R E S E A R C H
Open Access
Positive solutions for a second-order
p-Laplacian impulsive boundary value
problem
Youzheng Ding
1,3*and Donal O’Regan
2*Correspondence:
1Department of Mathematics,
Shandong Jianzhu University, Jinan, Shandong 250101, China
3School of Mathematics, Shandong
University, Jinan, Shandong 250100, China
Full list of author information is available at the end of the article
Abstract
In this work, we study the existence and multiplicity of positive solutions for a second-orderp-Laplacian boundary value problem involving impulsive effects. We establish our main resultsviaJensen’s inequality, the first eigenvalue of a relevant linear operator and the Krasnoselskii-Zabreiko fixed point theorem. Some examples are presented to illustrate the main results.
MSC: 34B15; 34B18; 34B37; 45G15; 45M20
Keywords: impulsive boundary value problem;p-Laplacian; fixed point theorem;
positive solution; eigenvalue; Jensen inequality
1 Introduction
Second-order differential equations with thep-Laplacian operator arise in modeling some physical and natural phenomena and can occur, for example, in non-Newtonian mechan-ics, nonlinear elasticity, glaciology, population biology, combustion theory, and nonlinear flow laws, see [, ]. Recently, many cases of the existence and multiplicity of positive solu-tions for boundary value problems of differential equasolu-tions with thep-Laplacian operator have appeared in the literature. For details, see, for example, [–] and the references therein.
In [], Lian and Ge investigated the Sturm-Liouville-like boundary value problem
⎧ ⎨ ⎩
(ϕp(u(t))+f(t,u(t)) = , <t< ,
u() –au(ξ) = , u() +βu(η) = , (.)
and by virtue of Krasonsel’skii’s fixed point theorem, they obtained the existence of pos-itive solutions and multiple pospos-itive solutions under suitable conditions imposed on the nonlinear termf ∈C([, ]×[, +∞), [, +∞)).
In [], Xuet al.studied the existence of multiple positive solutions for the following boundary value problem with thep-Laplacian operator and impulsive effects
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(φp(u(t))+q(t)f(t,u(t)) = , t=tk, <t< ,
u|t=tk=Ik(u(tk)), k= , , . . . ,m,
au() –bu() =li=αiu(ξi), u() =
l
i=βiu(ξi),
(.)
where the nonlinear term may be singular onu= . The main tools are fixed point index theorems for compact maps in Banach spaces. They stated the proofs by considering an approximating completely continuous operator.
In [], Feng studied an integral boundary value problem of fourth-orderp-Laplacian dif-ferential equations involving the impulsive effecty|t=tk= –Ik(y(tk)),k= , , . . . ,m. Using a suitably constructed cone and fixed point theory for cones, the existence of multiple pos-itive solutions was established. Furthermore, upper and lower bounds for these pospos-itive solutions were given.
Motivated by the above works, in this paper, we investigate the existence and multiplicity of positive solutions for the second-orderp-Laplacian boundary value problems involving impulsive effects
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(ϕp(u(t)))= –f(t,u(t)), <t< ,t=tk,k= , , . . . ,m,
u|t=tk=Ik(u(tk)), k= , , . . . ,m, u|t=tk= , k= , , . . . ,m,
u() =u() = ,
(.)
whereJ= [, ],J=J\ {t,t, . . . ,tm},tk(k= , , . . . ,m, wheremis a fixed positive integer)
are fixed points with <t<t<· · ·<tk<· · ·<tm< ;ϕp(t) is thep-Laplacian operator,i.e.,
ϕp(t) =|t|p–t,p> , (ϕp)–=ϕq,p–+q–= ;u|t=tkdenotes the jump ofu(t) att=tk,i.e., u|t=tk=u(t
+
k) –u(tk–), whereu(tk+) andu(t–k) represent the right-hand limit and left-hand
limit ofu(t) att=tk, respectively. In addition, we suppose thatIk∈C([, +∞), [, +∞)),
f ∈C([, ]×[, +∞), [, +∞)).
The main features of this paper are as follows. Firstly, we convert the boundary value problem (.) into an equivalent integral equation so that we can construct a special cone. Next, we consider impulsive effect as a perturbation to the corresponding problem without the impulsive terms, so that we can construct an integral operator for an appropriate linear Robin boundary value problem and obtain its first eigenvalue and eigenfunction, which are used in the proofs of main theorems by Jensen’s inequalities. Finally, employing the Krasnoselskii-Zabreiko fixed point theorem, we establish the existence and multiplicity of positive solutions of (.). Although our problem (.) merely involves Robin boundary conditions, our methods are different from those in [, , ], and our main results are optimal.
This paper is organized as follows. Section contains some preliminary results. Sec-tion is devoted to the existence and multiplicity of positive soluSec-tions for (.). SecSec-tion contains some illustrative examples.
2 Preliminaries
LetPC[J,R] :={u|u:J→Ris continuous att=tk,u(tk–) =u(tk) andu(tk+) exist,k= , , . . . ,
m}. ThenPC[J,R] is a Banach space with normu=maxt∈[,]|u(t)|. We denoteBr:=
{u∈PC[J,R] :u<r}forr> in the sequel.
Lemma .(see []) Let f and Ikbe as in (.). Then the problem (.) is equivalent to
u(t) =
t
ϕq
s
f τ,u(τ)dτ
ds+
tk<t
Ik u(tk)
. (.)
It is clear thatu(t) =ϕq(
t f(τ,u(τ)) dτ) > ,t∈JandIk> , which implies thatu(t) is
increasing on [, ]. Furthermore, for givens,s∈Jwiths≤s, we haveu(s)≤u(s).
Hence,u(t) is nonincreasing onJ, and thus
u() –u()
≤
u(t) –u()
t , t∈(, ],
i.e.,u(t)≥tu() =tu. Therefore,
u(t)≥tu, ∀t∈[, ], in particular, u(t)≥tu, ∀t∈[t,tm]. (.)
We denotePby
P:=u∈PC[J,R] :u(t)≥tu,t∈[, ]. (.)
ThenPis a cone onPC[J,R].
Define an operatorA:P→PC[J,R]
Au(t) :=
t
ϕq
s
f τ,u(τ)dτ
ds+
tk<t
Ik u(tk)
=
t
ϕq
s
f τ,u(τ)dτ
ds+
m
k=
H(t,tk)Ik u(tk)
,
where
H(t,tk) =
⎧ ⎨ ⎩
, tk<t,
, tk≥t.
Clearly, the operatorAis a completely continuous operator, and the existence of positive solutions for (.) is equivalent to that of positive fixed points ofA. Moreover, it is easy to seeA(P)⊂Pby (.).
In what follows, we consider the following eigenvalue problem:
⎧ ⎨ ⎩
–u=λu, t∈[, ],
u() =u() = , (.)
whereλis a parameter. We easily know that (.) has a nontrivial solution ifλ> . Fur-thermore, we have
wherec,care constants andc
+c= .u() =u() = implies thatcos √
λ= , and thus λ= (–π+kπ),k= , . . . .λ–
:= (–
π
+ ×π)=
π
andsin(
πt
) are called the first
eigen-value and the corresponding eigenfunction associated withλ, respectively. Consequently,
it is easy to have the following result:
Lemma . Ifψ(t) :=sin(πt), then
G(t,s)ψ(t) dt=
πψ(s), (.)
where G(t,s) =min{t,s}, t,s∈[, ].
Lemma .(see []) Let E be a real Banach space and W a cone of E. Suppose A: (BR\
Br)∩W→W is a completely continuous operator with <r<R. If either
() Auufor each∂Br∩W andAuufor each∂BR∩W or
() Auufor each∂Br∩W andAuufor each∂BR∩W,
then A has at least one fixed point in(BR\Br)∩W .
Lemma .(Jensen’s inequalities, see []) Letθ > , n≥, ai≥(i= , , . . . ,n), and
ϕ∈C([a,b], [, +∞)). Then b
a
ϕ(t) dt θ
≤(b–a)θ–
b a
ϕ(t)θdt and n
i= ai
θ
≤nθ–
n
i= aθ
i, ∀θ≥,
b a
ϕ(t) dt θ
≥(b–a)θ–
b a
ϕ(t)θdt and n
i= ai
θ
≥nθ–
n
i=
aθi, ∀ <θ≤.
3 Main results
Let p* :=max{,p– }, p* :=min{,p– }, κ
:= p*–, κ := (m)p*–, κ := p
*– , κ:=
(m)p*–,κ :=
pp* p––,κ
:= (m)p
*–
. We now list our hypotheses. (H) There existr> anda≥,a≥ satisfying
a
p* p–
κ+
π
a
p*
t
p*
κ
m
k=
cos
πtk
>π
,
such that
f(t,y)≥ayp–, Ik(y)≥ay, ∀t∈[, ], <y<r. (.)
(H) There existc> andb≥,b≥ satisfying
b+b= , b
p* p–
κ+
πbp*κ m
k=cos(
πtk
)
tp*
sin(πt)dt <
such that
f(t,y)≤byp–+c, Ik(y)≤by+c, ∀t∈[, ],y≥. (.)
(H) There existc> anda≥,a≥ satisfying
a
p* p–
κ+
π
a
p*
t
p*
κ
m
k=
cos
πtk
>π
,
such that
f(t,y)≥ayp––c, Ik(y)≥ay–c, ∀t∈[, ],y≥. (.)
(H) There existr> andb≥,b≥ satisfying
b+b= , b
p* p–
κ+
πbp*κ m
k=cos(
πtk
)
tp*
sin(πt
)dt
<π
,
such that
f(t,y)≤byp–, Ik(y)≤by, ∀t∈[, ], <y<r. (.)
(H) There existsρ> such that ≤y≤ρ, andt∈[, ] implies
f(t,y)≤ηp–ρp–, Ik(y)≤ηkρ,
whereη,ηk≥ and <η+
m
k=ηk≤.
(H) There existsρ> such thattρ≤y≤ρ, andt∈[, ] implies
f(t,y)≥ηp–ρp–, Ik(y)≥ηkρ,
whereη,ηk≥, (ηp–(p– )( –t)p/(p–)+
m
k=ηk) > .
Theorem . Suppose that (H)-(H) are satisfied. Then (.) has at least one positive solution.
Proof If (H) is satisfied, then we obtainuAufor allu∈P∩∂Br. Indeed, if the claim is
false, there is au∈P∩∂Brsuch thatu≥Au,i.e.,
u(t)≥
t
ϕq
s
f τ,u(τ)dτ
ds+
m
k=
H(t,tk)Ik u(tk)
.
Now apply Lemma . to obtain
up*(t)≥
t
ϕq
s
f τ,u(τ)dτ
ds+
m
k=
H(t,tk)Ik u(tk)
p*
≥κ t
ϕq
s
f τ,u(τ)dτ
ds p*
+κ m
k=
H(t,tk)Ik u(tk)
≥κ
The above and (H) imply that
Combining (.) and (.), we obtain
(π– a
Therefore, a
p*
, which contradicts (H). Thus we have
uAu, for anyu∈P∩∂Br. (.)
On the other hand, by (H), we shall prove that there exists a sufficiently large number
This, together with Lemma ., yields
Multiply both sides of the above bysin(πt
) and integrate over [, ] and use (.) to obtain
Combining this and (H), we get
Therefore,
Therefore, (.) and (.), together with Lemma ., guarantee that (.) has at least one
positive solution in (BR\Br)∩P.
Theorem . Suppose that (H)-(H) are satisfied. Then (.) has at least one positive solution.
Proof If (H) is satisfied, we will prove that there exists a sufficiently large numberR> such thatuAu,∀u∈P∩∂BR. Suppose there existsu∈P∩∂BRsuch thatu≥Au, and
Now, we consider two cases.
which contradicts (H). Thus
uAu, ∀u∈∂Br∩P. (.)
By Lemma ., (.) and (.) imply that (.) has at least one positive solution in (BR\
Br)∩P.
Theorem . Suppose that (H), (H) and (H) are satisfied. Then (.) has at least two positive solutions.
Proof Ifu∈P∩∂Bρ, it follows from (H) that
Au ≤
ϕq
s
f τ,u(τ)dτ
ds+
m
k=
Ik u(tk)
≤
ϕq
s
ηp–ρp–dτ
ds+
m
k=
ρηk
≤ρ
ηp–
p +
m
k=
ηk
<ρ
η+
m
k=
ηk
≤ u,
from which we obtain
uAu, ∀u∈∂Bρ∩P. (.)
On the other hand, by (H) and (H), we may take <r<ρandR>ρsuch thatuAu,
∀u∈P∩∂BranduAu,∀u∈P∩∂BR(see the proofs of Theorems . and .). Now
Lemma . guarantees that the operatorAhas at least two fixed points, one in (BR\Bρ)∩P
and the other in (Bρ\Br)∩P. The proof is completed.
Theorem . Suppose that (H), (H) and (H) are satisfied. Then (.) has at least two positive solutions.
Proof For anyu∈P∩∂Bρ,u(t)≥tu=tρfor allt∈[t, ]. It follows from (H) that
Au =
ϕq
s
f τ,u(τ)dτ
ds+
m
k=
Ik u(tk)
≥
t ϕq
s
f τ,u(τ)dτ
ds+
m
k=
Ik u(tk)
≥
t ϕq
s
ηp–ρp–dτ
ds+
m
k=
ηkρ
≥ρ
ηp–(p– )( –t)p/(p–)+
m
k=
ηk
>ρ=u. (.)
Thus,
On the other hand, by (H) and (H), we may take <r<ρandR>ρsuch thatuAu,
∀u∈P∩∂Br anduAu,∀u∈P∩∂BR(see the proofs of Theorems . and .). Thus
Lemma . indicates that the operatorAhas at least two fixed points, one in (BR\Bρ)∩P
and the other in (Bρ\Br)∩P. The proof is completed.
4 Examples
Example . Consider the impulsive boundary value problem
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(ϕp(u))= –f(t,u), t=, <t< ,
u|t= =I(u(
)),
u|t= = ,
u() =u() = ,
(.)
wherep=.
Case . Letf(t,u) =uα, <α<
,I(u) =u
α, <α
< . Then
lim
u→+
f(t,u)
up– =ulim→+
uα
up– =∞, ulim→+
I(u)
u =ulim→+u
α–=∞. (.)
From (.) we see that (H) is satisfied. In fact, sincep= / andt= /,p*= /,κ=κ=
/√. Takinga= ,a= , we get
a
p* p–
κ+
π a
p*
t
p*
κ
m
k=
cos
πtk
>π
.
Moreover,
lim
u→∞
f(t,u)
up– =ulim→∞
uα
up–= , ulim→∞
I(u)
u =ulim→∞u
α–= . (.)
From (.) we see that (H) is satisfied, for example takingb= ,b=. All the assump-tions in Theorem . are satisfied, and the problem (.) has at least one positive solution by Theorem ..
Case . Takef(t,u) =uβ,β>
,I(u) =u
β,β
> . One can easily verify conditions (H)
and (H). Thus the problem (.) has at least one positive solution by Theorem .. Case . Letf(t,u) = uα+uβ, <α<<β,I(u) =u. Thus, we get
lim
u→+
f(t,u)
up– =ulim→+
uα+uβ
up– =∞. (.)
From (.) we see that (H) is satisfied. Note
lim
u→∞
f(t,u)
up– =ulim→∞ uα+uβ
up– =∞. (.)
From (.) we see that (H) is satisfied. Take ρ= ,η=,η = in (H), and note for
≤u≤ρandt∈[, ] thatf(t,u)≤ρα+ρβ =< √
=η
,I(u) =u
≤
=η. As a result,
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The first author conceived of the study and carried out the proof. Authors read and approved the final manuscript.
Author details
1Department of Mathematics, Shandong Jianzhu University, Jinan, Shandong 250101, China.2Department of
Mathematics, National University of Ireland, Galway, Ireland. 3School of Mathematics, Shandong University, Jinan,
Shandong 250100, China.
Acknowledgements
The authors are grateful to the anonymous referee for his/her valuable suggestions. The first author thanks Prof. Zhongli Wei for his valuable help. This work is supported financially by the Shandong Provincial Natural Science Foundation (ZR2012AQ007) and Graduate Independent Innovation Foundation of Shandong University (yzc12063).
Received: 18 April 2012 Accepted: 28 August 2012 Published: 11 September 2012 References
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