On the prescribed variable exponent mean curvature impulsive system boundary value problems

R E S E A R C H

Open Access

On the prescribed variable exponent mean

curvature impulsive system boundary value

problems

Li Yin

_{, Xiaopin Liu}

_{, Jingjing Liu}

_{and Qihu Zhang}

Dedicated to Professor Xianling Fan on his 70 birthday.

*_{Correspondence:}

[email protected]

2_{College of Mathematics and}

Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China Full list of author information is available at the end of the article

Abstract

This paper investigates the existence of solutions for prescribed variable exponent mean curvature impulsive system, with periodic-like, Dirichlet, and Neumann boundary value conditions, respectively. The proof of our main result is based upon the Leray-Schauder degree. The sufficient conditions for the existence of solutions are given.

MSC: 34B37; 34B15

Keywords: variable exponent mean curvature; impulsive system; the Leray-Schauder degree

1 Introduction

In this paper, we consider the existence of solutions for the prescribed variable exponent mean curvature system

–ϕt,u+ft,u,u= , t∈(,T),t=ti, ()

whereu: [,T]→RN_{, with the following impulsive conditions:}

lim

t→t+

i

u(t) –lim

t→t–

i

u(t) =Ai

lim

t→t–

i

u(t),lim

t→t–

i

u(t)

, i= , . . . ,k, ()

lim

t→t+

i

ϕt,u(t)–lim

t→t–

i

ϕt,u(t)=Bi

lim

t→t–

i

u(t),lim

t→t–

i

u(t)

, i= , . . . ,k, ()

and one of the following boundary value conditions:

u() =u(T), and ϕ,u()=ϕT,u(T), ()

u() =u(T) = , ()

u() =u(T) = , ()

where

ϕ(t,x) = |x|

p(t)–_x

( +|x|q(t)p(t)₎q(t)

, ∀t∈[,T],x∈RN,

p,q∈C([,T],R+_{) are absolutely continuous; andp,qsatisfyp(t)≥,q(t)≥. –(ϕ(t,u))}

is called the variable exponent mean curvature operator;  <t<t<· · ·<tk<T;Ai,Bi∈ C(RN_×RN_,RN_).

Ifp() =p(T) andq() =q(T), thenϕ(,u()) =ϕ(T,u(T)) impliesu() =u(T), and () is the periodic boundary value condition. Thus we call () the periodic-like boundary value condition.

The system ()-() is called a prescribed variable exponent mean curvature impulsive system. It has three characteristics,i.e.impulsive, mean curvature and variable exponent. Let us simply introduce the three characteristics.

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments, such as mechanical systems with im-pact, biological systems such as heart beats, blood flows, population dynamics, theoret-ical physics, radiophysics, pharmacokinetics, mathemattheoret-ical economy, chemtheoret-ical technol-ogy, electric technoltechnol-ogy, metallurgy, ecoltechnol-ogy, industrial robotics, biotechnology processes, chemistry, engineering, control theory, medicine, and so on. There are many results on the Laplacian impulsive differential equations boundary value problems (see [–]). The results as regardsp-Laplacian impulsive differential equations boundary value problems are more difficult due to the nonlinearity ofp-Laplacian (see [–]). Because of the im-pulsive conditions and the non-continuity of solutions, this paper has more difficulties than []. In many papers about the usual Laplacian impulsive problems, the authors use the impulsive conditionlim_t→t+

i u

_{(t) –limt→t}–

i u _{(t) =B}

i(limt→t–_i u(t),limt→t_i–u(t)), but we

think that the condition () is better in this paper. Therefore, we should consider what kind of impulsive condition is suitable for prescribed variable exponent mean curvature problems, it is a difference between this paper with the usual Laplacian impulsive prob-lems.

Simultaneously, system () is a kind of mean curvature system. This kind of problems has attracted more and more attentions recently (see [–]). In [], the authors generalized the usual mean curvature systems to variable exponent mean curvature systems, and dis-cuss the existence of solutions of () with periodic-like boundary value condition (without impulsive conditions). In [], the authors dealt with the existence of solutions and non-negative solutions of ()-() with initial boundary value condition. This paper deals with the existence of solutions of ()-() with periodic-like boundary value condition, Neumann boundary value condition, or Dirichlet boundary value condition, respectively. This paper was motivated by [, ]. Similar to [, ], the proof of our main result is based upon the Leray-Schauder degree, but this paper is more difficult than [, ].

(a) If⊂RN_{is a bounded domain, the infimumλ}

p(·)of the eigenvalue ofp(x)-Laplacian

is zero in general, and thep(x)-Laplacian does not have the so-called first eigenfunction (see []); but the fact thatλp>  is very important in the study ofp-Laplacian problems,

and the first eigenfunction of the p-Laplacian was used to construct subsolutions of a

p-Laplacian problem successfully (see []).

(b) For variable exponent functionLp(·)(RN_{), the norm| · |}

p(·) is not invariant for the

translational coordinate transformation,i.e.under the usual Luxemburg norm| · |p(·), we have|u(x)|p(·)=|u(x+x)|p(·)in general (see []).

(c) In [], the author generalized the Picone identities for half-linear elliptic operators withp(x)-Laplacians and applications to Sturmian comparison theory, but the formula is different from the constant exponent case.

In this paper, we investigate the existence of solutions for the prescribed variable expo-nent mean curvature impulsive differential system boundary value problems, the proof of our main result is based upon the Leray-Schauder degree.

This paper is divided into four sections; in the second section, we will discuss the ex-istence of solutions of variable exponent mean curvature impulsive system periodic-like boundary value problems. In the third section, we will discuss the existence of solutions of variable exponent mean curvature impulsive system Dirichlet boundary value problems. Finally, in the fourth section, we will discuss the existence of solutions of variable exponent mean curvature impulsive system Neumann boundary value problems.

2 Periodic-like boundary value problems

In this section, we will discuss the existence of solutions of variable exponent mean cur-vature impulsive system periodic-like boundary value problems,i.e.the existence of solu-tions of ()-(). In order to do that, we give the following notasolu-tions and basic assumpsolu-tions: For any v∈RN, vj will denote the jth component of v; the inner product in RN will be denoted by ·,·; | · | will denote the absolute value and the Euclidean norm onRN_{. Denote J= [,T], J= [,T]\{t}

,t, . . . ,tk+}, J= [t,t],Ji= (ti,ti+], i= , . . . ,k,

where t= , tk+ =T. Denote Jio the interior of Ji, i= , , . . . ,k. Let PC(J,RN) ={x: J →RN _|x∈C(J

i,RN),i= , , . . . ,k, andx(ti+) exists fori= , . . . ,k}; PC(J,RN) ={x∈ PC(J,RN_)|x∈C(Jo

i,RN),limt→t+i u

_{(t) andlim} t→t–

i+u

_{(t) exist fori= , , . . . ,k}. For any} u(t) = (u(t), . . . ,uN(t))∈ PC(J,RN), we denote |ui| =sup{|ui(t)| |t ∈J}. Obviously, PC(J,RN_{) is a Banach space with the normu}

= (Ni=|ui|)



,PC(J,RN) is a Banach

space with the normu=u+u. In the following,PC(J,RN) andPC(J,RN) will

be simply denoted byPCandPC_{, respectively. DenoteL}_=L_(J,RN_{), and the norm inL}

isuL= [N_i=(T

 |ui(r)|dr)]

 _.

LetN≥, the functionf :J×RN ×RN→RN is assumed to be Caratheodory, and by this we mean:

(i) for almost everyt∈Jthe functionf(t,·,·)is continuous;

(ii) for each(x,y)∈RN×RN the functionf(·,x,y)is measurable onJ;

(iii) for eachR> there is aβR∈L(J,R)such that, for almost everyt∈Jand every

(x,y)∈RN_×RN _{with|x| ≤R,|y| ≤R, one has|f(t,x,y)| ≤β} R(t).

We say a functionu:J→RN is a solution of () ifu∈PCwithϕ(t,u) absolutely con-tinuous onJo

i,i= , , . . . ,k, which satisfies ()a.e.onJ.

2.1 Preliminary

In this subsection, we will make some preparations.

Lemma .(see []) ϕis a continuous function and satisfies the following. (i) For anyt∈J,ϕ(t,·)is strictly monotone,i.e.

ϕ(t,x) –ϕ(t,x),x–x > , for anyx,x∈RN,x=x.

(ii) For any fixedt∈J,ϕ(t,·)is a homeomorphism fromRNto

E=x∈RN| |x|< .

For anyt∈J, we denote byϕ–(t,·) the inverse operator ofϕ(t,·), then

ϕ–(t,x) = –|x|q(t)p(t–)q(t)|_x|p(t)–_{x, forx}∈_E\{_}_,ϕ–_{(t, ) = .}

Let us now consider the following simple problem:

ϕt,u(t)=g(t), t∈(,T),t=ti, ()

with the following impulsive boundary value conditions:

⎧ ⎪ ⎨ ⎪ ⎩

lim_t→t+

i u(t) –limt→ti–u(ti) =ai, i= , . . . ,k,

lim_t→t+

i ϕ(t,u

_{(t)) –limt→t}–

i ϕ(t,u _{(t)) =b}

i, i= , . . . ,k, u() =u(T), ϕ(,u()) =ϕ(T,u(T)),

()

whereai,bi∈RN,

k

i=|bi|< ;g∈Land satisfies

T

 g(t)dt+

k

i=bi= .

Denote

L_m=

v∈L

T



v(r)dr= 

.

Leth(t) =g(t) +_T k_i=bi, thenh∈Lm. Ifuis a solution of () with (), by integrating ()

from  tot, we find that

ϕt,u(t)=ϕ,u()+

ti<t

bi+

t



h(r) – 

T k

i= bi

dr, ∀t∈J. ()

Denoteρ=ϕ(,u()). Define operatorF:L−→PCas

F(g)(t) =

t



g(t)dt, ∀t∈J,∀g∈L.

From () and (ii) in Lemma ., we can see that

sup

t∈J

ϕt,u(t)=sup

t∈J ρ+

ti<t

bi+F

h– 

T k

i= bi

(t)

Denote

Db=

h∈L_mthere existsρ∈RN such that

sup t∈J ρ+

ti<t

bi+F

h– 

T k i= bi

(t)

< 

.

By (), we have

u(t) =u() +

ti<t

ai+F ϕ– t, ρ+

ti<t

bi+F

h– 

T k i= bi

(t), ∀t∈J,

the boundary value conditions imply that

k

i= ai+

T  ϕ– t, ρ+

ti<t

bi+F

h– 

T k i= bi

(t)

dt= .

Denotea= (a, . . . ,ak)∈RkN,b= (b, . . . ,bk)∈RkN. It is easy to see thatρis dependent

ona,bandh. For fixeda,b∈RkN_,h∈D

b, we define

(a,b,h)(ρ) = k

i= ai+

T  ϕ– t, ρ+

ti<t

bi+F

h– 

T k i= bi

(t)

dt.

Denote

Da,b=

h∈L_mthere existsρ∈RN such that

sup t∈J ρ+

ti<t

bi+F

h– 

T k i= bi

(t)

<  and(a,b,h)(ρ) = 

.

If () with () has a solution inPC, we must haveh=g(t) +_T k_i=bi∈Da,b.

DenoteWm=RkN×Lmwith the norm

wWm= k

i=

|ai|+ k

i=

|bi|+hL, ∀w= (a,b,h)∈W_m,

thenWmis a Banach space.

Denote

C∗=  Nmint∈J

ϕ–

t, 

N

, M=

(a, . . . ,ak)∈RkN

k

i=

|ai|<

 TC∗

, ()

M=

(b, . . . ,bk)∈RkN

k

i=

|bi|<

 N

. ()

(_{) For any fixed(a,b)∈M}

×M,∀h∈Da,bthe equation

w(ρ) = , wherew= (a,b,h)∈Wm,

has a unique solutionρ(w)∈RN_.

(_{) For any fixeda= (a}

, . . . ,ak)∈M,b= (b, . . . ,bk)∈M,Da,bcontains the open ball E(,_) :={u∈L_m|uL< 

}, and then one defines a mapping ρ:M ×M× E(,_)→RN_.

(_{) The function ρ : M}

 ×M ×E(,_)→RN, defined in (), is continuous and bounded.

(_{) For any fixeda= (a}

, . . . ,ak)∈M,b= (b, . . . ,bk)∈M,the setDa,bis open and un-bounded inL_m.

Proof (_{) From the definition ofD}

a,b, for any fixed (a,b)∈M×M, ∀h∈Da,b, the

equationw(ρ) =  has at least one solution. Sinceϕ(t,·) is strictly monotone, we can

see thatϕ–_{(t,·) is strictly monotone. Thus, the equation}

w(ρ) =  has a unique solution ρ(w)∈RN.

(_{) Leth∈E(,} 

), then ≤ hL<_. Denote

g=h– 

T k

i=

bi, A=

x∈RN|x|< 

.

Obviously, for any (a,b,h)∈M×M×E(,_), for anyρ∈A,(a,b,h)(ρ) is well defined, i.e.,

sup

t∈J

ρ+

ti<t

bi+F(g)(t) < .

Observe that ≤ |_t

i<tbi+F(g)(t)|<   and

 <|ρ+

ti<tbi+F(g)(t)|< 

 for any

(t,ρ)∈J×∂A. Since|ρ+_t

i<tbi+F(g)(t)|–|

ti<tbi+F(g)(t)|> 

for any (t,ρ)∈J×∂A, we have

w(ρ),ρ =

_k

i= ai,ρ

+

T

 ϕ–

t,

!

ρ+

ti<t

bi+F(g)(t) "

,ρ+

ti<t

bi+F(g)(t) #

dt

–

T

 ϕ–

t,

!

ρ+

ti<t

bi+F(g)(t) "

,

ti<t

bi+F(g)(t) #

dt

≥

T



 –ρ+

ti<t

bi+F(g)(t) q(t)

–

p(t)q(t)

×ρ+

ti<t

bi+F(g)(t)



p(t)+

dt– 

k

i=

|ai|

–

T



 –ρ+

ti<t

bi+F(g)(t) q(t)

–

×ρ+

ti<t

bi+F(g)(t)



p(t)

ti<t

bi+F(g)(t) dt = T 

 –ρ+

ti<t

bi+F(g)(t) q(t)

–

p(t)q(t)

ρ+

ti<t

bi+F(g)(t)



p(t)

×ρ+

ti<t

bi+F(g)(t)

–

ti<t

bi+F(g)(t) dt–



k

i=

|ai|

≥C∗

T



ρ+

ti<t

bi+F(g)(t)

–

ti<t

bi+F(g)(t) dt–



k

i=

|ai|

≥ TC∗–

 

k

i=

|ai|> .

This means that w(ρ),ρ>  for anyρ∈∂A.

For any fixed (a,b,x)∈M×M×E(,_), let us consider

(ρ,λ) =λ(a,b,x)(ρ) + ( –λ)ρ.

It is easy to see that (ρ,λ) =  has no solution on∂Afor anyλ∈[, ]. According

to the homotopy invariance property of the Brouwer degree, we can see thatw(ρ) = 

possesses a solution inA. ThusE(,_)⊂Da,b, and one then defines a mappingρ:M×

M×E(,_)→RN.

(_{) For any (a,b,h)∈M}

×M×E(,_), from the definition ofDa,b, we have

sup t∈J ρ+

ti<t

bi+F

h– 

T k i= bi

(t)

< .

Since_ti<bi+F(h–_T

k

i=bi)() = , we have

|ρ|=

ρ+

ti<

bi+F

h– 

T k i= bi () < .

Thus the mappingρis bounded.

Now, let us prove the continuity ofρ. Letwn= (an,bn,hn)∈M×M×E(,_) is a

convergent sequence inM×M×E(,_), andwn→w= (a,b,h)∈M×M× E(,_) asn→+∞, whereb= (b,, . . . ,bk,). Since{ρ(wn)} is a bounded sequence, it

contains a convergent subsequence{ρ(wnj)}. We may assume thatρ(wnj)→ρasj→+∞.

If w(ρ) is well defined,i.e.,supt∈J|ρ+

ti<tbi,+F(h–  T

k

i=bi,)(t)|< . Since w_nj(ρ(wnj)) = , lettingj→+∞, we havew(ρ) = . Combining with (), we getρ= ρ(w). This means thatρis continuous.

It only remains to prove thatw(ρ) is well defined.

Denote byE∗⊂RN_{an open ball which centered atρ(w}

) such thatw(·) is well defined

onE∗. According to Lemma ., we have

w(ρ),ρ–ρ(w) =

w(ρ) –w

ρ(w)

w(ρ),ρ–ρ(w) > , ∀ρ∈∂E∗.

It is easy to see that there exists a neighborhoodU⊂E(,_) ofhinLm,U⊂Mis

a neighborhood ofa, U⊂M is a neighborhood ofb, such that, for each (a,b,y)∈ U×U×U,(a,b,y)(·) is well defined onE∗. In addition, the mapping

(a,b,y)−→ inf

ρ∈∂E∗

(a,b,y)(ρ),ρ–ρ(w)

is easily seen to be continuous inU×U×U. Thus, there exists a neighborhoodV× V×V⊂U×U×Uofwsuch that

(a,b,x)(ρ),ρ–ρ(w) > , ∀ρ∈∂E∗,∀(a,b,x)∈V×V×V.

For any fixed (a,b,x)∈V×V×V, let us consider

(ρ,λ) =λ(a,b,x)(ρ) + ( –λ)

ρ–ρ(w)

.

Obviously, (ρ,λ) =  has no solution on∂E∗for anyλ∈[, ].

For any (a,b,x)∈V×V×V, according to the homotopy invariance property of the

Brouwer degree, we conclude that the equation (a,b,x)(ρ) =  has its (unique) solution

onE∗. Sincewnj→wasj→+∞, we havewnj ∈V×V×V whenjis large enough,

andρ(wnj)∈E∗. Sinceρ(wnj)→ρasj→+∞,ρ∈E∗. This means thatw(ρ) is well

defined. (_{) Letu}

n(t) = _NTn sinn_Tπt ·(, , . . . , ), thenun∈Lm. DenoteA={x∈RN | |x|< _}.

For anyn∈N+_,∀a∈M

,∀b∈Mand∀ρ∈A, we can see the following function is well

defined:

n(ρ) = k

i= ai+

T

 ϕ–

t,

ρ+

ti<t

bi+F

un–



T k

i= bi

(t)

dt.

If|ρ|=_, then there exists someρj_{such that|ρ}j_{| ≥} 

N. Without loss of generality, we

may assume thatρj_≥ 

N. Sincesupt∈J|F(un)(t)| ≤  Nπ and

k

i=|bi|<_N , we have

ti<t

bi+F

un–



T k

i= bi

(t)

<

 N +

 Nπ +

 N <



N, ∀t∈J.

Denotezn(t) =ρ+

ti<tbi+F(un–  T

k

i=bi). Obviously, _<|zn(t)|<_,∀t∈J.

Sinceρj_≥ 

N, thejth componentz j

nofznsatisfieszjn(t) >_N ,∀t∈J. Notice thatp(t)≥,

then we can see that thejth component (ϕ–_(t,z

n(t)))jofϕ–(t,zn(t)) satisfies

ϕ–t,zn(t) j

= –zn(t) q(t) –

p(t)q(t)_z

n(t)



p(t)–_zj

n(t)

≥

 –  N

q(t)

–

p(t)q(t)

__N



p(t)– _

N ≥C∗.

Consider(ρ,λ) =λn(ρ) + ( –λ)ρ. Since C∗= N mint∈Jϕ–(t,  N) and

k

i=|ai|< 

TC∗, we have j

(ρ,λ) has no solution on∂Afor anyλ∈[, ]. Thus

dB

(ρ, ),A, 

=dB

(ρ, ),A, 

= .

Thus  =(ρ, ) =n(ρ) has a solution onA. Therefore{un} ⊂Da,b, This means that Da,bis unbounded.

Similar to the proof of (_{), for anyw}

= (a,b,h)∈ {(a,b)} ×Da,b, there exists a

neigh-borhoodV⊆Uofh, and a neighborhoodE∗⊂RN ofρ(w) such that(a,b,x)(ρ) is well

defined for any (a,b,x)∈ {(a,b)} ×Vandρ∈E∗, which satisfies the requirement that, for any (a,b,x)∈ {(a,b)} ×V,(a,b,x)(ρ) =  has a solutionρ∈E∗. This means that the setDa,b

is open inL m.

This completes the proof.

We continue now with our argument previous to Lemma .. Let us define

P:PC→PC, u−→u(); Q:L→L, h−→  T

T



h(t)dt;

b:L→L, h−→(I–Q)h–



T k

i=

bi, for any fixedb∈RN.

We can splitL_asL_=L

m+F, whereF is theN-dimensional subspace of constant

mappings. The operatorQis a continuous projection fromL_ontoF

. Let us consider the

subset$Da,bofLwhich is given by

$

Da,b=Da,b+F,

and define the nonlinear operatorK(a,b):$Da,b→PC, as

K(a,b)(h)(t) =F

ϕ– !

t,

ρ+

ti<t

bi+F(h) "

(t), ∀t∈J.

We sayUis a closed equi-integrable set in$Da,b, if there existsβ∈L, such that, for any u∈U,

u(t)≤β(t) a.e. onJ.

Lemma . If(a,b)∈M×M,then the operator(K(a,b)◦b)(·)is continuous and sends closed equi-integrable subsets of$Da,binto relatively compact sets in PC.

Proof It is easy to check that (K(a,b)◦b)(h)(·)∈PC,∀h∈$Da,b. Since

(K(a,b)◦b)(h)(t) =ϕ–

t,

!

ρ+

ti<t

bi+F

b(h) "

, ∀t∈J,

Let nowUbe a closed equi-integrable set in$Da,b, then there existsβ∈L, such that, for

anyu∈U,

u(t)≤β(t) a.e. onJ.

We want to show that (K(a,b)◦b)(U)⊂PCis a compact set.

Let{un}is a sequence in (K(a,b)◦b)(U), then there exists a sequence{hn} ∈Usuch that un= (K(a,b)◦b)(hn). For anyr,r∈J, we have

Fb(hn)

(r) –F

b(hn)

(r)≤ r

r

β(t)dt+|r–r|



T

T



β(t)dt+

k

i= bi

.

Hence the sequence {F(b(hn))} is uniformly bounded and equi-continuous. By the

Ascoli-Arzela theorem, there exists a subsequence of {F(b(hn))} (which we rename

the same) is convergent inPC. According to the bounded continuity of the operatorρ, we can choose a subsequence of {ρ(a,b,b(hn)) +F(b(hn))} (which we still denote

{ρ(a,b,b(hn)) +F(b(hn))}) which is convergent inPC, then the sequence

ϕt, (K(a,b)◦b)(hn)(t)

=ρ+

ti<t

bi+F

b(hn)

is convergent according to the norm inPC, by which, combined with the continuous of

ϕ–, we can see

(K(a,b)◦b)(hn)(t) =ϕ– !

t,

ρ+

ti<t

bi+F

b(hn) "

(t), ∀t∈J,

is convergent according to the norm inPC. Since

(K(a,b)◦b)(hn)(t) =F

ϕ– !

t,

ρ+

ti<t

bi+F

b(hn) "

(t), ∀t∈J,

according to the continuity ofϕ–_{, we can see (K}

(a,b)◦b)(hn) is convergent inPC. Thus

we conclude that{un}is convergent inPC. This completes the proof.

We denote byNf(u) :PC→Lthe Nemytskii operator associated tof defined by

Nf(u)(t) =f

t,u(t),u(t), a.e. onJ. ()

Denote

#(Nf)(u) = (I–Q)Nf(u) –



T k

i= Bi,

K(u) =F

ϕ– !

t,

ρ+

ti<t

Bi+F

#(Nf)(u) "

and

∗_u(ρ) =

k

i= Ai+

T

 ϕ–

t,

!

ρ+

ti<t

Bi+

t



fr,u,udr "

dt,

whereAi=Ai(limt→t–

i u(t),limt→ti–u _(t)),B

i=Bi(limt→t–

i u(t),limt→t–i u

_{(t)),ρis a solution}

of∗_u(ρ) = . We assume

(H)

k

i=|Ai(x,y)|<_TC∗and

k

i=|Bi(x,y)|<_N ,∀(x,y)∈RN.

Lemma . If(H)is satisfied,then u is a solution of()-()if and only if u is a solution of the following abstract operator equation:

u=Pu+

ti<t

Ai+



T k

i=

Bi+QNf(u) +K(u). ()

Proof (i) Ifuis a solution of ()-(), by integrating () from  tot, we find that

ϕt,u(t)=ρ+

ti<t

Bi+

t



ft,u,udt. ()

From (), we have



T k

i= Bi+



T

T



ft,u,udt= 

T k

i=

Bi+QNf(u) = ,

thus#(Nf)(u) =Nf(u).

From (), we have

u=Pu+

ti<t

Ai+



T k

i=

Bi+QNf(u) +K(u),

whereρsatisfies

k

i= Ai+F

ϕ–

! t,

ρ+

ti<t

Bi+F

#(Nf)(u) "

(T) = .

Henceuis a solution of ().

(ii) Ifuis a solution of (), it is easy to see that () is satisfied. Lett= , we have

u() =Pu+ 

T k

i=

Bi+QNf(u),

then



T k

i=

Bi+QNf(u) = , ()

By the definition of the mappingρ, we have

k

i= Ai+F

ϕ–

! t,

ρ+

ti<t

Bi+F

#(Nf)(u) "

(T) = ,

thenu() =u(T). From () and (), we also have

ϕt,u=ρ+

ti<t

Bi+F

#(Nf)(u)

(t), t∈(,T),t=ti,

ϕt,u=ft,u,u, t∈(,T),t=ti.

()

From (), we find that () is satisfied. Sincek_i=Bi+F(#(Nf)(u))(T) = , we have

ϕ,u()=ϕT,u(T).

Henceuis a solution of ()-(). This completes the proof.

2.2 Existence of solutions

In this subsection, we will apply the Leray-Schauder degree to deal with the existence of solutions of ()-().

Theorem . If(H)is satisfied,⊂PCis open bounded such that the following condi-tions hold:

() For anyu∈,the mappingt→f(t,u,u)belongs to{u∈L| uL< 

}.

(_{) For eachλ∈(, ),the problem} ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(ϕ(t,u))=λf(t,u,u), t∈(,T),t=ti,

lim_t→t+

i u(t) –limt→t–i u(t)

=λAi(limt→t_i–u(t),limt→t–_i u(t)), i= , . . . ,k,

lim_t→t+

i ϕ(t,u

_{(t)) –limt→t}–

i ϕ(t,u _(t))

=λBi(limt→t–

i u(t),limt→t–i u

_{(t)), i= , . . . ,k,} u() =u(T), and ϕ(,u()) =ϕ(T,u(T))

()

has no solution on∂. () The equation

ω(l) := 

T

T



f(t,l, )dt+ 

T k

i=

Bi(l, ) =  ()

has no solution on∂∩F.

(_{) The Brouwer degreed}

B[ω,∩F, ]= . Then()-()has a solution on.

Proof Denote

Ai=Ai

lim

t→t–

i

u(t),lim

t→t–

i

u(t)

, Bi=Bi

lim

t→t–

i

u(t),lim

t→t–

i

u(t)

Let us consider the following impulsive equation:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(ϕ(t,u))=λNf(u) + ( –λ)[QNf(u) +_T k

i=Bi], t∈(,T),t=ti,

lim_t→t+

i u(t) –limt→ti–u(t) =λAi, i= , . . . ,k,

lim_t→t+

i ϕ(t,u

_{(t)) –lim} t→t–

i ϕ(t,u

_{(t)) =λB}

i, i= , . . . ,k, u() =u(T), ϕ(,u()) =ϕ(T,u(T)).

()

For anyλ∈(, ], observe that, ifuis a solution to () oruis a solution to (), we have necessarily

QNf(u) +



T k

i= Bi= .

This means that () and () have the same solutions forλ∈(, ]. We denoteN(·,·) :PC_{×[, ]→L}_{defined by}

N(u,λ) =λNf(u) + ( –λ)

QNf(u) +



T k

i= Bi

,

whereNf(u) is defined by (). Denote

λ:PC→PC, u−→(I–Q)N(u,λ) –

λ T

k

i= Bi,

Kλ(u)(t) =F

ϕ–

! t,

ρ+λ

ti<t

Bi+F

λ(u)

"

(t), ∀t∈J,

whereρ=ρ(λ,u) is the solution ofλk_i=Ai+Kλ(u) = .

Obviously

λ(u) = (I–Q)λNf(u) – λ T

k

i=

Bi, and (·) = .

From (_{), we can see that (I–Q)λN}

f(u)∈E(,_)⊂Lm,∀u∈,∀λ∈[, ]. Combining

(H) and Lemma ., we can see that there exists only oneρ=ρ(λ,u)∈Fsuch that

sup

t∈J

ρ+λ

ti<t

Bi+F

λ(u)<  and λ

k

i=

Ai+Kλ(u) = , ∀λ∈[, ],∀u∈.

Let

f(u,λ) :=Pu+λ

ti<t

Ai+λ



T k

i=

Bi+QN(u,λ) +Kλ(u)

=Pu+λ ti<t

Ai+



T k

i=

then the fixed point off(u, ) is a solution for ()-(). Also problem () can be written

in the equivalent form

u=f(u,λ). ()

Sincef is Caratheodory, it is easy to see thatN(·,·) is continuous and sends bounded

sets into equi-integrable sets. According to Lemma ., we can conclude thatf is

com-pact continuous on ×[, ]. From Lemma ., we can see that the problem ()-() is equivalent tou=f(u, ). We assume that forλ= , () does not have a solution on∂,

otherwise we complete the proof. Now, from hypothesis (_{), it follows that () has no}

so-lution for (u,λ)∈∂×(, ]. Forλ= , () is equivalent to the following usual differential equation boundary value problem:

–(ϕ(t,u))=QNf(u) +_T ki=Bi(limt→t–

i u(t),limt→ti–u

_{(t)), t∈(,T),} u() =u(T), and ϕ(,u()) =ϕ(T,u(T)),

and ifuis a solution to this problem, we must have

T



ft,u(t),u(t)dt+

k

i= Bi

lim

t→t–

i

u(t),lim

t→t–

i

u(t)

= . ()

Whenλ= , the problem is a usual differential equation boundary value problem. Hence

ϕ(t,u(t))≡c, wherec∈Fis a constant mapping. Sinceu() =u(T), for anyi∈ {, . . . ,N},

there exists r_i ∈(,T), such that (ui_)(ri

) = , hence (ui)≡, and we haveu≡l∈F.

Thus, by () we have

T



f(r,l, )dr+

k

i=

Bi(l, ) = ,

which, together with hypothesis (_{), implies thatu=l∈/ ∂. Thus we have proved that}

() has no solution (u,λ) on∂×[, ], and then we find that, for eachλ∈[, ], the Leray-Schauder degreedLS[I–f(·,λ),, ] is well defined. From the homotopy invariant

property of that degree, we have

dLS %

I–f(·, ),,  &

=dLS %

I–f(·, ),,  &

.

Whenλ= , we have(·) = , andK(u) = ,∀u∈PC. Thus

f(u, ) =Pu+



T k

i=

Bi+QNf(u) +K(u) =Pu+



T k

i=

Bi+QNf(u),

and then all the solutions ofu–f(u, ) =  belong toF. Thus

u–f(u, ) =u–Pu–



T k

i=

Bi–QNf(u) = –



T k

i=

By the properties of the Leray-Schauder degree, we have

dLS %

I–f(·, ),,  &

= (–)NdB[ω,∩F, ],

where the functionωis defined in () anddBdenotes the Brouwer degree. Since, by

hy-pothesis (_{), this last degree is different from zero. Thus ()-() has a solution. This}

com-pletes the proof.

In the following, we will give an application of Theorem .. Assume:

(H) f(t,u,v) =δg(t,u,v), whereδis a positive parameter, and

g(t,u,v) =τ(t)|u|q(t)–_u+μ(t)|_v|q(t)_+γ(t),

whereq,q∈C(I,R), and≤q(t) <q(t),∀t∈Jor≤q(t) <q(t),∀t∈J.

(H) μ = (μ, . . . ,μN) ∈ C(J,RN), τ ∈ C(J,R), and τ satisfies τ ≤ mint∈J|τ(t)| ≤

maxt∈J|τ(t)| ≤τ, whereτandτare positive constants.

Obviously, there exists a positive constantεsuch that

C#:=min t∈J

σ

N σ

ε

q(t)–μ|Nε|q(t)

> ,

whereσ=_T+T .

(H)

k

i=|Ai| ≤σ_ε,

k

i=|Bi| ≤min{_TδτC#,mint∈I |

ε

T+|p(t)

(+|Nε|q(t)p(t)₎q(t)}.

(H) Aji(x,y)yj≥,∀x,y∈RN,i= , . . . ,k,j= , . . . ,N, and k

i=Bi(l, ) = ,∀l∈RN.

(H) γ = (γ, . . . ,γN)∈C(J,RN)satisfies|γi|<_τC#, for anyj= , . . . ,N.

Denote

ε=

'

u∈PC max

≤j≤Nu j

+u j



<ε (

.

Obviously,εis an open subset ofPC.

Theorem . If(H)-(H)are satisfied,then problem()-()has at least one solution on ε,when the positive parameterδis small enough.

Proof We denoteN(·,·) :PC_{×[, ]→L}_{defined by}

N(u,λ) =λNf(u) + ( –λ)

QNf(u) +



T k

i= Bi

,

Let us consider the following problem:

u=f(u,λ) :=Pu+λ

ti<t

Ai+λ



T k

i=

Bi+QN(u,λ) +Kλ(u)

=Pu+λ ti<t

Ai+



T k

i=

Bi+QNf(u) +Kλ(u). ()

Obviously,uis a solution of ()-() if and only ifuis a solution of the abstract equation () whenλ= . We only need to prove that the conditions of Theorem . are satisfied.

(_{) When the positive parameterδis small enough, for anyu∈}

ε, we can see that the

mappingt−→δg(t,u,u)belongs to{u∈L| uL< 

}.

(_{) We shall prove that for eachλ∈(, )the problem} ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(ϕ(t,u))=λf(t,u,u), t∈(,T),t=ti,

lim_t→t+

i u(t) –limt→t–i u(t) =λAi(limt→ti–u(t),limt→t–i u

_{(t)), i= , . . . ,k,}

lim_t→t+

i ϕ(t,u

_{(t)) –limt→t}–

i ϕ(t,u _(t))

=λBi(limt→t–

i u(t),limt→t–i u

_{(t)), i= , . . . ,k,} u() =u(T), and ϕ(,u()) =ϕ(T,u(T))

has no solution on∂ε.

If it is false, then there exists aλ∈(, ), andu∈∂εis a solution of (). We have

ϕt,u=ρ+λ

ti<t

Bi+

t



λfr,u,udr, ∀t∈J.

From the boundary value condition (), we have

T



fr,u,udr+

k

i=

Bi= . ()

Sinceu∈∂ε, there exists anj∈ {, . . . ,N}such that|uj|+|(uj)|=ε.

(i) Suppose that|uj_|

≥σ ε, then|(uj)|≤( –σ)ε. Let  <r<r<T, according to ()

and (H), we have

uj(r) –uj(r)= _rr



uj(r)dr+

r<ti<r

Ai

≤

T



uj(r)dr+

r<ti<r

|Ai|

≤

T



( –σ)εdr+

k

i=

|Ai| ≤ σ

ε+

σ

ε=

σ

ε.

Since|uj_|

The condition (H) means thatτhas constant sign onJ. Thusτujalso has constant sign

onJ. Assume thatτuj_{is positive, then we have}

T



fjt,u,udt+

k

i= Bj_i

≥

T



δτ(t)u(t)q(t)–

uj(t)–μj_u(t)q(t)

–γj(t)dt–

k

i=

|Bi|

≥ 

TδτC#> .

This is a contradiction to ().

Assume thatτujis negative, from (H) and (H), then we have

T



fjt,u,udt+

k

i= Bj_i

≤

T



–δτ(t)u(t)q(t)–uj(t)–μj_u(t)q(t)–γj(t)dt+

k

i=

|Bi|

≤–

TδτC#< .

This is a contradiction to (). (ii) Suppose that|uj_|

<σ ε, then ( –σ)ε<|(uj)|≤ε. This implies that|(uj)(s)|> ( – σ)ε=_T+ εfor somes∈J. Without loss of generality, we may assume that (uj)(s) > ,

the discussion of the case of (uj_)(s

) <  is similar.

(a_{) Suppose thatinf}

t∈J(uj_{)(t) > . From (H}

), we have

Aj_i

lim

t→t–

i

u(t),lim

t→t–

i

u(t)

≥, i= , . . . ,k.

Since

uj(t) =uj() +

t



uj(t)dt+

ti<t

Aj_i,

we can see thatuj_{(t) is increasing, andu}j_{(T) >u}j_{(). This is a contradiction to ().}

(b_{) Suppose thatinf}

t∈J(uj_{)(t) < . Then there existsr}j

∈Jsuch that (uj)(r j

) < .

De-note

ϕjt,u(t) = |u

_|p(t)–_(uj_)(t)

( +|u|q(t)p(t)₎q(t)

.

We have

ϕjs,u

(s) –ϕj

r_j,urj_=λ s

rj

fjt,u,udt+λ rj_<ti<s

Bj_i+λ s<ti<rj

According to (H) and (H), when the positive parameterδis small enough, we have

| ε

T+| p(s)

( +|Nε|q(s)p(s)₎ 

q(s)

≤ϕjs,u

(s)≤ϕj

s,u

(s) –ϕj

tj_,ut_j

≤λ

T



fjt,u,udt+λ rj_<ti<s

Bj_i+λ s<ti<rj

Bj_i

≤δ

T

 τ

%

|Nε|q(r)_+μ

|Nε|q(t)+γ &

dt+

k

i=

|Bi|

< |

ε

T+| p(s)

( +|Nε|q(s)p(s)₎q(s)

.

This is a contradiction to ().

(c) Suppose thatinft∈J(uj)(t) = . Similar to the proof of (b), we can get a contradiction to ().

Summarizing this argument, for eachλ∈(, ), problem () has no solution on∂ε.

(_{) From (H}

), (H), and (H), it is easy to see that

ω(l) =

T



f(t,l, )dt+

k

i=

Bi(l, ) =

T



τ(t)|l|q(t)–_{l+γ(t)dt= }

has no solution on∂∩F.

(_{) Let}

h(t,l,λ) =λω(l) + ( –λ)τ(t) |τ(t)|l=λ

T



τ(t)|l|q(t)–_{l+γ(t)dt+ ( –λ)}τ(t)

|τ(t)|l.

According to (H), it is easy to see thatωi(l) and_|ττ(t)(t)|l

i_{have the same sign for any|l}i_|=ε.

Denote

(l,λ) =

T



h(t,l,λ)dt.

For anyλ∈[, ], we see that(l,λ) =  does not have solutions on∂ε∩F, then the

Brouwer degree

dB[ω,ε∩F, ] =dB %

(l, ),ε∩F,  &

=dB %

(l, ),ε∩F,  &

= .

Since () does not have solutions on∂ε, we have

dLS %

I–f(·, ),ε, 

&

=dLS %

I–f(·, ),ε, 

&

.

Since

dLS %

I–f(·, ),ε, 

&

= (–)NdB[,ε∩F, ] = (–)NdB %

(l, ),ε∩F,  &

we have

dLS %

I–f(·, ),ε, 

&

= (–)NdB[,ε∩F, ]= .

This completes the proof.

3 Dirichlet boundary value problems

In this section, we will discuss the existence of solutions of variable exponent mean curva-ture impulsive system Dirichlet boundary value problems,i.e.the existence of a solution of ()-() and ().

3.1 Preliminary

Let us now consider the following simple problem:

ϕt,u(t)=g(t), t∈(,T),t=ti, ()

with the following impulsive boundary value conditions:

⎧ ⎪ ⎨ ⎪ ⎩

lim_t→t+

i u(t) –limt→ti–u(t) =ai, i= , . . . ,k,

lim_t→t+

i ϕ(t,u

_{(t)) –limt→t}–

i ϕ(t,u _{(t)) =b}

i, i= , . . . ,k, u() =u(T) = ,

()

whereai,bi∈RN, k

i=|bi|< ;g∈L.

Ifuis a solution of () with (), by integrating () from  tot, we find that

ϕt,u(t)=ϕ,u()+

ti<t

bi+

t



g(t)dt, ∀t∈J. ()

Denoteρ=ϕ(,u()).

From the definition ofϕ, we can see that

sup

t∈J

ρ+

ti<t

bi+F(g)(t) < .

Denote

Ub=

h∈Lthere existsρ∈RN such thatsup

t∈J

ρ+

ti<t

bi+F(h)(t) < 

.

By (), we have

u(t) =

ti<t

ai+F

ϕ– !

t,

ρ+

ti<t

bi+F(g) "

(t), ∀t∈J.

The boundary value conditions imply that

k

i= ai+

T

 ϕ–

t,

!

ρ+

ti<t

bi+F(g)(t) "

Denotea= (a, . . . ,ak)∈RkN,b= (b, . . . ,bk)∈RkN. It is easy to see thatρis dependent

ona,b, andg. For fixedg∈Ub,a∈RkN, we define

(a,b,g)(ρ) = k

i= ai+

T

 ϕ–

t,

!

ρ+

ti<t

bi+F(g)(t) "

dt.

Denote

Ua,b=

h∈Ubthere existsρ∈RN such that

sup

t∈J

ρ+

ti<t

bi+F(h)(t)

<  and(a,b,h)(ρ) = 

.

If () with () has a solution inPC_{, we must haveg(·)∈U} a,b.

DenoteW=RkN_×L_{with the norm}

wW= k

i=

|ai|+ k

i=

|bi|+gL, ∀w= (a,b,g)∈W,

thenWis a Banach space.

Copying the proof of Lemma ., we have the following.

Lemma . The functionw(·)has the following properties:

(_{) For any fixed(a,b)∈M}

×M,∀h∈Ua,b,the equation

w(ρ) = , wherew= (a,b,h)∈W,

has a unique solutionρ(w)∈RN_,whereM

,Mare defined in()and().

() For any fixed a= (a, . . . ,ak)∈M, b= (b, . . . ,bk)∈M, Ua,b contains the open ball E#(,_) :={u∈L|uL < 

}, and then defines a mapping ρ:M×M× E#(,_)→RN.

() The function ρ :M ×M×E#(,_)→RN, defined in (), is continuous and bounded.

(_{) For any fixeda= (a}

, . . . ,ak)∈M,b= (b, . . . ,bk)∈M,the setUa,bis open and un-bounded inL.

We continue now with our argument previous to Lemma .. Define the nonlinear operatorK(a,b):Ua,b→PC, as

K(a,b)(h)(t) =F

ϕ– !

t,

ρ+

ti<t

bi+F(h) "

(t), ∀t∈J.

Similar to the proof of the Lemma ., we have the following.