivative f Δ 2 = ( f Δ ) Δ : T κ 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq79_HTML.gif.

Definition 2.5 ([[1], Definition 2.5])

For a function f : T R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq72_HTML.gif, we will talk about the second derivative f Δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq76_HTML.gif provided f Δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq77_HTML.gif is differentiable on T κ 2 = ( T κ ) κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq78_HTML.gif with derivative f Δ 2 = ( f Δ ) Δ : T κ 2 R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq80_HTML.gif.

The Δ-measure μ Δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq81_HTML.gif and Δ-integration are defined as those in [26].

Definition 2.6 ([[1], Definition 2.7])

Assume that f : T R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq72_HTML.gif is a function, f ( t ) = ( f 1 ( t ) , f 2 ( t ) , , f N ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq82_HTML.gif and let A be a Δ-measurable subset of T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq40_HTML.gif. f is integrable on A if and only if f i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq83_HTML.gif ( i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq11_HTML.gif) are integrable on A, and A f ( t ) Δ t = ( A f 1 ( t ) Δ t , A f 2 ( t ) Δ t , , A f N ( t ) Δ t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq84_HTML.gif.

Definition 2.7 ([[17], Definition 2.3])

Let B T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq85_HTML.gif. B is called a Δ-null set if μ Δ ( B ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq86_HTML.gif. Say that a property P holds Δ-almost everywhere (Δ-a.e.) on B, or for Δ-almost all (Δ-a.a.) t B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq87_HTML.gif if there is a Δ-null set E 0 B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq88_HTML.gif such that P holds for all t B E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq89_HTML.gif.

For p R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq90_HTML.gif, p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq91_HTML.gif, we set the space
L Δ p ( [ 0 , T ) T , R N ) = { u : [ 0 , T ) T R N : [ 0 , T ) T | f ( t ) | p Δ t < + } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equj_HTML.gif
with the norm
f L Δ p = ( [ 0 , T ) T | f ( t ) | p Δ t ) 1 p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equk_HTML.gif

We have the following theorem.

Theorem 2.1 ([[1], Theorem 2.1])

Let p R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq90_HTML.gif be such that p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq91_HTML.gif. Then the space L Δ p ( [ 0 , T ) T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq92_HTML.gif is a Banach space together with the norm L Δ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq93_HTML.gif. Moreover, L Δ 2 ( [ a , b ) T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq94_HTML.gif is a Hilbert space together with the inner product given for every ( f , g ) L Δ p ( [ a , b ) T , R N ) × L Δ p ( [ a , b ) T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq95_HTML.gif by
f , g L Δ 2 = [ a , b ) T ( f ( t ) , g ( t ) ) Δ t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equl_HTML.gif

where ( , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq96_HTML.gif denotes the inner product in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq97_HTML.gif.

Definition 2.8 ([[1], Definition 2.11])

A function f : [ a , b ] T R N , f ( t ) = ( f 1 ( t ) , f 2 ( t ) , , f N ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq98_HTML.gif. We say that f is absolutely continuous on [ a , b ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq57_HTML.gif (i.e., f A C ( [ a , b ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq99_HTML.gif) if for every ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq68_HTML.gif, there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq70_HTML.gif such that if { [ a k , b k ) T } k = 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq100_HTML.gif is a finite pairwise disjoint family of subintervals of [ a , b ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq57_HTML.gif satisfying k = 1 n ( b k a k ) < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq101_HTML.gif, then k = 1 n | f ( b k ) f ( a k ) | < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq102_HTML.gif.

Now, we recall the Sobolev space W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq103_HTML.gif on [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq7_HTML.gif defined in [1]. For the sake of convenience, in the sequel we let u σ ( t ) = u ( σ ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq104_HTML.gif.

Definition 2.9 ([[1], Definition 2.12])

Let p R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq90_HTML.gif be such that p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq105_HTML.gif and u : [ 0 , T ] T R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq106_HTML.gif. We say that u W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq107_HTML.gif if and only if u L Δ p ( [ 0 , T ) T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq108_HTML.gif and there exists g : [ 0 , T ] T κ R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq109_HTML.gif such g L Δ p ( [ 0 , T ) T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq110_HTML.gif and
[ 0 , T ) T ( u ( t ) , ϕ Δ ( t ) ) Δ t = [ 0 , T ) T ( g ( t ) , ϕ σ ( t ) ) Δ t , ϕ C T , r d 1 ( [ 0 , T ] T , R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ4_HTML.gif
(2.1)
For p R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq90_HTML.gif, p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq105_HTML.gif, we denote
V Δ , T 1 , p ( [ 0 , T ] T , R N ) = { x A C ( [ 0 , T ] T , R N ) : x Δ L Δ p ( [ 0 , T ) T , R N ) , x ( 0 ) = x ( T ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equm_HTML.gif
It follows from Remark 2.2 in [1] that
V Δ 1 , p ( [ 0 , T ] T , R N ) W Δ 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equn_HTML.gif

is true for every p R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq90_HTML.gif with p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq105_HTML.gif. These two sets are, as a class of functions, equivalent. It is the characterization of functions in W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq103_HTML.gif in terms of functions in V Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq111_HTML.gif too. That is the following theorem.

Theorem 2.2 ([[1], Theorem 2.5])

Suppose that u W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq107_HTML.gif for some p R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq90_HTML.gif with p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq105_HTML.gif, and that (2.1) holds for g L Δ p ( [ 0 , T ) T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq110_HTML.gif. Then there exists a unique function x V Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq112_HTML.gif such that the equalities
x = u , x Δ = g Δ -a.e. on [ 0 , T ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ5_HTML.gif
(2.2)
are satisfied and
[ 0 , T ) T g ( t ) Δ t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ6_HTML.gif
(2.3)

By identifying u W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq107_HTML.gif with its absolutely continuous representative x V Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq113_HTML.gif for which (2.2) holds, the set W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq103_HTML.gif can be endowed with the structure of a Banach space. That is the following theorem.

Theorem 2.3 ([[25], Theorem 2.21])

Assume p R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq90_HTML.gif and p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq105_HTML.gif. The set W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq103_HTML.gif is a Banach space together with the norm defined as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ7_HTML.gif
(2.4)
Moreover, the set H Δ , T 1 = W Δ , T 1 , 2 ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq114_HTML.gif is a Hilbert space together with the inner product
u , v H Δ , T 1 = [ 0 , T ) T ( u σ ( t ) , v σ ( t ) ) Δ t + [ 0 , T ) T ( u Δ ( t ) , v Δ ( t ) ) Δ t u , v H Δ , T 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equo_HTML.gif

The Banach space W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq103_HTML.gif has some important properties.

Theorem 2.4 ([[25], Theorem 2.23])

There exists C 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq115_HTML.gif such that the inequality
u C 1 u H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ8_HTML.gif
(2.5)

holds for all u H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq116_HTML.gif, where u = max t [ 0 , T ] T | u ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq117_HTML.gif.

Moreover, if [ 0 , T ) T u ( t ) Δ t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq118_HTML.gif, then
u C 1 u Δ L Δ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equp_HTML.gif

Theorem 2.5 ([[25], Theorem 2.25])

If the sequence { u k } k N W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq119_HTML.gif converges weakly to u in W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq120_HTML.gif, then { u k } k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq121_HTML.gif converges strongly in C ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq122_HTML.gif to u.

Theorem 2.6 ([[25], Theorem 2.27])

Let L : [ 0 , T ] T × R N × R N R , ( t , x , y ) L ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq123_HTML.gif be Δ-measurable in t for each ( x , y ) R N × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq124_HTML.gif and continuously differentiable in ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq125_HTML.gif for Δ-almost every t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif. If there exist a C ( R + , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq127_HTML.gif, b L Δ 1 ( [ 0 , T ] T , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq128_HTML.gif and c L Δ q ( [ 0 , T ] T , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq129_HTML.gif ( 1 < q < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq130_HTML.gif) such that for Δ-almost t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq18_HTML.gif and every ( x , y ) R N × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq124_HTML.gif, one has
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ9_HTML.gif
(2.6)
where 1 p + 1 q = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq131_HTML.gif, then the functional Φ : W Δ , T 1 , p ( [ 0 , T ] T , R N ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq132_HTML.gif defined as
Φ ( u ) = [ 0 , T ) T L ( σ ( t ) , u σ ( t ) , u Δ ( t ) ) Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equq_HTML.gif
is continuously differentiable on W Δ , T 1 , p ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq103_HTML.gif and
Φ ( u ) , v = [ 0 , T ) T [ ( L x ( σ ( t ) , u σ ( t ) , u Δ ( t ) ) , v σ ( t ) ) + ( L y ( σ ( t ) , u σ ( t ) , u Δ ( t ) ) , v Δ ( t ) ) ] Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ10_HTML.gif
(2.7)

3 Variational setting

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we make a variational structure. From this variational structure, we can reduce the problem of finding solutions of (1.1) to the one of seeking the critical points of a corresponding functional.

If u H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq116_HTML.gif, by identifying u H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq116_HTML.gif with its absolutely continuous representative x V Δ , T 1 , 2 ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq133_HTML.gif for which (2.2) holds, then u is absolutely continuous and u ˙ L 2 ( [ 0 , T ) T ; R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq134_HTML.gif. In this case, u Δ ( t + ) u Δ ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq135_HTML.gif may not hold for some t ( 0 , T ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq136_HTML.gif. This leads to impulsive effects.

Take v H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq137_HTML.gif and multiply the two sides of the equality
u Δ 2 ( t ) + A ( σ ( t ) ) u ( σ ( t ) ) + F ( σ ( t ) , u σ ( t ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equr_HTML.gif
by v σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq138_HTML.gif and integrate on [ 0 , T ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq139_HTML.gif, then we have
[ 0 , T ) T [ u Δ 2 ( t ) + A ( σ ( t ) ) u ( σ ( t ) ) + F ( σ ( t ) , u σ ( t ) ) ] v σ ( t ) Δ t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ11_HTML.gif
(3.1)
Moreover, combining u Δ ( 0 ) u Δ ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq140_HTML.gif, one has
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equs_HTML.gif
Combining (3.1), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equt_HTML.gif

Considering the above, we introduce the following concept solution for problem (1.1).

Definition 3.1 We say that a function u H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq116_HTML.gif is a weak solution of problem (1.1) if the identity
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equu_HTML.gif

holds for any v H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq137_HTML.gif.

Consider the functional φ : H Δ , T 1 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq141_HTML.gif defined by
φ ( u ) = 1 2 [ 0 , T ) T | u Δ ( t ) | 2 Δ t + j = 1 p i = 1 N 0 u i ( t j ) I i j ( t ) d t 1 2 [ 0 , T ) T ( A σ ( t ) u σ ( t ) , u σ ( t ) ) Δ t + J ( u ) = ψ ( u ) + ϕ ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ12_HTML.gif
(3.2)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equv_HTML.gif
and
ϕ ( u ) = j = 1 p i = 1 N 0 u i ( t j ) I i j ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equw_HTML.gif
Lemma 3.1 The functional φ is continuously differentiable on H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif and
φ ( u ) , v = [ 0 , T ) T ( u Δ ( t ) , v Δ ( t ) ) Δ t + j = 1 p i = 1 N I i j ( u i ( t j ) ) v i ( t j ) [ 0 , T ) T [ ( A σ ( t ) u σ ( t ) , v σ ( t ) ) ( F ( σ ( t ) , u σ ( t ) ) , v σ ( t ) ) ] Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ13_HTML.gif
(3.3)
Proof Set L ( t , x , y ) = 1 2 | y | 2 1 2 ( A ( t ) x , x ) F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq143_HTML.gif for all x , y R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq144_HTML.gif and t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq18_HTML.gif. Then L ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq145_HTML.gif satisfies all assumptions of Theorem 2.6. Hence, by Theorem 2.6, we know that the functional ψ is continuously differentiable on H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif and
ψ ( u ) , v = [ 0 , T ) T [ ( u Δ ( t ) , v Δ ( t ) ) ( A σ ( t ) u σ ( t ) , v σ ( t ) ) ( F ( σ ( t ) , u σ ( t ) ) , v σ ( t ) ) ] Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equx_HTML.gif

for all u , v H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq146_HTML.gif.

On the other hand, by the continuity of I i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq147_HTML.gif, i Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq148_HTML.gif, j Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq149_HTML.gif, one has that ϕ C 1 ( H Δ , T 1 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq150_HTML.gif and
ϕ ( u ) , v = j = 1 p i = 1 N I i j ( u i ( t j ) ) v i ( t j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equy_HTML.gif

for all u , v H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq146_HTML.gif. Thus, φ is continuously differentiable on H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif and (3.3) holds. □

By Definition 3.1 and Lemma 3.1, the weak solutions of problem (1.1) correspond to the critical points of φ.

Moreover, we need more preliminaries. For any u H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq116_HTML.gif, let
q ( u ) = 1 2 [ 0 , T ) T [ | u σ ( t ) | 2 ( A σ ( t ) u σ ( t ) , u σ ( t ) ) ] Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equz_HTML.gif
We see that
q ( u ) = 1 2 u 2 1 2 [ 0 , T ) T ( ( A σ ( t ) + I N × N ) u σ ( t ) , u σ ( t ) ) Δ t = 1 2 ( I K ) u , u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equaa_HTML.gif
where K : H Δ , T 1 H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq151_HTML.gif is the bounded self-adjoint linear operator defined, using the Riesz representation theorem, by
K u , v = [ 0 , T ) T ( ( A σ ( t ) + I N × N ) u σ ( t ) , v σ ( t ) ) Δ t , u , v H Δ , T 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equab_HTML.gif
I N × N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq152_HTML.gif and I denote an N × N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq6_HTML.gif identity matrix and an identity operator, respectively. By (3.2), φ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq153_HTML.gif can be rewritten as
φ ( u ) = q ( u ) + ϕ ( u ) + J ( u ) = 1 2 ( I K ) u , u + ϕ ( u ) + J ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ14_HTML.gif
(3.4)
The compact imbedding of H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif into C ( [ 0 , T ] T , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq154_HTML.gif implies that K is compact. By classical spectral theory, we can decompose H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif into the orthogonal sum of invariant subspaces for I K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq155_HTML.gif
H Δ , T 1 = H H 0 H + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equac_HTML.gif
where H 0 = ker ( I K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq156_HTML.gif and H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq157_HTML.gif, H + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq158_HTML.gif are such that, for some δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq70_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ15_HTML.gif
(3.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ16_HTML.gif
(3.6)

Remark 3.1 K has only finitely many eigenvalues λ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq159_HTML.gif with λ i > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq160_HTML.gif since K is compact on H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq161_HTML.gif. Hence H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq157_HTML.gif is finite dimensional. Notice that I K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq155_HTML.gif is a compact perturbation of the self-adjoint operator I. By a well-known theorem, we know that 0 is not in the essential spectrum of I K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq155_HTML.gif. Hence, H 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq162_HTML.gif is a finite dimensional space too.

To prove our main results, we need the following definitions and theorems.

Definition 3.2 ([[30], P 81 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq163_HTML.gif])

Let X be a real Banach space and I C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq164_HTML.gif. I is said to be satisfying (PS) condition on X if any sequence { x n } X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq165_HTML.gif for which I ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq166_HTML.gif is bounded and I ( x n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq167_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq168_HTML.gif, possesses a convergent subsequence in X.

Firstly, we state the local linking theorem.

Let X be a real Banach space with a direct decomposition X = X 1 X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq169_HTML.gif. Consider two sequences of a subspace
X 0 1 X 1 1 X 1 , X 0 2 X 1 2 X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equad_HTML.gif
such that
dim X n 1 < + , dim X n 2 < + , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equae_HTML.gif
and
X 1 = n N X n 1 ¯ , X 2 = n N X n 2 ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equaf_HTML.gif

For every multi-index α = ( α 1 , α 2 ) N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq170_HTML.gif, we denote by X α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq171_HTML.gif the space X α 1 X α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq172_HTML.gif. We say α β α 1 β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq173_HTML.gif, α 2 β 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq174_HTML.gif. A sequence ( α n ) N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq175_HTML.gif is admissible if, for every α N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq176_HTML.gif, there is m 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq177_HTML.gif such that n m 0 α n α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq178_HTML.gif.

Definition 3.3 ([[31], Definition 2.2])

Let I C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq164_HTML.gif. The functional I satisfies the ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq179_HTML.gif condition if every sequence ( u α n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq180_HTML.gif such that α n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq181_HTML.gif is admissible and
u α n X α n , sup | I ( u α n ) | < , ( 1 + u α n ) I ( u α n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equag_HTML.gif

contains a subsequence which converges to a critical point of I.

Theorem 3.1 [[31], Theorem 2.2]

Suppose that I C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq164_HTML.gif satisfies the following assumptions:

(I1) X 1 { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq182_HTML.gif and I has a local linking at 0 with respect to ( X 1 , X 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq183_HTML.gif; that is, for some M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq184_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equah_HTML.gif

(I2) I satisfies ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq179_HTML.gif condition.

(I3) I maps bounded sets into bounded sets.

(I4) For every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq185_HTML.gif, I ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq186_HTML.gif as u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq187_HTML.gif, u X n 1 X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq188_HTML.gif.

Then I has at least two critical points.

Remark 3.2 Since I C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq164_HTML.gif, by the condition (I1) of Theorem 3.1, 0 is the critical point of I. Thus, under the conditions of Theorem 3.1, I has at least one nontrivial critical point.

Secondly, we state another three critical point theorems.

Theorem 3.2 ([[32], Theorem 5.29])

Let E be a Hilbert space with E = E 1 E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq189_HTML.gif and E 2 = E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq190_HTML.gif. Suppose I C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq191_HTML.gif, satisfies (PS) condition, and

(I5) I ( u ) = 1 2 L u , u + b ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq192_HTML.gif, where L u = L 1 P 1 u + L 2 P 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq193_HTML.gif and L κ : E κ E κ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq194_HTML.gif is bounded and self-adjoint, κ = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq195_HTML.gif,

(I6) b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq196_HTML.gif is compact, and

(I7) there exist a subspace E ˜ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq197_HTML.gif and sets S E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq198_HTML.gif, Q E ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq199_HTML.gif and constants α > ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq200_HTML.gif such that
  1. (i)

    S E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq201_HTML.gif and I | S α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq202_HTML.gif,

     
  2. (ii)

    Q is bounded and I | Q ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq203_HTML.gif,

     
  3. (iii)

    S and ∂Q link.

     

Then I possesses a critical value c α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq204_HTML.gif.

Theorem 3.3 ([[32], Theorem 9.12])

Let E be a Banach space. Let I C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq205_HTML.gif be an even functional which satisfies the (PS) condition and I ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq206_HTML.gif. If E = V W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq207_HTML.gif, where V is finite dimensional, and I satisfies

(I8) there are constants ρ , α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq208_HTML.gif such that I | B ρ W α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq209_HTML.gif, where B ρ = { x E : x < ρ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq210_HTML.gif,

(I9) for each finite dimensional subspace E ˜ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq211_HTML.gif, there is an R = R ( E ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq212_HTML.gif such that I 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq213_HTML.gif on E ˜ B R ( E ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq214_HTML.gif,

then I possesses an unbounded sequence of critical values.

In order to state another critical point theorem, we need the following notions. Let X and Y be Banach spaces with X being separable and reflexive, and set E = X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq215_HTML.gif. Let S X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq216_HTML.gif be a dense subset. For each s S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq217_HTML.gif, there is a semi-norm on E defined by
p s : E R , p s ( u ) = | s ( x ) | + y for  u = x + y X Y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equai_HTML.gif

We denote by T S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq218_HTML.gif the topology on E induced by a semi-norm family { p s } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq219_HTML.gif, and let w and w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq220_HTML.gif denote the weak-topology and weak*-topology, respectively.

For a functional Φ C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq221_HTML.gif, we write Φ a = { u E : Φ ( u ) a } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq222_HTML.gif. Recall that Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq223_HTML.gif is said to be weak sequentially continuous if, for any u k u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq224_HTML.gif in E, one has lim k Φ ( u k ) v Φ ( u ) v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq225_HTML.gif for each v E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq226_HTML.gif, i.e., Φ : ( E , w ) ( E , w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq227_HTML.gif is sequentially continuous. For c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq228_HTML.gif, we say that Φ satisfies the ( C ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq229_HTML.gif condition if any sequence { u k } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq230_HTML.gif such that Φ ( u k ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq231_HTML.gif and ( 1 + u k ) Φ ( u k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq232_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq233_HTML.gif contains a convergent subsequence.

Suppose that

( Φ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq234_HTML.gif) for any c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq228_HTML.gif, Φ c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq235_HTML.gif is T S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq218_HTML.gif-closed, and Φ : ( Φ c , T S ) ( E , w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq236_HTML.gif is continuous;

( Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq237_HTML.gif) there exists ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq238_HTML.gif such that κ : = inf Φ ( B ρ Y ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq239_HTML.gif, where
B ρ = { u E : u < ρ } ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equaj_HTML.gif
( Φ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq240_HTML.gif) there exist a finite dimensional subspace Y 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq241_HTML.gif and R > ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq242_HTML.gif such that c ¯ : = sup Φ ( E 0 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq243_HTML.gif and sup Φ ( E 0 S 0 ) < inf Φ ( B ρ Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq244_HTML.gif, where
E 0 : = X Y 0 , and S 0 = { u E 0 : u R } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equak_HTML.gif

Theorem 3.4 ([33])

Assume that Φ is even and ( Φ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq234_HTML.gif)-( Φ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq240_HTML.gif) are satisfied. Then Φ has at least m = dim Y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq245_HTML.gif pairs of critical points with critical values less than or equal to c ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq246_HTML.gif provided Φ satisfies the ( C ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq229_HTML.gif condition for all c [ κ , c ¯ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq247_HTML.gif.

Remark 3.3 In our applications, we take S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq248_HTML.gif= X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq249_HTML.gif so that T S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq218_HTML.gif is the product topology on E = X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq215_HTML.gif given by the weak topology on X and the strong topology on Y.

4 Main results

Lemma 4.1 ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq250_HTML.gif is compact on H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif.

Proof Let { u k } H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq251_HTML.gif be any bounded sequence. Since H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif is a Hilbert space, we can assume that u k u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq224_HTML.gif. Theorem 2.5 implies that u k u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq252_HTML.gif. By (2.5), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equal_HTML.gif

The continuity of I i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq147_HTML.gif and this imply that ϕ ( u k ) ϕ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq253_HTML.gif in H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif. The proof is complete. □

First of all, we give two existence results.

Theorem 4.1 Suppose that (A) and the following conditions are satisfied.

(F1) lim | x | F ( t , x ) | x | 2 = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq254_HTML.gif uniformly for Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq18_HTML.gif,

(F2) lim | x | 0 F ( t , x ) | x | 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq255_HTML.gif uniformly for Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq18_HTML.gif,

(F3) there exist λ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq256_HTML.gif and β > λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq257_HTML.gif such that
lim sup | x | F ( t , x ) | x | λ < uniformly for Δ -a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equam_HTML.gif
and
lim inf | x | ( F ( t , x ) , x ) 2 F ( t , x ) | x | β > 0 uniformly for Δ -a.e. t [ 0 , T ] T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equan_HTML.gif
(F4) there exists r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq258_HTML.gif such that
F ( t , x ) 0 , | x | r , and Δ -a.e. t [ 0 , T ] T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equao_HTML.gif
(F5) there exist a i j , b i j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq259_HTML.gif and ξ i j [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq260_HTML.gif such that
| I i j ( t ) | a i j + b i j | t | ξ i j for every t R , i Γ , j Λ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equap_HTML.gif

(F6) 0 t I i j ( s ) d s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq261_HTML.gif for every t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq262_HTML.gif, i Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq148_HTML.gif, j Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq263_HTML.gif,

(F7) there exists ζ i j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq264_HTML.gif such that
2 0 t I i j ( s ) d s I i j ( t ) t 0 for all i Γ , j Λ and | t | ζ i j , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equaq_HTML.gif
and
lim t 0 I i j ( t ) t = 0 for all i Γ , j Λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equar_HTML.gif

Then problem (1.1) has at least two weak solutions. The one is a nontrivial weak solution, the other is a trivial weak solution.

Proof By Lemma 3.1, φ C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq265_HTML.gif. Set X = H Δ , T 1 , X 1 = H + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq266_HTML.gif with ( e n ) n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq267_HTML.gif being its Hilbertian basis, X 2 = H H 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq268_HTML.gif and define
X n 1 = span { e 1 , e 2 , , e n } , n N , X n 2 = X 2 , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equas_HTML.gif
Then we have
X 0 1 X 1 1 X 1 , X 0 2 X 1 2 X 2 , X 1 = n N X n 1 ¯ , X 2 = n N X n 2 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equat_HTML.gif
and
dim X n 1 < + , dim X n 2 < + , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equau_HTML.gif

We divide our proof into four parts in order to show Theorem 4.1.

Firstly, we show that φ satisfies the ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq179_HTML.gif condition.

Let { u α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq269_HTML.gif be a sequence in H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif such that α n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq181_HTML.gif is admissible and
u α n X α n , sup | φ ( u α n ) | < + , ( 1 + u α n ) φ ( u α n ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equav_HTML.gif
then there exists a constant C 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq270_HTML.gif such that
| φ ( u α n ) | C 2 , ( 1 + u α n ) φ ( u α n ) C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ17_HTML.gif
(4.1)
for all large n. On the other hand, by (F3), there are constants C 3 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq271_HTML.gif and ρ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq272_HTML.gif such that
F ( t , x ) C 3 | x | λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ18_HTML.gif
(4.2)
for all | x | ρ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq273_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif. By (A) one has
| F ( t , x ) | max s [ 0 , ρ 1 ] a ( s ) b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ19_HTML.gif
(4.3)
for all | x | ρ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq274_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif. It follows from (4.2) and (4.3) that
| F ( t , x ) | max s [ 0 , ρ 1 ] a ( s ) b ( t ) + C 3 | x | λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ20_HTML.gif
(4.4)
for all x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq21_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif. Since d l m L ( [ 0 , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq275_HTML.gif for all l , m = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq9_HTML.gif, there exists a constant C 4 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq276_HTML.gif such that
| [ 0 , T ) T ( A σ ( t ) u σ ( t ) , u σ ( t ) ) Δ t | C 4 [ 0 , T ) T | u σ ( t ) | 2 Δ t , u H Δ , T 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ21_HTML.gif
(4.5)
From (F5) and (2.5), we have that
| ϕ ( u ) | j = 1 p i = 1 N 0 | u i ( t j ) | ( a i j + b i j | t | ξ i j ) d t a ¯ p N u + b ¯ j = 1 p i = 1 N u ξ i j + 1 a ¯ p N C 1 u + b ¯ C 1 j = 1 p i = 1 N u ξ i j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ22_HTML.gif
(4.6)
for all u H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq116_HTML.gif, where a ¯ = max i Γ , j Λ { a i j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq277_HTML.gif, b ¯ = max i Γ , j Λ { b i j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq278_HTML.gif. Combining (4.4), (4.5), (4.6) and Hölder’s inequality, we have
1 2 u α n 2 = φ ( u α n ) ϕ ( u α n ) + 1 2 [ 0 , T ) T | u α n σ ( t ) | 2 Δ t + 1 2 [ 0 , T ) T ( A σ ( t ) u α n σ ( t ) , u α n σ ( t ) ) Δ t J ( u α n ) C 2 + a ¯ p N C 1 u α n + b ¯ C 1 j = 1 p i = 1 N u α n ξ i j + 1 + C 4 [ 0 , T ) T | u α n σ ( t ) | 2 Δ t + C 3 [ 0 , T ) T | u α n σ ( t ) | λ Δ t + max s [ 0 , ρ 1 ] a ( s ) [ 0 , T ) T b σ ( t ) Δ t C 2 + a ¯ p N C 1 u α n + b ¯ C 1 j = 1 p i = 1 N u α n ξ i j + 1 + C 4 T λ 2 λ ( [ 0 , T ) T | u α n σ ( t ) | λ Δ t ) 2 λ + C 3 [ 0 , T ) T | u α n σ ( t ) | λ Δ t + C 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ23_HTML.gif
(4.7)
for all large n, where C 5 = max s [ 0 , ρ 1 ] a ( s ) [ 0 , T ) T b σ ( t ) Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq279_HTML.gif. On the other hand, by (F3), there exist C 6 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq280_HTML.gif and ρ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq281_HTML.gif such that
( F ( t , x ) , x ) 2 F ( t , x ) C 6 | x | β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ24_HTML.gif
(4.8)
for all | x | ρ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq282_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif. By (A),
| ( F ( t , x ) , x ) 2 F ( t , x ) | C 7 b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ25_HTML.gif
(4.9)
for all | x | ρ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq283_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif, where C 7 = ( 2 + ρ 2 ) max s [ 0 , ρ 2 ] a ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq284_HTML.gif. Combining (4.8) and (4.9), one has
( F ( t , x ) , x ) 2 F ( t , x ) C 6 | x | β C 6 ρ 2 β C 7 b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ26_HTML.gif
(4.10)
for all x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq21_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif. According to (F7), there exists C 8 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq285_HTML.gif such that
2 0 t I i j ( s ) d s I i j ( t ) t C 8 for all  i Γ , j Λ  and  t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ27_HTML.gif
(4.11)
Thus, by (4.1), (4.10) and (4.11), we obtain
3 C 2 2 φ ( u α n ) φ ( u α n ) , u α n = 2 ϕ ( u α n ) ϕ ( u α n ) , u α n + [ 0 , T ) T [ ( F ( σ ( t ) , u α n σ ( t ) ) , u α n σ ( t ) ) 2 F ( σ ( t ) , u α n σ ( t ) ) ] Δ t = j = 1 p i = 1 N ( 2 0 u α n i ( t j ) I i j ( t ) d t I i j ( u α n i ( t j ) ) u α n i ( t j ) ) + [ 0 , T ) T [ ( F ( σ ( t ) , u α n σ ( t ) ) , u α n σ ( t ) ) 2 F ( σ ( t ) , u α n σ ( t ) ) ] Δ t p N C 8 + C 6 [ 0 , T ) T | u α n σ | β Δ t C 6 ρ 2 β T C 7 [ 0 , T ) T b σ ( t ) Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ28_HTML.gif
(4.12)
for all large n. From (4.12), [ 0 , T ) T | u α n σ | β Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq286_HTML.gif is bounded. If β > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq287_HTML.gif, by Hölder’s inequality, we have
[ 0 , T ) T | u α n σ ( t ) | λ Δ t T β λ β ( [ 0 , T ) T | u α n σ ( t ) | β Δ t ) λ β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ29_HTML.gif
(4.13)
Since ξ i j [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq288_HTML.gif for all i Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq148_HTML.gif, j Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq149_HTML.gif, by (4.7) and (4.13), { u α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq269_HTML.gif is bounded in H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif. If β λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq289_HTML.gif, by (2.5), we obtain
[ 0 , T ) T | u α n σ ( t ) | λ Δ t = [ 0 , T ) T | u α n σ ( t ) | β | u α n σ ( t ) | λ β Δ t u α n λ β [ 0 , T ) T | u α n σ ( t ) | β Δ t C 1 λ β u α n λ β [ 0 , T ) T | u α n σ ( t ) | β Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ30_HTML.gif
(4.14)
Since ξ i j [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq260_HTML.gif, λ β < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq290_HTML.gif, by (4.7) and (4.14), { u α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq269_HTML.gif is also bounded in H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif. Hence, { u α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq269_HTML.gif is also bounded in H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif. Going if necessary to a subsequence, we can assume that u α n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq291_HTML.gif in H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif. From Theorem 2.5, we have u α n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq292_HTML.gif and [ 0 , T ) T | u α n σ ( t ) u σ ( t ) | 2 Δ t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq293_HTML.gif. Since
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equaw_HTML.gif

This implies [ 0 , T ) T | u α n Δ ( t ) u Δ ( t ) | 2 Δ t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq294_HTML.gif, and hence u α n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq295_HTML.gif. Therefore, u α n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq296_HTML.gif in H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif. Hence φ satisfies the ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq179_HTML.gif condition.

Secondly, we show that φ maps bounded sets into bounded sets.

It follows from (3.2), (4.4), (4.5) and (4.6) that
| φ ( u ) | = 1 2 [ 0 , T ) T | u Δ ( t ) | 2 Δ t + j = 1 p i = 1 N 0 u i ( t j ) I i j ( t ) d t 1 2 [ 0 , T ) T ( A σ ( t ) u σ ( t ) , u σ ( t ) ) Δ t + J ( u ) 1 2 [ 0 , T ) T | u Δ ( t ) | 2 Δ t + C 4 2 [ 0 , T ) T | u σ ( t ) | 2 Δ t + a ¯ p N C 1 u + b ¯ C 1 j = 1 p i = 1 N u ξ i j + 1 + C 3 [ 0 , T ) T | u σ ( t ) | λ Δ t + max s [ 0 , ρ 1 ] a ( s ) [ 0 , T ) T b σ ( t ) Δ t 1 2 C 4 u 2 + a ¯ p N C 1 u + b ¯ C 1 j = 1 p i = 1 N u ξ i j + 1 + C 3 T u λ + C 5 1 2 C 4 u 2 + a ¯ p N C 1 u + b ¯ C 1 j = 1 p i = 1 N u ξ i j + 1 + C 3 T C 1 λ u λ + C 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equax_HTML.gif

for all u H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq116_HTML.gif. Thus, φ maps bounded sets into bounded sets.

Thirdly, we claim that φ has a local linking at 0 with respect to ( X 1 , X 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq183_HTML.gif.

Applying (F2), for ϵ 1 = δ 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq297_HTML.gif, there exists ρ 3 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq298_HTML.gif such that
| F ( t , x ) | ϵ 1 | x | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ31_HTML.gif
(4.15)
for all | x | ρ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq299_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif. By (F7), for ϵ 2 = δ 4 p N C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq300_HTML.gif, there exists ρ 4 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq301_HTML.gif such that
| I i j ( t ) | ϵ 2 | t | , | t | ρ 4 , i Γ , j Λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ32_HTML.gif
(4.16)
Let ρ 5 = min { ρ 3 , ρ 4 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq302_HTML.gif. For u X 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq303_HTML.gif with u r 1 ρ 5 C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq304_HTML.gif, by (2.5), (3.2), (3.6), (4.15) and (4.16), we have
φ ( u ) = q ( u ) + j = 1 p i = 1 N 0 u i ( t j ) I i j ( t ) d t [ 0 , T ) T F ( σ ( t ) , u σ ( t ) ) Δ t δ u 2 j = 1 p i = 1 N 0 | u i ( t j ) | | I i j ( t ) | d t ϵ 1 [ 0 , T ) T | u σ ( t ) | 2 Δ t δ u 2 j = 1 p i = 1 N 0 | u i ( t j ) | ϵ 2 | t | d t ϵ 1 [ 0 , T ) T | u σ ( t ) | 2 Δ t δ u 2 ϵ 2 j = 1 p i = 1 N u 2 ϵ 1 [ 0 , T ) T | u σ ( t ) | 2 Δ t δ u 2 ϵ 2 p N C 1 u 2 ϵ 1 u 2 δ u 2 δ 4 u 2 δ 4 u 2 = δ 2 u 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equay_HTML.gif
This implies that
φ ( u ) 0 , u X 1  with  u r 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equaz_HTML.gif
On the other hand, it follows from (F6) that
ϕ ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ33_HTML.gif
(4.17)
for all u H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq116_HTML.gif. Let u = u + u 0 X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq305_HTML.gif satisfy u r 2 r C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq306_HTML.gif. Using (F4), (2.5), (3.2), (3.5) and (4.17), we obtain
φ ( u ) = q ( u ) + ϕ ( u ) [ 0 , T ) T F ( σ ( t ) , u σ ( t ) ) Δ t δ u 2 [ 0 , T ) T F ( σ ( t ) , u σ ( t ) ) Δ t δ u 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equba_HTML.gif
This implies that
φ ( u ) 0 , u X 2  with  u r 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equbb_HTML.gif

Let M = min { r 1 , r 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq307_HTML.gif. Then φ satisfies the condition ( I 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq308_HTML.gif of Theorem 3.1.

Finally, we claim that for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq309_HTML.gif,
φ ( u ) as  u , u X n 1 X 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equbc_HTML.gif
For given n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq309_HTML.gif, since X n 1 X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq310_HTML.gif is a finite dimensional space, there exists C 9 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq311_HTML.gif such that
u C 9 u L 2 , u X n 1 X 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ34_HTML.gif
(4.18)
By (F1), there exists ρ 6 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq312_HTML.gif such that
F ( t , x ) C 9 2 ( C 4 + δ ) | x | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ35_HTML.gif
(4.19)
for all | x | ρ 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq313_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif. From (A), we get
| F ( t , x ) | max s [ 0 , ρ 6 ] a ( s ) b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ36_HTML.gif
(4.20)
for all | x | ρ 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq314_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif. Equations (4.19) and (4.20) imply that
F ( t , x ) C 9 2 ( C 4 + δ ) | x | 2 C 10 max s [ 0 , ρ 6 ] a ( s ) b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ37_HTML.gif
(4.21)
for all x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq21_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq126_HTML.gif, where C 10 = C 9 2 ( C 4 + δ ) ρ 6 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq315_HTML.gif. Using (3.2), (3.6), (4.5), (4.17), (4.18) and (4.21), we have, for u = u + + u 0 + u X n 1 X 2 = X n 1 H 0 H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq316_HTML.gif,
φ ( u ) = 1 2 [ 0 , T ) T | u Δ ( t ) | 2 Δ t + j = 1 p i = 1 N 0 u i ( t j ) I i j ( t ) d t 1 2 [ 0 , T ) T ( A σ ( t ) u σ ( t ) , u σ ( t ) ) Δ t [ 0 , T ) T F ( σ ( t ) , u σ ( t ) ) Δ t δ u 2 + 1 2 [ 0 , T ) T | ( u + ) Δ ( t ) | 2 Δ t 1 2 [ 0 , T ) T ( A σ ( t ) ( u + ) σ ( t ) , ( u + ) σ ( t ) ) Δ t [ 0 , T ) T F ( σ ( t ) , u σ ( t ) ) Δ t δ u 2 + 1 2 [ 0 , T ) T | ( u + ) Δ ( t ) | 2 Δ t + C 4 2 [ 0 , T ) T | ( u + ) σ ( t ) | 2 Δ t [ 0 , T ) T F ( σ ( t ) , u σ ( t ) ) Δ t δ u 2 + C 4 2 u + 2 C 9 2 ( C 4 + δ ) u σ L 2 2 + C 10 T + max s [ 0 , ρ 6 ] a ( s ) [ 0 , T ) T b σ ( t ) Δ t δ u 2 + C 4 u + 2 ( C 4 + δ ) u 2 + C 10 T + C 11 = δ u 2 + C 4 u + 2 ( C 4 + δ ) u + + u 0 + u 2 + C 10 T + C 11 δ u 2 + C 4 u + 2 ( C 4 + δ ) u + 2 δ u 0 + u 2 + C 10 T + C 11 δ u 2 + C 4 u + 2 ( C 4 + δ ) u + 2 δ u 0 2 + C 10 T + C 11 = δ u 2 + C 10 T + C 11 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equbd_HTML.gif

where C 11 = max s [ 0 , ρ 6 ] a ( s ) [ 0 , T ) T b σ ( t ) Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq317_HTML.gif. Hence, for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq309_HTML.gif, φ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq318_HTML.gif as u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq187_HTML.gif and X n 1 X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq310_HTML.gif.

Thus, by Theorem 3.1, problem (1.1) has at least one nontrivial weak solution. The proof is complete. □

Example 4.1 Let T = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq31_HTML.gif, T = π 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq319_HTML.gif, N = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq320_HTML.gif, t 1 = π 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq321_HTML.gif. Consider the second-order Hamiltonian system with impulsive effects
{ u ¨ ( t ) + A ( t ) u ( t ) + F ( t , x ) = 0 , a.e.  t [ 0 , π 2 ] ; u ( 0 ) u ( π 2 ) = u ˙ ( 0 ) u ˙ ( π 2 ) = 0 , Δ u ˙ ( t 1 ) = u ˙ ( t 1 + ) u ˙ ( t 1 ) = I ( u ( t 1 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ38_HTML.gif
(4.22)
where A ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq322_HTML.gif,
F ( t , x ) = { | x | 4 , | x | 5 , 625 5 3 2 x 1875 2 5 3 2 , 3 2 < x < 5 , 0 , | x | 3 2 , 625 3 2 5 x + 1875 2 3 2 5 , 5 x < 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Eqube_HTML.gif
for all x R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq323_HTML.gif and t [ 0 , π 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq324_HTML.gif,
I ( t ) = { 0 , t 4 , 6 ( t 4 ) 3 , 3 t < 4 , 6 t 12 , 1 < t < 3 , 6 t 3 , | t | 1 , 6 t + 12 , 3 < t < 1 , 6 ( t + 4 ) 3 , 4 < t 3 , 0 , t 4 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equbf_HTML.gif
then all conditions of Theorem 4.1 hold. According to Theorem 4.1, problem (4.22) has at least one nontrivial weak solution. In fact,
u ( t ) = { 3 2 cos t , t [ 0 , π 4 ] ; 3 2 sin t , t [ π 4 , π 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equbg_HTML.gif

is the solution of problem (4.22).

Theorem 4.2 Assume that (A), (F5), (F6), (F7) and the following conditions are satisfied.

(F8) lim sup | x | 0 F ( t , x ) | x | 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq325_HTML.gif uniformly for Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq326_HTML.gif,

(F9) there exist constants μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq327_HTML.gif and r 3 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq328_HTML.gif such that ( F ( t , x ) , x ) μ F ( t , x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq329_HTML.gif for all t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq18_HTML.gif and | x | r 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq330_HTML.gif,

(F10) F ( t , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq331_HTML.gif for all x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq21_HTML.gif and Δ-a.e. t [ 0 , T ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq18_HTML.gif.

Then problem (1.1) has at least one nontrivial weak solution.

Proof Set E 1 = H + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq332_HTML.gif, E 2 = H H 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq333_HTML.gif and E = H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq334_HTML.gif. Then E is a real Hilbert space, E = E 1 E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq189_HTML.gif, E 2 = E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq190_HTML.gif and dim ( E 2 ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq335_HTML.gif.

Firstly, we prove that φ satisfies the (PS) condition. Indeed, let { u k } H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq251_HTML.gif be a sequence such that | φ ( u k ) | C 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq336_HTML.gif and φ ( u k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq337_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq233_HTML.gif. As the proof of Theorem 4.1, it suffices to show that { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq338_HTML.gif is bounded in H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif. By (F9) there exist positive constants C 13 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq339_HTML.gif, C 14 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq340_HTML.gif such that
F ( t , x ) C 13 | x | μ C 14 , t [ 0 , T ] T , x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ39_HTML.gif
(4.23)
(see [34]). By (F9), (4.11) and (4.23), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ40_HTML.gif
(4.24)
for large k, where C 15 = ( r 3 + μ ) max s [ 0 , r 3 ] a ( s ) [ 0 , T ) T b σ ( t ) Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq341_HTML.gif. Equation (4.24) implies that there exists C 16 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq342_HTML.gif such that
[ 0 , T ) T | u k σ ( t ) | μ Δ t C 16 ( 1 + u k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ41_HTML.gif
(4.25)
Combining (3.2), (4.6), (4.11) and (4.25), we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ42_HTML.gif
(4.26)

for large k. Since μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq327_HTML.gif, ξ i j [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq288_HTML.gif, by (4.26), { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq338_HTML.gif is bounded in H Δ , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq142_HTML.gif.

For any small ϵ 3 = δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq343_HTML.gif, by (F8) we know that there is a ρ 7 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq344_HTML.gif such that
F ( t , x ) ϵ 3 | x | 2 , for  | x | < ρ 7 Δ -a.e.  t [ 0 , T ] T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ43_HTML.gif
(4.27)
By (F7), for ϵ 4 = δ 8 p N C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq345_HTML.gif, there exists ρ 8 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq346_HTML.gif such that
| I i j ( t ) | ϵ 4 | t | , | t | ρ 8 , i Γ , j Λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_Equ44_HTML.gif
(4.28)
Let ρ 9 = 1 2 min { ρ 7 , ρ 8 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq347_HTML.gif. For u E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq348_HTML.gif with u r 1 ρ 9 C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-148/MediaObjects/13661_2012_Article_254_IEq349_HTML.gif, by (2.5), (3.2), (3.6), (4.27) and (4.28), we have
φ ( u ) = q ( u ) + j = 1 p i = 1 N 0 u i ( t j ) I i j ( t ) d t [ 0 , T ) T F ( σ ( t ) , u σ ( t ) ) Δ t δ u 2 j = 1 p i = 1 N 0 | u i ( t j ) | | I i j ( t ) | d t ϵ 3 [ 0 , T ) T | u σ ( t ) | 2 Δ t δ u 2 j = 1 p i = 1 N 0 | u i ( t j ) | ϵ 4 | t | d t ϵ 3 [ 0 , T ) T | u σ ( t ) | 2 Δ t δ u 2 ϵ 4 j = 1 p i = 1 N u 2 ϵ 3 [ 0 , T ) T | u σ ( t ) | 2 Δ t δ u 2 ϵ 4 p N C 1 u 2 ϵ 3 u 2 δ u 2 δ 8 u 2 δ 2 | u 2 =