R E S E A R C H
Open Access
Deviating arguments, impulsive effects,
and positive solutions for second order
singular
p
-Laplacian equations
Xuemei Zhang
1*and Meiqiang Feng
2*Correspondence: [email protected] 1Department of Mathematics and
Physics, North China Electric Power University, Beijing, 102206, People’s Republic of China
Full list of author information is available at the end of the article
Abstract
Using a new method for dealing with second order singularp-Laplacian equations with impulsive effects and deviating arguments, several new and more general results are obtained for the existence of at least single, twin or triple positive solutions by using Krasnosel’skii and Zabreiko’s fixed point theorem, the fixed point theorem due to Avery and Henderson, and Leggett-Williams’ fixed point theorem. We discuss our problems under two cases when the deviating arguments are delayed or advanced. Our results cover equations without deviating arguments and are compared with some recent results by Kajikiya, Lee, and Sim.
Keywords: deviating argument; impulse effect; second order singularp-Laplacian equation; cone and partial ordering
1 Introduction
Second order differential equations withp-Laplacian arise naturally in non-Newtonian mechanics, nonlinear elasticity, glaciology, population biology, combustion theory, and nonlinear flow laws; see [, ]. In recent years many cases of the existence, multiplicity, and uniqueness of positive solution of differential equations withp-Laplacian have attracted considerable attention [–].
In [], Kajikiya et al. investigated the following one-dimensional p-Laplacian prob-lem:
(ϕp(u))+λω(t)f(u) = , t∈(, ), u() =u() = ,
and by virtue of the global bifurcation theory, they obtained the existence, nonexistence, uniqueness, and multiplicity of positive solutions as well as sign-changing solutions under suitable conditions imposed on the nonlinear termf.
In [], employing the shooting method, Iturriagaet al.obtained both existence and the exact number of positive solutions of the problem
λ(ϕp(u))+f(u) = in (, ), u() =u() = .
Recently, using global bifurcation theory, Dai and Ma [] showed the existence of nodal solutions for the following problem:
(ϕp(u))+f(t,u) = , t∈(, ), u() =u() = .
In [], Ding and O’Regan studied a second orderp-Laplacian differential equation in-volving the impulsive effectu|t=tk=Ik(u(tk)),k= , , . . . ,m. Using Jensen’s inequality, the first eigenvalue of a relevant linear operator and the Krasnoselskii-Zabreiko fixed point theorem, the existence and multiplicity of positive solutions were established.
At the same time, a class ofp-Laplacian differential equations with deviating arguments both of an advanced or delayed type have received much attention. For example, in [], Jankowski considered the following third orderp-Laplacian differential equation:
⎧ ⎪ ⎨ ⎪ ⎩
(ϕp(u(t)))+h(t)f(t,u(t),u(α(t))) = , t∈J, u(t+
k) =u(tk–) +Qk(u(tk)), k= , , . . . ,m,
βu() –γu() = , δu() +ηu() = , u() = ,
whereα(t)≡tonJ. The author obtained the existence of at least three positive solutions. The main tool is a fixed point theorem due to Avery [], which is a generalization of the Leggett-Williams fixed point theorem.
Of course, a natural question is the following.
Q. Can the existence of positive solutions for a second orderp-Laplacian differential equation with deviating arguments both of an advanced or delayed type be proved?
Remark . In [], by means of the properties of Green’s function, Jankowski obtain the inequality
min
t∈[ξ,]u(t)≥ρu,
whereξ∈(, ) andρ∈(, ).
In fact, the calculation ofρ is very difficult whent∈[ξ, ]. This is probably the main reason that there is almost no paper to study the existence of positive solutions for a class of second orderp-Laplacian impulsive differential equations with two parameters and de-viating arguments both of an advanced or delayed type. In [], Jankowski obtained a constant numberρ by means of the properties of Green’s function. However, it is well known that there is not any Green’s function whatsoever in one-dimensionalp-Laplacian boundary value problems of second order differential equations. This implies the follow-ing question.
Q. Can a similar inequality be obtained if there is no Green’s function whent∈[ξ, ]?
For this one needs to use a new technique to deal with second orderp-Laplacian equa-tions with deviating arguments, especially for second orderp-Laplacian equations with impulsive effects.
Consider the one-dimensional p-Laplacian differential equation with deviating argu-ments
–φp
subject to one of the following impulsive and boundary conditions:
–u|t=tk=Ik(u(tk)), k= , , . . . ,n,
u() = , u() = g(t)u(t)dt (.)
or
u|t=tk=Ik(u(tk)), k= , , . . . ,n,
u() = h(t)u(t)dt, u() = , (.)
whereϕp(s) =|s|p–s,p> , (ϕp)–=ϕq,
p+
q= ,ωmay be singular att= and/ort= ,tk (k= , , . . . ,n, wherenis fixed positive integer) are fixed points with <t<t<· · ·<tk<
· · ·<tn< ,u|t=tkdenotes the jump ofu(t) att=tk,i.e.,
u|t=tk=u
tk+–ut–k,
whereu(t+k) andu(tk–) represent the right-hand limit and left-hand limit ofu(t) att=tk, respectively.
Throughout this paper we assume thatk= , , . . . ,nandα(t)≡tonJ= [, ]. In addi-tion,ω,f,α,Ik,g, andhsatisfy
(H) ω∈C((, ), [, +∞))with < ω(s)ds<∞andωdoes not vanish on any
subinter-val of(, );
(H) f ∈C([, ]×[, +∞), [, +∞)),α∈C(J,J);
(H) Ik∈C([, +∞), [, +∞));
(H) g,h∈L[, ]are nonnegative andσ∈[, ),ς∈[, ), where
σ=
g(s)ds, ς=
h(s)ds. (.)
Some special cases of problem (.)-(.) or problem (.)-(.) have been investigated. For example, Kajikiya et al.[] considered problem (.) in the case that Ik≡,k= , , . . . ,n, andα(t)≡tonJ. By using the properties of eigenfunctions and global bifurca-tion theory, the authors proved the existence, uniqueness, nonexistence, and multiplicity results of positive solutions as well as sign-changing solutions, specially when the nonlin-ear term isp-linear near andp-sublinear at∞.
Remark . There are almost no papers except [, ] studying one-dimensional p-Laplacian differential equations with deviating arguments both of an advanced or de-layed type using fixed pointed theory. However, in [, ], Jankowski only dealt with third orderp-Laplacian equations and fourth orderp-Laplacian equations. To our knowledge, it is the first paper where positive solutions have been investigated for the class of second order singularp-Laplacian equations with impulsive effects and deviating arguments both of an advanced or delayed type.
and Leggett-Williams’ fixed point theorem. In Section , we use a fixed point theorem of cone expansion and compression to obtain the existence of at least one or two positive so-lutions of problem (.)-(.) with advanced argumentα. Section will further discuss the existence of twin positive solutions of problem (.)-(.) with advanced argumentα. Two new results will be presented by a new fixed point theorem due to Avery and Henderson. Section is for developing existence criteria for (at least) three positive solutions of prob-lem (.)-(.). Finally, in Section , we formulate sufficient conditions under which de-layed problem (.)-(.) has at least one, two, and three positive solutions. In particular, our results in these sections are new whenα(t)≡tont∈J. To the best of the authors’ knowledge, it is the first paper where the fixed point theories are applied top-Laplacian boundary value problems of second order impulsive differential equations with deviating arguments.
2 Preliminaries
LetJ=J\{t,t, . . . ,tn}andEbe the Banach space
E=u|u:J→Ris continuous att=tk,ut–k=u(tk) andutk+exist,k= , , . . . ,n withu=max≤t≤|u(t)|. We denote
r:=
u∈E:u<r for allr> in the sequel.
In our main results, we will make use of the following definitions and lemmas.
Definition .(See []) LetE be a real Banach space overR. A nonempty closed set P⊂Eis said to be a cone provided that
(i) au+bv∈Pfor allu,v∈Pand alla≥,b≥, (ii) u, –u∈Pimpliesu= .
Every coneP⊂Einduces an ordering inEgiven byx≤yif and only ify–x∈P.
Definition . The mapβis said to be a nonnegative continuous concave functional on a conePof a real Banach spaceEprovided thatβ:P→[,∞) is continuous and
βtx+ ( –t)y≥tβ(x) + ( –t)β(y) for allx,y∈Pand ≤t≤.
Definition . A functionu∈E∩C(, ) is called a solution of (.)-(.) if it satisfies
(.)-(.). Ifu(t)≥ andu(t)≡ onJ, thenuis called a positive solution of (.)-(.).
Lemma . Assume that(H)-(H)hold.Then u∈E∩C(, )is a solution of problem
(.)-(.)if and only if u∈E is a solution of the following equations:
u(t) = –σ
g(t)
t
φq
s
ω(r)fr,uα(r)dr
ds
dt
+
g(t) t≤tk Ik
u(tk)
dt
+ see by integration of (.) that
–φp
By the boundary condition, we have
u(t) = –φq
Repeating the above procession, fort∈J, we have
Then we have
Then the proof of sufficiency is complete. Conversely, we see ifu∈Eis a solution of (.). Direct differentiation of (.) implies, fort=tk,
This together with the boundary conditionu() = g(t)u(t)dtimplies that
u()≥
ξ( –t)g(t)dt
–ξ– ( –ξ) ξg(t)dt+ ξ( –t)g(t)dtu(ξ). (.)
It is clear that (.) holds whent= . Noticing that
u() –u()
≤
u(ξ) –u() –ξ ,
we have
u(ξ)≥ξu() + ( –ξ)u()≥( –ξ)u(). (.) It follows from (.) and (.) that (.) and (.) hold.
Define a coneKinEby
K=
u∈E:u≥, min
t∈[ξ,]u(t)≥δu
,
whereδis defined in (.). It is easy to see thatKis a closed convex cone ofE. DefineT:K→Eby
(Tu)(t) = –σ
g(t)
t
φq
s
ω(r)fr,uα(r)dr
ds
dt
+
g(t) t≤tk Ik
u(tk)
dt
+
t
φq
s
ω(r)fr,uα(r)dr
ds+ t≤tk Ik
u(tk)
. (.)
Obviously, (Tu)(t)≥ fort∈J.
From the definition of T and the proof of Lemma ., we claim that for each u∈K, Tu∈Kand satisfies (.), and (Tu)() is the maximum value of (Tu)(t) onJ.
From (.) and Lemma ., it is also easy to obtain the following results.
Lemma . Assume that(H)-(H)hold.Then problem(.)-(.)is equivalent to the fixed
point problem of T in K.
Lemma . Assume that(H)-(H)hold.Then T:K→K is completely continuous. Lemma .(Fixed point theorem of cone expansion and compression; see []) Let
andbe two bounded open sets in Banach space E,such that∈and¯⊂.Let
P be a cone in E and let operator A:P∩(¯\)→P be completely continuous.Suppose
that one of the following two conditions is satisfied:
(i) Axx,∀x∈P∩∂;Axx,∀x∈P∩∂
(ii) Axx,∀x∈P∩∂;Axx,∀x∈P∩∂.
Then A has at least one fixed point in P∩(\ ¯).
Remark . To the best of our knowledge, it is the first paper where the fixed point the-orem of cone expansion and compression is applied top-Laplacian differential equations.
Given a nonnegative continuous functionalψ on a conePof a real Banach space, we define, for eachr> , the set
P(ψ,r) =u∈P:ψ(u) <r.
Lemma .(See []) Let P be a cone in a real Banach space.Letαandγ be increasing nonnegative continuous functional on P,and letθbe a nonnegative continuous functional on P withθ() = such that,for some c> and H> ,
γ(x)≤θ(x)≤α(x) and x ≤Hγ(x)
for all x∈P(γ,c).Suppose there exist a completely continuous operator A:P(γ,c)→P and <a<b<c such that
θ(λx)≤λθ(x) for≤λ≤and x∈∂P(θ,b), and
(i) γ(Ax) >cfor allx∈∂P(γ,c); (ii) θ(Ax) <bfor allx∈∂P(θ,b);
(iii) P(α,a)=∅andα(Ax) >aforx∈∂P(α,a).
Then A has at least two positive solutions xand xbelonging to P(γ,c)satisfying
a<α(x) withθ(x) <b and b<θ(x) withγ(x) <c.
Let <a<bbe given and letβbe a nonnegative continuous concave functional on the coneK. Define the convex setsKa,K(β,a,b) by
Ka=
x∈K:x<a,
K(β,a,b) =x∈K:a≤β(x),x ≤b.
Finally we state the Leggett-Williams fixed point theorem [].
Lemma .(See []) Let K be a cone in a real Banach space E,A:K¯a→ ¯Kabe completely continuous andβ be a nonnegative continuous concave functional on K withβ(x)≤ x for all x∈Ka.Suppose there exist <d<a<b≤c such that
(i) {x∈K(β,a,b) :β(x) >a} =∅andβ(Ax) >aforx∈K(β,a,b); (ii) Ax<dforx ≤d;
(iii) β(Ax) >aforx∈K(β,a,c)withAx>b.
Then A has at least three positive solutions,x,x,x,satisfying
3 Single or twin solutions of problem (1.1)-(1.2) for the case
α
(t)≥tonJFor convenience we introduce the following notations:
fδρρ=min ρ<δρsuch that the following conditions hold:
(H) fρ<φp(l),I
Next, we prove that
+
Next, we prove that
+
t
g(t)I
u(t)
+I
u(t)
+· · ·+In
u(tn)
dt
=
–σ tn
g(t)In
u(tn)
dt
+
tn–
g(t)In–
u(tn–)
dt+· · ·+
t
g(t)I
u(t)
dt
≥
–σ tn
g(t)In
u(tn)
dt
≥
–σ tn
ξ
g(t)In
u(tn)
dt
=
–σLρσ
=ρ=u.
This is also a contraction. Hence (.) holds.
Applying (i) of Lemma . to (.) and (.) shows thatT has a fixed pointu∈K∩ (ρ\ ¯ρ). Thus, it follows that problem (.)-(.) has one positive solutionusatisfying
ρ<u ≤ρ. This gives the proof of the theorem.
Noticing thatL>l, it follows from the proof of Theorem . that:
Corollary . Suppose(H)-(H)hold,α(t)≥t on J and there existρ,ρ∈(,∞)with
Lρ<lρsuch that(H)and(H)hold.Then problem(.)-(.)has at least one positive
solution u withρ<u ≤ρ.
Corollary . Suppose(H)-(H)hold,α(t)≥t on J and there existρ,ρ∈(,∞)with ρ<ρsuch that the following conditions hold:
(H∗) fρ
<φp(l),I
ρ
(k) <l.
(H∗) Iρ δρ(k)≥L.
Then problem(.)-(.)has at least one positive solution u withρ<u<ρ.
Corollary . Suppose(H)-(H)hold andα(t)≥t on J.Assume that one of the following
conditions holds:
(H) f<φp(l),I(k) <ql,I∞(k)≥Lδ.
(H) I(k)≥Lδ,f∞<φp(l),I∞(k) <l.
Then problem(.)-(.)has at least one positive solution.
Proof We show that (H) implies (H) and (H). Suppose that (H) holds. Then there exists
<ρ<δρsuch that
f(t,u)
φp(u)
<φp(l), Ik(u)
u <l fort∈J, <u≤ρ and
Ik(u)
u ≥
L
Hence, we obtain
f(t,u) <φp(l)φp(u)≤φp(l)φp(ρ) =φp(lρ), Ik(u) <lu≤lρ fort∈J, <u≤ρ
and
Ik(u)≥L
δu≥
L
δδρ=Lρ foru≥δρ.
Noticingδρ<ρ, it follows that
Ik(u)≥Lρ forδρ≤u≤ρ.
Therefore, (H) and (H) hold. Hence, by Theorem ., problem (.)-(.) has at least
one positive solution.
Similarly, (H) implies that (H∗) and (H∗) hold.
Remark . It is easy to see that Corollary . holds if we change (H) into the following
condition:
(H∗) f= ,I(k) = , andI
∞(k) =∞.
However, we give no information whatsoever on the existence of positive solution for problem (.)-(.) if we change (H) into the following condition:
(H∗) f∞= ,I∞(k) = , andI(k) =∞.
Theorem . Suppose(H)-(H),α(t)≥t on J and one of the following conditions holds:
(H) There existρ,ρ,ρ∈(,∞),withρ<δρandρ<ρsuch that
fρ
<φp(l), I
ρ
(k) <lρ, I
ρ
δρ(k) >φp(L),
fρ
<φp(l), I
ρ
(k) <lρ.
(H) There existρ,ρ,ρ∈(,∞),withρ<ρ<ρsuch that
Iγρρ (k) >φp(L), f ρ
<φp(l), I
ρ
(k) <lρ, Iγρρ (k) >φp(L).
Then problem(.)-(.)has at least two positive solutions u,uwith u∈K∩(ρ\ ¯ρ),
u∈K∩(ρ\ ¯ρ).
Proof We only consider the condition (H). If (H) holds, then the proof is similar to that
of the case when (H) holds.
In fact, it follows from the proof of Theorem . that
Tuu, u∈K, u=ρ. (.)
Applying Lemma . to (.)-(.) shows thatThas two fixed pointsuandusuch that
u∈K∩(ρ\ ¯ρ),u∈K∩(ρ\ ¯ρ). These are the desired distinct positive solutions
Corollary . Suppose(H)-(H),α(t)≥t on J,and there existsρ∈(,∞)such that one
of the following conditions holds:
(H) ≤f<φp(l),≤I(k) <l,Iδρρ >L,and≤f∞<φp(l),≤I∞(k) <l.
(H) φp(L) <f≤ ∞,L<I(k)≤ ∞,f<φp(l),I(k) <l,and φp(L) <f∞≤ ∞, L<I∞(k)≤ ∞.
Then problem(.)-(.)has at least two positive solutions in K.
Proof We show that (H) implies (H). It is easy to verify that ≤f<φp(l), ≤I(k) <l
imply that there existsρ∈(,δρ) such that
fρ
<l, I
ρ
(k) <l.
Letη∈(f∞,φp(l)). Then there existsr>ηsuch thatmaxt∈Jf(t,u)≤ηφp(u) foru∈[r,∞) since ≤f∞<φp(u). Let
β=maxmax
t∈J f(t,u) : ≤u≤r
and ρ>max
φq
β φp(l) –η
,ρ
.
Then we have
max
≤t≤f(t,u)≤ηφp(u) +β≤ηφp(ρ) +β<φp(l)φp(ρ), ∀u∈[,ρ].
This implies thatfρ
≤φp(l). Similarly, from ≤Ik∞<l, we haveI
ρ
(k)≤lρ. Hence, (H)
implies (H). Similarly (H) implies (H), and the corollary is proved.
Theorem . can be generalized and we obtain many solutions.
Theorem . Suppose that(H)-(H)hold andα(t)≥t on J.Then we have the following
assertions.
() If there exists{ρi}i=m⊂(,∞)withρ<δρ<ρ<ρ<δρ<· · ·<ρmsuch that
fρm–
<φp(l), I
ρm–
(k) <lρm–, Iδρρmm(k) >L, m= , , . . . ,m,
then problem(.)-(.)has at leastmsolutions inK.
() If there exists{ρi}i=m⊂(,∞)withρ<ρand ρ<δρ<ρ<ρ<δρ<· · ·<ρm+such that
Iρm–
δρm–(k) >L, f ρm
<φp(l), I
ρm
<lρm, m= , , . . . ,m,
then problem(.)-(.)has at leastm– solutions inK.
4 Further results on twin solutions for the case of
α
(t)≥tonJForu∈K, we define the nonnegative increasing continuous functionalsθ,θ, andθby
θ= min
t∈[ξ,]u(t) =u(),
θ=max
t∈[,ξ]u(t) =u()
and
θ=max
t∈[,]u(t) =u().
It is easy to see that, for eachu∈K,
θ(u)≤θ(u) =θ(u). (.)
In addition, for eachu∈K,θ(u) =u()≥δu, which implies that
u ≤
δθ(u) for allu∈K. (.)
Finally, we also note that
θ(λu) =λθ(u), ≤λ≤ andu∈∂K(θ,r) forr> .
For notational convenience, we denote
N= σ
–σ, N=
n+γ
–σ.
We now present the results in this section.
Theorem . Suppose that there exist constants <a<b<c such that
<a< N N
b<δN N
c.
Assume f and Iksatisfy the following conditions:
(C) Ik(u) >Nc foru∈[c,cδ];
(C) f(t,u)≤φp(Nb)for(t,u)∈J×[,δb],Ik(u)≤
b
Nforu∈[,
b
δ];
(C) Ik(u) >Na foru∈[δa,a].
Then problem(.)-(.)has at least two positive solutions uand usuch that
a<θ(u) withθ(u) <b and b<θ(u) withθ(u) <c.
Proof By the definition of the operatorT and its properties, it suffices to show that the conditions of Lemma . hold with respect toT.
which implies that
δa≤u(t)≤a, t∈[ξ, ]. Using assumption (C), we have
Ik
Thus, by Lemma ., there exist at least two fixed points ofTwhich are positive solutions uandu, belonging toK(θ,c), of problem (.)-(.) such that
a<θ(u) withθ(u) <b and b<θ(u) withθ(u) <c.
We remark that the condition (C) in Theorem . can be replaced by the following
conditions:
(C) fb≤φp(N),Ib(k)≤N;
(C) f≤φp(N),I(k)≤N.
Corollary . If the condition(C)in Theorem.is replaced by(C)and(C),
Proof It follows from the proof of Theorem . that Corollary . holds.
5 Triple solutions for the case of
α
(t)≥tonJIn this section, we shall study the existence of three positive solutions of problem (.)-(.). Let the nonnegative continuous concave functionalψ(u) :K→[,∞) be defined by
ψ(u) = min
t∈[ξ,]u(t) =u(), u∈K.
Note that foru∈K,ψ(u)≤ u.
Theorem . Suppose that there exist constants <d<a<a
δ ≤c such that
Proof By the definition of the operatorT and its properties, it suffices to show that the conditions of Lemma . hold with respect toT.
= Therefore, it follows from (A) that
Finally, we assert that ifu∈K(ψ,a,c) andTu>b, thenψ(Tu) >a. Supposeu∈K(ψ,a,c) andTu>b, then
ψ(Tu) = min
t∈[ξ,](Tu)(t) = (Tu)()
≥δTu >δb=a.
To sum up, the hypotheses of Lemma . hold. Therefore, an application of Lemma . implies problem (.)-(.) has at least three positive solutionsu,u, andusuch that
u<d, a<ψ(u) and u>d withψ(u) <a.
We remark that the condition (A) in Theorem . can be replaced by the following
conditions:
(A) fc≤φp(N),I
c
(k)≤N;
(A) f≤φp(N),I(k)≤N.
Corollary . If the condition(A)in Theorem.is replaced by(A)and(A),
respec-tively,then the conclusion of Theorem.also holds.
Proof It follows from the proof of Theorem . that Corollary . holds.
6 Case of
α
(t)≤tonJNow we deal with problem (.)-(.) for the case ofα(t)≤tonJ. LetEbe as defined in Section . We define a coneK∗inEby
K∗=
u∈E:u≥, min
t∈[,ξ∗]u(t)≥δ ∗u,
whereξ∗∈(tn, ) and
δ∗= ξ
∗ ξ∗
th(t)dt ξ∗–ξ∗ς+ ξ∗th(t)dt.
It is easy to see thatK∗is a closed convex cone ofE. DefineT∗:K∗→Eby
T∗u(t) = –ς
g(t)
t
ϕq
s
ω(r)fr,uα(r)dr
ds
dt
+
g(t) tk<t
Ik
u(tk)
dt
+
t
ϕq
s
ω(r)fr,uα(r)dr
ds+ tk<t Ik
u(tk)
. (.)
It is clear thatu(t) is a positive solution of problem (.)-(.) if and only ofuis a solution of operator equation (.).
Lemma . Assume that(H)-(H)hold.Then problem(.)-(.)is equivalent to the fixed
point problem of T∗in K∗.
Lemma . Assume that(H)-(H)hold.Then T∗:K∗→K∗is completely continuous.
Similar to the proofs in Sections -, we have the following results. For convenience we introduce the following notations:
fδρ∗ρ=min
min t∈[,ξ∗]
f(t,u)
φp(ρ):u∈
δ∗ρ,ρ, Iδρ∗ρ(k) =min
Ik(u)
ρ :u∈
δ∗ρ,ρ,
σ∗=
ξ∗
t
g(s)ds, N∗= σ
∗
–σ,
L∗ =
σ∗
–σ.
Single or twin solutions of problem (.)-(.) for the caseα(t)≤tonJ is treated in the following theorems.
Theorem . Suppose(H)-(H)hold,α(t)≤t on J,and there existρ,ρ∈(,∞)with ρ<δ∗ρsuch that the following conditions hold:
(H) fρ<φp(l),I
ρ
(k) <l.
(H)∗∗ Iδρ∗ρ(k)≥L∗.
Then problem(.)-(.)has at least one positive solution u withρ<u ≤ρ.
Proof If (H) holds, then similar to the proof of (.) we have
T∗uu, u∈K∗, u=ρ. (.)
Considering (H)∗∗, we haveIk(x)≥L∗ρ,k= , , . . . ,n, forδ∗ρ≤x≤ρ,t∈[,ξ∗].
Since ≤α(t)≤ξ∗onJ, it follows fromδ∗ρ≤x(t)≤ρonJthat
δ∗ρ≤x
α(t)≤ρ.
Next, we prove that
T∗uu, u∈K∗, u=ρ. (.)
In fact, if there existsu∈K∗,u=ρsuch thatT∗u≤u, then we have
≤t∈J ⇒ u(t)≥
T∗u
(t)
=
–ς
g(t)
t
ϕq
s
ω(r)fr,u
α(r)dr
ds
dt
+
g(t) tk<t
Ik
u(tk)
dt
+
t
ϕq
s
ω(r)fr,u
α(r)dr
ds+ tk<t
Ik
u(tk)
≥
This is also a contraction. Hence (.) holds.
Applying (i) of Lemma . to (.) and (.) shows thatT∗ has a fixed pointu∈K∗∩ (ρ\ ¯ρ). Thus, it follows that problem (.)-(.) has one positive solutionusatisfying ρ<u(t)≤ρ,t∈J. This gives the proof of the theorem. Remark . The method to deal with the impulsive term in the proof of Theorem . is different from that in the proof of Theorem ..
Theorem . Suppose(H)-(H),α(t)≤t on J and one of the following conditions holds:
Theorem . Assume that(H)-(H)andα(t)≤t on J.Moreover,suppose that there exist
constants <a<b<c such that
<a<N
∗
N
b<δ
∗N∗
N
c.
Let f and Iksatisfy the following conditions:
(C)∗ Ik(u) >Nc∗foru∈[c,δc∗];
(C)∗ f(t,u)≤φp(Nb)for(t,u)∈J×[,δb∗],Ik(u)≤
b
Nforu∈[,
b
δ∗];
(C)∗ Ik(u) >Na∗foru∈[δ∗a,a].
Then problem(.)-(.)has at least two positive solutions uand usuch that
a<θ(u) withθ(u) <b and b<θ(u) withθ(u) <c.
Finally we consider triple solutions of problem (.)-(.) for the case ofα(t)≥tonJ.
Theorem . Assume that(H)-(H)andα(t)≤t on J.Moreover,suppose that there exist
constants <d<a<δa∗≤c such that
(A) f(t,u)≤φp(Nd)for(t,u)∈J×[,d],Ik(u)≤Ndforu∈[,d]; (A)∗ Ik(u)≥Na∗ foru∈[a,δa∗];
(A) f(t,u)≤φp(Nc)for(t,u)∈J×[,c],Ik(u)≤Nc foru∈[,c].
Then problem(.)-(.)has at least three positive solutions u,u,and usuch that
u<d, a<ψ(u) and u>d withψ(u) <a.
Remark . Minor adjustments to the proof of Theorem . are only needed to prove that Theorems .-. hold, so we leave the details to the reader.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
XZ completed the main study and carried out the results of this article. MF checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
Author details
1Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, People’s Republic of
China.2School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, People’s
Republic of China.
Acknowledgements
This work is sponsored by the project NSFC (11301178), the Fundamental Research Funds for the Central Universities (2014ZZD10, 2014MS58) and the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201511232018). The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.
Received: 28 October 2014 Accepted: 14 April 2015
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