Existence of Positive Solutions for m-Point Boundary Value Problems on Time Scales

Volume 2009, Article ID 189768,12pages doi:10.1155/2009/189768

Research Article

Existence of Positive Solutions for m-Point Boundary Value Problems on Time Scales

Ying Zhang and ShiDong Qiao

Department of Mathematics, Shanxi Datong University, Datong, Shanxi 037009, China

Correspondence should be addressed to Ying Zhang,[email protected] Received 27 August 2008; Revised 24 November 2008; Accepted 14 January 2009 Recommended by Binggen Zhang

We study the one-dimensional p-Laplacian m-point boundary value problem ϕpu^Δt^Δ atft, ut 0,t∈0,1T,u0 0,u1 _m−2

i1 aiuξi, whereTis a time scale,ϕps |s|^p−2s, p > 1, some new results are obtained for the existence of at least one, two, and three positive solution/solutions of the above problem by using Krasnoselsklls fixed point theorem, new fixed point theorem due to Avery and Henderson, as well as Leggett-Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensionalp-Laplacianm- point boundary value problem on time scales has been studied.

Copyrightq2009 Y. Zhang and S. Qiao. 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.

1. Introduction

With the development of p-Laplacian dynamic equations and theory of time scales, a few authors focused their interest on the study of boundary value problems for p-Laplacian dynamic equations on time scales. The readers are referred to the paper1–7.

In 2005, He1considered the following boundary value problems:

ϕp

u^Δt_∇

atfut 0, t∈0, T_T, u0−B₀u^Δη 0, u^ΔT 0 or

u^Δ0 0, uT B₁u^Δη 0,

1.1

whereT is a time scales,ϕps |s|^p−2s, p >1, η∈0, ρt_T.The author showed the existence of at least two positive solutions by way of a new double fixed point theorem.

In 2004, Anderson et al. 2used the virtue of the fixed point theorem of cone and obtained the existence of at least one solution of the boundary value problem:

g

u^Δt_∇

ctfu 0, a < t < b,

ua−B₀u^Δγ 0, u^Δb 0. 1.2

In 2007, Geng and Zhu3used the Avery-Peterson and another fixed theorem of cone and obtained the existence of three positive solutions of the boundary value problem:

ϕ_p

u^Δt_∇

atfut 0, t∈0, T_T,

u0−B₀u^Δη 0, u^ΔT 0. 1.3

Also, in 2007, Sun and Li4discussed the existence of at least one, two or three positive solutions of the following boundary value problem:

ϕ_p

u^Δt_Δ

htfu^σt 0, t∈a, b_T,

ua−B0u^Δa 0, u^Δσb 0. 1.4

In this paper, we are concerned with the existence of multiple positive solutions to the m-point boundary value problem for the one dimension p-Laplcaian dynamic equation on time scaleT

ϕ_p

u^Δt_Δ

atft, ut 0, t∈0,1_T, u0 0, u1 ^m−2

i1

aiuξi, 1.5

where T is a time scale, ϕ_ps |s|^p−2s, p > 1,0 < ξ₁ < ξ₂ < · · · < ξ_m−2 < 1,0 ≤ a_i, i 1,2, . . . , m−3, a_m−2>0,and

H1_m−2

i1 a_iξ_i<1;

H2f ∈C_rd0,1_T×0,∞,0,∞;

H3a∈Crd0,1_T,0,∞and there existst0∈ξm−2,1such that at0>0.

In this paper, we have organized the paper as follows. InSection 2, we give some lemmas which are needed later. InSection 3, we apply the Krassnoselskiifs 8fixed point theorem to prove the existence of at least one positive solution to the MBVP1.5. InSection 4, conditions for the existence of at least two positive solutions to the MBVP1.5are discussed by using Avery and Henderson9fixed point theorem. InSection 5, to prove the existence of at least three positive solutions to the MBVP 1.5 are discussed by using Leggett and Williams10fixed point theorem.

For completeness, we introduce the following concepts and properties on time scales.

A time scaleTis a nonempty closed subset ofR, assume thatT has the topology that it inherits from the standards topology onR.

Definition 1.1. LetT be a time scale, fort ∈ T, one defines the forward jump operator σ : T → T byσt inf{s ∈ T : s > t}, and the backward jump operator ρ : T → T by ρt sup{s ∈ T : s < t},while the graininess function μ : T → 0,∞is defined by μt σt−t. If σt > t,one says that is right-scattered, while ifρt < t,one says that tis left-scattered. Also, ift <supT andσt t,thentis called right-dense, and ift >infT andρt t,thentis called left-dense. One also needs below the setT^k as follows: ifT has a left-scattered maximumm,thenT^k T−m,otherwiseT^k T.For instance, if supT ∞, thenT^kT.

Definition 1.2. Assumef :T → Ris a function and lett∈T.Then , one definesf^Δtto be the numberprovided it existswith the property that any givenε >0,there is a neighborhood Uoftsuch that

f

σt

−fs

−f^Δtσt−s≤εσt−s, 1.6 for alls∈U.One says thatfis delta differentiableor in short: differentiableonT provided f^Δtexist for allt∈T.

IfTR,thenf^Δt ft,ifT Z,thenf^Δt Δft.

A functionf :T → R.

iIffis continuous , thenfis rd-continuous.

iiThe jump operatorσis rd-continuous.

iiiIffis rd-continuous, then so isf^σ.

A functionF :T → Ris called an antidervative off :T → R, providedF^Δt ft holds for allt∈T^k. One defines the definite integral by

b

aftΔtFb−Fa. 1.7

For alla, b∈T. Iff^Δt≥0,thenfis nondecreasing.

2. The Preliminary Lemmas

Lemma 2.1see5,6. Assume that (H1)–(H₃) hold. Thenutis a solution of the MBVP1.5on 0,1_Tif and only if

ut − ^t

0

ϕ_q ^s

0

aτf τ, uτ

Δτ

Δs

−t· _m−2

i1 a_iξi

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i t·

₁

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

,

2.1

whereϕ_qs |s|^q−2s,1/p 1/q 1,andq >1.

Lemma 2.2. Assume that conditions (H1)–(H3) are satisfied, then the solution of the MBVP1.5on 0,1_Tsatisfies

ut≥0, t∈0,1_T. 2.2

Lemma 2.3see5. If the conditions (H1)–(H₃) are satisfied, then

ut≥γ u , t∈ ξ_m−2,1

, 2.3

where

u sup

t∈0,1T

ut, γ min

a_m−21−ξ_m−2

1−a_m−2ξ_m−2 , a_m−2ξ_m−2, ξ₁

.

2.4

Lemma 2.4see6. mint∈ξm−2,1Aut min{Au1, Auξm−2}.

LetEdenote the Banach spaceCrd0,1_Twith the norm u sup_t∈0,1T|ut|.

Define the coneP ⊂E,by

P

u∈E|ut≥0, t∈ ξ_m−2,1

minut≥γ u , uis concave

. 2.5

The solutions of MBVP1.5are the points of the operatorAdefined by

Aut − ^t

0

ϕq s 0

aτf τ, uτ

Δτ

Δs

−t· _m−2

i1 ai_ξi

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

t· ₁

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi ut.

2.6

So,AP ⊂P.It is easy to check thatA:P → Pis completely continuous.

3. Existence of at least One Positive Solutions

Theorem 3.1see8. LetEbe a Banach space, and letP ⊂ Ebe a cone. AssumeΩ1andΩ2 are open boundary subsets ofEwith 0∈Ω1,Ω1⊂Ω2,and letA:P∩Ω2\Ω1 → Pbe a completely continuous operator such that either

i Au ≤ u foru∈P∩∂Ω1, Au ≥ u foru∈P∩∂Ω2; or ii Au ≥ u foru∈P∩∂Ω1, Au ≤ u foru∈P∩∂Ω2hold.

Then A has a fixed point inP∩Ω2\Ω1.

Theorem 3.2. Assume conditions (H1)–(H3) are satisfied. In addition, suppose there exist numbers 0 < r < R < ∞ such thatft, u ≤ ϕ_pmϕpr,if t ∈ 0, σ1,0 ≤ u ≤ r, and ft, u ≥ ϕpMγϕpR,ift∈ξm−2,1, R≤u <∞,where

M 1−_m−2

i1 aiξi

γξ_m−21

ξ_m−2ϕq_s

ξ_m−2aτΔτΔs, m 1−_m−2

i1 aiξi

₁

0ϕ_qs

0aτΔτΔs.

3.1

Then the MBVP1.5has at least one positive solution.

Proof. Define the conePas in2.5, define a completely continuous integral operatorA:P → Pby

Aut − ^t

0

ϕq s 0

aτf τ, uτ

Δτ

Δs

−t· _m−2

i1 ai_ξi

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

t· ₁

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

.

3.2

FromH1–H3, Lemmas2.1and2.2,AP⊂P. Ifu∈Pwith u r,then we get

Aut − ^t

0

ϕ_q ^s

0

aτf τ, uτ

Δτ

Δs

−t· _m−2

i1 ai_ξi

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i t·

₁

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

≤t· ₁

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

≤ϕ_qϕpmϕpr· ₁

0ϕ_qs

0aτΔτΔs 1−_m−2

i1 a_iξ_i

≤rm· ₁

0ϕ_qs

0aτΔτΔs 1−_m−2

i1 a_iξ_i r u .

3.3

This implies that Au ≤ u .So, if we setΩ1{u∈C_rd0,1| u < r},then Au ≤ u , foru∈P∩∂Ω1.

Let us now setΩ2{u∈C_rd0,1| u < R}.

Then for u ∈ P with u < R, by Lemma 2.4we have ut ≥ γ u , t ∈ ξm−2,1.

Therefore, we have Aut≥Au

ξ_m−2 − ^ξm−2

0

ϕq s 0

aτf

τ, uτ Δτ

Δs

−ξ_{m−2 m−2}

i1 a_iξi

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

ξ_m−2· ₁

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

ξ_m−21

0ϕq_s

0aτfτ, uτΔτΔs−_ξm−2

0 ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i 1

1−_m−2

i1 aiξi m−2

i1

ai

ξi

ξ_m−2 0

ϕq s 0

aτf τ, uτ

Δτ

Δs

−ξ_m−2 ^ξi

0

ϕq s 0

aτf

τ, uτ Δτ

Δs

≥ ξ_m−21

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

− _ξm−2

0 ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

≥ ξ_m−21

ξ_m−2ϕq_s

ξ_m−2aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

≥ϕ_q

ϕ_pMγϕpRξ_m−21

ξ_m−2ϕq_s

ξ_m−2aτΔτΔs 1−_m−2

i1 a_iξ_i

≥ MγR

1−_m−2

i1 aiξi

·ξ_m−2 ¹

ξm−2

ϕ_q ^s

ξm−2

aτΔτ

Δs u .

3.4

Hence, Au ≥ u for u ∈ P ∩∂Ω2. Thus by the Theorem 3.1, A has a fixed point u in P∩Ω2\Ω1.Therefore, the MBVP1.5has at least one positive solution.

4. Existence of at least Two Positive Solutions

In this section, we apply the Avery-Henderson fixed point theorem9to prove the existence of at least two positive solutions to the nonlinear MBVP1.5.

Theorem 4.1see Avery and Henderson9. LetPbe a cone in a real Banach spaceE.Set

P Φ, ρ3

u∈P|Φu< ρ3

. 4.1

If ν and Φ are increasing, nonnegative continuous functionals on P, let θ be a nonnegative continuous functional on P withθ0 0 such that, for some positive constants ρ₃andM >0,Φu≤θu≤νuand u ≤MΦu,for allu∈PΦ, ρ3. Suppose that there exist positive numbersρ₁< ρ₂< ρ₃such thatθλu λθufor all 0≤λ≤1 andu∈∂Pθ, ρ2.

IfA:PΦ, ρ3 → P is a completely continuous operator satisfying i ΦAu> ρ₃for allu∈∂PΦ, ρ3;

iiθAu< ρ₂for allu∈∂Pθ, ρ2;

iiiPν, ρ1/φandνAu> ρ1for allu∈∂Pν, ρ1,thenAhas at least two fixed points u₁andu₂such thatρ₁ < νu1withθu1< ρ₂andρ₃<u2withΦu2< ρ₃. Letl∈0,1_Tand 0< ξ_m−2 < l <1.Define the increasing, nonnegative and continuous functionalsΦ, θ,andνonP,byΦu uξm−2, θu uξm−2,andνu ul.

FromLemma 2.4, for eachu∈P,Φu θu≤νu.

In addition, for eachu∈P,Lemma 2.3impliesΦu uξm−2≥γ u . Thus,

u < 1

γΦu, ∀u∈P. 4.2

We also see thatθ0 0 andθλu λθufor all 0≤λ≤1 andu∈∂Pθ, q.

Theorem 4.2. Assume (H1)–(H₃) hold, suppose there exist positive numbersρ₁ < ρ₂< ρ₃,such that the functionfsatisfies the following conditions:

B1ft, u> ϕ_pmγϕpρ1,fort∈ξ_m−2, landu∈γρ1, ρ₁; B2ft, u< ϕ_pmϕpρ2,fort∈ξ_m−2,1andu∈0, ρ2;

B3ft, u> ϕpMγϕpρ3,fort∈ξm−2, landu∈ρ3,1/γρ3.

Then the MBVP1.5has at least two positive solutionsu₁andu₂such thatu₁t> ρ₁ withu₁l< ρ₂andu₂l> ρ₂withu₂l< ρ₃.

Proof. We now verify that all of the conditions ofTheorem 4.1are satisfied.

Define the cone P as2.5, define a completely continuous integral operatorA:P → Pby

Aut − ^t

0

ϕ_q ^s

0

aτf τ, uτ

Δτ

Δs

−t· _m−2

i1 a_iξi

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i t·

₁

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i .

4.3

Mandmas in3.1. To verify that conditioniofTheorem 4.1holds, we chooseu∈

∂PΦ, ρ3, thenΦu ρ₃.This impliesρ₃ ≤ u ≤ 1/γΦu. Note that u ≤1/γΦu

1/γρ3.We haveρ3 ≤ ut≤ 1/γρ3,fort∈ ξm−2,1_T.As a consequence ofB3,ft, u >

ϕ_pMγϕpρ3,fort∈ξ_m−2, l_T.SinceAu∈P,we have fromLemma 2.2, ΦAu Au

ξ_m−2 − ^ξm−2

0

ϕ_q ^s

0

aτf τ, uτ

Δτ

Δs

−ξ_m−2· _m−2

i1 a_iξi

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

ξ_m−2· ₁

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

ξ_m−21

0ϕq_s

0aτfτ, uτΔτΔs−_ξm−2

0 ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i 1

1−^m−2

i1a_iξ_i

m−2

i1

ai

ξi

ξm−2

0

ϕq s 0

aτf

τ, uτ Δτ

Δs

−ξ_m−2 ^ξi

0

ϕ_q ^s

0

aτf τ, uτ

Δτ

Δs

≥ ξ_m−21

0ϕ_qs

0aτfτ, uτΔτΔs−_ξm−2

0 ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

≥ ξ_m−21

ξm−2ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

≥ϕq

ϕpMγϕp ρ3

ξ_m−21

ξm−2ϕ_qs

0aτΔτΔs 1−_m−2

i1 aiξi

≥ Mγρ₃ξ_m−2 1−_m−2

i1 a_iξ_i

1

ξ_m−2ϕ_q ^s

0

aτΔτ

Δs≥ρ₃.

4.4

Then conditioniofTheorem 4.1holds.

Letu ∈ ∂Pθ, ρ2. Thenθu ρ2. This implies 0 ≤ ut ≤ u ≤ 1/γρ2,fort ∈ ξ_m−2,1.FromB2, we have

θAu Au

ξ_m−2

≤ξ_m−2· ₁

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

≤ξ_m−2· ₁

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

≤mρ2· ₁

0ϕ_qs

0aτΔτΔs 1−_m−2

i1 aiξi ρ2 u .

4.5

Hence conditioniiofTheorem 4.1holds.

If we first defineut ρ1/2,fort∈0,1_T, thenνu ρ1/2< ρ1.SoPν, ρ1/φ.

Now, letu∈∂Pν, ρ1,thenνu ul ρ₁.This mean thatρ₁/γ ≤ut≤ u ≤ ρ₁. FromB1andLemma 2.4, we get

νAu Aul≥Au

ξ_m−2 − ^ξm−2

0

ϕq s 0

aτf τ, uτ

Δτ

Δs

−ξ_m−2· _m−2

i1 a_iξi

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

ξ_m−2· ₁

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

ξ_m−21

0ϕq_s

0aτfτ, uτΔτΔs−_ξm−2

0 ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i 1

1−_m−2

i1 aiξi m−2

i1

ai

ξi

ξ_m−2 0

ϕq s 0

aτf τ, uτ

Δτ

Δs

−ξ_m−2 ^ξi

0

ϕq s 0

aτf τ, uτ

Δτ

Δs

≥ ξ_m−21

0ϕq_s

0aτfτ, uτΔτΔs−_ξm−2

0 ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

≥ ξ_m−21

ξ_m−2ϕq_s

ξ_m−2aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

≥ϕ_q

ϕ_pmγϕp

ρ₁ξ_m−21

ξ_m−2ϕq_s

ξ_m−2aτΔτΔs 1−_m−2

i1 a_iξ_i mγρ1

1−_m−2

i1 aiξi

·ξ_m−2 ¹

ξm−2

ϕ_q ^s

ξm−2

aτΔτ

Δs≥ρ₁.

4.6

Then conditioniiiofTheorem 4.1holds.

Since all conditions of Theorem 4.1 are satisfied, the MBVP 1.5 has at least two positive solutions u₁ and u₂ such that u₁t > ρ₁ with u₁l < ρ₂ and u₂l > ρ₂ with u2l< ρ3.

5. Existence of at least Three Positive Solutions

We will use the Leggett-Williams fixed point theorem10to prove the existence of at least three positive solutions to the nonlinear MBVP1.5.

Theorem 5.1see Leggett and Williams10. LetPbe a cone in the real Banach spaceE.Set Pr

x∈P | x < r , PΨ, a, b

x∈P |a≤Ψx, x ≤b

. 5.1

Suppose A : Pr → Pr be a completely continuous operator and be a nonnegative continuous concave functional onP withΨu ≤ u for allu ∈Pr. If there exists 0< ρ₁ <

ρ2<1/γρ2≤ρ3such that the following condition hold:

i{u∈PΨ, ρ2,1/γρ2|Ψu> ρ₂}/φandΨAu> ρ₂for allu∈PΨ, ρ2,1/γρ2; ii Au < ρ₁for u ≤ρ₁;

iii ΨAu> ρ₂foru∈PΨ, ρ2,1/γρ2with Au >1/γρ2,thenAhas at least three fixed pointsu₁, u₂ andu₃ in Pr satisfying u1 < ρ₁,Ψu2 > ρ₂, ρ₁ < u3 with Ψu2< ρ₂.

Theorem 5.2. Assume (H1)–(H3) hold . Suppose that there exist constants 0< ρ1< ρ2 <1/γρ2 ≤ ρ₃such that

C1ft, u≤ϕ_pmϕpρ3,fort∈ξ_m−2, landu∈0, ρ3;

C2ft, u> ϕpMγϕpρ2,fort∈ξm−2, landu∈ρ2,1/γρ2; C3ft, u< ϕpmϕpρ1,fort∈ξm−2,1andu∈0, ρ1.

Then the MBVP 1.5 has at least three positive solutions u₁, u₂,and u₃ such that u₁ξ< ρ₁, u₂l> ρ₂, u₃ξ> ρ₁withu₃l< ρ₂.

Proof. The conditions ofTheorem 5.1will be shown to be satisfied. Define the nonnegative continuous concave functionalΨ:P → 0,∞to beΨu uξm−2,the coneP as in2.5, Mandmas in3.1. We haveΨu≤ u for allu∈P.Ifu∈ P_ρ3, then u ≤ ρ₃,and from assumptionC1, then we have

Aut − ^t

0

ϕq s 0

aτf τ, uτ

Δτ

Δs

−t· _m−2

i1 ai_ξi

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i t·

₁

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

≤t· ₁

0ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

≤ϕ_q

ϕ_pmϕp ρ₃

· ₁

0ϕq_s

0aτΔτΔs 1−_m−2

i1 a_iξ_i

≤mρ₃· ₁

0ϕq_s

0aτΔτΔs

1−_m−2

i1 a_iξ_i ρ₃.

5.2

This implies that Au ≤ ρ3. Thus, we have A : Pρ3 → Pρ3. Since 1/γρ2 ∈ PΨ, ρ2,1/γρ2 and Ψ1/γρ2 1/γρ2 > ρ₂,{u ∈ PΨ, ρ2,1/γρ2|Ψu > ρ₂}/φ.

For u ∈ PΨ, ρ2,1/γρ2we haveρ2 ≤ uξm−2 ≤ u ≤ 1/γρ2.Using assumptionC2, ft, u> ϕ_pMγϕpρ2,we obtain

ΨAu Au ξ_m−2

− ^ξm−2

0

ϕ_q ^s

0

aτf τ, uτ

Δτ

Δs

−ξ_m−2· _m−2

i1 a_iξi

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

ξ_m−2· ₁

0ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

ξ_m−21

0ϕq_s

0aτfτ, uτΔτΔs−_ξm−2

0 ϕq_s

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

1 1−_m−2

i1 aiξi m−2

i1

a_i ξ_i ^ξm−2

0

ϕ_q ^s

0

aτf

τ, uτ Δτ

Δs

−ξ_m−2 ^ξi

0

ϕq s 0

aτf τ, uτ

Δτ

Δs

≥ ξ_m−21

0ϕ_qs

0aτfτ, uτΔτΔs−_ξm−2

0 ϕ_qs

0aτfτ, uτΔτΔs 1−_m−2

i1 aiξi

≥ ξ_m−21

ξ_m−2ϕq_s

ξ_m−2aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

≥ϕ_q

ϕ_pMγϕp

ρ₂ξ_m−21

ξ_m−2ϕq_s

ξ_m−2aτΔτΔs 1−_m−2

i1 a_iξ_i

≥ Mγρ2

1−_m−2

i1 aiξi ·ξ_m−2 ¹

ξm−2

ϕq s ξm−2

aτΔτ

Δs≥ρ2.

5.3

Hence, conditioniofTheorem 5.1holds.

If u ≤ρ1,from assumptionC3, we obtain

Aut≤ϕ_q

ϕ_pmϕp ρ₁

· ₁

0ϕ_qs

0aτΔτΔs

1−_m−2

i1 a_iξ_i

≤mρ₁· ₁

0ϕ_qs

0aτΔτΔs 1−_m−2

i1 a_iξ_i ρ₁.

5.4

This implies that Au ≤ρ1.

Consequently, conditioniiofTheorem 5.1holds.

We suppose thatu∈PΨ, ρ2, ρ3,with Au >1/γρ2.Then we get ΨAu Au

ξ_m−2

≥ ξ_m−21

ξm−2ϕq_s

ξm−2aτfτ, uτΔτΔs 1−_m−2

i1 a_iξ_i

≥ϕ_q

ϕ_pMγϕp

ρ₂ξ_m−21

ξm−2ϕ_qs

ξm−2aτΔτΔs 1−_m−2

i1 aiξi

≥ Mγρ₂ 1−_m−2

i1 aiξi ·ξ_m−2 ¹

ξm−2

ϕq s ξm−2

aτΔτ

Δs≥ρ₂.

5.5

Hence, conditioniiiofTheorem 5.1holds.

Because all of the hypotheses of the Leggett-Williams fixed point theorem are satisfied, the nonlinear MBVP1.5has at least three positive solutionsu₁, u₂,andu₃such thatu₁ξ<

ρ1, u2l> ρ2,andu3ξ> ρ1withu3l< ρ2.

Acknowledgment

This work is supported by the Research and Development Foundation of College of Shanxi Provinceno. 200811043.

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