Advances in Difference Equations Volume 2010, Article ID 746106,20pages doi:10.1155/2010/746106
Research Article
Time-Scale-Dependent Criteria for
the Existence of Positive Solutions to p-Laplacian Multipoint Boundary Value Problem
Wenyong Zhong
1and Wei Lin
21School of Mathematics and Computer Sciences, Jishou University, Hunan 416000, China
2Shanghai Key Laboratory of Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Correspondence should be addressed to Wei Lin,wlin@fudan.edu.cn Received 1 May 2010; Revised 23 July 2010; Accepted 30 July 2010 Academic Editor: Alberto Cabada
Copyrightq2010 W. Zhong and W. Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By virtue of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem, we analytically establish several sufficient criteria for the existence of at least two or three positive solutions in thep-Laplacian dynamic equations on time scales with a particular kind ofp-Laplacian andm-point boundary value condition. It is this kind of boundary value condition that leads the established criteria to be dependent on the time scales. Also we provide a representative and nontrivial example to illustrate a possible application of the analytical results established. We believe that the established analytical results and the example together guarantee the reliability of numerical computation of thosep-Laplacian andm-point boundary value problems on time scales.
1. Introduction
The investigation of dynamic equations on time scales, originally attributed to Stefan Hilger’s seminal work1,2two decades ago, is now undergoing a rapid development. It not only unifies the existing results and principles for both differential equations and difference equations with constant time stepsize but also invites novel and nontrivial discussions and theories for hybrid equations on various types of time scales3–11. On the other hand, along with the significant development of the theories, practical applications of dynamic equations on time scales in mathematical modeling of those real processes and phenomena, such as the population dynamics, the economic evolutions, the chemical kinetics, and the neural signal processing, have been becoming richer and richer12,13.
As one of the focal topics in the research of dynamic equations on time scales, the study of boundary value problems for some specific dynamic equations on time scales recently has elicited a great deal of attention from mathematical community14–33. In particular, a series of works have been presented to discuss the existence of positive solutions in the boundary value problems for the second-order equations on time scales14–21. More recently, some analytical criteria have been established for the existence of positive solutions in some specific boundary value problems for thep-Laplacian dynamic equations on time scales22,33.
Concretely, He25investigated the following dynamic equation:
φp
uΔt∇
htfu 0, t∈0, TT, 1.1
with the boundary value conditions
uΔ0 0, uT B0 uΔ
η
0. 1.2
Here and throughout,Tis supposed to be a time scale; that is, T is any nonempty closed subset of real numbers inRwith order and topological structure defined in a canonical way.
The closed interval inTis defined asa, bT a, b∩T. Accordingly, the open interval and the half-open interval could be defined, respectively. In addition, it is assumed that 0, T ∈T, η∈0, ρTT,f∈Cld0,∞,0,∞,h∈Cld0, TT,0,∞, andbx B0xbxfor some positive constantsbandb. Moreover,φpuis supposed to be thep-Laplacian operator, that is,φpu |u|p−2uandφp−1φq, in whichp >1 and 1/p 1/q1. With these configurations and with the aid of the Avery-Henderson fixed point theorem34, He established the criteria for the existence of at least two positive solutions in 1.1 fulfilling the boundary value conditions1.2.
Later on, Su and Li 24 discussed the dynamic equation 1.1 which satisfies the boundary value conditions
uΔ0 0, uT B0
m−2
i1
biuΔξi
0, 1.3
whereξ ∈ 0, T, 0 < ξ1 < ξ2 < · · · < ξm−2 < T, andbi ∈ 0,∞fori 1,2, . . . ,m−2. By virtue of the five functionals fixed point theorem35, they proved that the dynamic equation 1.1with conditions1.3has three positive solutions at least. Meanwhile, He and Li in26, studied the dynamic equation1.1satisfying either the boundary value conditions
u0−B0 uΔ0
0, uΔT 0, 1.4
or the conditions
uΔ0 0, uT B0 uΔT
0. 1.5
In the light of the five functionals fixed point theorem, they established the criteria for the existence of at least three solutions for the dynamic equation 1.1 either with conditions 1.4or with conditions1.5.
More recently, Yaslan27,28investigated the dynamic equation:
uΔ∇t htft, ut 0, t∈t1, t3T⊂T, 1.6 which satisfies either the boundary value conditions
αut1−β0uΔt1 uΔt2, uΔt3 0, 1.7
or the conditions
uΔt1 0, αut3 βuΔt3 uΔt2. 1.8
Here, 0 t1 < t2 < t3,α >0,β0 0, andβ > 1. Indeed, Yaslan analytically established the conditions for the existence of at least two or three positive solutions in these boundary value problems by virtue of the Avery-Henderson fixed point theorem and the Leggett-Williams fixed point theorem36. It is worthwhile to mention that these theoretical results are novel even for some special cases on time scales, such as the conventional difference equations with fixed time stepsize and the ordinary differential equations.
Motivated by the aforementioned results and techniques in coping with those boundary value problems on time scales, we thus turn to investigate the possible existence of multiple positive solutions for the following one-dimensionalp-Laplacian dynamic equation:
φp
uΔt∇
htft, ut 0, t∈0, TT, 1.9
with thep-Laplacian andm-point boundary value conditions:
φp uΔ0
m−2
i1
aiφp uΔξi
, uT βB0 uΔT
m−2
i1
B uΔξi
. 1.10
In the following discussion, we implement three hypotheses as follows.
H1One hasai0 fori1, . . . , m−2, 0< ξ1< ξ2<· · ·< ξm−2< T, andd01− m−2i1 ai>0.
H2One has thath: 0, σTT → 0,∞is left dense continuousld-continuous, and there exists at0∈0, TTsuch thatht0/0. Alsof :0, σTT×0,∞→0,∞is continuous.
H3BothB0andBare continuously odd functions defined onR. There exist two positive numbersbandbsuch that, for anyv >0,
bvB0v, Bvbv 1.11
and that
βb−m−2b−μT0. 1.12
It is clear that, together with conditions 1.10 and the above hypotheses H1–H3, the dynamic equation 1.9 not only covers the corresponding boundary value problems in the literature, but even nontrivially generalizes these problems to a much wider class of boundary value problems on time scales. Also it is valuable to mention that condition1.12 in hypothesis H3 is necessarily relevant to the graininess operator μ : T → 0, ∞ around the time instant T. Such kind of condition has not been required in the literature, to the best of authors’ knowledge. Thus, this paper analytically establishes some new and time-scale-dependent criteria for the existence of at least double or triple positive solutions in the boundary value problems 1.9 and 1.10 by virtue of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem. Indeed, these obtained criteria significantly extend the results existing in26–28.
The remainder of the paper is organized as follows.Section 2preliminarily provides some lemmas which are crucial to the following discussion.Section 3analytically establishes the criteria for the existence of at least two positive solutions in the boundary value problems 1.9and 1.10with the aid of the Avery-Henderson fixed point theorem. Section 4gives some sufficient conditions for the existence of at least three positive solutions by means of the five functionals fixed point theorem. More importantly,Section 5provides a representative and nontrivial example to illustrate a possible application of the obtained analytical results on dynamic equations on time scales. Finally, the paper is closed with some concluding remarks.
2. Preliminaries
In this section, we intend to provide several lemmas which are crucial to the proof of the main results in this paper. However, for concision, we omit the introduction of those elementary notations and definitions, which can be found in 11, 12, 33 and references therein.
The following lemmas are based on the following linear boundary value problems:
φp
uΔt∇
gt 0, t∈0, TT,
φp
uΔ0
m−2
i1
aiφp
uΔξi
, uT βB0
uΔT
m−2
i1
B uΔξi
.
2.1
Lemma 2.1. Assume thatd01− m−2i1 ai/0. Then, forg ∈Cld0, TT, the linear boundary value problems2.1have a unique solution satisfying
ut T
t
φq
s
0
gτ∇τ 1
d0
m−2
i1
ai
ξi
0
gτ∇τ
Δs
βB0
φq T
0
gτ∇τ 1
d0
m−2
i1
ai ξi
0
gτ∇τ
−m−2
i1
B
φq
ξi
0
gτ∇τ 1
d0 m−2
i1
ai
ξi
0
gτ∇τ
,
2.2
for allt∈0, σTT.
Proof. According to the formulat
aft, sΔsΔ fσt, t t
aft, sΔsintroduced in12, we have
uΔt φq
− t
0
gτ∇τ− 1 d0
m−2
i1
ai ξi
0
gτ∇τ
. 2.3
Thus, we obtain that
φp
uΔt
− t
0
gτ∇τ− 1 d0
m−2
i1
ai
ξi
0
gτ∇τ, 2.4
and that
φp
uΔt∇
−gt. 2.5
To this end, it is not hard to check thatutsatisfies2.2, which implies thatutis a solution of the problems2.1.
Furthermore, in order to verify the uniqueness, we suppose that bothu1tandu2t are the solutions of the problems2.1. Then, we have
φp
uΔ1t∇
− φp
uΔ2t∇
0, t∈0, TT, 2.6
φp uΔ10
−φp uΔ20
m−2
i1
ai φp
uΔ1ξi
−φp
uΔ2ξi
, 2.7
u1T−u2T βB0
uΔ1T
−βB0
uΔ2T
m−2
i1
B
uΔ1ξi
−B
uΔ2ξi
. 2.8
According to Theorem A.5 in37,2.6further yields
φp
uΔ1t
−φp
uΔ2t
c, t∈0, TT. 2.9
Hence, from2.7and2.9, the assumptiond0 1− m−2i1 ai/0, and the definition of the p-Laplacian operator, it follows that
uΔ1t−uΔ2t≡0, t∈0, TT. 2.10
This equation, together with2.8, further implies
u1t≡u2t, t∈0, σTT, 2.11
which consequently leads to the completion of the proof, that is,utspecified in2.2is the unique solution of the problems2.1.
Lemma 2.2. Assume thatd0 1− m−2i1 ai > 0 and that βb−m−2b−μT 0. If g ∈ Cld0, σTT,0,∞, then the unique solution of the problems2.1satisfies
ut0, t∈0, σTT. 2.12
Proof. By2.2specified inLemma 2.1, we get
uΔt −φq
t
0
gτ∇τ 1
d0
m−2
i1
ai ξi
0
gτ∇τ
0, t∈0, TT. 2.13
Thus,utis nonincreasing in the interval0, σTT. In addition, notice that
uσT T
σTφq
s 0
gτ∇τ 1
d0
m−2
i1
ai
ξi
0
gτ∇τ
Δs
βB0
φq T
0
gτ∇τ 1
d0 m−2
i1
ai ξi
0
gτ∇τ
−m−2
i1
B
φq
ξi
0
gτ∇τ 1
d0
m−2
i1
ai
ξi
0
gτ∇τ
−μTφq
T
0
gτ∇τ 1
d0
m−2
i1
ai ξi
0
gτ∇τ
βB0
φq
T
0
gτ∇τ 1
d0 m−2
i1
ai
ξi
0
gτ∇τ
−m−2
i1
B
φq ξi
0
gτ∇τ 1
d0
m−2
i1
ai ξi
0
gτ∇τ
βb−m−2b−μT φq
T
0
gτ∇τ 1
d0 m−2
i1
ai ξi
0
gτ∇τ
.
2.14
The last term in the above estimation is no less than zero because of the assumptions. Thus, from the monotonicity ofut, we get
utuσT0, t∈0, σTT, 2.15
which completes the proof.
Now, denote thatE Cld0, σTT and that u supt∈0,σT
T|ut|, whereu ∈ E.
Thus, it is easy to verify thatEendowed with · becomes a Banach space. Furthermore, define a cone, denoted byP, through,
P
u∈ E |ut0 fort∈0, σTT,
uΔt0 fort∈0, TT, uΔ∇t0 fort∈0, σTT .
2.16
Also, for a given positive real numberr, define a function setPr by
Pr {u∈ P | u< r}. 2.17
Naturally, we denote thatPr {u∈ P | u r}and that∂Pr {u ∈ P | u r}. With these settings, we have the following properties.
Lemma 2.3. Ifu ∈ P,theni ut T −t/Tufor anyt ∈ 0, TT,ii T −sut T−tusfor any pair ofs, t∈0, TTwithts.
The proof of this lemma, which could be found in26,28, is directly from the specific construction of the setP. Next, let us construct a mapA:P → Ethrough
Aut T
t
φq s
0
hτfτ, uτ∇τ 1
d0
m−2
i1
ai ξi
0
hτft, uτ∇τ
Δs
βB0
φq T
0
hτfτ, uτ∇τ 1
d0 m−2
i1
ai ξi
0
hτfτ, uτ∇τ
−m−2
i1
B
φq
ξi
0
hτfτ, uτ∇τ 1 d0
m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
,
2.18
for anyu∈ P. Then, through a standard argument33, it is not hard to validate the following properties on this map.
Lemma 2.4. Assume that the hypothesesH1–H3are all fulfilled. Then,AP⊂ P, andA:Pr → Pis completely continuous.
3. At Least Two Positive Solutions in Boundary Value Problems
In this section, we aim to adopt the well-known Avery-Henderson fixed point theorem to prove the existence of at least two positive solutions in the boundary value problems1.9 and1.10. For the sake of self-containment, we first state the Avery-Henderson fixed point theorem as follows.
Theorem 3.1see34. LetPbe a cone in a real Banach spaceE. For eachd > 0, setPψ, d {x∈ P |ψx< d}. Letαandγbe increasing, nonnegative continuous functionals onP, and letθbe a nonnegative continuous functional onPwithθ0 0 such that, for somec >0 andH >0,
γxθxαx, xHγx, 3.1
for allx∈ Pγ, c. Suppose that there exist a completely continuous operatorA:Pγ, c → Pand three positive numbers 0< a < b < csuch that
θλxλθx, 0λ1, x∈∂Pθ, b, 3.2
andiγAx> cfor allx∈∂Pγ, c,iiθAx< bfor allx∈∂Pθ, b, andiiiPα, a/∅and αAx> afor allx∈∂Pα, a. Then, the operatorAhas at least two fixed points, denoted byx1and x2, belonging toPγ, cand satisfyinga < αx1withθx1< bandb < θx2withγx2< c.
Now, sett min{t∈T|T/2tT}and selectt∈Tsatisfying 0< t < t. Denote, respectively, that
M T−t T
t
0
φq s
0
hτ∇τ
Δs,
N
T βb
·φq
1 d0
T
0
hτ∇τ
,
L T−t T
T
t
φq
s t
hτ∇τ
Δs,
L0
T−t βb−m−2b
·φq
1 d0
T
0
hτ∇τ
.
3.3
Hence, we are in a position to obtain the following results.
Theorem 3.2. Assume that the hypotheses H1–H3 all hold and that there exist positive real numbersa,b,csuch that
0< a < b < c, a < L
Nb <LT−t
TL c. 3.4
In addition, assume thatfsatisfies the following conditions:
C1ft, u> φpc/Mfort∈0, tTandu∈c,T/T−tc;
C2ft, u< φpb/Nfort∈0, TTandu∈0,T/T−tb;
C3ft, u> φpa/Lfort∈t, TTandu∈0, a.
Then, the boundary value problems1.9and 1.10have at least two positive solutionsu1 andu2
such that
a < max
t∈t,TTu1t with max
t∈t,TTu1t< b, b < max
t∈t,TTu1t with min
t∈t,tTu2t< c. 3.5
Proof. Construct the conePand the operatorAas specified in2.16and2.18, respectively.
In addition, define the increasing, nonnegative, and continuous functionalsγ,θ, andαonP, respectively, by
γu min
t∈t,tT
ut ut, θu max
t∈t,TT
ut ut,
αu max
t∈t,TT
ut ut. 3.6
Evidently,γu θuαufor eachu∈ P.
In addition, for eachu∈ P,Lemma 2.3manifests thatγu utT−t/Tu.
Thus, we have
u T
T−tγu, 3.7
for eachu∈ P. Also, notice thatθλu λθuforλ∈0,1andu∈∂Pθ, b. Furthermore, fromLemma 2.4, it follows that the operatorA:Pγ, c → Pis completely continuous.
In what follows, we are to verify that all the conditions ofTheorem 3.1are satisfied with respect to the operatorA.
Letu∈∂Pγ, c. Then,γu mint∈t,tTut ut c. This implies thatutcfor t∈0, tT, which, combined with3.7, yields
cut T
T−tc, 3.8
fort∈0, tT. Because of assumptionC1,ft, ut> φpc/Mfort∈0, tT. According to the specific form in2.18,Lemma 2.3, and the propertyAu∈ P, we obtain that
γAu
Aut T−t
T Au T−t
T Au0
T−t T
T
0
φq
s
0
hτfτ, uτ∇τ 1 d0
m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
Δs
βB0
φq T
0
hτfτ, uτ∇τ 1
d0
m−2
i1
ai ξi
0
hτfτ, uτ∇τ
−m−2
i1
B
φq
ξi
0
ft, uτ∇τ 1
d0 m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
T−t T
T
0
φq s
0
hτfτ, uτ∇τ 1
d0
m−2
i1
ai ξi
0
hτfτ, uτ∇τ
Δs βb−m−2b
φq T
0
hτfτ, uτ∇τ 1 d0
m−2
i1
ai ξi
0
hτfτ, uτ∇τ
T−t T
T
0
φq
s 0
hτfτ, uτ∇τ
Δs
T−t T
t
0
φq s
0
hτfτ, uτ∇τ
Δs
T−t T
t
0
φq s
0
hτfτ, uτ∇τ
Δs
> T−t T · c
M· t
0
φq s
0
hτ∇τ
Δsc.
3.9
Thus, conditioniinTheorem 3.1is satisfied.
Next, consideru ∈∂Pθ, b. In such a case, we haveγu θu maxt∈t,TTut ut b, which implies that 0utbfort∈t, TT. Analogously, it follows from3.7 that, for allu∈ P,
u T
T−tγu T
T−tb. 3.10
Therefore, we obtain 0utT/T−tbfort∈0, TT. This, combined with assumption C2, givesft, ut< φpb/Nfor allt∈0, TT. Thus, we have
θAu max
t∈t,TTAut AutAu0
T
0
φq
s
0
hτfτ, uτ∇τ 1
d0
m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
Δs
βB0
φq
T
0
hτfτ, uτ∇τ 1 d0
m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
−m−2
i1
B
φq
ξi
0
ft, uτ∇τ 1
d0 m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
T
0
φq 1
d0 T
0
hτfτ, uτ∇τ
Δs βbφq 1
d0 T
0
hτfτ, uτ∇τ
< b N
T
0
φq
1 d0
T
0
hτ∇τ
Δs βbφq
1 d0
T
0
hτ∇τ
b T βb
N ·φq
1 d0
T
0
hτ∇τ
b,
3.11
which consequently implies the validity of conditioniiinTheorem 3.1.
Finally, notice that the constant functions 1/2a ∈ Pα, a, so that Pα, a/∅. Let u ∈ ∂Pα, a. Then, we getαu maxt∈t,TTut ut a. This with assumptionC3 implies that 0utaandft, u> φpa/Lfor allt∈t, TT. Similarly, we have
αAu AutT−t
T Au0
T−t T
T
0
φq s
0
hτfτ, uτ∇τ 1 d0
m−2
i1
ai ξi
0
hτfτ, uτ∇τ
Δs
βφq
T
0
hτfτ, uτ∇τ 1
d0 m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
−m−2
i1
biφq ξi
0
ft, uτ∇τ 1
d0
m−2
i1
ai ξi
0
hτfτ, uτ∇τ
T−t T
T
t
φq s
0
hτfτ, uτ∇τ
Δs
T−t T
T
t
φq s
t
hτfτ, uτ∇τ
Δs
> a L ·T−t
T ·
T
t
φq s
t
hτ∇τ
Δsa.
3.12
Indeed, the validity of conditioniiiinTheorem 3.1is verified.
According to Theorem 3.1, we consequently approach the conclusion that the boundary value problems1.9and1.10possess at least two positive solutions, denoted by u1andu2, satisfyinga < αu1withθu1< bandb < θu2withγu2< c, respectively.
4. At Least Three Positive Solutions in Boundary Value Problems
In this section, we are to prove the existence of at least three positive solutions in the boundary value problems1.9and 1.10by using the five functionals fixed point theorem which is attributed to Avery35.
Letγ, β, θbe nonnegative continuous convex functionals onP.αandψare supposed to be nonnegative continuous concave functionals onP. Thus, for nonnegative real numbers h,a,b,c, andd, define five convex sets, respectively, by
P γ, c
x∈ P |γx< c , P
γ, α, a, c
x∈ P |aαx, γxc , Q
γ, β, d, c
x∈ P |βxd, γxc , P
γ, θ, α, a, b, c
x∈ P |aαx, θxb, γxc , Q
γ, β, ψ, h, d, c
x∈ P |hψx, βxd, γxc .
4.1
Theorem 4.1 see 35. LetP be a cone in a real Banach space E. Suppose that α and ψ are nonnegative continuous concave functionals onP, and thatγ,β, andθare nonnegative continuous convex functionals onPsuch that, for some positive numberscandM,
αxβx, xMγx, 4.2
for all x ∈ Pγ, c. In addition, suppose that A : Pγ, c → Pγ, cis a completely continuous operator and that there exist nonnegative real numbersh, d, a, bwith 0< d < asuch that
i{x∈ Pγ, θ, α, a, b, c|αx> a}/∅andαAx> aforx∈ Pγ, θ, α, a, b, c;
ii{x∈ Qγ, β, ψ, h, d, c|βx< d}/∅andβAx< dforx∈ Qγ, β, ψ, h, d, c;
iiiαAx> aforx∈ Pγ, α, a, cwithθAx> b;
ivβAx< dforx∈ Qγ, β, d, cwithψAx< h.
Then the operator Aadmits at least three fixed points x1,x2,x3 ∈ Pγ, c satisfying βx1 < d, a < αx2, andd < βx3withαx3< a, respectively.
With this theorem, we are now in a position to establish the following result on the existence of at least three solutions in the boundary value problems1.9and1.10.
Theorem 4.2. Suppose that the hypotheses H1–H3 are all fulfilled. Assume that there exist positive real numbersa,b,csuch that
0< a < b < c, a <T−t
T b < T−tT−t
T2 c, Nb < Mc. 4.3
Also assume thatfsatisfies the following conditions:
C1ft, u< φpc/Nfort∈0, TTandu∈0,T/T−tc;
C2ft, u> φpb/Mfort∈0, tTandu∈b,T2/T−t2b;
C3ft, u< φpa/L0fort∈0, TTandu∈0,T/T−ta.
Then, the boundary value problems1.9and1.10admit at least three solutionsu1t,u2t, and u3t, defined on0, σTT, satisfying, respectively,
t∈tmax,TTu1t< a, b < min
t∈0,tTu2t, a < max
t∈t,TTu3t with min
t∈0,tTu3t< b. 4.4
Proof. Let the coneP be as constructed in 2.16 and the operatorA as defined in 2.18.
Define, respectively, the nonnegative continuous concave functionals on thePas follows:
γu θu max
t∈t,TT
ut ut,
αu min
t∈0,tT
ut ut,
βu max
t∈t,TT
ut ut,
ψu min
t∈0,tT
ut ut.
4.5
Thus, we getαu βuforu∈ P. Moreover, fromLemma 2.3, it follows that
u T
T−tγu, 4.6
foru∈ P. Next, we intend to verify that all the conditions inTheorem 4.1hold with respect to the operatorA.
To this end, arbitrarily pick up a functionu ∈ Pγ, c. Then,γu maxt∈t,TTut ut c, which, combined with4.6, implies that 0 ut T/T−tcfort ∈ 0, TT andu ∈ P. Thus, we haveft, ut < φpc/Nfor t ∈ 0, TT, owing to assumptionC1. Moreover, sinceAu∈ P, we have
γAu AutAu0
T
0
φq s
0
hτfτ, uτ∇τ 1 d0
m−2
i1
ai ξi
0
hτfτ, uτ∇τ
Δs
βB0
φq
T 0
hτfτ, uτ∇τ 1
d0
m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
−m−2
i1
B
φq ξi
0
ft, uτ∇τ 1
d0 m−2
i1
ai ξi
0
hτfτ, uτ∇τ
T
0
φq 1
d0
T
0
hτfτ, uτ∇τ
Δs βbφq 1
d0
T
0
hτfτ, uτ∇τ
< c N
T
0
φq
1 d0
T
0
hτ∇τ
Δs βbφq
1 d0
T
0
hτ∇τ
c T βb
N ·φq
1 d0
T
0
hτ∇τ
c.
4.7
This, withLemma 2.4, clearly manifests that the operatorA:Pγ, c→ Pγ, cis completely continuous.
Moreover, the set
u∈ P
γ, θ, α, b, T T−tb, c
|αu> b
4.8
is not empty, because the constant functionut ≡ 2T−t/2T −tbbelongs to the set {u∈ Pγ, θ, α, b,T/T−tb, c|αu> b}. Analogously, the set
u∈ Q
γ, β, ψ,T−t T a, a, c
|βu< a
4.9
is nonempty, sinceut≡T t/2Ta∈ {u∈ Qγ, β, ψ,T−t/Ta, a, c|βu< a}. For particularu∈ Pγ, θ, α, b,T/T−tb, c, a utilization of4.6produces
b min
t∈0,tT
ut utut T
T−tγu T
T−tθu T2
T−t2b, 4.10
fort∈0, tT. According to assumptionC2, we thus obtain
ft, ut> φp
b M
, 4.11
for allt∈0, tT. Hence, it follows from4.11andLemma 2.3that αAu Aut T−t
T Au0
T−t
T T
0
φq
s
0
hτfτ, uτ∇τ 1
d0 m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
Δs
βB0
φq T
0
hτfτ, uτ∇τ 1 d0
m−2
i1
ai ξi
0
hτfτ, uτ∇τ
−m−2
i1
B
φq
ξi
0
ft, uτ∇τ 1
d0 m−2
i1
ai
ξi
0
hτfτ, uτ∇τ
T−t
T t
0
φq
s 0
hτfτ, uτ∇τ
Δs
> b
M·T−t
T ·
t
0
φq
s
0
hτ∇τ
Δsb.
4.12
This definitely verifies the validity of conditioniinTheorem 4.1.
Next, let us consideru∈ Qγ, β, ψ,T−t/Ta, a, c. In this case, we get
0ut T
T−ta, 4.13
for t ∈ 0, TT. Thus, an adoption of the assumption C3 yields ft, ut < φpa/L0. Furthermore, we have
βAu Aut
T
t
φq
s
0
hτfτ, uτ∇τ 1
d0 m−2
i1
ai
ξi
0
hτft, uτ∇τ
Δs
βB0
φq T
0
hτfτ, uτ∇τ 1
d0
m−2
i1
ai ξi
0
hτfτ, uτ∇τ
−m−2
i1
B
φq ξi
0
hτfτ, uτ∇τ 1 d0
m−2
i1
ai ξi
0
hτfτ, uτ∇τ
≤
T−t βb−m−2b φq
1 d0
T
0
hτfτ, uτ∇τ
< a L0
T−t βb−m−2b
·φq
1 d0
T
0
hτ∇τ
a.
4.14
Accordingly, the validity of conditioniiinTheorem 4.1is verified.
Aside from conditionsiandii, we are finally to verify the validity of conditions iii and iv. For this purpose, on the one hand, consider u ∈ Pγ, α, b, cwith θAu >
T/T−tb. Thus, we have
αAu AutAut θAu> T
T−tb > b. 4.15
On the other hand, consideru∈ Qγ, β, a, cwithψAu<T −t/Ta. In such a case, we obtain that
βAu AutT−t
T−tAut T−t
T−tψAu< T−t
T a < a. 4.16
Therefore, both conditionsiiiandivinTheorem 4.1are satisfied. Consequently, by virtue of Theorem 4.1, the boundary value problems 1.9and 1.10 have at least three positive solutions circumscribed on0, σTTsatisfying maxt∈t,TTu1t< a,b <mint∈0,tTu2t, and a <maxt∈t,TTu3twith mint∈0,tTu3t< b.
5. A Specific Example
In this section, we provide a representative and nontrivial example to clearly illustrate the feasibility of the time-scale-dependent results of dynamic equations with boundary value conditions that are obtained in the preceding section.
Construct a nontrivial time-scale set as
T
1− 1
2 N0
∪1,2∪ {T}. 5.1
Take all the parameters in problems1.9and1.10as follows:T 2, 2< T 3,p 3/2, q3,m4,a1a2 1/4,b1/2,b1,β6,ξ1 1/2,ξ2 1,t1, andt1/2, so that d01/2. For simplicity but without loss of generality, setht≡1. Also notice that there exist countable right-scattered pointsti1−1/2i, i0,1,2, . . .. Then, it is easy to validate the condition
βb−m−2b−μT0, 5.2
which is dependent on the time scale property around the time instant T. Furthermore, implementing the integral formula38:
b
a
fsΔs
b
a
fsds
ti∈a,bT
σti
ti
fti−fs
ds, 5.3
we concretely obtain that
M T−t T
t
0
φq s
0
hτ∇τ
Δs
1
0
s2Δs 1
0
s2ds ∞ i1
σti
ti
t2i −s2
ds 5 21,
N
T βb
·φq 1
d0 T
0
hτ∇τ
128,
L0
T−t βb−m−2b
·φq 1
d0
T
0
hτ∇τ
104.
5.4
Particularly, take the function in dynamic equation as
ft, u 23u2
16 t u u2, t∈0,2T, u0. 5.5
This kind of function is omnipresent in the mathematical modeling of biological or chemical processes. Then it allows us to properly set the other parameters asa1/416,b105, and c1078N. It is evident that these parameters satisfy
0< a < T−t
T b < T−tT−t
T2 c, Nb < Mc. 5.6
Now, we can verify the validity of conditions C1–C3 in Theorem 4.2. Indeed, direct computations yield:
ft, u<23c N
1/2
φp
c N
, 5.7
ast∈0, TTandu∈0,2c,
ft, u 23b2
16 t b b2 >21φp b
M
, 5.8
ast∈0, tTandu∈b,4b, and
ft, u23u2
16 23a2<2a 1 112 φp
a L0
, 5.9
ast∈0, TTandu∈0,4a. Hence, conditionsC1–C3inTheorem 4.2are satisfied for the above specified functions and parameters. Therefore, in the light ofTheorem 4.2, we conclude that the dynamic equation on the specified time scales
uΔ1/2∇
23u2
16 t u u2 0, t∈0,2T, 5.10
with the boundary value conditions
uΔ01/2 1
4
uΔ 1
2
1/2 1 4
uΔ11/2 , u2 6uΔ2 1
2uΔ 1
2 1
2uΔ1,
5.11
has at least three positive solutions defined on 0, TTsatisfying maxt∈t,TTu1t < a,b <
mint∈0,tTu2t, anda <maxt∈t,TTu3twith mint∈0,tTu3t< b.
6. Concluding Remarks
In this paper, some novel and time-scale-dependent sufficient conditions are established for the existence of multiple positive solutions in a specific kind of boundary value problems
on time scales. This kind of boundary value problems not only includes the problems discussed in the literature but also is adapted to more general cases. The well-known Avery- Henderson fixed point theorem and the five functionals fixed point theorem are adopted in the arguments.
It is valuable to mention that the writing form of the ending point of the interval on time scales should be accurately specified in dealing with different kind of boundary value conditions. Any inaccurate expression may lead to a problematic or incomplete discussion. Also it is noted that some other fixed point theorems and degree theories may be adapted to dealing with various boundary value problems on time scales. In addition, future directions for further generalization of the boundary value problem on time scales may include the generalization of thep-Laplacian operator to increasing homeomorphism and homeomorphism, which has been investigated in39for the nonlinear boundary value of ordinary differential equations; the allowance of the functionf to change sign, which has been discussed in31and needs more detailed and rigorous investigations.
Acknowledgments
This paper was supported by the NNSF of ChinaGrants nos. 10501008 and 60874121and by the Rising-Star Program Foundation of Shanghai, ChinaGrant no. 07QA14002. The authors are grateful to the referee and editors for their very helpful suggestions and comments.
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