Multi parameter fourth order impulsive integral boundary value problems with one dimensional m Laplacian and deviating arguments

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R E S E A R C H

Open Access

Multi-parameter fourth order impulsive

integral boundary value problems with

one-dimensional

m-Laplacian and deviating

arguments

Meiqiang Feng

*

_{and Jingliang Qiu}

*_{Correspondence:}

[email protected] School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, People’s Republic of China

Abstract

Using inequality techniques and ﬁxed point theories, several new and more general existence and multiplicity results are derived in terms of diﬀerent values of

λ

> 0 and

μ

> 0 for a fourth order impulsive integral boundary value problem with

one-dimensionalm-Laplacian and deviating arguments. We discuss our problems under two cases when the deviating arguments are delayed and advanced. Moreover, the nonexistence of a positive solution is also studied. In this paper, our results cover fourth order boundary value problems without deviating arguments and impulsive eﬀect and are compared with some recent results by Jankowski.

Keywords: multi-parameter; impulsive integral boundary value problems with advanced and delayed arguments; inequality techniques and ﬁxed point theories; one-dimensionalm-Laplacian; existence and nonexistence of positive solutions

1 Introduction

Functional differential equations with impulse effect occur in many applications, such as population dynamics, biology, biotechnology, industrial robotic, pharmacokinetics, opti-mal control,etc., and can be expressed by functional differential equations with impulses, see [–]. Functional differential equations with impulses are characterized by sudden changing of their state and by the fact that the processes under consideration depend on their prehistory at each moment of time. Therefore, the study of impulsive functional dif-ferential equations has gained prominence and it is a rapidly growing field, see Zhang and Feng [], Nieto and Rodríguez-López [], Yan and Shen [], Li and Shen [], Yang and Shen [], YS Liu [], YJ Liu [], He and Yu [] and Dinget al.[] and the references therein. We note that the difficulties dealing with such problems are that they have deviat-ing arguments and their states are discontinuous. Therefore, the results of impulsive func-tional differential equations, especially for higher order impulsive funcfunc-tional differential equations, are fewer than those of differential equations without impulses and deviating arguments.

Moreover, owing to its importance in modeling the stationary states of the deﬂection of an elastic beam, fourth order boundary value problems have attracted much attention from many authors; for example, see Sun and Wang [], Yao [], O’Regan [], Yang [],

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Zhang [], Gupta [], Agarwal [], Bonanno and Bella [] and Han and Xu []. In particular, we would like to mention some results of Zhang and Liu [] and Feng []. In [], Zhang and Liu studied the following fourth order four-point boundary value problem without impulsive eﬀect:

⎧ ⎪ ⎨ ⎪ ⎩

(φp(x(t)))=w(t)f(t,x(t)), t∈[, ],

x() = , x() =ax(ξ),

x() = , x() =bx(η),

where  <ξ,η< , ≤a<b< . By using the upper and lower solution method, ﬁxed point theorems and the properties of Green’s functionG(t,s) andH(t,s), the authors give suﬃcient conditions for the existence of one positive solution.

Recently, Feng [] studied a fourth order boundary value problem with impulses and integral boundary conditions

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(φp(y(t)))=f(t,y(t)), t∈J,t=tk,k= , , . . . ,n,

y|t=tk= –Ik(y(tk)), k= , , . . . ,n, y() =y() =_g(s)y(s)ds,

φp(y()) =φp(y()) =



h(s)φp(y(s))ds.

Using a suitably constructed cone and ﬁxed point theory for cones, the existence of mul-tiple positive solutions was established. Furthermore, upper and lower bounds for these positive solutions were given.

However, to the best of our knowledge, no paper has considered the existence, multi-plicity and nonexistence of positive solutions for fourth order impulsive diﬀerential equa-tions with one-dimensionalm-Laplacian, multiple parameters and deviating arguments till now; for example, see [–] and the references therein.

In this paper, we investigate a fourth order impulsive integral boundary value problem with one-dimensionalm-Laplacian and deviating arguments

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(φm(y(t)))=λω(t)f(t,y(α(t))), t∈J,t=tk,k= , , . . . ,n,

y|t=tk= –μIk(tk,y(tk)), k= , , . . . ,n, ay() –by() =_g(s)y(s)ds,

ay() +by() =_g(s)y(s)ds,

φp(y()) =φp(y()) =



h(t)φp(y(t))dt,

(.)

whereλ>  andμ>  are two parameters,a,b> ,J= [, ],φm(s) is anm-Laplace op-erator,i.e.,φm(s) =|s|m–_s_,_m_{> , (}_φ_m)–₌_φ

m∗, _m +_m∗ = ,tk(k= , , . . . ,n) (wherenis a ﬁxed positive integer) are ﬁxed points with  =t<t<t<· · ·<tk<· · ·<tn<tn+= ,

y|t=tk =y(t

+

k) –x(tk–), wherey(tk+) andy(tk–) represent the right-hand limit and the left-hand limit ofy(t) att=tk, respectively. In addition,ω,f,Ik,gandhsatisfy

(H) ω∈Lp[, ]for some≤p≤+∞, and there existsη> such thatω(t)≥ηa.e. onJ;

(H) f :J×R+→R+is continuous withf(t,y) > for allt∈Jandy> ,α∈C(J,J)with

R+= [, +∞);

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(H) g,h∈L[, ]are nonnegative andξ∈[,a),ν∈[, ), where

ξ=





g(t)dt, ν=





h(t)dt. (.)

Some special cases of (.) have been investigated. For example, Jankowski [] consid-ered problem (.) withλ= ,Ik=  andω∈C[, ], notω∈Lp[, ] for some ≤p≤+∞. By using a ﬁxed point theorem for cones due to Avery and Peterson, the author proved the existence results of positive solutions for problem (.).

Motivated by the results mentioned above, in this paper we study the existence, multi-plicity and nonexistence of positive solutions for problem (.) by using different meth-ods from that of the proof of Theorem . and Theorem . in [] to overcome difficul-ties arising from the appearances ofα(t)≡tandω(t) isLp_{-integrable. The arguments are} based upon a fixed point theorem due to Krasnoselskii which deals with fixed points of a cone-preserving operator defined on an ordered Banach space.

The organization of this paper is as follows. In Section , we present the expression and properties of Green’s function associated with problem (.). In Section , we present some definitions and lemmas which are needed throughout this paper. In Section , we use a fixed point theorem to obtain the existence, multiplicity and nonexistence of positive solutions for problem (.) with advanced argumentα. In Section , we formulate sufficient conditions under which delayed problem (.) has positive solutions. In particular, our results in these sections are new whenα(t)≡tont∈J. Finally, in Section , we offer some remarks and comments of the associated problem (.).

2 Expression and properties of Green’s function

We shall reduce problem (.) to an integral equation. To this goal, ﬁrstly by means of the transformation

φm

y(t) = –x(t), (.)

we convert problem (.) into

x(t) +λω(t)f(t,y(α(t))) = , t∈J,

x() =x() =_h(t)x(t)dt (.)

and

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

y(t) = –φm∗(x(t)), t∈J,t=tk,

y|t=tk= –μIk(tk,y(tk)), k= , , . . . ,n, ay() –by() =_g(s)y(s)ds,

ay() +by() =_g(s)y(s)ds.

(.)

Theorem . If(H), (H)and(H)hold,then problem(.)has a unique solution x given

by

x(t) =λ





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where

H(t,s) =G(t,s) +   –ν





G(s,τ)h(τ)dτ, (.)

G(t,s) =

t( –s), ≤t≤s≤,

s( –t), ≤s≤t≤. (.)

Proof The proof of Theorem . is similar to that of Lemma . in [].

Writee(t) =t( –t). Then from (.) and (.) we can prove thatH(t,s) andG(t,s) have the following properties.

Theorem . Let G and H be given as in Theorem..If(H)holds,then

H(t,s) > , G(t,s) > , ∀t,s∈(, ), (.)

H(t,s)≥, G(t,s)≥, ∀t,s∈J, (.)

e(t)e(s)≤G(t,s)≤G(t,t) =t( –t) =e(t)≤ ¯e=max

t∈J e(t) = 

, ∀t,s∈J, (.)

ρe(s)≤H(t,s)≤γs( –s) =γe(s)≤

γ, ∀t,s∈J, (.)

where

γ = 

 –ν, ρ=



e(τ)h(τ)dτ

 –ν . (.)

Theorem . If(H), (H)and(H)hold,then problem(.)has a unique solution y

ex-pressed in the form

y(t) =





H(t,s)φm∗

x(s) ds+μ

n

k=

H(t,tk)Ik

tk,y(tk), (.)

where

H(t,s) =G(t,s) + 

a–ξ





G(s,τ)g(τ)dτ, (.)

G(t,s) = 

d

(b+as)(b+a( –t)), if≤s≤t≤,

(b+at)(b+a( –s)), if≤t≤s≤ (.)

and

d=a(b+a).

Proof The proof of Theorem . is similar to that of Lemma . in [].

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Theorem . Letζ∈(,t),Gand Hbe given as in Theorem..If(H)holds,then we

have



db

_≤_G

(t,s)≤G(s,s)≤ 

d(b+a)

_, _∀_t_,_s_∈_J_, _(.)

ρ≤H(t,s)≤

a

a–ξG(s,s)≤ρ, ∀t,s∈J, (.)

G(t,s)≥δG(s,s)≥

b_δ

d , H(t,s)≥

δa

a–ξG(s,s)≥δρ, ∀t∈[ζ, ],s∈J, (.)

where

δ= b

a+b, ρ= b_γ



a+ b, ρ=

γ(b+a)

a+ b , γ=



a–ξ. (.)

Proof It follows from the deﬁnition ofG(t,s) andH(t,s) that (.) and (.) hold. Now, we show that (.) also holds.

In fact, fort∈[ζ, ] ands∈J, we have that

G(t,s)

G(s,s)

=b+a( –t)

b+a( –s)≥

b

b+a fors≤t,

G(t,s)

G(s,s)

=b+at

b+as ≥ b+aζ

b+a fort≤s.

This and (.) show that

H(t,s)≥δG(s,s)

 + 

a–ξ





g(τ)dτ

≥ aδ

a–ξG(s,s), t∈[ζ, ],s∈J.

This together with (.) and (.) ﬁnishes the proof of (.).

From Theorem . and Theorem ., we have the following result.

Theorem . Assume that(H)-(H)hold. Then problem(.)has a unique solution y

given by

y(t) =





H(t,s)φm∗

λ





H(s,τ)ω(τ)fτ,yα(τ) dτ

ds

+μ

n

k=

H(t,tk)Ik

tk,y(tk).

3 Preliminaries for the case

α

(t)≥tonJ

We begin by introducing the notations

f₌_{lim sup} y→+ maxt∈J

f(t,y)

φm(y)

, f∞=lim sup

y→∞

max

t∈J

f(t,y)

φm(y),

f=lim inf y→+ min_t_∈_J

f(t,y)

φm(y), f∞=lim infy→∞ mint∈J

f(t,y)

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I(k) =lim sup

y→+ max

t∈J

Ik(t,y)

y , I

∞₍_k_{) =}_{lim sup} y→∞

max

t∈J

Ik(t,y)

y ,

I(k) =lim inf y→+ min_t_∈_J

Ik(t,y)

y , I∞(k) =lim infy→∞ mint∈J

Ik(t,y)

y , k= , , . . . ,n.

We will also need the functions

f∗(u) =max

max

t∈J f(t,y),y∈[,u]

, I_k∗(u) =max

max

t∈J Ik(t,y),y∈[,u]

,

and let

f_∗= lim

u→+ f∗(u)

φm(u)

, f_∞∗ = lim

u→∞

f∗(u)

φm(u) ,

I_∗(k) = lim

u→+ I_k∗(u)

u , I

∗

∞(k) =_ulim_→∞

I_k∗(u)

u ,

wherek= , , . . . ,n.

Our ﬁrst lemma gives some relationships between the functionsf andf∗andI∗_kandIk.

Lemma .(See []) Assume that(H)holds.Then

f_∗=f, f_∞∗ =f∞, I_∗(k) =I(k), I_∞∗(k) =I∞(k),

where k= , , . . . ,n.

Proof The proof of Lemma . is similar to that of Lemma . in [].

The following ﬁxed point theorem of a cone is crucial in the proofs of our results.

Lemma .(See []) Let P be a cone in a real Banach space E.Assume thatand

are bounded open sets in E with∈,¯⊂.If

A:P∩(¯\)→P

is completely continuous such that either

(i) Ax ≤ x,∀x∈P∩∂andAx ≥ x,∀x∈P∩∂,or

(ii) Ax ≥ x,∀x∈P∩∂andAx ≤ x,∀x∈P∩∂,

then A has at least one ﬁxed point in P∩(¯\). LetJ=J\{t,t, . . . ,tm}, and

PC[, ] =y∈C[, ] :y|(tk,tk+)∈C(tk,tk+),y

_t– k ,y

t+_k exist,k= , , . . . ,m.

ThenPC_{[, ] is a real Banach space with the norm}

y_PC=max

y∞,y_∞, (.)

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A functiony∈PC_{[, ]}_∩_C₍_J_{) with}_ϕ_p(_y₎_∈_C_{(, ) is called a solution of problem} (.) if it satisﬁes (.).

Deﬁne a cone inPC_{[, ] by}

K=

y∈PC[, ] :y(t)≥ onJand min

t∈[ζ,]y(t)≥δ

ρ

ρ

yPC

, (.)

whereδ,ρandρare deﬁned in (.) and (.), respectively. Forr> , let

r=

y∈K:yPC<r

. (.)

Deﬁne an operatorTλμ:K→PC[, ] by

Tλμy(t) =





H(t,s)φm∗

λ





ds +μ n k=

H(t,tk)Ik

tk,y(tk) . (.)

It follows from (.) and Theorem . that the following lemma holds.

Lemma . Assume that(H)-(H)hold.Then y∈K is a positive ﬁxed point of Tλμif and

only if y is a positive solution of problem(.).

Lemma . Suppose that(H)-(H)hold.Then Tλμ(K)⊂K and T μ

λ :K→K is completely

continuous.

Proof For allu∈K, thenTλu≥ onJand it follows from (.) and (.) that

T_λμy(t) =





H(t,s)φm∗

λ





ds +μ n k=

H(t,tk)Ik

tk,y(tk)

≤ρ





φm∗

λ





ds+μ

n

k=

Ik

tk,y(tk)

,

t∈J, (.)

_Tμ λy

(t)≤





_H

t(t,s)φm∗

λ





H(s,τ)ω(τ)fτ,y(τ) dτ

ds +μ m k=

H__t(t,tk)Ik

tk,y(tk)

≤ρ





φm∗

λ





ds+μ

n

k=

Ik

tk,y(tk)

,

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whereρ=_da(a+b),

H__t(t,s) =G__t(t,s) = 

d

–a(b+as), if ≤s≤t≤,

a(b+a( –s)), if ≤t≤s≤

and

max

t,s∈J,t=s

H__t(t,s)= max

t,s∈J,t=s

G__t(t,s)= 

da(b+a).

It follows from (.) and (.) that

TλμyPC≤ρ





φm∗

λ





ds

+μ

n

k=

Ik

tk,y(tk)

, (.)

where

ρ=max{ρ,ρ}. (.)

Noticing (.), (.) and (.), we have

min

t∈[ζ,]

T_λμy(t)

= min

t∈[ζ,]





H(t,s)φm∗

λ





ds

+μ

n

k=

H(t,tk)Ik

tk,y(tk)

≥δρ





φm∗

λ





ds+μ

n

k=

Ik

tk,y(tk)

=δρ ρ

ρ





φm∗

λ





ds+μ

n

k=

Ik

tk,y(tk)

≥δρ ρ

T_λμy_PC.

Thus,Tλμ(K)⊂K.

Finally, similar to the proof of Lemma . in [], one can prove thatTλμ:K→K is completely continuous. This gives the proof of Lemma ..

To obtain some of the norm inequalities in Lemma . and Lemma ., we employ Hölder’s inequality.

Lemma .(Hölder) Let f ∈Lp_[_a_,_b_]_{with p}_{> ,}_g_∈_Lq_[_a_,_b_]_{with q}_{> ,}_and p+



q= .Then

fg∈L[a,b]and

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Let f ∈L_[_a_,_b_],_g_∈_L∞_[_a_,_b_]._{Then fg}_∈_L_[_a_,_b_]_and

fg≤ fg∞.

Next, we consider the following cases forω∈Lp_{[, ]:}_p_{> ,}_p_{= ,}_p₌_∞_{. Case}_p_{>  is} treated in Lemma . and Lemma ..

Lemma . Assume that(H)-(H)hold,α(t)≥t on J and let r> be given.If there exist

ε> andε> such that f∗(r)≤εφm(r)and Ik∗(r)≤εr(k= , , . . . ,n),then

T_λμy_PC≤ρ

φm∗

λγ εeqωp +μnε yPC, y∈∂r. (.)

Proof By the deﬁnition off∗(r) andI_k∗, iff∗(r)≤εφm(r) andIk∗(r)≤εr(k= , , . . . ,n), then

f(t,y)≤εφm(r), Ik(t,y)≤εr fort∈Jand ≤y≤r.

Since ≤t≤α(t)≤ onJ, it follows from ≤y(t)≤ronJthat ≤y(α(t))≤r. Therefore, we havef(t,y(α(t)))≤εφm(r) fort∈J, and it follows from (.) that

Tλμy_PC≤ρ





φm∗

λ





ds+μ

n

k=

Ik

tk,y(tk)

≤ρ

φm∗

λ





γe(τ)ω(τ)fτ,yα(τ) dτ

+μ

n

k=

Ik

tk,y(tk)

≤ρ

φm∗

λγ





e(τ)ω(τ)εφm(r)dτ

+μ

n

k=

εr

=ρ

φm∗

λγ εφm(r)





e(τ)ω(τ)dτ

+μ

n

k=

εr

≤ρ

φm∗

λγ εφm(r)eqωp +μnεr =ρ

φm∗

λγ εeqωp +μnεr yPC, ∀y∈∂r.

This completes the proof.

The following result deals with the casep= .

Corollary . Assume that(H)-(H)hold,α(t)≥t on J and let r> be given.If there

existε> andε> such that f∗(r)≤εφm(r)and Ik∗(r)≤εr(k= , , . . . ,n),then

φm∗

λγ εe∞ω +μnεr yPC, y∈∂r.

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Corollary . Assume that(H)-(H)hold,α(t)≥t on J and let r> be given.If there

existε> andε> such that f∗(r)≤εφm(r)and Ik∗(r)≤εr(k= , , . . . ,n),then

φm∗

λγ εeω∞ +μnεr yPC, y∈∂r.

Proof By Lemma ., leteω∞replaceeqωpand repeat the argument above. Lemma . Assume that(H)-(H)hold,α(t)≥t on J and let l> and l be given.If

f(t,y)≥lφm(y)and Ik(t,y)≥ly(k= , , . . . ,n)for t∈J and y∈K,then

T_λμy_PC≥δ

ρ_ ρ

(lλρη)m

∗_–

β+μnl yPC, (.)

whereβ=φm∗(



ζ e(τ)dτ).

Proof By the deﬁnition ofK, ify∈K, then we have

y(t)≥ onJ and min

t∈[ζ,]y(t)≥δ

ρ

ρ

yPC.

Since ≤t≤α(t)≤ onJ, it follows fromy(t)≥ onJthaty(α(t))≥.

Similarly, sinceζ≤t≤α(t)≤ on [ζ, ], it follows frommint∈[ζ,]y(t)≥δρρyPCthat

min

t∈[ζ,]y

α(t) ≥δρ ρ

y_PC.

Therefore,f(t,y(α(t)))≥lφm(y) fort∈J, and it follows from the deﬁnition ofTλμand (.) that

Tλμy_PC≥ρφm∗

λ





ρe(τ)ω(τ)fτ,yα(τ) dτ

+μρ n

k=

Ik

tk,y(tk)

≥ρ(λρη)m

∗_–

φm∗





e(τ)fτ,yα(τ) dτ

+μρ n

k=

Ik

tk,y(tk)

≥ρ(λρη)m

∗_–

φm∗





e(τ)lφm

yα(τ) dτ

+μρ n

k=

ly(tk)

≥ρ(lλρη)m

∗_–

φm∗



ζ

e(τ)φm

yα(τ) dτ

+μρl n

k=

y(tk)

≥ρ(lλρη)m

∗_–

φm∗



ζ

e(τ)φm

δρ ρ

yPC

dτ

+μρl n

k=

δρ ρ

yPC

=δρ

 

ρ

(lλρη)m

∗_–

β+μnl yPC.

This completes the proof.

Lemma . Assume that(H)-(H)hold andα(t)≥t on J.If y∈∂r,r> ,then

Tλμy_PC≤ρ

φm∗

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where

Mr= max t∈J,≤y≤r

f(t,y)> , M∗=maxM_k∗,k= , , . . . ,n> ,

M∗_k= max

t∈J,≤y≤r

Ik(t,y), k= , , . . . ,n.

Proof Ify∈∂r, then ≤y(t)≤rfort∈J.

Since ≤t≤α(t)≤ onJ, it follows from ≤y≤rthat ≤yα(t) ≤r.

Therefore, fromf(t,y)≤Mrfort∈Jandy∈∂r, we have

ft,yα(t) ≤Mr. So, fory∈∂r, we have





φm∗

λ





ds+μ

n

k=

Ik

tk,y(tk)

≤ρ

φm∗

λ





+μ

n

k=

Ik

tk,y(tk)

≤ρ

φm∗

λγ





e(τ)ω(τ)Mrdτ

+μ

n

k=

M∗

=ρ

φm∗

λγMr





e(τ)ω(τ)dτ

+μnM∗

≤ρ

φm∗

λγMreqωp +μnM∗ .

This gives the proof.

Corollary . When p= ,assume that(H)-(H)hold andα(t)≥t on J.If y∈∂r,r> ,

then

φm∗

λγMre∞ω +μnM∗ .

Proof By Lemma ., lete∞ωreplaceeqωpand repeat the argument above. Corollary . When p=∞,assume that(H)-(H)hold andα(t)≥t on J. If y∈∂r,

r> ,then

φm∗

λγMreω∞ +μnM∗ .

Proof By Lemma ., leteω∞replaceeqωpand repeat the argument above. Lemma . Assume that(H)-(H)hold andα(t)≥t on J.If y∈∂r,r> ,then

Tλμy_PC≥ρ(σrλρη)m

∗_–

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where

σr= min t∈J,≤y≤r

f(t,y)> , σ∗=min{mk,k= , , . . . ,n}> ,

σk= min t∈J,≤y≤r

Ik(t,y), k= , , . . . ,n.

Proof Ify∈∂r, then ≤y(t)≤rfort∈J.

Since ≤t≤α(t)≤ onJ, it follows from ≤y(t)≤rfort∈Jthat ≤yα(t) ≤r.

Therefore, fromf(t,y)≥σrfort∈Jand ≤y≤r, we have

ft,yα(t) ≥σr. So, fory∈∂r, we have

TλμyPC≥ρφm∗

λ





ρe(τ)ω(τ)fτ,yα(τ) dτ

+μρ n

k=

Ik

tk,y(tk)

≥ρ(λρη)m

∗_–

φm∗





e(τ)fτ,yα(τ) dτ

+μρ n

k=

Ik

tk,y(tk)

≥ρ(λρη)m

∗_–

φm∗





e(τ)σrdτ

+μρ n

k=

σ_r∗

≥ρ(σrλρη)m

∗_–

φm∗



ζ

e(τ)dτ

+μρnσr∗ =ρ(σrλρη)m

∗_–

β+μρnσr∗.

This ﬁnishes the proof.

4 Main results for the case

α

(t)≥tonJ

In this section, we apply Lemma . to establish the existence, multiplicity and nonex-istence of positive solutions for problem (.). We consider the following three cases for

ω∈Lp_{[, ]:}_p_{> ,}_p_{=  and}_p₌_∞_{. Case}_p_{>  is treated in the following theorem.} Theorem . Assume that(H)-(H)hold andα(t)≥t on J.Then:

(a) Iff_{= }_and_I₍_k_{) = }_or_f∞_{= }_and_I∞₍_k_{) = }_,_{then there exist}_λ

> andμ> 

such that problem(.)has a positive solution forλ>λandμ>μ.

(b) Iff=∞andI(k) =∞orf∞=∞andI∞(k) =∞,then there existλ> andμ> 

such that problem(.)has a positive solution for <λ<λand <μ<μ.

(c) Iff=f∞= andI(k) =I∞(k) = ,then there existλ> andμ> such that

problem(.)has at least two positive solutions forλ>λandμ>μ.

(d) Iff=f∞=∞andI∞(k) =I∞(k) =∞,then there existλ> andμ> such that

problem(.)has at least two positive solutions for <λ<λand <μ<μ.

(e) Iff_<_∞_,_I₍_k_{) <}_∞_,_f∞_<_∞_and_I∞_<_∞_,_{then there exist}_λ

> andμ> such

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(f ) Iff> ,I(k) > ,I∞> andf∞> ,then there existλ> andμ> such that

problem(.)has no positive solution forλ>λandμ> .

Proof Part (a). Choose a numberr> . By Lemma ., we haveT μ

λyPC>y_PC for y∈∂r,λ>λandμ>μ, where

λ=

 ρβ

r

m–

(σrρη)–> , μ=  ρnσr∗

r> .

Iff=  andI(k) = , then from Lemma . we havef_∗=  andI_∗(k) = , and so we can chooser∈(,r) so thatf∗(r)≤εrandI_k∗(r)≤εr, whereε>  andε>  respec-tively satisfy

ρφm∗

λγ εeqωp < , ρμnε< . (.)

Then Lemma . shows that

φm∗

λγ εeqωp +μnε yPC<y_PC fory∈∂_r_.

Iff∞=  andI∞(k) = , then from Lemma .,f_∞∗ =  andI∗_∞(k) = . Hence, there exists

r∈(r,∞) such thatf∗(r)≤εrandI∗k(r)≤εr, whereε>  andε>  satisﬁes (.). Thus

φm∗

λγ εeqωp +μnε yPC<y_PC fory∈∂_r_.

Therefore, it follows from Lemma . thatTλμhas a ﬁxed point in¯r\r or¯r\r,

according to whetherf_{=  and}_I₍_k_{) =  or}_f∞_{=  and}_I∞₍_k_{) = , respectively.} Conse-quently, problem (.) has a positive solution forλ>λandμ>μ.

Part (b). Choose a numberr> . By Lemma ., there existsλ>  such that

_Tμ

λyPC<yPC fory∈∂_r_,  <λ<λ_and  <μ<μ_,

where

λ=

 ρ

m–

γMreqωp –

, μ=  ρnM∗

.

Iff=∞andI(k) =∞, there existsr∈(,r) such thatf(t,y)≥lφm(y) andIk(t,y)≥ly fort∈Jand ≤y≤r, wherel>  andl>  are chosen so that

δρ

 

ρ

(lλρη)m

∗_–

β> , δρ

 

ρ

μnl> . (.)

Obviously,

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Then, from Lemma ., we can obtain

Tλμy_PC≥δ

ρ_ ρ

(lλρη)m

∗_–

β+μnl yPC>y_PC fory∈∂_r_.

Iff∞=∞andI∞(k) =∞, then there existsNˆ >  such thatf(t,y)≥lφm(y),Ik(t,y)≥ly (k= , , . . . ,n) fort∈Jandy≥ ˆN, andl>  andl>  satisfy (.).

Letr=max{r,Nˆρ/δρ}. Ify∈∂r, then

min

t∈[ζ,]y(t)≥

δρ

ρ

y_PC≥ ˆN.

So,

f(t,y)≥lφm(y), Ik(t,y)≥ly (k= , , . . . ,n) fort∈Jandy∈∂r.

From Lemma ., we can get

T_λμy_PC≥δ

ρ_ ρ

(lλρη)m

∗_–

β+μnl yPC>y_PC fory∈∂_r_.

Therefore, it follows from Lemma . thatT_λμhas a ﬁxed point in¯r\r or¯r\r,

according to whetherf=∞andI(k) =∞orf∞=∞andI∞(k) =∞, respectively. Con-sequently, problem (.) has a positive solution for  <λ<λand  <μ<μ.

Part (c). Choose two numbers  <r<r. By Lemma ., there existλ>  andμ>  such that

TλμyPC>yPC fory∈∂_r_i,i= , .

Sincef₌_f∞_{=  and}_I₍_k_{) =}_I∞₍_k_{) = , from the proof of part (a), it follows that we can} chooser∈(,r/) andr∈(r,∞) such that

Tλμy_PC<yPC fory∈∂_r_i,i= , .

It follows from Lemma . thatTλμhas two ﬁxed pointsyandysuch thaty∈ ¯r\r

andy∈ ¯r\r. These are the desired distinct positive solutions of problem (.) for

λ>λandμ>μsatisfying

r≤ yPC≤r_<r_≤ y__PC≤r_. (.)

Part (d). Choose two numbers  <r<r. By Lemma ., there existλ>  andμ>  such that

_Tμ

λyPC<yPC for  <λ<λ_,  <μ<μ_andy∈∂_r_i,i= , .

Sincef=∞andf∞=∞andI∞(k) =I∞(k) =∞, from the proof of part (b), we know that we can chooser∈(,r/) andr∈(r,∞) such that

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It follows from Lemma . thatTλμhas two ﬁxed pointsyandysuch thaty∈ ¯r\r

andy∈ ¯r\r. These are the desired distinct positive solutions of problem (.) for

 <λ<λand  <μ<μsatisfying (.).

Part (e). Sincef_<_∞_,_I₍_k_{) <}_∞_,_f∞_<_∞_and_I∞₍_k_{) <}_∞_{, there exist positive numbers}

li>  (i= , , , ),h>  andh>  such thath<hand fort∈J,  <y≤h, we have

f(t,y)≤lφm(y), Ik(t,y)≤ly and fort∈J,y≥h, we have

f(t,y)≤lφm(y), Ik(t,y)≤ly. Let

l=max

l,l,max

f(t,y)

φm(y):t∈J,h≤y≤h

> ,

l∗=max

l,l,max

Ik(t,y)

y :t∈J,h≤y≤h

> .

Thus, we have

f(t,y)≤lφm(y), Ik(t,y)≤l∗y fort∈Jandy∈[,∞).

Since ≤t≤α(t)≤ onJ, it follows from ≤y(t)≤h,y(t)≥handh≤y(t)≤hon

Jthat ≤y(α(t))≤h,y(α(t))≥handh≤y(α(t))≤honJ, respectively.

Assume thatyis a positive solution of problem (.). We will show that this leads to a contradiction for

 <λ<λ=

 ρ

m–

γleqωp –

and

 <μ<μ=  ρnl∗

.

Since (T_λμy)(t) =y(t) fort∈J, by Lemma . we have that

y_PC=T_λμy

PC≤ρ

φm∗

λγleqωp +μnl∗ yPC<y_PC,

which is a contradiction.

Part (f ). Sincef> ,I(k) > ,I∞>  andf∞> , there exist positive numbersli>  (i= , , , ),h>  andh>  such thath<hand fort∈J, ≤y≤h, we have

f(t,y)≥lφm(y), Ik(t,y)≥ly, and fort∈J,y≥h, we have

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Let

l∗∗=min

l,l,min

f(t,y)

φm(y):t∈J,h≤y≤h

> ,

l∗∗∗=min

l,l,min

Ik(t,y)

y :t∈J,h≤y≤h

> .

Then

f(t,y)≥l∗∗φm(y), Ik(t,y)≥l∗∗∗y fort∈Jandy∈[,∞).

Assume thatyis a positive solution of problem (.). We will show that this leads to a contradiction for

λ>λ=

ρ δρ_β

m–

l∗∗ρη –,

μ>μ=

ρ δρ_nl∗∗∗.

Since (T_λμy)(t) =y(t) fort∈J, by Lemma . we have that

yPC=T_λμy

PC≥δ

ρ_ ρ

l∗∗λρη m∗–β+μnl∗∗∗ yPC>y_PC,

which is a contradiction.

The results of the following theorem deal with the casep= .

Corollary . Assume that(H)-(H)hold andα(t)≥t on J.Then:

> andμ> 

(c) Iff₌_f∞_{= }_and_I₍_k_{) =}_I∞₍_k_{) = }_,_{then there exist}_λ

> andμ> such that

that problem(.)has no positive solution for <λ<λand <μ<μ.

Proof It follows from the proofs of Corollary . and Corollary . that Corollary .

holds.

Finally we consider the case ofp=∞.

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> andμ> 

Proof It follows from the proofs of Corollary . and Corollary . that Corollary .

holds.

5 Positive solutions of problem (1.1) for the case of

α

(t)≤tonJ

Now we deal with problem (.) for the case ofα(t)≤tonJ. Similarly as Theorem . and Lemmas .-., we can prove the following results.

Lemma . Letζ∗∈(tn, ),Gand Hbe given as in Theorem..If(H)holds,then we

have

G(t,s)≥δG(s,s)≥

b_δ

d , H(t,s)≥

δa

a–ξG(s,s)≥δρ, ∀t∈

,ζ∗,s∈J, (.)

where d is deﬁned in Theorem.,δandρare deﬁned in(.).

Proof In fact, fort∈[,ζ∗] ands∈J, we have that

G(t,s)

G(s,s)

=b+a( –t)

b+a( –s)≥

b+a( –ζ∗)

b+a fors≤t,

G(t,s)

G(s,s)

=b+at

b+as ≥ b

b+a fort≤s.

This and (.) show that

H(t,s)≥δG(s,s)

 + 

a–ξ





g(τ)dτ

≥ aδ

a–ξG(s,s), t∈

,ζ∗,s∈J.

This together with (.) and (.) ﬁnishes the proof of (.).

LetPC_{[, ] be as deﬁned in Section . We deﬁne a cone}_K∗_in_PC_{[, ] by}

K∗=

y∈PC[, ]y(t)≥ onJand min

t∈[,ζ∗]≥δ

ρ

ρ

yPC

,

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Deﬁne∗Tλμ:K∗→PC[, ] by

∗

Tλμy(t) =





H(t,s)φm∗

λ





ds +μ n k=

H(t,tk)Ik

tk,y(tk) . (.)

It is clear thatyis a positive solution of problem (.) if and only ofyis a ﬁxed point of∗T_λμ.

Lemma . Assume that(H)-(H)hold.Then y∈K∗ is a positive ﬁxed point of∗Tλμ if

and only if y is a positive solution of problem(.).

Lemma . Assume that(H)-(H)hold.Then∗Tλμ(K∗)⊂K∗and∗T μ

λ :K∗→K∗is

com-pletely continuous.

Letf∗andI_k∗be deﬁned as in Section . Similar to the proof of that in Lemmas .-., we have the following results. Here, we only consider the casem>  and only give the proof of Lemma ..

Lemma . Assume that(H)-(H)hold,α(t)≤t on J and let r> be given.If there exist

ε> andε> such that f∗(r)≤εφm(r)and I_k∗(r)≤εr(k= , , . . . ,n),then

∗_Tμ λy≤ρ

φm∗

λγ εeqωp +μnε yPC, y∈∂r. (.) Proof By the deﬁnition off∗(r) andI_k∗, iff∗(r)≤εφm(r) andIk∗(r)≤εr(k= , , . . . ,n), then

f(t,y)≤εφm(r), Ik(t,y)≤εr fort∈Jand ≤y≤r.

Since ≤α(t)≤t≤ onJ, it follows from ≤y(t)≤ronJthat ≤y(α(t))≤r. Therefore, we havef(t,y(α(t)))≤εφm(r) fort∈J, and it follows from (.) and (.) that

∗_Tμ

λyPC≤ρ





φm∗

λ





ds+μ

n

k=

Ik

tk,y(tk)

≤ρ

φm∗

λ





+μ n k= Ik tk,y(tk)

≤ρ

φm∗

λγ





e(τ)ω(τ)εφm(r)dτ

+μ

n

k=

εr

=ρ

φm∗

λγ εφm(r)





e(τ)ω(τ)dτ

+μ

n

k=

εr

≤ρ

φm∗

λγ εφm(r)eqωp +μnεr =ρ

φm∗

λγ εeqωp +μnεr yPC, ∀y∈∂r.

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Lemma . Assume that(H)-(H)hold,α(t)≤t on J and let l> and l be given.If

f(t,y)≥lφm(y)and Ik(t,y)≥ly for t∈J and y∈K∗,then

∗_Tμ

λy_PC≥δ

ρ_ ρ

(lλρη)m

∗_–

β+μnl yPC.

Letˆr={y∈K∗:yPC<r}.

Lemma . Assume that(H)-(H)hold andα(t)≤t on J.If y∈∂ˆr,r> ,then

∗_Tμ

λyPC≤ρ

φm∗

λγMreqωp +μnM∗ ,

where Mrand M∗are deﬁned in Lemma..

Lemma . Assume that(H)-(H)hold andα(t)≤t on J.If y∈∂ˆr,r> ,then

∗_T

λu_PC≥ρ(σrλρη)m

∗_–

β+μρnσr∗,

whereσrandσ∗are deﬁned in Lemma.. Letf_,_f∞_,_f

,f∞,I(k),I∞(k),I(k) andI∞(k) be deﬁned as in Section . Similar to the proof of Theorem ., we have the following results.

Theorem . Assume that(H)-(H)hold andα(t)≤t on J.Then:

> andμ> 

(e) Iff<∞,I(k) <∞,f∞<∞andI∞<∞,then there existλ> andμ> such

6 Remarks and comments

In this section, we oﬀer some remarks and comments of the associated problem (.).

Remark . The idea of deviating arguments for problem (.) is from Jankowski [],

but the method and conclusion are quite diﬀerent, and Jankowski only considered the caseλ= ,μ=  andω∈C[, ], notω(t) isLp-integrable.

Remark . Generally, it is diﬃcult to study the existence of positive solutions for