p-Laplacian Generalized Sturm-Liouville Nonlocal Boundary Value Problems

Volume 2012, Article ID 831960,12pages doi:10.1155/2012/831960

Research Article

On the Nonhomogeneous Fourth-Order

p-Laplacian Generalized Sturm-Liouville Nonlocal Boundary Value Problems

Jian Liu

and Zengqin Zhao

1School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Shandong, Jinan 250014, China

2School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China

Correspondence should be addressed to Jian Liu,[email protected] Received 3 August 2012; Accepted 12 September 2012

Academic Editor: Yanbin Sang

Copyrightq2012 J. Liu and Z. Zhao. 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.

We study the nonlinear nonhomogeneous n-point generalized Sturm-Liouville fourth-order p- Laplacian boundary value problem by using Leray-Schauder nonlinear alternative and Leggett- Williams fixed-point theorem.

1. Introduction

In this paper, we prove the existence of one and multiple positive solutions of the following differential equations:

φ_p

ut

−k²φ_p ut

gtft, ut, t∈0,1, u0 φ_qa, u1 φ_qb,

αu0−βu0 ⁿ⁻²

i1

a_iuξi,

γu1 δu1 ⁿ⁻²

i1

biuξi,

1.1

whereφpisp-Laplacian operator, that is,φpu |u|^p−2u,p >1,φ_p⁻¹φq, 1/p1/q1m,k /0, α, β, γ, δ0,ξ_i∈0,1,a, b, a_i, b_i ∈0,∞ i1,2, . . . , n−2,f ∈C0,1×0,∞,0,∞, ft,0/≡0,gt∈C0,1,0,∞.

Recently, much attention has been paid to the existence of positive solutions for nonlocal nonlinear boundary value problems BVPs for short, see 1–4 and references therein. Such problems have potential applications in physics, biology, chemistry, and so forth. For example, a second-order three-point is used as a model for the membrane response of a spherical cap in nonlinear diffusion generated by nonlinear sources and in chemical reactor theory.

At the same time, the boundary value problems withp-Laplacian operator have been discussed extensively, for example, see1–3,5–7.

In1, Feng et al. researched the boundary value problem φp

u

t qtft, ut, 0t1, u0 ^m

i1

aiuξi, u1 ^m

i1

biuξi; 1.2

they obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity offby applying Krasnoselskii fixed-point theorem.

Zhou and Ma studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem withp-Laplacian operator in3:

φ_p u

t qtft, ut, 0t1, u0 ^m

i1

α_iuξi, u η

0, u1 ⁿ

i1

β_iuθi; 1.3

they established a corresponding iterative scheme for1.4by employing the monotone itera- tive technique.

We would also like to mention the work of Zhang and Liu in7, in which they con- sidered the existence of positive solutions for

φp

ut

ft, ut, 0< t <1, u0 ⁿ⁻²

i1

a_iuξi, u1 0, u0 ⁿ⁻²

i1

b_iuξi, u1 0,

1.4

by virtue of monotone iterative techniques, and they established a necessary and sufficient condition of positive solutions for their problem.

However, to the best of our knowledge, there are not many results concerning about the existence and multiple solutions of fourth-orderp-Laplacian generalized Sturm-Liouville n-point boundary value problems. In this paper, motivated by the study of4,8, we commit- ted to consider the fourth-orderp-Laplacian generalized Sturm-Liouville nonlocal boundary value problem without assuming any monotonicity condition on the nonlinearityf.

The rest of the paper is arranged as follows. We state some definitions and several preliminary results inSection 2that we will use in the sequel. Then inSection 3we present

the existence of one positive solution of BVP1.1by Leray-Schauder nonlinear alternative.

InSection 4we get three solutions by Leggett-Williams fixed-point theorem.

2. Preliminaries and Some Lemmas

The basic space used in this paper isEC0,1. It is well known thatEis a real Banach space with the normumax_t∈0,1|ut|.

Denote

ϕt βαt, ψt γδ−γt, t∈0,1, ραγβγαδ,

Δ

−ⁿ⁻²

i1a_iϕξi ρ−ⁿ⁻²

i1a_iψξi ρ−ⁿ⁻²

i1biϕξi −ⁿ⁻²

i1biψξi .

2.1

Definition 2.1. A functionuis said to be a solution of the boundary value problem1.1if u∈C²0,1satisfies1.1andφ_pu∈C²0,1. In addition,uis said to be a positive solution ifut>0 fort∈0,1, anduis a solution of BVP1.1.

Throughout the paper, we assume the following condition is satisfied:

H0ρ > 0, ρ−_n−2

i1 aiψξi > 0, ρ−_n−2

i1 biϕξi > 0,Δ < 0, ft, ut k²ab/

min_t∈0,1gt.

Letyt −φput, then BVP1.1is divided into the following two parts:

−yk²ygtft, ut, t∈0,1,

y0 a, y1 b, 2.2

uφ_q y

0, t∈0,1, αu0−βu0 ⁿ⁻²

i1

aiuξi, γu1 δu1 ⁿ⁻²

i1

biuξi. 2.3

It is not difficult that we can transform2.2into the following differential equations:

−yk²ygtft, ut−k²a1−t−k²bt, t∈0,1,

y0 0, y1 0. 2.4

By routine calculations we can get the following three Lemmas.

Lemma 2.2. The BVP2.4has a unique solution

yt ₁

0

G₁t, s

gsfs, us−k²a1−s−k²bs ds, 2.5

where

G₁t, s 1 ρ

⎧⎪

⎪⎨

⎪⎪

⎩

sinhks·sinhk1−t

ksinhk , 0st1, sinhkt·sinhk1−s

ksinhk 0ts1.

2.6

Lemma 2.3. The BVP2.3has a unique solution

ut ₁

0

G2t, sφq

ys

dsAϕt Bψt, 2.7

where

G2t, s 1 ρ

ψtϕs, 0st1, ψsϕt, 0ts1,

A 1 Δ

n−2

i1

a_{i 1}

0

G₂ξi, sφq

ys

ds ρ−ⁿ⁻²

i1

a_iψξi

n−2

i1

bi

₁

0

G2ξi, sφq

ys

ds −ⁿ⁻²

i1

aiψξi ,

B 1 Δ

−ⁿ⁻²

i1

a_iϕξi ⁿ⁻²

i1

a_{i 1}

0

G₂ξi, sφq

ys ds

ρ−ⁿ⁻²

i1

a_iϕξi ⁿ⁻²

i1

b_{i 1}

0

G₂ξi, sφq

ys ds

.

2.8

The proof ofLemma 2.3is similar to that of Lemma 5.5.1 in8, so we omit it here.

From Lemmas2.2and2.3we can get thatutis a solution of BVP1.1if and only if

ut ₁

0

G2t, sφq

₁

0

G1s, τ

gτfτ, uτ−k²a1−τ−k²bτ ds A

φq

y

ϕt B φq

y ψt,

2.9

where A

φ_q y

1 Δ

n−2 i1

ai

₁

0

G2ξi, sφq

₁

0

G1s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds ρ−ⁿ⁻²

i1

aiψξi

n−2

i1

bi

₁

0

G2ξi, sφq

₁

0

G1s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds −ⁿ⁻²

i1

aiψξi ,

B φq

y

1 Δ

−ⁿ⁻²

i1

a_iϕξi ⁿ⁻²

i1

a_{i 1}

0

G₂ξi, sφq

₁

0

G₁s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds

ρ−ⁿ⁻²

i1

a_iϕξi ⁿ⁻²

i1

b_{i 1}

0

G₂ξi, sφq

₁

0

G₁s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds .

2.10 Lemma 2.4. Consider,

G_it, sG_is, s, t, s∈0,1, i1,2, G1t, sΛ0G1s, s, t∈,1−, s∈0,1,

G₂t, sΛG2s, s, t∈,1−, s∈0,1,

2.11

where

Λ0 sinhk sinhk , ∈

0,1

2

, Λ min

ϕ

ϕ1,ψ1− ψ0

, ∈

0,1

2

.

2.12

Denote

Λ1maxϕ,ψ, Λ2min

t∈,1−min ϕt, min

t∈,1−ψt

, λmin

Λ,Λ2

Λ1

. 2.13

Lemma 2.5. Let K

u|u∈C0,1, ut0,∀t∈0,1, min

t∈,1−utλu

,

Tut ₁

0

G₂t, sφq

₁

0

G₁s, τ

gτfτ, uτ−k²a1−τ−k²bτ ds A

φ_q y

ϕt B φ_q

y ψt,

2.14

then

TK⊆K. 2.15

Proof. Firstly, we prove thatTut0. Forf ∈C0,1×0,∞,0,∞,gt∈C0,1,0,∞, then we can getφ_qy φ_q1

0G₁s, τgτfτ, uτ−k²a1−τ−k²bτdτ > 0, for all u∈ K. Furthermore, conditionH0leads toAφqy 0, andBφqy0, thus, we get Tut0.

Secondly, fort∈,1−, we can get

t∈,1−min Tut

min

t∈,1−

₁

0

G₂t, sφq

₁

0

G₁s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds

A φq

y

ϕt B φq

y ψt

Λ

₁

0

G₂s, sφq

₁

0

G₁s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds

A φq

y

ϕt B φq

y ψt

Λ

₁

0

G2s, sφq

₁

0

G1s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds

Λ2

Λ1Λ1

A φq

y B

φq

y

λ

₁

0

G₂s, sφq

₁

0

G₁s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds

Λ1

A φ_q

y B

φ_q

y

λTu.

2.16

Thus we can get that min_t∈,1−TutλTu, which meansTK⊆K.

We present here several definitions.

Given a coneKin a real Banach spaceE, a mapαis said to be a nonnegative continuous concaveresp., convexfunctional onKprovided thatα:K → 0,∞is continuous and

α

tx 1−ty

tαx 1−tα y

, resp., α

tx 1−ty

tαx 1−tα y

,

2.17

for all,x, y∈Kandt∈0,1.

Let 0< a < bbe given, and letαbe a nonnegative continuous concave functional on K. Define the convex setsP_randPα, a, bby

P_r {x∈K| x< r}, Pα, a, b {x∈K|aαx,xb}. 2.18

For the convenience of the reader, we present here the Leggett-Williams fixed-point theorem and the Leray-Schauder nonlinear alternative theorem.

Lemma 2.6see9, Leggett-Williams fixed-point theorem. LetA :Pc → Pcbe a completely continuous operator, and letαbe a nonnegative continuous concave functional onKsuch thatαx x for allx∈Pc. Suppose there exist 0< a < b < dcsuch that

A1{x∈Pα, b, d:αx> b}/∅, andαAx> bforx∈Pα, b, d;

A2Ax< aforxa;

A3αAx> bforx∈Pα, b, cwithAx> d.

ThenAhas at least three fixed pointsx1,x2, andx3and such thatx1< a,b < αx2andx3> a, withαx3< b.

Now we cite the Leray-Schauder nonlinear alternative.

Lemma 2.7see10. LetFbe a Banach space andΩa bounded open subset ofF, 0∈Ω.T :Ω → Fbe a completely continuous operator. Then, either there existsx∈∂Ω,λ >1 such thatTx λx, or there exists a fixed pointx^∗∈Ω.

3. Results of One Nontrivial Solution

In this section, we study the existence of one nontrivial solution of BVP 1.1 by Leray- Schauder nonlinear alternative.

Denote

H1φq

₁

0

G1τ, τgτpτdτ

,

H2φq

₁

0

G1τ, τgτrτdτ

,

A 1 Δ

n−2

i1

a_i ρ−ⁿ⁻²

i1

a_iψξi n−2

i1

bi −ⁿ⁻²

i1

aiψξi ,

B 1 Δ

−ⁿ⁻²

i1

a_iϕξi ⁿ⁻²

i1

a_i

ρ−ⁿ⁻²

i1

a_iϕξi ⁿ⁻²

i1

b_i ,

NM·2^q−1

1AΛ 1BΛ 1

, l NH2

1−NH₁, Ω {u∈C0,1,u< l}.

3.1

Theorem 3.1. AssumeNH₁ <1,ft,0/≡0, and there exist nonnegative functionsp, r ∈L¹0,1 such that|ft, u|pt|u|^p−1rt, a.e.t, u∈0,1×0,∞, then BVP1.1has a nontrivial solutionu^∗∈Ω.

Proof. If there exist two nonnegative functionsp, r ∈L¹0,1such that|ft, u|pt|u|^p−1 rt, a.e.t, u∈0,1×0,∞, we can get that

φq

₁

0

G1s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

φ_{q 1}

0

G₁s, τgτfτ, uτdτ

φq

1 0

G1τ, τgτ

pτ|u|^p−1rτ dτ

2^q−1

uφq

₁

0

G₁τ, τgτpτdτφ_{q 1}

0

G₁τ, τaτrτ

dτ

2^q−1uH1H₂,

3.2

thus, we get ₁

0

G2ξi, sφq

₁

0

G1s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds

₁

0

G₂ξi, sφq

₁

0

G₁s, τgτfτ, uτdτ

ds

M·2^q−1uH1H₂.

3.3

In the same way, we obtain

A φ_q

y

M·2^q−1uH1H₂A, B

φ_q y

M·2^q−1uH1H₂B.

3.4

Thus we have Tu max

0t1|Tut|

max

0t1

₁

0

G₂t, sφq

₁

0

G₁s, τ

gτfτ, uτ−k²a1−τ−k²bτ ds

A φq

y

ϕt B φq

y ψt

max

0t1

₁

0

G2t, sφq

1 0

G1s, τgτfτ, uτdτ

ds

A φq

y

ϕt B φq

y ψt

M·2^q−1uH1H2 M·2^q−1uH1H2

AΛ 1BΛ 1

M·2^q−1uH1H2

1AΛ 1BΛ 1

NuH1H2.

3.5

Suppose that there existsμ >1 such that

Tuμu, u∈∂Ω. 3.6

Therefore,

μlμuTul

NH1NH2

l

, 3.7

which leads toμ NH1NH2/l 1, and this contradicts μ > 1, then by Lemma 2.7,T has a fixed pointu^∗∈Ω; sinceft,0/≡0, the BVP1.1has a nontrivial solutionu^∗∈Ω. This completes the proof ofTheorem 3.1.

4. Results of Multiple Positive Solutions

In the following parts, we will study the existence of multiple positive solutions of BVP1.1 by using Leggett-Williams fixed-point theorem.

Denote

Pc{u∈K| u< c}. 4.1

Define the nonnegative continuous concave functional onKby

αu min

t1−ut. 4.2

It is obvious that for eachu∈K, αuu.

LetM max_0t11

0G2t, sds,m ₁₋

G2s, sds,h φq₁₋

k²abG1τ, τdτ, andA, B, Λ,Λ0,Λ1be defined in Sections2and3.

We list the following three hypotheses:

H1ft, u< φpc/M1 Λ1A Λ1B/ ₁

0G1τ, τgτdτ, for allt∈0,1, 0uc;

H2ft, u< φ_pa/M1 Λ1A Λ1B/ ₁

0G₁τ, τgτdτ, for allt∈0,1, 0ua;

H3ft, u> φp2^q−1b/mΛ1Λ1AΛ 1B h/Λ0

₁₋

G1τ, τgτdτ, for allt∈0,1, bub/λ.

Theorem 4.1. AssumeH1–H3hold, then BVP1.1has at least three positive solutionsu1, u2, andu₃, such thatu1< a,b <min_,1−u₂t, andu3> a, with min_,1−u₃t< b.

Proof. Firstly, we prove thatT :P_c → P_c. The operatorTis completely continuous.

From conditionH1, we can get ₁

0

G₂ξi, sφq

₁

0

G₁s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds

Mφ_q

₁

0

G₁τ, τgτfτ, uτdτ

c

1 Λ1A Λ1B.

4.3

Hence, Tu max

0t1|Tut|

max

0t1

₁

0

G₂t, sφq

₁

0

G₁s, τ

gτfτ, uτ−k²a1−τ−k²bτ dτ

ds

A φ_q

y

ϕt B φ_q

y ψt

max

0t1

₁

0

G2t, sφq

₁

0

G1s, τ

gτfτ, uτdτ ds A

φq

y

ϕt B φq

y ψt

c

1 Λ1A Λ1B c

1 Λ1A Λ1BΛ1A c

1 Λ1A Λ1BΛ1Bc.

4.4

Thus we getTuc; therefore,T :P_c → P_c. The operatorTis completely continuous by an application of Ascoli-Arzela theorem.

In the same way, conditionH2implies that conditionA2ofLemma 2.6is satisfied.

In the following, we show that conditionA1ofLemma 2.6is satisfied.

Let

u0t b

λ, t∈0,1, 4.5

then

u₀∈P

α, b,b λ

, αu0

b

λ > b, 4.6

thus,{u∈Pα, b, b/λ|αu> b}/∅.

Ifu∈Pα, b, b/λ, thenbusb/λ,s∈,1−. By conditionH3, we obtain

₁

0

G₂ξi, sφq

₁

0

G₁s, τ gτf

τ, uτ−k²a1−τ−k²bτ dτ

ds

Λ

₁₋

G₂s, s 1

2^q−1φ_{q 1−}

G₁s, τgτfτ, uτ dτ ds

−φq

₁₋

G1s, τ

k²a1−τ k²bτ dτ ds

Λ

₁₋

G2s, s 1

2^q−1φq

₁₋

G1s, τgτfτ, uτ dτ ds

−φq

₁₋

G1τ, τ

k²a1−τ k²bτ dτ ds

Λ

₁₋

G2s, s 1

2^q−1φq

₁₋

G1s, τgτfτ, uτ dτ

−φq

₁₋

G1τ, τ

k²ak²b dτ ds

b

1 Λ1A Λ1B.

4.7

Thus we get

αTut min

t∈,1−Tut

min

t∈,1−

₁₋

G2s, sφq

₁

0

G1s, τ gτf

τ, uτ−k²a1−τ−k²bτ dτ

ds A

φq

y

ϕt B φq

y ψt

b

1 Λ2A Λ2B b

1 Λ2A Λ2BΛ2A b

1 Λ2A Λ2BΛ2Bb.

4.8 Therefore, conditionA1ofLemma 2.6is satisfied.

Finally, we show that conditionA3ofLemma 2.6is satisfied.

Ifu∈Pα, b, c, andTu> b/λ, thenαTut min_t1−TutλTu> b.

Therefore, condition A3 of Lemma 2.6is also satisfied. By Lemma 2.6, there exist three positive solutionsu1,u2, andu3such thatu1< a, b <min_t∈,1−u2t, andu3> a, with min_t∈,1−u₃t< b. Thus we completed the proof.

Acknowledgments

The first author is supported by the Natural Science Foundation of Shandong Province of ChinaZR2012AQ024and the University Science and Technology Foundation of Shandong Provincial Education Department J10LA62. The second author is supported by the National Natural Science Foundation of China10871116, the Natural Science Foundation of Shandong Province of ChinaZR2010AM005, and the Doctoral Program Foundation of Education Ministry of China200804460001.

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