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We obtain the existence of positive solutions by using fixed-point theorem in cones.. Recently, the boundary value problems with p-Laplacian operator have also been discussed extensively

Volume 2009, Article ID 169321, 14 pages

doi:10.1155/2009/169321

Research Article

The Existence of Positive Solutions for Third-Order

p-Laplacian m-Point Boundary Value Problems with

Sign Changing Nonlinearity on Time Scales

Fuyi Xu and Zhaowei Meng

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Correspondence should be addressed to Fuyi Xu,xfy 02@163.com

Received 25 February 2009; Revised 10 April 2009; Accepted 2 June 2009

Recommended by Alberto Cabada

We study the following third-order p-Laplacian m-point boundary value problems on time scales

φ p uΔ∇∇  atft, ut  0, t ∈ 0, T Tκ , u0  m−2

i1 b i uξ i , uΔT  0, φ p uΔ∇0 

m−2

i1 c i φ p uΔ∇ξ i , where φ p s is p-Laplacian operator, that is, φ p s  |s| p−2 s, p > 1, φ−1p 

φ q , 1/p  1/q  1, 0 < ξ1 < · · · < ξ m−2 < ρT We obtain the existence of positive solutions by

using fixed-point theorem in cones In particular, the nonlinear term ft, u is allowed to change

sign The conclusions in this paper essentially extend and improve the known results

Copyrightq 2009 F Xu and Z Meng 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

The theory of time scales was initiated by Hilger 1 as a mean of unifying and extending theories from differential and difference equations The study of time scales has lead to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models, see, for example 2 6 Recently, the boundary value

problems with p-Laplacian operator have also been discussed extensively in literature; for

example, see 7 18 However, to the best of our knowledge, there are not many results

concerning the higher-order p-Laplacian mutilpoint boundary value problem on time scales.

A time scaleT is a nonempty closed subset of R We make the blanket assumption that

0, T are points in T By an interval 0, TT, we always mean the intersection of the real interval

0, T with the given time scale; that is 0, T ∩ T.

In 19 , Anderson considered the following third-order nonlinear boundary value problemBVP:

xt  ft, xt, t1≤ t ≤ t3,

x t1  xt2  0, γx t3  δxt3  0. 1.1

author studied the existence of solutions for the nonlinear boundary value problem by using Krasnoselskii’s fixed point theorem and Leggett and Williams fixed point theorem, respectively

In9,10 , He considered the existence of positive solutions of the p-Laplacian dynamic

equations on time scales



φ p uΔ∇ atfut  0, t ∈ 0, T T, 1.2

satisfying the boundary conditions

u 0 − B0



uΔ

η

 0, uΔT  0, 1.3 or

uΔ0  0, uT − B1



uΔ

η

where η ∈ 0, ρT He obtained the existence of at least double and triple positive solutions

of the problems by using a new double fixed point theorem and triple fixed point theorem, respectively

In18 , Zhou and Ma firstly studied the existence and iteration of positive solutions for

the following third-order generalized right-focal boundary value problem with p-Laplacian

operator



φ p



u

t  qtft, ut, 0 ≤ t ≤ 1,

u0 m

i1

α i u ξ i , un  0, u1 n

i1

β i uθ i . 1.5

They established a corresponding iterative scheme for the problem by using the monotone iterative technique

All the above works were done under the assumption that the nonlinear term is nonnegative The key conditions used in the above papers ensure that positive solution

is concave down If the nonlinearity is negative somewhere, then the solution needs no longer to be concave down As a result, it is difficult to find positive solutions of the p-Laplacian equation when the nonlinearity changes sign In particular, little work has been

done on the existence of positive solutions for higher order p-Laplacian m-point boundary value problems with nonlinearity f being nonnegative on time scales Therefore, it is a natural problem to consider the existence of positive solution for higher order p-Laplacian

equations with sign changing nonlinearity on time scales This paper attempts to fill this gap

in literature

In this paper, by using different method, we are concerned with the existence

of positive solutions for the following third-order p-Laplacian m-point boundary value

problems on time scales:



φ p uΔ∇∇ atft, ut  0, t ∈ 0, T Tκ ,

u0 m−2

i1

b i u ξ i , uΔT  0, φ p



uΔ∇0m−2

i1

c i φ p



uΔ∇ξ i,

1.6

where φ p s is p-Laplacian operator, that is, φ p s  |s| p−2 s, p > 1, φ−1p  φ q , 1/p  1/q  1,

and b i , c i , a, f satisfy

H1 b i , c i ∈ 0, ∞, 0 < ξ1< · · · < ξ m−2 < ρT, 0 <m−2

i1 b i < 1, 0 <m−2

i1 c i < 1;

H2 f : 0, T Tκ × 0, ∞ → −∞, ∞ is continuous, a ∈ C ld 0, T Tκ , 0, ∞, and

there exists t0∈ 0, TTκ such that at0 > 0.

2 Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in1 5

r > inf T, define the forward jump operator σ and backward jump operator ρ, respectively,

by

σ t  inf{τ ∈ T | τ > t} ∈ T,

ρ r  sup{τ ∈ T | τ < r} ∈ T 2.1

for all t, r ∈ T If σt > t, t is said to be right scattered, if ρr < r, r is said to be left scattered;

if σt  t, t is said to be right dense, and if ρr  r, r is said to be left dense If T has a right

scattered minimumm, define T k  T − {m}; otherwise set T k  T If T has a left scattered

maximum M, define T k  T − {M}; otherwise set T k T.

be the number fΔt provided that it exists, with the property that for each  > 0, there is a neighborhood U of t such that



fσt − fs − fΔtσt − s ≤ |σt − s| 2.2

for all s ∈ U.

For f : T → R and t ∈ T k , the nabla derivative of f at t, denoted by f∇t provided it

exists with the property that for each  > 0, there is a neighborhood U of t such that



fρ t− fs − f∇tρ t − s ≤ ρ t − s 2.3

for all s ∈ U.

Definition 2.3 A function f is left-dense continuous i.e., ld-continuous, if f is continuous at

each left-dense point inT and its right-sided limit exists at each right-dense point in T.

Definition 2.4 If φΔt  ft, then we define the delta integral by

b

a

f tΔt  φb − φa. 2.4

If F∇t  ft, then we define the nabla integral by

b

a

f t∇t  Fb − Fa. 2.5

Lemma 2.5 If condition H1 holds, then for h ∈ C ld 0, T Tκ , the boundary value problem (BVP)

uΔ∇ ht  0, t ∈ 0, T Tκ ,

u0 m−2

i1

b i u ξ i , uΔT  0 2.6

has the unique solution

u t 

t

0

T − shs∇s 

m−2

i1 b i ξ0i T − shs∇s

1−m−2 i1 b i

Proof By caculating, we can easily get2.7 So we omit it

Lemma 2.6 If condition H1 holds, then for h ∈ C ld 0, T Tκ , the boundary value problem (BVP)



φ p



uΔ∇∇

 ht  0, t ∈ 0, T Tκ ,

u0 m−2

i1

b i u ξ i , uΔT  0, φ p



uΔ∇0m−2

i1

c i φ p



uΔ∇ξ i

2.8

has the unique solution

u t 

t

0

T − sφ q

s

0

h r∇r  C

∇s 

m−2

i1 b i ξ0i T − sφ q s0h r∇r  C∇s

1−m−2 i1 b i

, 2.9

i1 c i ξ0i hr∇r/1 −m−2

i1 c i .

Proof Integrating both sides of equation in2.8 on 0, t , we have

φ p



uΔ∇t φ p



uΔ∇0−

t

0

h r∇r. 2.10

So,

φ p



uΔ∇ξ i φ p



uΔ∇0−

ξ i

0

h r∇r. 2.11

By boundary value condition φ p uΔ∇0 m−2

i1 c i φ p uΔ∇ξ i, we have

φ p



uΔ∇0 −

m−2

i1 c i ξ0i h r∇r

1−m−2 i1 c i

By2.10 and 2.12 we know

uΔ∇t  −φ q

⎛

⎝

m−2

i1 c i ξ0i h r∇r

1−m−2 i1 c i

 t

0

h r∇r

⎞

⎠. 2.13 This together withLemma 2.5implies that

u t 

t

0

T − sφ q

s

0

h r∇r  C

∇s 

m−2

i1 b i ξ0i T − sφ q s0h r∇r  C∇s

1−m−2 i1 b i

, 2.14

where C m−2

i1 c i ξ0i hr∇r/1 −m−2

i1 c i The proof is complete

Lemma 2.7 Let condition H1 holds If h ∈ C ld 0, T Tκ and ht ≥ 0, then the unique solution ut

of 2.8 satisfies

u t ≥ 0, t ∈ 0, T Tκ 2.15

Proof By uΔ∇t  −φ qm−2

i1 c i ξ0i hr∇r/1 −m−2

i1 c i  t

the graph of ut is concave down on 0, TTκ , and uΔt is nonincreasing on 0, T Tκ This

together with the assumption that the boundary condition uΔT  0 implies that uΔt ≥ 0 for t ∈ 0, T Tκ This implies that

min

So we only prove u0 ≥ 0 By condition H1 we have

u0 

m−2

i1 b i ξ0i T − sφ q s0h r∇r  C∇s

1−m−2 i1 b i

≥ 0. 2.17

The proof is completed

Lemma 2.8 Let condition H1 hold If h ∈ C ld 0, T Tκ and ht ≥ 0, then the unique positive solution ut of (BVP) 2.8 satisfies

inf

where σ1m−2

i1 b i ξ i /T −m−2

i1 b i T − ξ i t∈0,T Tκ |ut|.

Proof By uΔ∇t  −φ qm−2

i1 c i ξ0i hr∇r/1 −m−2

i1 c i  t

the graph of ut is concave down on 0, TTκ , and uΔt is nonincreasing on 0, T Tκ This

together with the assumption that the boundary condition uΔT  0 implies that uΔt ≥ 0 for t ∈ 0, T Tκ This implies that

min

For all i ∈ {1, 2, , m − 2}, we have from the concavity of u that

u ξ i  − u0

ξ i ≥ u T − u0

that is,

u ξ i  − u0  ξ i

T u0 ≥ ξ i

T u T. 2.21

This together with the boundary condition u0 m−2

i1 b i uξ i implies that

min

t∈0,T Tκ u t ≥

m−2

i1 b i ξ i

i1 b i T − ξ iu T. 2.22 This completes the proof

Let E  C ld 0, T Tκ be endowed with the ordering x ≤ y if xt ≤ yt for all t ∈ 0, T Tκ ,

and t∈0,T Tκ |ut| is defined as usual by maximum norm Clearly, it follows that

For the convenience, let

ψ s  φ q

⎛

⎝ s

0

a r∇r 

m−2

i1 c i ξ0i a r∇r

1−m−2 i1 c i

⎞

We define two cones by

P  {u : u ∈ E, ut ≥ 0, t ∈ 0, TTκ },

K 



min

t∈0,T Tκ u



,

2.24

where σ  σ1σ2, σ1is defined inLemma 2.8and

σ2

m−2

i1 b i ξ0i ψ s∇s



1−m−2

i1 b i T0T − sψT∇s m−2

i1 b i ξ0i ψ T∇s/1−m−2

i1 b i

. 2.25

Define the operators F : P → E and S : K → E by setting

Fut 

t

0

T − sφ q

s

0

a rfr, ur∇r  A

∇s



m−2

i1 b i ξ0i T − sφ q s0a rfr, ur∇r  A∇s

1−m−2 i1 b i

,

2.26

where A m−2

i1 c i ξ0i arfr, ur∇r/1 −m−2

i1 c i,

Sut 

t

0

T − sϕs∇s 

m−2

i1 b i ξ0i ϕ s∇s

1−m−2 i1 b i

, 2.27

where ϕs  φ q s

0arfr, ur∇r   A,  A m−2

i1 c i ξ0i arfr, ur∇r/1 −m−2

i1 c i, and

ft, ut  max{ft, ut, 0} Obviously, u is a solution of the BVP1.6 if and only if u is a fixed point of operator F.

Lemma 2.9 S : K → K is completely continuous.

Lebesgue dominated convergence theorem, we can easily prove that operator S is completely

continuous

Lemma 2.10 see 20,21  Let K be a cone in a Banach space X Let D be an open bounded subset

x /  Ax for x ∈ ∂D K Then the following results hold.

K , then i K A, D K   1.

2 If there exists x0 ∈ K \ {0} such that x / Ax  λx0 for all x ∈ ∂D K and all λ > 0, then

i K A, D K   0.

3 Let U be open in X such that U ⊂ D K If i K A, D K   1 and i K A, U K   0, then A has a

fixed point in D K \ U K The same result holds if i K A, D K   0 and i K A, U K   1, where

i K A, D K  denotes fixed point index.

We define



u t ∈ K : min

t∈0,T Tκ u t < σρ



. 2.28

Lemma 2.11 see 20  Ωρ defined above has the following properties:

a K σρ⊂ Ωρ ⊂ K ρ;

b Ωρ is open relative to K;

c u ∈ ∂Ω ρ if and only if min t∈0,T Tκ ut  σρ;

d if u ∈ ∂Ω ρ , then σρ ≤ ut ≤ ρ for t ∈ 0, T Tκ

For the convenience, we introduce the following notations:

1

T

0

T − sψT∇s 

m−2

i1 b i ξ0i ψ T∇s

1−m−2 i1 b i

m−2

i1 b i ξ0i ψ s∇s

1−m−2 i1 b i

. 2.29

Remark 2.12 By H1 we can know that 0 < m, M < ∞, Mσ  Mσ1σ2 mσ1< m.

Lemma 2.13 If f satisfies the following condition :

f t, u ≤ φ p



, t, u ∈ 0, T Tκ×0, ρ

then

i K



S, K ρ



Proof For u ∈ ∂K ρ, then from2.30 we have

s

0

a rfr, ur∇r   A 

s

0

a rfr, ur∇r 

m−2

i1 c i ξ0i a rfr, ur∇r

1−m−2 i1 c i

≤ φ p



mρ⎛⎝ T

0

a r∇r 

m−2

i1 c i ξ0i a r∇r

1−m−2 i1 c i

⎞

⎠.

2.32

So that

ϕ s  φ q

s

0

a rfr, ur∇r   A

≤ mρψT. 2.33

Therefore,

Su t ≤

T

0

T − sϕs∇s 

m−2

i1 b i ξ0i ϕ s∇s

1−m−2 i1 b i

≤ mρ

⎛

⎝ T

0

T − sψT∇s 

m−2

i1 b i ξ0i ψ T∇s

1−m−2 i1 b i

⎞

⎠  ρ.

2.34

This implies that ρ Hence byLemma 2.101 it follows that i K S, K ρ  1

Lemma 2.14 If f satisfies the following condition:

f t, u ≥ φ p



, t, u ∈ 0, T Tκ×σρ, ρ

then

i K



S, Ω ρ



Proof Let et ≡ 1 for t ∈ 0, T Tκ Then e ∈ ∂K1 We claim that

u /  Su  λe, u ∈ ∂Ω ρ , λ > 0. 2.37

In fact, if not, there exist u0∈ ∂Ω ρ and λ0> 0 such that u0 Su0 λ0e By ft, u0 ≥ φ p Mσρ,

we have

s

0

a rfr, u0r∇r   A 

s

0

a rfr, u0r∇r 

m−2

i1 c i ξ0i a rfr, u0r∇r

1−m−2 i1 c i

≥ φ p



Mσρ⎛⎝ s

0

a r∇r 

m−2

i1 c i ξ0i a r∇r

1−m−2 i1 c i

⎞

⎠.

2.38

So that

ϕ s  φ q

s

0

a rfr, u0r∇r   A

≥ Mσρφ q

⎛

⎝ s

0

a r∇r 

m−2

i1 c i ξ0i a r∇r

1−m−2 i1 c i

⎞

⎠

 Mσρψs.

2.39

For t ∈ 0, T Tκ, then

u0t  Su0t  λ0e t

≥ Su00  λ0



m−2

i1 b i 0ξ i ϕ s∇s

1−m−2 i1 b i

 λ0

≥ Mσρ

1−m−2 i1 b i

m−2

i1

b i

ξ i

0

ψ s∇s  λ0

 σρ  λ0.

2.40

This together withLemma 2.11c implies that

a contradiction Hence byLemma 2.102 it follows that i K S, Ω ρ  0

3 Main Results

We now give our results on the existence of positive solutions of BVP1.6

Theorem 3.1 Suppose that conditions H1 and H2 hold, and assume that one of the following

conditions holds.

H3 There exist ρ1, ρ2∈ 0, ∞ with ρ1< σρ2such that

i ft, u ≤ φ p mρ1, t, u ∈ 0, T Tκ × 0, ρ1 ;

ii ft, u ≥ 0, t, u ∈ 0, T Tκ × σρ1, ρ2 , moreover ft, u ≥ φ p Mσρ2, t, u ∈

0, T Tκ × σρ2, ρ2

H4 There exist ρ1, ρ2∈ 0, ∞ with ρ1< ρ2such that

i ft, u ≤ φ p mρ2, t, u ∈ 0, T Tκ × 0, ρ2 ;

ii ft, u ≥ φ p Mσρ1, t, u ∈ 0, T Tκ × σ2ρ1, ρ2

Then, the BVP1.6 has at least one positive solution.

Proof Assume that H3 holds, we show that S has a fixed point u1inΩρ2\ K ρ1 By ft, u ≤

φ p mρ1 andLemma 2.13, we have that

i K



S, K ρ1



By ft, u ≥ φ p Mσρ2 andLemma 2.14, we have that

i K



S, Ω ρ2



ByLemma 2.11a and ρ1 < σρ2, we have K ρ1 ⊂ K σρ2 ⊂ Ωρ2 It follows fromLemma 2.103

that S has a fixed point u1inΩρ2\ K ρ1 Clearly,

which implies that σρ1≤ u1t ≤ ρ2, t ∈ 0, T Tκ By conditionH3ii, we have ft, u1t ≥ 0,

t ∈ 0, T Tκ , that is, ft, u1t  ft, u1t Hence,

Fu1 Su1. 3.4

This means that u1is a fixed point of operator F.

When conditionH4 holds, by ft, u ≤ φ p mρ2 andLemma 2.13, we have that

i K



S, K ρ2



By ft, u ≥ φ p Mσρ1 andLemma 2.14, we have that

i K



S, Ω ρ1



ByLemma 2.11a and ρ1 < ρ2, we have K σρ1 ⊂ Ωρ1 ⊂ K ρ2 It follows fromLemma 2.103

that S has a fixed point u2in K ρ2\ Ωρ1 Obviously,

which implies that σ2ρ1≤ u2t ≤ ρ2, t ∈ 0, T Tκ By conditionH4ii, we have ft, u2t ≥

0, t ∈ 0, T Tκ , that is, ft, u2t  ft, u2t Hence,

Fu2 Su2. 3.8

This means that u2 is a fixed point of operator F Therefore, the BVP 1.6 has at least one positive solution

Theorem 3.1 Suppose that conditions H1 and H2 hold, and assume that one of. ..

Then, the BVP1.6 has at least one positive solution.