Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 879140, 9 pages
doi:10.1155/2008/879140
Research Article
Eigenvalue Problems for p-Laplacian Functional
Dynamic Equations on Time Scales
Changxiu Song
School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China
Correspondence should be addressed to Changxiu Song, scx168@sohu.com
Received 29 February 2008; Accepted 25 June 2008
Recommended by Johnny Henderson
This paper is concerned with the existence and nonexistence of positive solutions of the p-Laplacian
functional dynamic equation on a time scale, φp x t∇ λatfxt, xut 0, t ∈ 0, T ,
x0 t ψt, t ∈ −τ, 0, x0 − B0 x 0 0, x T 0. We show that there exists a λ∗ > 0
such that the above boundary value problem has at least two, one, and no positive solutions for
0 < λ < λ∗ , λ λ∗ and λ > λ∗ , respectively.
Copyright q 2008 Changxiu Song. 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
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 please
see 1, 2. For notation, we will use the convention that, for each interval J of R, J will denote
time-scale interval, that is, J : J ∩ T.
In this paper, let T be a time scale such that −τ, 0, T ∈ T. We are concerned with the
existence of positive solutions of the p-Laplacian dynamic equation on a time scale
Δ ∇
φp x t
λatf xt, x μt 0,
x0 t ψt,
t ∈ −τ, 0,
t ∈ 0, T ,
x0 − B0 xΔ 0 0,
xΔ T 0,
1.1
where φp u is the p-Laplacian operator, that is, φp u |u|p−2 u, p > 1, φp −1 u φq u,
where 1/p 1/q 1.
H1 The function f : R 2 →R is continuous and nondecreasing about each element;
f0, 0 ≥ c > 0.
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Advances in Difference Equations
H2 The function a : T→R is left dense continuous i.e., a ∈ Cld T, R and does not
vanish identically on any closed subinterval of 0, T . Here Cld T, R denotes the set
of all left dense continuous functions from T to R .
H3 ψ : −τ, 0→R is continuous and τ > 0.
H4 μ : 0, T →−τ, T is continuous, μt ≤ t for all t.
H5 B0 : R→R is continuous and nondecreasing; B0 ks kB0 s, k ∈ R and satisfies that
there exist β ≥ δ > 0 such that
δs ≤ B0 s ≤ βs
for s ∈ R .
1.2
H6 limx→∞ fx, ψs/xp−1 ∞ uniformly in s ∈ −τ, 0.
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 pLaplacian problems on time scales, especially for p-Laplacian functional dynamic equations
on time scales.
The motivations for the present work stems from many recent investigations in 5–10
and references therein. Especially, Kaufmann and Raffoul 7 considered a nonlinear functional
dynamic equation on a time scale and obtained sufficient conditions for the existence of
positive solutions, Li and Liu 10 studied the eigenvalue problem for second-order nonlinear
dynamic equations on time scales. In this paper, our results show that the number of positive
solutions of 1.1 is determined by the parameter λ. That is to say, we prove that there exists a
λ∗ > 0 such that 1.1 has at least two, one, and no positive solutions for 0 < λ < λ∗ , λ λ∗ and
λ > λ∗ , respectively.
For convenience, we list the following well-known definitions which can be found in
11–13 and the references therein.
Definition 1.1. For t < sup T and r > inf T, define the forward jump operator σ and the
backward jump operator ρ, respectively, as
σt inf{τ ∈ T | τ > t} ∈ T,
ρr sup{τ ∈ T | τ < r} ∈ T ∀t, r ∈ T.
1.3
If σt > t, t is said to be right scattered, and 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
minimum m, define Tκ T − {m}; otherwise set Tκ T. If T has a left-scattered maximum M,
define Tκ T − {M}; otherwise set Tκ T.
Definition 1.2. For x : T→R and t ∈ Tκ , define the deltaderivative of xt, xΔ t, to be the
number when it exists, with the property that, for any ε > 0, there is a neighborhood U of t
such that
x σt − xs − xΔ t σt − s < εσt − s ∀s ∈ U.
1.4
For x : T→R and t ∈ Tκ , define the nabla derivative of xt, x∇ t, to be the number when it
exists, with the property that, for any ε > 0, there is a neighborhood V of t such that
x ρt − xs − x∇ t ρt − s < ερt − s ∀s ∈ V.
1.5
If T R, then xΔ t x∇ t x t. If T Z, then xΔ t xt 1 − xt is forward
difference operator while x∇ t xt − xt − 1 is the backward difference operator.
Changxiu Song
3
t
Definition 1.3. If F Δ t ft, then define the delta integral by a fsΔs Ft−Fa. If Φ∇ t t
ft, then define the nabla integral by a fs∇s Φt − Φa.
The following lemma is crucial to prove our main results.
Lemma 1.4 14. Let E be a Banach space and let P be a cone in E. For r > 0, define Pr {x ∈ P :
||x|| < r}. Assume that F : P r →P is completely continuous such that Fx /
x for x ∈ ∂Pr {x ∈ P :
||x|| r}.
i If ||Fx|| ≥ ||x|| for x ∈ ∂Pr , then iF, Pr , P 0.
ii If ||Fx|| ≤ ||x|| for x ∈ ∂Pr , then iF, Pr , P 1.
2. Positive solutions
We note that xt is a solution of 1.1 if and only if
⎧ T
⎪
⎪
⎪
φ
λarf
xr,
x
μr
∇r
B
⎪
0
q
⎪
⎪
⎪
0
⎪
⎨
t T
xt ⎪
φ
λarf
xr,
x
μr
∇r
Δs, t ∈ 0, T ,
⎪
q
⎪
⎪
⎪
0
s
⎪
⎪
⎪
⎩ψt,
t ∈ −τ, 0.
2.1
Let E Cld 0, T , R be endowed with the norm ||x|| maxt∈0,T |xt| and define the
cone of E by
P
x ∈ E : xt ≥
δ
x for t ∈ 0, T .
T β
2.2
Clearly, E is a Banach space with the norm x. For each x ∈ E, extend xt to −τ, T with xt ψt for t ∈ −τ, 0.
Define Fλ : P →E as
T
Fλ xt B0 φq
λarf xr, x μr ∇r
t
0
0
φq
T
s
λarf xr, x μr ∇r Δs,
2.3
t ∈ 0, T .
We seek a fixed point, x1 , of Fλ in the cone P . Define
xt ⎧
⎨x1 t, t ∈ 0, T ,
⎩ψt,
t ∈ −τ, 0.
Then xt denotes a positive solution of BVP 1.1.
2.4
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It follows from 2.3 that the following lemma holds.
Lemma 2.1. Let Fλ be defined by 2.3. If x ∈ P , then
i Fλ P ⊂ P.
ii Fλ : P →P is completely continuous.
The proof of Lemma 2.1 can be found in 15.
We need to define further subsets of 0, T with respect to the delay μ. Set
Y1 : t ∈ 0, T : μt < 0 ;
Y2 : t ∈ 0, T : μt ≥ 0 .
2.5
Throughout this paper, we assume Y1 /
∅ and φq Y 1 ar∇r > 0.
Lemma 2.2. Suppose that (H1)–(H5) hold. Then there exists a λ∗ > 0 such that the operator Fλ has a
fixed point x∗ ∈ P \ {θ} at λ∗ , where θ is the zero element of the Banach space E.
Proof. Set
t T
T
ar∇r
φq
ar∇r Δs,
et B0 φq
0
0
s
t ∈ 0, T .
2.6
We know that e ∈ P. Let λ∗ Mf−1e , where
Mfe max f er, e μr ≥ c > 0,
r∈0,T T
λ∗ arf xr, x μr ∇r
Fλ∗ x t B0 φq
t
0
2.7
0
φq
T
s
λ arf xr, x μr ∇r Δs,
∗
t ∈ 0, T .
From above, we have
et ≥ Fλ∗ e t.
2.8
Let x0 t et and xn t Fλ∗ xn−1 t, n 1, 2, . . . , t ∈ 0, T . Then
q−1
x0 t ≥ x1 t ≥ · · · ≥ xn t ≥ · · · ≥ cλ∗
et.
2.9
By the Lebesgue dominated convergence theorem 16 together with H3, it follows that
∗
{Fλn∗ x0 }∞
n0 decreases to a fixed point x ∈ P \ {θ} of the operator Fλ∗ . The proof is
complete.
{xn }∞
n0
Lemma 2.3. Suppose that (H1)–(H6) hold and that I ⊂ b, ∞ for some b > 0. Then there exists a
constant CI > 0 such that for all λ ∈ I and all possible fixed points x of Fλ at λ, one has ||x|| < CI .
Proof. Set
S {x ∈ P : Fλ x x, λ ∈ I}.
2.10
Changxiu Song
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We need to prove that there exists a constant CI > 0 such that x < CI for all x ∈ S. If the
number of elements of S is finite, then the result is obvious. If not, without loss of generality,
we assume that there exists a sequence {xn }∞
n0 such that limn→∞ xn ∞, where xn ∈ P is
the fixed point of the operator Fλ defined by 2.3 at λn ∈ I n 1, 2, . . ..
Then
xn t ≥
δ xn ,
T β
t ∈ 0, T .
2.11
We choose J > 0 such that
Jbq−1 δ 2
φq
T β
Y1
ar∇r
> 1,
2.12
x > L, s ∈ −τ, 0.
2.13
L > 0 such that
f x, ψs ≥ Jxp−1 ,
In view of H6 there exists an N sufficiently large such that xN > L. For t ∈ 0, T , we have
xN Fλ xN N
FλN xN T ≥ δφq
T
λN arf xN r, xN μr ∇r
0
≥ δφq
Y1
λN arf xN r, ψ μr ∇r
> δJbq−1 min φq
t∈Y1
Jbq−1 δ 2 xN φq
T β
> xN ,
Y1
2.14
p−1
arxN r∇r
≥
Y1
ar∇r
which is a contradiction. The proof is complete.
Lemma 2.4. Suppose that (H1)–(H5) hold and that the operator Fλ has a positive fixed point x in P at
λ > 0. Then for every λ∗ ∈ 0, λ the operator Fλ has a fixed point x∗ ∈ P \ {θ} at λ∗ , and x∗ < x.
Proof. Let xt be the fixed point of the operator Fλ at λ. Then
T
t T
λarf xr, x μr ∇r
φq
λarf xr, x μr ∇r Δs
xt B0 φq
0
s
0
t T
T
> B0 φq
λ∗ arf xr, x μr ∇r
φq
λ∗ arf xr, x μr ∇r Δs,
0
0
s
2.15
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Advances in Difference Equations
where 0 < λ∗ < λ. Set
T
t T
λ∗ arf xr, x μr ∇r
φq
λ∗ arf xr, x μr ∇r Δs,
Fλ∗ x t B0 φq
0
s
0
2.16
x0 t xt, and xn Fλ∗ xn−1 Fλn∗ x0 t. Then
q−1
et ≤ xn1 ≤ xn ≤ · · · ≤ x1 t ≤ x0 t,
cλ∗
2.17
decreases to a fixed point
where et is also defined by 2.6, which implies that {Fλn∗ x}∞
n0
x∗ ∈ P \ {θ} of the operator Fλ∗ , and x∗ < x. The proof is complete.
Lemma 2.5. Suppose that (H1)–(H6) hold. Let ∧ {λ > 0 : Fλ have at least one fixed point at λ in P }.
Then ∧ is bounded above.
Proof. Suppose to the contrary that there exists a fixed point sequence {xn }∞
n0 ⊂ P of Fλ at λn
such that limn→∞ λn ∞. Then we need to consider two cases:
i there exists a constant H > 0 such that xn ≤ H, n 0, 1, 2 . . . ;
ii there exists a subsequence {xnk }∞
k1 such that limk→∞ ||xnk || ∞ which is impossible
by Lemma 2.3.
Only i is considered. We can choose M > 0 such that f0, 0 > MH, and further
fxn , xn μ > MH. For t ∈ 0, T , we have
T
t T
xn t B0 φq
λn arf xn r, xn μr ∇r
λn arf xn r, xn μr ∇r Δs.
φq
0
0
s
2.18
Now we consider 2.18.Assume that the case i holds. Then
T
t T
H ≥ xn t ≥ B0 φq
λn arMH∇r
φq
λn arMH∇r Δs
0
λn MH
q−1
0
s
2.19
et
q−1 δ
≥ λn MH
e
T β
leads to
q−1 q−2 δ
1 ≥ λn M
H
e
T β
for t ∈ 0, T ,
which is a contradiction. The proof is complete.
Lemma 2.6. Let λ∗ sup ∧. Then ∧ 0, λ∗ , where ∧ is defined just as in Lemma 2.5.
2.20
Changxiu Song
7
Proof. In view of Lemma 2.4, it follows that 0, λ∗ ⊂ ∧. We only need to prove λ∗ ∈ ∧. In
fact, by the definition of λ∗ , we may choose a distinct nondecreasing sequence {λn }∞
n1 ⊂ ∧
such that limn→∞ λn λ∗ . Let xn ∈ P be the positive fixed point of Fλ at λn , n 1, 2, . . . .
∞
By Lemma 2.3, {xn }∞
n1 is uniformly bounded, so it has a subsequence denoted by {xn }n1 ,
converging to xλ∗ ∈ P. Note that
T
t T
xn t B0 φq
λn arf xn r, xn μr ∇r
λn arf xn r, xn μr ∇r Δs.
φq
0
s
0
2.21
Taking the limitation n→∞ to both sides of 2.21, and using the Lebesgue dominated
convergence theorem 16, we have
T
t T
xλ∗ B0 φq
λ∗ arf xλ∗ r, xλ∗ μr ∇r
λ∗ arf xλ∗ r, xλ∗ μr ∇r Δs,
φq
0
0
s
2.22
which shows that Fλ has a positive fixed point xλ∗ at λ λ∗ . The proof is complete.
Theorem 2.7. Suppose that (H1)–(H6) hold. Then there exists a λ∗ > 0 such that 1.1 has at least two,
one, and no positive solutions for 0 < λ < λ∗ , λ λ∗ and λ > λ∗ , respectively.
Proof. Assume that H1–H5 hold. Then there exists a λ∗ > 0 such that Fλ has a fixed point
xλ∗ ∈ P \ {θ} at λ λ∗ . In view of Lemma 2.4, Fλ also has a fixed point xλ < xλ∗ , xλ ∈ P \ {θ}
and 0 < λ < λ∗ . Note that f is continuous on R 2 . For 0 < λ < λ∗ , there exists a δ0 > 0 such
that
∗
λ
f xλ∗ r δ, xλ∗ μr δ − f xλ∗ r, xλ∗ μr ≤ f0, 0
−1
for r ∈ 0, T , 0 < δ ≤ δ0 .
λ
2.23
Hence,
λarf xλ∗ r δ, xλ∗ μr δ − λ∗ arf xλ∗ r, xλ∗ μr
λar f xλ∗ r δ, xλ∗ μr δ − f xλ∗ r, xλ∗ μr
− λ∗ − λ arf xλ∗ r, xλ∗ μr
≤ λ∗ − λ arf0, 0 − λ∗ − λ f xλ∗ r, xλ∗ μr
2.24
λ∗ − λ ar f0, 0 − f xλ∗ r, xλ∗ μr
≤ 0,
∀r ∈ 0, T .
From above, we have
Fλ xλ∗ δ ≤ Fλ∗ xλ∗ xλ∗ < xλ∗ δ.
2.25
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Advances in Difference Equations
Set R1 ||xλ∗ t δ|| for t ∈ 0, T and PR1 {x ∈ P : ||x|| < R1 }. We have Fλ x /
x for x ∈ ∂R1 .
By Lemma 2.1, iFλ , PR1 , P 1. In view of H6, we can choose L > R1 > 0 such that
f x, ψs ≥ Jxp−1 ,
Jλq−1 δ 2
φq
T β
Y1
ar∇r
> 1 for x > L, s ∈ −τ, 0.
2.26
Set
R2 PR2 x ∈ P : x < R2 .
T β
L 1,
δ
2.27
Similar to Lemma 2.3, it is easy to obtain that
Fλ x Fλ x T ≥ δφq
T
λarf xr, x μr ∇r
0
≥ δφq
> δJλ
Y1
q−1
λarf xr, ψ μr ∇r
min xt φq
t∈Y1
Jλq−1 δ 2
xφq
≥
T β
> x
Y1
2.28
ar∇r
Y1
ar∇r
for x ∈ ∂PR2 .
In view of Lemma 2.1, iFλ , PR2 , P 0. By the additivity of fixed point index,
i Fλ , PR2 \ P R1 , P i Fλ , PR2 , P − i Fλ , PR1 , P −1.
2.29
So, Fλ has at least two fixed points in P . The proof is complete.
Acknowledgments
This work was supported by Grant 10571064 from NNSF of China, and by a grant from NSF of
Guangdong.
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