Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 279752, 11 pages pdf

1.2 p-Laplacian problems with two-, three-, m-point boundary conditions for ordinary diﬀerential equations and finite diﬀerence equations have been studied extensively, for example see 1

Trang 1

Volume 2011, Article ID 279752, 11 pages

doi:10.1155/2011/279752

Research Article

Functional Dynamic Equations on Time Scales

Changxiu Song and Xuejun Gao

School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China

Correspondence should be addressed to Changxiu Song,scx168@sohu.com

Received 31 March 2010; Revised 8 December 2010; Accepted 9 December 2010

Academic Editor: Daniel Franco

Copyrightq 2011 C Song and X Gao 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

The authors study the boundary value problems for a p-Laplacian functional dynamic equation on

a time scale,φ p xΔ∇t∇ atfxt, xμt 0, t ∈ 0, T, x0t ψt, t ∈ −r, 0, xΔ0

xΔ∇0 0, xT B0xΔη 0 By using the twin fixed-point theorem, suﬃcient conditions are

established for the existence of twin positive solutions

1 Introduction

Let T be a closed nonempty subset of R, and let T have the subspace topology inherited from the Euclidean topology on R In some of the current literature, T is called a time scale or measure chain For notation, we shall use the convention that, for each interval of J of R, J

will denote time scales interval, that is, J : J ∩ T.

In this paper, let T be a time scale such that−r, 0, T ∈ T We are concerned with the existence of positive solutions of the p-Laplacian dynamic equation on a time scale

φ p

xΔ∇t∇ atfxt, xμt 0, t ∈ 0, T,

x0t ψt, t ∈ −r, 0, xΔ0 xΔ∇0 0, xT B0

xΔ

η

 0, 1.1

where φ p u is the p-Laplacian operator, that is, φ p u |u| p−2 u, p > 1, φ p−1u φ q u, where 1/p 1/q 1; η ∈ 0, ρT and

H1 the function f : R2 → Ris continuous,

H2 the function a : T → Ris left dense continuousi.e., a ∈ CldT, R and does not vanish identically on any closed subinterval of0, T Here, CldT, R denotes the set of all left dense continuous functions from T to R,

Trang 2

H3 ψ : −r, 0 → Ris continuous and r > 0,

H4 μ : 0, T → −r, T is continuous, μt ≤ t for all t,

H5 B0: R → R is continuous and satisfies that there are β ≥ δ ≥ 0 such that

δs ≤ B0s ≤ βs, for s ∈ R. 1.2

p-Laplacian problems with two-, three-, m-point boundary conditions for ordinary

diﬀerential equations and finite diﬀerence equations have been studied extensively, for example see 1 4 and references therein However, there are not many concerning the p-Laplacian problems on time scales, especially for p-p-Laplacian functional dynamic equations

on time scales

The motivations for the present work stems from many recent investigations in5

8 and references therein Especially, Kaufmann and Raﬀoul 8 considered a nonlinear functional dynamic equation on a time scale and obtained suﬃcient conditions for the existence of positive solutions In this paper, we apply the twin fixed-point theorem to obtain

at least two positive solutions of boundary value problemBVP for short 1.1 when growth

conditions are imposed on f Finally, we present two corollaries, which show that under the assumptions that f is superlinear or sublinear, BVP 1.1 has at least two positive solutions

Given a nonnegative continuous functional γ on a cone P of a real Banach space E, we define for each d > 0 the sets

P

γ, d

x ∈ P : γx < d ,

∂P

γ, d

x ∈ P : γ x d ,

P γ, d 

x ∈ P : γx ≤ d .

1.3

The following twin fixed-point lemma due to9 will play an important role in the proof of our results

Lemma 1.1 Let E be a real Banach space, P a cone of E, γ and α two nonnegative increasing

continuous functionals, θ a nonnegative continuous functional, and θ0 0 Suppose that there are two positive numbers c and M such that

γx ≤ θx ≤ αx, x ≤ Mγx, for x ∈ Pγ, c. 1.4

F : P γ, c → P is completely continuous There are positive numbers 0 < a < b < c such that

θλx ≤ λθx, ∀λ ∈ 0, 1, x ∈ ∂Pθ, b, 1.5

and

i γFx > c for x ∈ ∂Pγ, c,

ii θFx < b for x ∈ ∂Pθ, b,

iii αFx > a and Pα, a / ∅ for x ∈ ∂Pα, a.

Trang 3

Then, F has at least two fixed points x1and x2∈ Pγ, c satisfying

a < αx1, θx1 < b, b < θx2, γx2 < c. 1.6

2 Positive Solutions

We note that xt is a solution of 1.1 if and only if

xt 

⎧

⎪

T

0

T − sφ q

s

0

arfxr, xμr∇r

∇s

−B0

η

0

φ q

s

0− arfxr, xμr∇r

∇s

t

0

t − sφ q

s

0

− arfxr, xμr∇r

∇s, t ∈ 0, T,

2.1

Let E CΔld0, T, R be endowed with the norm x max t∈0,T |xt| and P {x ∈ E : x is

concave and nonnegative valued on0, T, and xΔ0 0}

Clearly, E is a Banach space with the norm x and P is a cone in E For each x ∈ E, extend xt to −r, T with xt ψt for t ∈ −r, 0.

Define F : P → E as

Fxt 

T

0

T − sφ q

s

0

∇s

− B0

η

0

φ q

s

0

−arfxr, xμr∇r

∇s

t

0

t − sφ q

s

0

∇s, t ∈ 0, T.

2.2

We seek a fixed point, x1, of F in the cone P Define

xt 

⎧

⎨

⎩

x1t, t ∈ 0, T, ψt, t ∈ −r, 0. 2.3

Then, xt denotes a positive solution of BVP 1.1

It follows from2.2 that

Lemma 2.1 Let F be defined by 2.2 If x ∈ P, then

i FP ⊂ P.

ii F : P → P is completely continuous.

Trang 4

iii xt ≥ T − t/Tx, t ∈ 0, T.

iv xt is decreasing on 0, T.

The proof is similar to the proofs of Lemma 2.3 and Theorem 3.1 in7, and is omitted

Fix l ∈ T such that 0 < l < η < T, and set

Y1 :t ∈ 0, T : μt < 0 , Y2:t ∈ 0, T : μt ≥ 0 , Y3: Y1∩ 0, l. 2.4

Throughout this paper, we assume Y3/ ∅ andY3φ qs

0ar∇r∇s > 0.

Now, we define the nonnegative, increasing, continuous functionals γ, θ, and α on P

by

γx max

t∈l,ηxt xl,

θx min

t∈ 0,l xt xl, αx max

t∈η,Txt x

η

.

2.5

We have

γx θx ≤ αx, x ∈ P, θx γx xl ≥ T − l

T x, αx xη

≥ T − η

T x, for each x ∈ P. 2.6

Then,

x ≤ T

T − l γx, x ≤ T

T − η αx, for each x ∈ P. 2.7

We also see that

θλx λθx, ∀λ ∈ 0, 1, x ∈ ∂Pθ, b. 2.8

For the notational convenience, we denote σ1, σ2and ρ1, ρ2by

σ β

Y3

φ q

s

0

ar∇r

∇s; ρ T2T δφ q

T

0

ar∇r

. 2.9

Theorem 2.2 Suppose that there are positive numbers a < b < c such that

0 < a < σ

ρ b <

T − lσ

Trang 5

Assume f satisfies the following conditions:

A fx, ψs > φ p c/σ for c ≤ x ≤ T/T − lc, uniformly in s ∈ −r, 0,

B fx, ψs < φ p b/ρ for 0 ≤ x ≤ T/T − lb, uniformly in s ∈ −r, 0,

fx1, x2 < φ p

b ρ

, for 0 ≤ x i≤ T

T − l b, i 1, 2, 2.11

C fx, ψs > φ p a/σ for a ≤ x ≤ T/T − ηa, uniformly in s ∈ −r, 0.

Then, BVP1.1 has at least two positive solutions of the form

xt 

⎧

⎨

⎩

ψt, t ∈ −r, 0,

x i t, t ∈ 0, T, i 1, 2, 2.12 where a < max t∈η,T x1t, min t∈0,l x1t < b and b < min t∈0,l x2t, max t∈l,η x2t < c.

Proof By the definition of operator F and its properties, it suﬃces to show that the conditions

ofLemma 1.1hold with respect to F.

First, we verify that x ∈ ∂P γ, c implies γFx > c.

Since γx xl c, one gets xt ≥ c for t ∈ 0, l Recalling that 2.7, we know c ≤ x ≤

T/T − lc for t ∈ 0, l Then, we get

γFx 

T

0

T − sφ q

s

0

∇s

− B0

η

0

φ q

s

0

−arfxr, xμr∇r∇s

l

0

l − sφ q

s

0

∇s

≥ −B0

η

0

φ q

s

0−arfxr, xμr∇r

∇s

≥ β

l

0

φ q

s

0

∇s

≥ β

Y3

φ q

s

0

arfxr, ψμr∇r

∇s

> β

Y3

φ q

s

0

ar∇r

∇s c

σ c.

2.13

Secondly, we prove that x ∈ ∂P θ, b implies θFx < b.

Since θx b implies xl b, it holds that b ≤ xt ≤ x ≤ T/T − lθx T/T − lb for t ∈ 0, l, and for all x ∈ ∂P θ, b implies

0≤ xt ≤ b, for t ∈ l, T. 2.14

Trang 6

0≤ xt ≤ T

T − l b, t ∈ 0, T. 2.15

So, we have

θFx 

T

0

T − sφ q

s

0

∇s

− B0

η

0

φ q

s

0

∇s

l

0

l − sφ q

s

0

∇s

<

T

0

Tφ q

T

0

∇s δ

T

0

φ q

T

0

∇s

T

0

Tφ q

T

0

∇s

 T2T δφ q

Y1

arfxr, ψμr∇r 

Y2

< b

ρ T 2T δφ q

T

0

ar∇r

 b.

2.16 Finally, we show that

P α, a / ∅, αFx > a, ∀x ∈ ∂Pα, a. 2.17

It is obvious that P α, a / ∅ On the other hand, αx xη a and 2.7 imply

a ≤ x ≤ T

T − η a, for t ∈

0, η

Thus,

αFx 

T

0

T − sφ q

s

0

∇s

− B0

η

0

φ q

s

0−arfxr, xμr∇r

∇s

η

0

η − s

φ q

s

0

∇s

Trang 7

≥ −B0

η

0

φ q

s

0

∇s

≥ β

l

0

φ q

s

0

∇s

≥ β

Y3

φ q

s

0

arfxr, ψμr∇r∇s

> β

Y3

φ q

s

0

ar∇r

∇s a

σ a.

2.19

ByLemma 1.1, F has at least two diﬀerent fixed points x1and x2satisfying

a < αx1, θx1 < b, b < θx2, γx2 < c. 2.20 Let

xt 

⎧

⎨

⎩

ψt, t ∈ −r, 0,

x i t, t ∈ 0, T, i 1, 2, 2.21

which are twin positive solutions of BVP1.1 The proof is complete

In analogy toTheorem 2.2, we have the following result

Theorem 2.3 Suppose that there are positive numbers a < b < c such that

0 < a < T − η

T b <

T − η

σ

Assume f satisfies the following conditions:

A’ fx, ψs < φ p c/ρ for 0 ≤ x ≤ T/T − lc, uniformly in s ∈ −r, 0,

fx1, x2 < φ p

c ρ

, for 0 ≤ x i≤ T

T − l c, i 1, 2, 2.23

B’ fx, ψs > φ p b/σ for b ≤ x ≤ T/T − lb, uniformly in s ∈ −r, 0,

C’ fx, ψs < φ p a/ρ for 0 ≤ x ≤ T/T − ηa, uniformly in s ∈ −r, 0,

fx1, x2 < φ p

a ρ

, for 0 ≤ x i≤ T

T − η a, i 1, 2. 2.24

Trang 8

xt 

⎧

⎨

⎩

ψt, t ∈ −r, 0,

x i t, t ∈ 0, T, i 1, 2. 2.25

Now, we give theorems, which may be considered as the corollaries of Theorems2.2

and2.3

Let

f0 lim

x → 0

f

x, ψs

x p−1 , f∞ lim

x → ∞

f

x, ψs

x p−1 , f00 lim

x1 → 0 ;x2 → 0

fx1, x2 max

x p−11 , x2p−1 ,

2.26

and choose k1, k2, k3such that

k1σ > 1, k2σ > 1, 0 < k3ρ < T − η

T . 2.27

From above, we deduce that 0 < k3ρ < l/T.

Theorem 2.4 If the following conditions are satisfied:

D f0> k1p−1 , f∞> k2p−1 , uniformly in s ∈ −r, 0,

E there exists a p1> 0 such that for all 0 ≤ x ≤ T/T − lp1, one has

f

x, ψ s<

p1

ρ

p−1

, uniformly in s ∈ −r, 0,

f x1, x2 <

p1

ρ

p−1

, for 0 ≤ x i≤ T

T − l p1, i 1, 2.

2.28

xt 

⎧

⎨

⎩

ψt, t ∈ −r, 0,

x i t, t ∈ 0, T, i 1, 2. 2.29 Proof First, choose b p1, one gets

f

x, ψs< φ p

b ρ

, for 0≤ x ≤ T

T − l b, uniformly in s ∈ −r, 0, fx1, x2 < φ p

b ρ

, for 0≤ x i≤ T

T − l b, i 1, 2.

2.30

Trang 9

Secondly, since f0> k1p−1 , there is R1> 0 suﬃciently small such that

f

x, ψ s> k1x p−1 , for 0≤ x ≤ R1. 2.31

Without loss of generality, suppose R1 ≤ T − ησ/Tρb Choose a > 0 so that a <

T − η/TR1 For a ≤ x ≤ T/T − ηa, we have x ≤ R1and a < σ/ρb Thus,

f

x, ψs> k1x p−1 ≥ k1a p−1

> φ p

a σ

, for a ≤ x ≤ T

T − η a. 2.32

Thirdly, since f∞> k2p−1 , there is R2> 0 suﬃciently large such that

f

x, ψs> k2x p−1

, for x ≥ R2. 2.33

Without loss of generality, suppose R2> T/T − lb Choose c ≥ R2 Then,

f

x, ψs> k2x p−1 ≥ k2c p−1

> φ p

c σ

, for c ≤ x ≤ T

T − l c. 2.34

We get now 0 < a < σ/ρb < T − lσ/Tρc, and then the conditions inTheorem 2.2

are all satisfied ByTheorem 2.2, BVP1.1 has at least two positive solutions The proof is complete

Theorem 2.5 If the following conditions are satisfied:

F f0< k3p−1 , uniformly in s ∈ −r, 0; f00 < k3p−1 ,

G there exists a p2> 0 such that for all 0 ≤ x ≤ T/T − lp2, one has

f

x, ψs>

p2

σ

p−1

, uniformly in s ∈ −r, 0. 2.35

xt 

⎧

⎨

⎩

ψt, t ∈ −r, 0,

x i t, t ∈ 0, T, i 1, 2. 2.36

The proof is similar to that ofTheorem 2.4and we omitted it

The following Corollaries are obvious

Corollary 2.6 If the following conditions are satisfied:

D’ f0 ∞, f∞ ∞, uniformly in s ∈ −r, 0,

Trang 10

E there exists a p1> 0 such that for all 0 ≤ x ≤ T/T − lp1, one has

f

x, ψs<

p1

ρ

p−1

, uniformly in s ∈ −r, 0,

f x1, x2 <

p1

ρ

p−1

, for 0 ≤ x i≤ T

T − l p1, i 1, 2.

2.37

xt 

⎧

⎨

⎩

ψt, t ∈ −r, 0,

x i t, t ∈ 0, T, i 1, 2. 2.38

Corollary 2.7 If the following conditions are satisfied:

F’ f0 0, uniformly in s ∈ −r, 0, f00  0;

G there exists a p2> 0 such that for all 0 ≤ x ≤ T/T − lp2, one has

f

x, ψs>

p2

σ

p−1

, uniformly in s ∈ −r, 0. 2.39

xt 

⎧

⎨

⎩

ψt, t ∈ −r, 0,

x i t, t ∈ 0, T, i 1, 2. 2.40

3 Example

Example 3.1 Let T −1/2, 0 ∪ {1/2 n : n ∈ N0}, at ≡ 1, r 1/2, η 1/2, p 3, B0x x.

We consider the following boundary value problem:

xΔ∇txΔ∇t∇ 104x3t

x3t x3t − 1/2 1 0, t ∈ 0, 1,

x0t ψt ≡ 0, t ∈

−1

2, 0

, xΔ0 xΔ∇0 0, x1 xΔ

1 2

 0,

3.1

where μ : 0, 1 → −1/2, 1 and μt t − 1/2; fx, ψs 6x3/x3 1, fx1, x2

6x31/x31 x3

2 1

Choosing a 1/2 × 1010, b 1, c 103, l 1/4, direct calculation shows that

Y1

0,1

2

, Y2

1

2, 1

, Y3

0,1

4

, σ  4√2

224 , ρ 3. 3.2

Trang 11

Consequently, 0 < a < T − η/Tb < T − ησ/Tρc and f satisfies

A’ fx, ψs < φ p c/ρ 106/9 for 0 ≤ x ≤ 4 × 103/3, uniformly in s ∈ −1/2, 0,

fx1, x2 < φ p

c ρ

106

9 , for 0≤ x i≤ 4× 103

3 , i 1, 2, 3.3

B’ fx, ψs > φ p b/σ 1/σ2for 1≤ x ≤ 4/3, uniformly in s ∈ −1/2, 0,

C’ fx, ψs < φ p a/ρ 1/36 × 1020for 0≤ x ≤ 1/1010, uniformly in s ∈ −1/2, 0,

fx1, x2 < φ p

a ρ

1

36× 1020, for 0≤ x i≤ 1

1010, i 1, 2. 3.4

Then all conditions ofTheorem 2.3hold Thus, withTheorem 2.3, the BVP3.1 has at least two positive solutions

Acknowledgment

This paper is supported by Grants nos.10871052 and 10901060 from the NNSF of China, and by Grantno 10151009001000032 from the NSF of Guangdong

References

1 R Avery and J Henderson, “Existence of three positive pseudo-symmetric solutions for a

one-dimensional p-Laplacian,” Journal of Mathematical Analysis and Applications, vol 277, no 2, pp 395–404,

2003

2 Y Liu and W Ge, “Twin positive solutions of boundary value problems for finite diﬀerence equations

with p-Laplacian operator,” Journal of Mathematical Analysis and Applications, vol 278, no 2, pp 551–561,

2003

3 A Cabada, “Extremal solutions for the diﬀerence φ-Laplacian problem with nonlinear functional boundary conditions,” Computers & Mathematics with Applications, vol 42, no 3–5, pp 593–601, 2001.

4 F.-H Wong, “Existence of positive solutions for m-Laplacian boundary value problems,” Applied Mathematics Letters, vol 12, no 3, pp 11–17, 1999.

5 E R Kaufmann, “Positive solutions of a three-point boundary-value problem on a time scale,”

Electronic Journal of Diﬀerential Equations, vol 82, pp 1–11, 2003.

6 Z He, “Double positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol 182, no 2, pp 304–315,

2005

7 L Bian, X He, and H Sun, “Multiple positive solutions of m-point BVPs for third-order p-Laplacian dynamic equaitons on time scales,” Advance in Diﬀerence Equations, vol 2009, Article ID 262857, 12

pages, 2009

8 E R Kaufmann and Y N Raﬀoul, “Positive solutions for a nonlinear functional dynamic equation on

a time scale,” Nonlinear Analysis: Theory, Methods & Applications, vol 62, no 7, pp 1267–1276, 2005.

9 R I Avery, C J Chyan, and J Henderson, “Twin solutions of boundary value problems for ordinary diﬀerential equations and finite diﬀerence equations,” Computers & Mathematics with Applications, vol

42, no 3–5, pp 695–704, 2001

Định dạng
Số trang	11
Dung lượng	492,84 KB