Solutions and nonnegative solutions for a weighted variable exponent impulsive integro-differential system with multi-point and integral mixed boundary value problems

R E S E A R C H

Open Access

Solutions and nonnegative solutions for a

weighted variable exponent impulsive

integro-differential system with multi-point

and integral mixed boundary value problems

Rong Dong and Qihu Zhang

*_{Correspondence:}

[email protected]; [email protected] Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China

Abstract

This paper investigates the existence of solutions for a weightedp(t)-Laplacian impulsive integro-differential system with multi-point and integral mixed boundary value problems via Leray-Schauder’s degree; sufficient conditions for the existence of solutions are given. Moreover, we get the existence of nonnegative solutions.

MSC: 34B37

Keywords: weightedp(t)-Laplacian; impulsive integro-differential system; Leray-Schauder’s degree

1 Introduction

In this paper, we consider the existence of solutions and nonnegative solutions for the following weightedp(t)-Laplacian integro-differential system:

–p(t)u+ft,u,w(t)p(t)–_{u,S(u),T(u)= , t}∈_{(, ),t=t}

i, ()

whereu: [, ]→RN,f(·,·,·,·,·) : [, ]×RN×RN×RN×RN →RN,ti∈(, ),i= , . . . ,k,

with the following impulsive boundary value conditions:

lim

t→t+_iu(t) –tlim→t–i

u(t) =Ai

lim

t→ti–

u(t),lim

t→t–i

w(t)



p(t)–_u(t)

, i= , . . . ,k, ()

lim

t→t+_iw(t)

up(t)–u(t) – lim

t→t–_iw(t)u p(t)–

u(t)

=Bi

lim

t→t–_iu(t),tlim→t–_i

w(t)



p(t)–_u(t)

, i= , . . . ,k, ()

u() = 



g(t)u(t)dt, u() =

m–

=

αu(ξ) – 



h(t)u(t)dt, ()

where p∈C([, ],R) and p(t) > , –p(t)u:= –(w(t)|u|p(t)–_{u) is called the weighted}

p(t)-Laplacian;  <t<t<· · ·<tk< ,  <ξ<· · ·<ξm–< ;α≥ (= , . . . ,m– );

g∈L_{[, ] is nonnegative,} 

g(t)dt=σ∈[, ];h∈L[, ], 

h(t)dt=δ;Ai,Bi∈C(RN×

RN_,RN_{); T and S are linear operators defined by (Su)(t) =} 

h∗(t,s)u(s)ds, (Tu)(t) = t

k∗(t,s)u(s)ds,t∈[, ], wherek∗,h∗∈C([, ]×[, ],R).

Ifσ<  andm–_= α–δ= , we say the problem is nonresonant, but ifσ=  or

m–

= α–

δ= , we say the problem is resonant.

Throughout the paper,o() means functions which are uniformly convergent to  (as n→+∞); for anyv∈RN_,vj_{will denote thejth component ofv; the inner product inR}N

will be denoted by ·,·,| · |will denote the absolute value and the Euclidean norm on

RN_{. DenoteJ= [, ],J= (, )\{t}

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

tk+= . Denote byJiothe interior ofJi,i= , , . . . ,k. Let

PCJ,RN=

x:J→RN

x∈C(Ji,RN),i= , , . . . ,k

andlim_t→t+

i x(t) exists fori= , . . . ,k

,

w∈PC(J,R) satisfy  <w(t),∀t∈(, )\{t, . . . ,tk}, and (w(t))

–

p(t)– ∈_L_{(, ),}

PCJ,RN= ⎧ ⎨ ⎩x∈PC

J,RN

x

_∈C(Jo

i,RN),limt→t+

i(w(t))



p(t)–_x(t)

and limt→t–

i+(w(t)) 

p(t)–_{x(t) exists fori= , , . . . ,k}

⎫ ⎬ ⎭.

For anyx= (x_{, . . . ,x}N_)∈PC(J,RN_{), denote|x}i_|=sup{|xi_{(t)| |t∈J}.}

Obviously,PC(J,RN) is a Banach space with the normx= (N_i=|xi|_)_{, andPC}_(J,

RN_{) is a Banach space with the normx=x+(w(t))}p(t)–_x_{. DenoteL}_=L_(J,RN₎

with the norm

xL=

_N

i=

_xi L

 

, ∀x∈L, wherexi_L=





_xi_(t)dt.

In the following,PC(J,RN_{) andPC}_(J,RN_{) will be simply denoted byPCandPC}_,

re-spectively. We denote

ut_i+= lim

t→t_i+u(t), u

t_i–=lim

t→t–_iu(t),

w()up()–u() = lim

t→+w(t)u

p(t)–

u(t),

w()up()–u() = lim

t→–w(t)u

p(t)–

u(t),

Ai=Ai

lim

t→t–_iu(t),tlim→t–_i

w(t)p(t)–_u(t)

, i= , . . . ,k,

Bi=Bi

lim

t→t–_iu(t),tlim→t–_i

w(t)p(t)–_u(t)

, i= , . . . ,k.

[–]. Ifp(t)≡p(a constant), ()-() becomes the well-knownp-Laplacian problem. Ifp(t) is a general function, one can see easily –p(t)cu=cp(t)–_{(–p(t)u) in general, but}

–pcu=cp–(–pu), so –p(t) represents a homogeneity and possesses more non-linearity, thus –p(t)is more complicated than –p. For example:

(a) If⊂RN is a bounded domain, the Rayleigh quotient

λp(x)= inf u∈W,p(x)()\{}

 p(x)|∇u|

p(x)_dx



p(x)|u|p(x)dx

is zero in general, and only under some special conditionsλp(x)>  (see []), when⊂R

(N= ) is an interval, the results show thatλp(x)>  if and only ifp(x) is monotone. But the

property ofλp>  is very important in the study ofp-Laplacian problems, for example, in

[], the authors use this property to deal with the existence of solutions.

(b) Ifw(t)≡ andp(t)≡p(a constant) and –pu> , thenuis concave, this property is used extensively in the study of one-dimensionalp-Laplacian problems (see []), but it is invalid for –p(t). It is another difference between –pand –p(t).

In recent years, many results have been devoted to the existence of solutions for the Laplacian impulsive differential equation boundary value problems; see, for example, [– ]. There are some methods to deal with these problems, for example, sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree. Because of the nonlinear property of –p, results on the existence of solutions forp-Laplacian impul-sive differential equation boundary value problems are rare (see [–]). In [], using the coincidence degree method, the present author investigates the existence of solutions forp(r)-Laplacian impulsive differential equation with multi-point boundary value condi-tions, when the problem is nonresonant. Integral boundary conditions for evolution prob-lems have various applications in chemical engineering, thermo-elasticity, underground water flow and population dynamics. There are many papers on the differential equations with integral boundary value problems; see, for example, [–].

In this paper, whenp(t) is a general function, we investigate the existence of solutions and nonnegative solutions for the weightedp(t)-Laplacian impulsive integro-differential system with integral and multi-point boundary value conditions. Results on these kinds of problems are rare. Our results contain both of the cases of resonance and nonresonance. Our method is based upon Leray-Schauder’s degree. The homotopy transformation used in [] is unsuitable for this paper. Moreover, this paper will consider the existence of () with (), () and the following impulsive condition:

lim

t→t+

i

w(t)



p(t)–_{u(t) – lim}

t→t–_i

w(t)



p(t)–_u(t)

=Di

lim

t→t_i–u(t),tlim→t–_i

w(t)



p(t)–_u(t)

, i= , . . . ,k, ()

whereDi∈C(RN×RN,RN), the impulsive condition () is called a linear impulsive

Moreover, since the Rayleigh quotientλp(x)=  in general and thep(t)-Laplacian is

non-homogeneity, when we deal with the existence of solutions of variable exponent impulsive problems like ()-(), we usually need the nonlinear term that satisfies the sub-(p–_{– )}

growth condition, but for the p-Laplacian impulsive problems, the nonlinear term only needs to satisfy the sub-(p– ) growth condition.

LetN≥, the functionf :J×RN_×RN_×RN_×RN_→RN_{is assumed to be Caratheodory,}

by which we mean:

(i) For almost everyt∈J, the functionf(t,·,·,·,·)is continuous;

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

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

(x,y,s,z)∈RN_×RN_×RN_×RN_{with|x| ≤R,|y| ≤R,|s| ≤R,|z| ≤R, one has}

f(t,x,y,s,z)≤αR(t).

We say a functionu:J→RNis a solution of () ifu∈PCwithw(t)|u|p(t)–uabsolutely continuous onJo

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

In this paper, we always useCito denote positive constants, if it cannot lead to confusion.

Denote

z–=inf

t∈Jz(t), z +_=sup

t∈J

z(t) for anyz∈PC(J,R).

We sayf satisfies the sub-(p–– ) growth condition iff satisfies

lim

|u|+|v|+|s|+|z|→+∞

f(t,u,v,s,z)

(|u|+|v|+|s|+|z|)q(t)–=  fort∈Juniformly,

whereq(t)∈PC(J,R) and  <q–≤q+<p–.

We will discuss the existence of solutions for system ()-() or () with (), () and () in the following three cases:

Case (i):σ< ,m–_= α–δ= ; Case (ii):σ= ,m–= α–δ= ; Case (iii):σ< ,m–_= α–δ< .

This paper is organized as five sections. In Section , we present some preliminaries and give the operator equation which has the same solutions of ()-() in the three cases, re-spectively. In Section , we give the existence of solutions for system ()-() or () with (), () and () whenσ< ,m–= α–δ= . In Section , we give the existence of solutions for system ()-() or () with (), () and () whenσ= ,m–_= α–δ= . Finally, in Section , we give the existence of solutions and nonnegative solutions for system ()-() or () with (), () and () whenσ< ,m–= α–δ< .

2 Preliminary

For any (t,x)∈J×RN_{, denoteϕ(t,x) =|x|}p(t)–_{x. Obviously,ϕhas the following properties.}

(i) For anyt∈[, ],ϕ(t,·)is strictly monotone,i.e.,

ϕ(t,x) –ϕ(t,x),x–x

>  for anyx,x∈RN,x=x.

(ii) There exists a functionα: [, +∞)→[, +∞),α(s)→+∞ass→+∞such that

ϕ(t,x),x≥α|x||x| for allx∈RN.

It is well known that ϕ(t,·) is a homeomorphism fromRN _{to R}N _{for any fixedt∈J.}

Denote

ϕ–(t,x) =|x|

–p(t)

p(t)–_{x forx}∈RN\{_}_,ϕ–_{(t, ) = ,}∀_t∈_J.

It is clear thatϕ–(t,·) is continuous and sends bounded sets to bounded sets.

In this section, we will do some preparation and give the operator equation which has the same solutions of ()-() in three cases, respectively. At first, let us now consider the following simple impulsive problem with boundary value condition ():

(w(t)ϕ(t,u(t)))=f(t), t∈(, ),t=ti,

lim_t→t+

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

limt→t+_i w(t)|u|p(t)–u(t) –limt→t–_i w(t)|u|p(t)–u(t) =bi, i= , . . . ,k,

⎫ ⎪ ⎬ ⎪

⎭ ()

whereai,bi∈RN;f ∈L.

Denotea= (a, . . . ,ak),b= (b, . . . ,bk). Obviously,a,b∈RkN.

We will discuss it in three cases, respectively.

2.1 Case (i)

Suppose thatσ<  andm–= α–δ= . Ifuis a solution of () with (), we have

w(t)ϕt,u(t)=w()ϕ,u()+

ti<t bi+

t



f(s)ds, ∀t∈J. ()

Denoteρ=w()ϕ(,u()). It is easy to see thatρis dependent ona,bandf(·). Define

the operatorF:L→PCas

F(f)(t) = t



f(s)ds, ∀t∈J,∀f ∈L.

By solving foruin () and integrating, we find

u(t) =u() +

ti<t ai+F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(f)(t)

(t), ∀t∈J,

which together with boundary value condition () implies

u() = 

( –σ)





g(t)

F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(f)(t)

(t) +

ti<t ai

and m– = α

ti<ξ ai+

ξ

 ϕ–

t,w(t)–

ρ+

ti<t

bi+F(f)(t)

dt – k i=

ai–



 ϕ–

t,w(t)–

ρ+

ti<t

bi+F(f)(t)

dt

–





h(t)

F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(f)(t)

(t) +

ti<t ai

dt= .

DenoteW=RkN_×L_{with the norm}

ω=

k

i= |ai|+

k

i=

|bi|+ϑL, ∀ω= (a,b,ϑ)∈W,

thenWis a Banach space. For anyω∈W, we denote

ω(ρ) = m–

= α

ti<ξ ai+

ξ

 ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

dt – k i=

ai–



 ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

dt

– 



h(t)

F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

(t) +

ti<t ai

dt.

Denoteξm–= . Then

ω(ρ) = – m–

= α

ξ≤ti ai+



ξ

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

dt +  

h(t)



t ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

dt+

ti≥t ai dt = – m– =

α– ξ+

ξ

h(t)dt



ξ

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

dt – m– =

ξ+

ξ h(t)

t

ξ

ϕ–

s,w(s)–

ρ+

si<s

bi+F(ϑ)(s)

ds dt

+ ξ



h(t)



t ϕ–

s,w(s)–

ρ+

si<s

bi+F(ϑ)(s)

ds dt – m– = α

ξ≤ti ai+





h(t)

Throughout the paper, we denote

E=

ξ



h(t) 

t

w(s)

–

p(s)– _{ds dt+}

m–

=

ξ+

ξ h(t)

t

ξ

w(s)

–

p(s)– _{ds dt}

+

m–

=

α– ξ+

ξ

h(t)dt



ξ

w(s)

–

p(s)–_ds,

δ∗=

m–

= α+





h(t)dt.

Lemma . Suppose that h(t)≥on[ξ, ],α≥ ξ+

ξ h(t)dt(= , . . . ,m– )and h(t)≤

on[,ξ].Then the functionω(·)has the following properties:

(i) For any fixedω∈W,the equation

ω(ρ) =  ()

has a unique solutionρ(ω)∈RN.

(ii) The functionρ:W→RN,defined in(i),is continuous and sends bounded sets to bounded sets.Moreover,for anyω= (a,b,ϑ)∈W,we have

ρ(ω)≤N

(N)p+

δ∗E+ 

E

k

i= |ai|

p#_– +

k

i=

|bi|+ϑL ,

where the notationMp#–_means

Mp#–=

Mp+–_{, M> ,}

Mp––_{, M≤.}

Proof (i) From Lemma ., it is immediate that

ω(x) –ω(x),x–x

<  forx=x,∀x,x∈RN,

and hence, if () has a solution, then it is unique. SetR= N[(N)p

+

(δ∗E+_E k_i=|ai|)p#–₊k

i=|bi|+ϑL].

Suppose that|ρ|>R, it is easy to see that there exists somej∈ {, . . . ,N}such that the

absolute value of thejth componentρ

j

 ofρsatisfies

ρj

 ≥ |ρ|

N > 

(N)p+

δ∗E+ 

E

k

i= |ai|

p#_– +

k

i=

|bi|+ϑL .

Thus thejth component ofρ+

ti<tbi+F(ϑ)(t) keeps sign onJ, namely, for anyt∈J, we have

ρj

 +

ti<t

bj_i+F(ϑ)j_(t)≥|ρ|

N > (N)

p+

δ∗E+ 

E

k

i= |ai|

p#–

+

k

i=

Obviously, we have

ρ+

ti<t

bi+F(ϑ)(t)

≤|ρ|

 ≤N

ρj

 +

ti<t

bj_i+F(ϑ)j_(t),

then it is easy to see that thejth component ofω(ρ) keeps the same sign ofρ

j

 . Thus,

ω(ρ)= .

Let us consider the equation

λω(ρ) + ( –λ)ρ= , λ∈[, ]. ()

According to the preceding discussion, all the solutions of () belong tob(R+ ) ={x∈ RN_{| |x|<R}

+ }. Therefore

dB

!

ω(ρ),b(R+ ), 

" =dB

!

I,b(R+ ), 

"

= ,

it means the existence of solutions ofω(ρ) = .

In this way, we define a functionρ(ω) :W→RN, which satisfiesω(ρ(ω)) = .

(ii) By the proof of (i), we also obtainρsends bounded sets to bounded sets, and

ρ(ω)≤N

(N)p+

δ∗E+ 

E

k

i= |ai|

p#–

+

k

i=

|bi|+ϑL .

It only remains to prove the continuity ofρ. Let{ωn}be a convergent sequence inW

andωn→ω, asn→+∞. Since{ρ(ωn)}is a bounded sequence, it contains a convergent

subsequence{ρ(ωnj)}. Suppose thatρ(ωnj)→ρasj→+∞. Sinceωnj(ρ(ωnj)) = , lettingj→+∞, we haveω(ρ) = , which together with (i) impliesρ=ρ(ω), it means

ρis continuous. This completes the proof.

Now we denote byNf(u) : [, ]×PC→L the Nemytskii operator associated tof

defined by

Nf(u)(t) =f

t,u(t),w(t)



p(t)–_{u(t),S(u),T(u) onJ. ()}

We defineρ:PC→RN as

ρ(u) =ρ(A,B,Nf)(u), ()

whereA= (A, . . . ,Ak),B= (B, . . . ,Bk).

It is clear thatρ(·) is continuous and sends bounded sets ofPCto bounded sets ofRN,

and hence it is compact continuous. Ifuis a solution of () with (), we have

u(t) =u() +

ti<t ai+F

ϕ–

t,w(t)–

ρ(ω) +

ti<t

bi+F(f)(t)

For fixeda,b∈RkN_{, we denoteK}

(a,b):L→PCas

K(a,b)(ϑ)(t) =F

ϕ–

t,w(t)–

ρ(a,b,ϑ) +

ti<t

bi+F(ϑ)(t)

(t), ∀t∈J.

DefineK:PC→PCas

K(u)(t) =F

ϕ–

t,w(t)–

ρ(u) +

ti<t Bi+F

Nf(u)

(t)

(t), ∀t∈J.

Lemma . (i)The operator K(a,b) is continuous and sends equi-integrable sets in L to

relatively compact sets in PC.

(ii)The operator Kis continuous and sends bounded sets in PCto relatively compact

sets in PC_.

Proof (i) It is easy to check thatK(a,b)(ϑ)(·)∈PC,∀ϑ∈L,∀a,b∈RkN. Since (w(t))

–

p(t)– ∈

L_and

K(a,b)(ϑ)(t) =ϕ–

t,w(t)–

ρ(a,b,ϑ) +

ti<t

bi+F(ϑ)

, ∀t∈[, ],

it is easy to check thatK(a,b)(·) is a continuous operator fromLtoPC.

Let nowUbe an equi-integrable set inL_{, then there existsα∈L}_{such that}

u(t)≤α(t) a.e. inJfor anyu∈L.

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

Let {un}be a sequence in K(a,b)(U), then there exists a sequence{ϑn} ∈U such that

un=K(a,b)(ϑn). For anyt,t∈J, we have

F(ϑn)(t) –F(ϑn)(t)=

_tϑn(t)dt–

t



ϑn(t)dt

=

t

t

ϑn(t)dt

≤

t

t

α(t)dt.

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

Ascoli-Arzela theorem, there exists a subsequence of{F(ϑn)}(which we rename the same) which

is convergent in PC. According to the bounded continuity of the operator ρ, we can

choose a subsequence of{ρ(a,b,ϑn) +F(ϑn)}(which we still denote{ρ(a,b,ϑn) +F(ϑn)})

which is convergent inPC, thenw(t)



p(t)–_K

(a,b)(ϑn)(t) =ϕ–(t,ρ(a,b,ϑn) +

ti<tbi+F(ϑn)) is convergent inPC.

Since

K(a,b)(ϑn)(t) =F

ϕ–

t,w(t)–

ρ(a,b,ϑn) +

ti<t

bi+F(ϑn)

(t), ∀t∈[, ],

it follows from the continuity ofϕ–and the integrability ofw(t)

–

p(t)– _inL_thatK (a,b)(ϑn)

is convergent inPC. Thus{un}is convergent inPC. (ii) It is easy to see from (i) and Lemma ..

Let us defineP:PC→PCas

P(u) = 

g(t)[K(u)(t) +

ti<tAi]dt

 –σ .

It is easy to see thatPis compact continuous.

Lemma . Suppose thatσ< ,m–_= α–δ= ;h(t)≥on[ξ, ],α≥ ξ+

ξ h(t)dt(= , . . . ,m– )and h(t)≤on[,ξ].Then u is a solution of()-()if and only if u is a solution

of the following abstract operator equation:

u=P(u) +

ti<t

Ai+K(u). ()

Proof Suppose thatuis a solution of ()-(). By integrating () from  tot, we find that

w(t)ϕt,u(t)=ρ(u) +

ti<t Bi+F

Nf(u)

(t), ∀t∈(, ),t=t, . . . ,tk. ()

It follows from () and () that

u(t) =u() +

ti<t Ai

+F

ϕ–

t,w(t)–

ρ(u) +

ti<t Bi+F

Nf(u)

(t), ∀t∈[, ],

u() = 

( –σ)

×





g(t)

F

ϕ–

t,w(t)–

ρ(u) +

ti<t Bi+F

Nf(u)

(t) +

ti<t Ai

dt

=



g(t)[K(u)(t) +

ti<tAi]dt

 –σ =P(u). ()

Combining the definition ofρ, we can see

u=P(u) +

ti<t

Ai+K(u).

Conversely, ifuis a solution of (), then () is satisfied. It is easy to check that

u() =P(u) = 

g(t)[K(u)(t) +

ti<tAi]dt

 –σ ,

u() =σu() + 



g(t)

K(u)(t) +

ti<t Ai

dt=





g(t)u(t)dt, ()

and

u() =P(u) + k

i=

By the condition of the mappingρ, we have m– = α

ti<ξ Ai+

ξ

 ϕ–

t,w(t)–

ρ+

ti<t Bi+F

Nf(u)

(t)

dt – k i=

Ai–



 ϕ–

t,w(t)–

ρ+

ti<t Bi+F

Nf(u)

(t)

dt

–





h(t)

F

ϕ–

t,w(t)–

ρ+

ti<t Bi+F

Nf(u)

(t)

(t) +

ti<t Ai

dt= .

Thus

u() =

m–

=

αu(ξ) – 



h(t)u(t)dt. ()

It follows from () and () that () is satisfied. From (), we have

w(t)ϕt,u(t)=ρ(u) +

ti<t Bi+F

Nf(u)

(t), t∈(, ),t=ti, ()

w(t)ϕt,u=Nf(u)(t), t∈(, ),t=ti.

It follows from () that () is satisfied.

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

2.2 Case (ii)

Suppose thatσ=  andm–= α–δ= . Ifuis a solution of () with (), we have

w(t)ϕt,u(t)=w()ϕ,u()+

ti<t bi+

t



f(s)ds, ∀t∈J.

Denoteρ=w()ϕ(,u()). It is easy to see thatρis dependent ona,bandf(·).

Bound-ary value condition () implies that





g(t)

F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(f)(t)

(t) +

ti<t ai

dt= ,

u() =

m–

= α{

ti<ξai+ ξ

 ϕ

–_{[t, (w(t))}–_(ρ +

ti<tbi+F(f)(t))]dt}  –m–_i= α+δ

–

k

i=ai+ _ϕ–[t, (w(t))–(ρ+

ti<tbi+F(f)(t))]dt  –m–_= α+δ

–



h(t)(F{ϕ

–_{[t, (w(t))}–_(ρ +

ti<tbi+F(f)(t))]}(t) +

ti<tai)dt  –m–_= α+δ

.

For anyω∈W, we denote

ω(ρ) =





g(t)

F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

(t) +

ti<t ai

Throughout the paper, we denoteE= _(w(t))

–

p(t)– _dt.

Lemma . The functionω(·)has the following properties:

(i) For any fixedω∈W,the equationω(ρ) = has a unique solutionρ(ω)∈RN. (ii) The functionρ:W→RN,defined in(i),is continuous and sends bounded sets to

bounded sets.Moreover,for anyω= (a,b,ϑ)∈W,we have

ρ(ω)≤N

(N)p+

E+ 

E k

i= |ai|

p#–

+

k

i=

|bi|+ϑL ,

where the notationMp#–means

Mp#–=

Mp+–_{, M> ,}

Mp––_{, M≤.}

Proof Similar to the proof of Lemma ., we omit it here.

We define ρ : PC → RN as ρ(u) = ρ(A,B,Nf)(u), where A = (A, . . . ,Ak), B =

(B, . . . ,Bk).

It is clear thatρ(·) is continuous and sends bounded sets ofPCto bounded sets ofRN,

and hence it is compact continuous. For fixeda,b∈RkN_{, we denoteK}∗

(a,b):L→PCas

K_(a,b)∗ (ϑ)(t) =F

ϕ–

t,w(t)–

ρ(a,b,ϑ) +

ti<t

bi+F(ϑ)(t)

(t), ∀t∈J.

DefineK:PC→PCas

K(u)(t) =F

ϕ–

t,w(t)–

ρ(u) +

ti<t Bi+F

Nf(u)

(t)

(t), ∀t∈J.

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

Lemma . (i)The operator K_(a,b)∗ is continuous and sends equi-integrable sets in L _to

relatively compact sets in PC.

(ii)The operator Kis continuous and sends bounded sets in PC to relatively compact

sets in PC_.

Let us define P:PC→PCas

P(u) =

m–

= α[

ti<ξAi+K(u)(ξ)] –

k

i=Ai

 –m–= α+δ

–K(u)() +



h(t)[K(u)(t) +

ti<tAi]dt  –m–_= α+δ

.

Lemma . Suppose thatσ= ,m–= α–δ= ,then u is a solution of()-()if and only

if u is a solution of the following abstract operator equation:

u=P(u) +

ti<t

Ai+K(u).

Proof Similar to the proof of Lemma ., we omit it here.

2.3 Case (iii)

Suppose thatσ<  andm–= α–δ< . Ifuis a solution of () with (), we have

w(t)ϕt,u(t)=w()ϕ,u()+

ti<t bi+

t



f(s)ds, ∀t∈J.

Denoteρ=w()ϕ(,u()). It is easy to see thatρis dependent ona,bandf(·).

Fromu() = _g(t)u(t)dt, we have

u() = 

( –σ)

×





g(t)

F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(f)(t)

(t) +

ti<t ai

dt. ()

Fromu() =m–_= αu(ξ) – _h(t)u(t)dt, we obtain

u() =

m–

= α{

ti<ξai+ ξ

 ϕ–[t, (w(t))–(ρ+

ti<tbi+F(f)(t))]dt}  –m–= α+δ

–

k

i=ai+ _ϕ–[t, (w(t))–(ρ+

ti<tbi+F(f)(t))]dt  –m–= α+δ

–



h(t)(F{ϕ–[t, (w(t))–(ρ+

ti<tbi+F(f)(t))]}(t) +

ti<tai)dt  –m–= α+δ

. ()

For fixedω∈W, we denote

ϒω(ρ) =

 ( –σ)





g(t)

F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

(t) +

ti<t ai

dt

–

m–

= α{

ti<ξai+ ξ

 ϕ–[t, (w(t))–(ρ+

ti<tbi+F(ϑ)(t))]dt}  –m–= α+δ

+

k

i=ai+ _ϕ–[t, (w(t))–(ρ+

ti<tbi+F(ϑ)(t))]dt  –m–= α+δ

+



h(t)(F{ϕ–[t, (w(t))–(ρ+

ti<tbi+F(ϑ)(t))]}(t) +

ti<tai)dt  –m–= α+δ

,

∀ρ∈RN.

Obviously,ϒω(ρ) can be rewritten as

ϒω(ρ) =

 ( –σ)





g(t)

F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

(t) +

ti<t ai

dt

+

m–

= α{

ξ≤tiai+



ξϕ

–_{[t, (w(t))}–_(ρ

+ti<tbi+F(ϑ)(t))]dt}  –m–= α+δ

+( –

m–

= α) _ϕ–[t, (w(t))–(ρ+

ti<tbi+F(ϑ)(t))]dt  –m–_= α+δ

+

k

i=ai( –

m–

= α)  –m–_= α+δ

+



h(t)(F{ϕ–[t, (w(t))–(ρ+

ti<tbi+F(ϑ)(t))]}(t) +

ti<tai)dt  –m–_= α+δ

.

Denoteξm–= . Moreover, we also have

ϒω(ρ)

= 

( –σ)





g(t)

F

ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

(t) +

ti<t ai dt + m– = α

ξ≤tiai  –m–= α+δ

+

m–

= (α– ξ+

ξ h(t)dt)



ξϕ

–_{[t, (w(t))}–_(ρ +

ti<tbi+F(ϑ)(t))]dt  –m–_= α+δ

+

m–

=

ξ+

ξ h(t)

t

ξϕ

–_{[s, (w(s))}–_(ρ +

si<sbi+F(ϑ)(s))]ds dt  –m–_= α+δ

– ξ

 h(t) 

t ϕ–[s, (w(s))–(ρ+

si<sbi+F(ϑ)(s))]ds dt+

 h(t)

ti≥taidt  –m–_= α+δ

+ 

 ϕ–

t,w(t)–

ρ+

ti<t

bi+F(ϑ)(t)

dt+

k

i=

ai.

Lemma . Suppose thatα,g,h satisfy one of the following:

(₎ m–

= α≤,g(t)( –

m–

= α+δ) +h(t)( –σ)≥;

() h(t)≥on[ξ, ],α≥ ξ+

ξ h(t)dt(= , . . . ,m– )andh(t)≤on[,ξ].

Then the functionϒω(·)has the following properties:

(i) For any fixedω∈W,the equationϒω(ρ) = has a unique solutionρ(ω)∈RN. (ii) The functionρ:W→RN,defined in(i),is continuous and sends bounded sets to

bounded sets.Moreover,for anyω= (a,b,ϑ)∈W,we have

ρ(ω)≤N

(N)p+

E+ 

( –σ)E

+δ∗+  E+ 

( –m–_= α+δ)E

k

i= |ai|

p#–

+

k

i=

|bi|+ϑL

where the notationMp#–_means

Mp#–=

Mp+–_{, M> ,}

Mp––_{, M≤.}

Proof Similar to the proof of Lemma ., we omit it here.

We define ρ : PC → RN as ρ(u) = ρ(A,B,Nf)(u), where A = (A, . . . ,Ak), B =

(B, . . . ,Bk).

It is clear thatρ(·) is continuous and sends bounded sets ofPCto bounded sets ofRN,

and hence it is compact continuous. For fixeda,b∈RkN_{, we denoteK}∗∗

(a,b):L→PCas

K_(a,b)∗∗ (ϑ)(t) =F

ϕ–

t,w(t)–

ρ(a,b,ϑ) +

ti<t

bi+F(ϑ)(t)

(t), ∀t∈J.

DefineK:PC→PCas

K(u)(t) =F

ϕ–

t,w(t)–

ρ(u) +

ti<t Bi+F

Nf(u)

(t)

(t), ∀t∈J.

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

Lemma . (i)The operator K_(a,b)∗∗ is continuous and sends equi-integrable sets in L to

relatively compact sets in PC_.

(ii)The operator Kis continuous and sends bounded sets in PC to relatively compact

sets in PC_.

Let us define P:PC→PCas

P(u) = 

g(t)[K(u)(t) +

ti<tAi]dt

 –σ .

It is easy to see that Pis compact continuous.

Lemma . Suppose thatσ< ,m–_= α–δ< andα,g,h satisfy one of the following:

() m–= α≤,g(t)( –

m–

= α+δ) +h(t)( –σ)≥;

(_{) h(t)≥on[ξ}

, ],α≥ ξ+

ξ h(t)dt(= , . . . ,m– )andh(t)≤on[,ξ].

Then u is a solution of()-()if and only if u is a solution of the following abstract operator

equation:

u=P(u) +

ti<t

Ai+K(u).

3 Existence of solutions in Case (i)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for system ()-() or () with (), () and () whenσ< ,m–= α–δ= .

Whenf satisfies the sub-(p–– ) growth condition, we have the following theorem.

Theorem . Suppose thatσ < ,m–_= α–δ= ;h(t)≥on[ξ, ],α≥ ξ+

ξ h(t)dt (= , . . . ,m– )and h(t)≤on[,ξ];f satisfies the sub-(p–– )growth condition;and

operators A and B satisfy the following conditions:

k

i=

Ai(u,v)≤C

 +|u|+|v| q+– p+–_,

k

i=

Bi(u,v)≤C

 +|u|+|v|q+–, ∀(u,v)∈RN×RN,

()

then problem()-()has at least a solution.

Proof First we consider the following problem:

(S)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

–p(t)u=λNf(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–(w(t))



p(t)–_{u(t)), i= , . . . ,k,}

limt→t_i+w(t)|u|p(t)–u(t) –limt→t_i–w(t)|u|p(t)–u(t)

=λBi(limt→t–

i u(t),limt→ti–(w(t))



p(t)–_{u(t)), i= , . . . ,k,}

u() = _g(t)u(t)dt, u() =m–_= αu(ξ) – _h(t)u(t)dt.

Denote

ρ_,λ(u) =ρ(λA,λB,λNf)(u),

K,λ(u) =F

ϕ–

t,w(t)–

ρ_,λ(u) +λ ti<t

Bi+F

λNf(u)

(t)

,

P,λ(u) =



g(t)[K,λ(u)(t) +

ti<tλAi]dt

 –σ ,

f(u,λ) =P,λ(u) +λ

ti<t

Ai+K,λ(u),

whereNf(u) is defined in ().

Obviously, (S) has the same solution as the following operator equation whenλ= :

u=f(u,λ). ()

It is easy to see that the operatorρ_,λis compact continuous for anyλ∈[, ]. It follows from Lemma . and Lemma . thatf(·,λ) is compact continuous fromPCtoPCfor

anyλ∈[, ].

We claim that all the solutions of () are uniformly bounded forλ∈[, ]. In fact, if it is false, we can find a sequence of solutions{(un,λn)}for () such thatun→+∞as

From Lemma ., we have

ρ_,λ(u)≤C

_k

i= |Ai|

p#–

+

k

i=

|Bi|+##Nf(u)##_L ≤C

 +uq_+–.

Thus

ρ_,λ(u) +

ti<t

λBi+F(λNf)

≤ρ_,λ(u)+

ti<t Bi

+F(Nf)≤C

 +uq_+–. ()

From (S), we have

w(t)u_n(t)p(t)–u_n(t) =ρ_,λ(un) +

ti<t

λBi+

t



λNf(un)(s)ds, ∀t∈J.

It follows from () and Lemma . that

w(t)u_n(t)p(t)–≤ρ_,λ(un)+ k

i= |Bi|+





Nf(un)(s)ds≤C+Cunq

+_–

 , ∀t∈J.

Denoteα=q_p+––_–. If the above inequality holds then

##w(t)p(t)–_u

n(t)##≤Cun

α

, n= , , . . . . ()

It follows from (), () and () that

un()≤Cunα, whereα=

q+_{– }

p–_{– }.

For anyj= , . . . ,N, we have

uj_n(t)=uj_n() +

ti<t Ai+

t



uj_n(s)ds

≤uj_n()+

ti<t Ai

+

t



w(s)p(–s)– _sup

t∈(,)

w(t)p(t)–_uj

n

(t)ds

≤ unα_[C+CE] +

ti<t Ai

≤Cunα_, ∀t∈J,n= , , . . . ,

which implies that

uj_n≤Cunα

, j= , . . . ,N;n= , , . . . .

Thus

un≤NCunα_, n= , , . . . . ()

Thus, we can choose a large enough R>  such that all the solutions of ()

be-long to B(R) ={u ∈PC | u <R}. Therefore the Leray-Schauder degree dLS[I – f(·,λ),B(R), ] is well defined forλ∈[, ], and

dLS

!

I–f(·, ),B(R), 

" =dLS

!

I–f(·, ),B(R), 

" .

It is easy to see thatuis a solution ofu=f(u, ) if and only ifuis a solution of the

following usual differential equation:

(S)

–p(t)u= , t∈(, ),

u() = _g(t)u(t)dt, u() =m–_= αu(ξ) –



h(t)u(t)dt.

Obviously, system (S) possesses a unique solutionu. Sinceu∈B(R), we have

dLS

!

I–f(·, ),B(R), 

" =dLS

!

I–f(·, ),B(R), 

"

= ,

which implies that ()-() has at least one solution. This completes the proof.

Theorem . Suppose thatσ < ,m–_= α–δ= ;h(t)≥on[ξ, ],α≥ ξ+

ξ h(t)dt (= , . . . ,m– )and h(t)≤on[,ξ];f satisfies the sub-(p–– )growth condition;and

operators A and D= (D, . . . ,Dk)satisfy the following conditions:

k

i=

Ai(u,v)≤C

 +|u|+|v| q+– p+–_,

k

i=

Di(u,v)≤C

 +|u|+|v|αi+_, ∀_(u,v)∈RN_×RN_,

whereαi≤ q

+_–

p(ti)–,and p(ti) – ≤q

+_–α

i,i= , . . . ,k.

Then problem()with(), ()and()has at least a solution.

Proof Obviously,Bi(u,v) =ϕ(ti,v+Di(u,v)) –ϕ(ti,v).

From Theorem ., it suffices to show that

k

i=

Bi(u,v)≤C

 +|u|+|v|q+–, ∀(u,v)∈RN×RN. ()

(a) Suppose that|v| ≤M∗|Di(u,v)|, whereM∗is a large enough positive constant. From

the definition ofD, we have

Bi(u,v)≤CDi(u,v) p(ti)–_≤

C

 +|u|+|v|αi(p(ti)–).

Sinceαi< q

+_–

p(ti)–, we haveαi(p(ti) – )≤q

+_{– . Thus () is valid.}

(b) Suppose that|v|>M∗|Di(u,v)|, we can see that

Bi(u,v)≤C|v|p(ti)–|

Di(u,v)| |v| =C|v|

p(ti)–_D