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
Trang 1Volume 2008, Article ID 879140, 9 pages
doi:10.1155/2008/879140
Research Article
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 thep-Laplacian
functional dynamic equation on a time scale,φ p xt∇ λatfxt, xut 0, t ∈ 0, T,
x0t ψt, t ∈ −τ, 0, x0 − B0x0 0, xT 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∗ andλ > 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
LetT be a closed nonempty subset of R, and let T have the subspace topology inherited from the Euclidean topology onR In some of the current literature, T is called a time scaleplease see1,2 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 thep-Laplacian dynamic equation on a time scale
φ p
xΔt∇ λatfxt, xμt 0, t ∈ 0, T,
x0t ψt, t ∈ −τ, 0, x0 − B0
xΔ0 0, xΔT 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.
H1 The function f : R2→R is continuous and nondecreasing about each element;
f0, 0 ≥ c > 0.
Trang 2H2 The function a : T→R is left dense continuous i.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 fromT to R
H3 ψ : −τ, 0→Ris continuous andτ > 0.
H4 μ : 0, T→−τ, T is continuous, μt ≤ t for all t.
H5 B0:R→R is continuous and nondecreasing; B0ks kB0s, k ∈ Rand satisfies that there existβ ≥ δ > 0 such that
H6 limx→∞ fx, ψs/x p−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 p-Laplacian 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 Liu10 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 of1.1 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 operatorρ, respectively, as
σt inf{τ ∈ T | τ > t} ∈ T, ρr sup{τ ∈ T | τ < r} ∈ T ∀t, r ∈ T. 1.3
Ifσt > 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 ifρr r, r is said to be left dense If T has a right-scattered
minimumm, define T κ T − {m}; otherwise set T κ T If T has a left-scattered maximum M,
defineTκ T − {M}; otherwise set T κ T.
Definition 1.2 For x : T→R and t ∈ T κ , define the deltaderivative of xt, xΔt, to be the
numberwhen it exists, with the property that, for any ε > 0, there is a neighborhood U of t
such that
xσt − xs − xΔtσt − s < εσt − s ∀s ∈ U. 1.4 Forx : T→R and t ∈ T κ , 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
xρt − xs − x∇tρt − s < ερt − s ∀s ∈ V. 1.5
IfT R, then xΔt x∇t x t If T Z, then xΔt xt 1 − xt is forward
difference operator while x∇t xt − xt − 1 is the backward difference operator.
Trang 3Definition 1.3 If FΔt ft, then define the delta integral bya t fsΔs Ft−Fa If Φ∇t
ft, then define the nabla integral bya t 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 ∈ ∂P r {x ∈ P :
||x|| r}.
i If ||Fx|| ≥ ||x|| for x ∈ ∂P r , then iF, P r , P 0.
ii If ||Fx|| ≤ ||x|| for x ∈ ∂P r , then iF, P r , P 1.
2 Positive solutions
We note thatxt is a solution of 1.1 if and only if
xt
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
B0
φ q T
0
λarfxr, xμr∇r
t
0
φ q T
s λarfxr, xμr∇r
Δs, t ∈ 0, T,
2.1
Let E Cld0, T, R be endowed with the norm ||x|| max t∈0,T |xt| and define the
cone ofE by
P
x ∈ E : xt ≥ T β δ x for t ∈ 0, T
Clearly,E is a Banach space with the norm x For each x ∈ E, extend xt to −τ, T
withxt ψt for t ∈ −τ, 0.
DefineF λ :P→E as
F λ xt B0
φ q T
0
λarfxr, xμr∇r
t
0
φ q T
s λarfxr, xμr∇r
Δs, t ∈ 0, T.
2.3
We seek a fixed point,x1, ofF λ in the coneP Define
xt
⎧
⎨
⎩
x1t, t ∈ 0, T,
Thenxt denotes a positive solution of BVP 1.1
Trang 4It follows from2.3 that the following lemma holds.
Lemma 2.1 Let F λ be defined by2.3 If x ∈ P, then
i F λ P ⊂ P.
ii F λ :P→P is completely continuous.
The proof ofLemma 2.1can be found in15
We need to define further subsets of0, 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 assumeY1/ ∅ and φ qY1ar∇r > 0.
Lemma 2.2 Suppose that (H1)–(H5) hold Then there exists a λ∗ > 0 such that the operator F λ has a fixed point x∗∈ P \ {θ} at λ∗, where θ is the zero element of the Banach space E.
Proof Set
et B0
φ q T
0
ar∇r
t
0
φ q T
s ar∇r
Δs, t ∈ 0, T. 2.6
We know thate ∈ P Let λ∗ M−1
f e , where
M f e max
r∈0,T fer, eμr≥ c > 0,
F λ∗xt B0
φ q T
0
λ∗arfxr, xμr∇r
t
0
φ q T
s λ∗arfxr, xμr∇r
Δs, t ∈ 0, T.
2.7
From above, we have
Letx0t et and x n t F λ∗x n−1 t, n 1, 2, , t ∈ 0, T Then
x0t ≥ x1t ≥ · · · ≥ x n t ≥ · · · ≥cλ∗q−1
By the Lebesgue dominated convergence theorem16 together with H3, it follows that
{x n}∞n0 {F n
λ∗x0}∞n0 decreases to a fixed pointx∗ ∈ P \ {θ} of the operator F λ∗ The proof is
complete
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 F λ at λ, one has ||x|| < CI Proof Set
S {x ∈ P : F λ x x, λ ∈ I}. 2.10
Trang 5We need to prove that there exists a constant CI > 0 such that x < CI for all x ∈ S If the
number of elements ofS is finite, then the result is obvious If not, without loss of generality,
we assume that there exists a sequence{x n}∞n0such that limn→∞ x n ∞, where x n ∈ P is
the fixed point of the operatorF λdefined by2.3 at λn ∈ I n 1, 2, .
Then
x n t ≥ T β δ x n , t ∈ 0,T. 2.11
We chooseJ > 0 such that
Jb q−1 δ2
T β φ q Y1ar∇r
L > 0 such that
fx, ψs≥ Jx p−1 , x > L, s ∈ −τ, 0. 2.13
In view ofH6 there exists an N sufficiently large such that x N > L For t ∈ 0, T, we have
x N F λ N x N
F λ N x N
T
≥ δφ q T
0
λ N arfx N r, x Nμr∇r
≥ δφ q
Y1
λ N arfx N r, ψμr∇r
> δJb q−1min
t∈Y1
φ q
Y1
arx N p−1 r∇r
≥ Jb T β q−1 δ2x N φ q
Y1
ar∇r
> x N ,
2.14
which is a contradiction The proof is complete
Lemma 2.4 Suppose that (H1)–(H5) hold and that the operator F λ has a positive fixed point x in P at
λ > 0 Then for every λ∗∈ 0, λ the operator F λ has a fixed point x∗∈ P \ {θ} at λ∗, and x∗< x Proof Let xt be the fixed point of the operator F λatλ Then
xt B0
φ q
T
0
λarfxr, xμr∇r
t
0
φ q T
s λarfxr, xμr∇r
Δs
> B0
φ q T
0
λ∗arfxr, xμr∇r
t
0
φ q T
s λ∗arfxr, xμr∇r
Δs,
2.15
Trang 6where 0< λ∗< λ Set
F λ∗xt B0
φ q T
0
λ∗arfxr, xμr∇r
t
0
φ q T
s λ∗arfxr, xμr∇r
Δs,
2.16
x0t xt, and x n F λ∗x n−1 F n
λ∗x0t Then
cλ∗q−1 et ≤ x n1 ≤ x n ≤ · · · ≤ x1t ≤ x0t, 2.17 where et is also defined by 2.6, which implies that {Fn
λ∗x}∞n0 decreases to a fixed point
x∗∈ P \ {θ} of the operator F λ∗, andx∗< x The proof is complete.
Lemma 2.5 Suppose that (H1)–(H6) hold Let ∧ {λ > 0 : F λ 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 {x n}∞n0 ⊂ P of F λ atλ n
such that limn→∞ λ n ∞ Then we need to consider two cases:
i there exists a constant H > 0 such that x n ≤ H, n 0, 1, 2 ;
ii there exists a subsequence {x n k}∞k1 such that limk→∞ ||x n k|| ∞ which is impossible
byLemma 2.3
Only i is considered We can choose M > 0 such that f0, 0 > MH, and further
fx n , x n μ > MH For t ∈ 0, T, we have
x n t B0
φ q
T
0
λ n arfx n r, x n
μr∇r
t
0
φ q T
s λ n arfx n r, x n
μr∇r
Δs.
2.18 Now we consider2.18.Assume that the case i holds Then
H ≥ x n t ≥ B0
φ q T
0
λ n arMH∇r
t
0
φ q T
s λ n arMH∇r
Δs
λ n MHq−1 et
≥λ n MHq−1 T β δ e
2.19
leads to
1≥λ n Mq−1 H q−2 δ
T β e for t ∈ 0, T, 2.20
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
Trang 7Proof 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 {λ n}∞n1 ⊂ ∧ such that limn→∞ λ n λ∗ Let x n ∈ P be the positive fixed point of F λ at λ n , n 1, 2,
By Lemma 2.3, {x n}∞n1 is uniformly bounded, so it has a subsequence denoted by {x n}∞n1 ,
converging tox λ∗ ∈ P Note that
x n t B0
φ q
T
0
λ n arfx n r, x n
μr∇r
t
0
φ q T
s λ n arfx n r, x n
μr∇r
Δs.
2.21 Taking the limitation n→∞ to both sides of 2.21, and using the Lebesgue dominated convergence theorem16, we have
x λ∗ B0
φ q
T
0
λ∗arfx λ∗r, x λ∗
μr∇r
t
0
φ q T
s λ∗arfx λ∗r, x λ∗
μr∇r
Δs,
2.22 which shows thatF λhas a positive fixed pointx λ∗atλ 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 F λ has a fixed point
x λ∗ ∈ P \ {θ} at λ λ∗ In view ofLemma 2.4,F λ also has a fixed pointx λ < x λ∗, x λ ∈ P \ {θ}
and 0 < λ < λ∗ Note that f is continuous on R2 For 0 < λ < λ∗, there exists a δ0 > 0 such
that
fx λ∗r δ, x λ∗
μr δ− fx λ∗r, x λ∗
μr≤ f0, 0
λ∗
λ − 1
forr ∈ 0, T, 0 < δ ≤ δ0.
2.23 Hence,
λarfx λ∗r δ, x λ∗
μr δ− λ∗arfx λ∗r, x λ∗
μr
λarfx λ∗r δ, x λ∗
μr δ− fx λ∗r, x λ∗
μr
−λ∗− λarfx λ∗r, x λ∗
μr
≤λ∗− λarf0, 0 −λ∗− λfx λ∗r, x λ∗
μr
λ∗− λarf0, 0 − fx λ∗r, x λ∗
μr
≤ 0, ∀r ∈ 0, T.
2.24
From above, we have
F λ
x λ∗ δ≤ F λ∗
x λ∗
x λ∗ < x λ∗ δ. 2.25
Trang 8SetR1 ||x λ∗t δ|| for t ∈ 0, T and P R1 {x ∈ P : ||x|| < R1} We have F λ x / x for x ∈ ∂R1.
ByLemma 2.1,iF λ , P R1, P 1 In view of H6, we can choose L > R1> 0 such that
fx, ψs≥ Jx p−1 ,
Jλ q−1 δ2
T β φ q Y1ar∇r
> 1 for x > L, s ∈ −τ, 0. 2.26
Set
R2 T β δ L 1, P R2 x ∈ P : x < R2
. 2.27
Similar toLemma 2.3, it is easy to obtain that
F λ x F λ xT
≥ δφ q
T
0
λarfxr, xμr∇r
≥ δφ q
Y1
λarfxr, ψμr∇r
> δJλ q−1min
t∈Y1
xtφ q
Y1
ar∇r
≥ Jλ T β q−1 δ2 x φ q
Y1
ar∇r
> x for x ∈ ∂P R2.
2.28
In view ofLemma 2.1,iF λ , P R2, P 0 By the additivity of fixed point index,
iF λ , P R2 \ P R1, P iF λ , P R2, P− iF λ , P R1, P −1. 2.29
So,F λhas at least two fixed points inP 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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