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,[email protected]
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,φpxt∇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∗, λλ∗andλ > λ∗, respectively.
Copyrightq2008 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
LetTbe a closed nonempty subset ofR, and letThave the subspace topology inherited from the Euclidean topology onR. In some of the current literature,Tis called a time scaleplease see1,2. For notation, we will use the convention that, for each intervalJofR, 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
φpxΔt∇λatfxt, xμt0, t∈0, T,
x0t ψt, t∈−τ,0, x0−B0
xΔ00, xΔT 0, 1.1
where φpu is the p-Laplacian operator, that is, φpu |u|p−2u, p > 1, φp−1u φqu,
where 1/p1/q1.
2 Advances in Difference Equations
H2The function a : T→R is left dense continuous i.e.,a ∈ CldT,R and does not
vanish identically on any closed subinterval of0, T. HereCldT,Rdenotes the set
of all left dense continuous functions fromTtoR.
H3ψ :−τ,0→Ris continuous andτ >0.
H4μ:0, T→−τ, Tis continuous,μt≤tfor allt.
H5B0:R→Ris continuous and nondecreasing;B0ks kB0s, k∈Rand satisfies that
there existβ≥δ >0 such that
δs≤B0s≤βs fors∈R. 1.2
H6limx→∞fx, ψs/xp−1∞uniformly ins∈−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 Raffoul7considered a nonlinear functional dynamic equation on a time scale and obtained sufficient conditions for the existence of positive solutions, Li and Liu10studied the eigenvalue problem for second-order nonlinear dynamic equations on time scales. In this paper, our results show that the number of positive solutions of1.1is determined by the parameterλ. That is to say, we prove that there exists a λ∗>0 such that1.1has at least two, one, and no positive solutions for 0< λ < λ∗, λλ∗and λ > λ∗,respectively.
For convenience, we list the following well-known definitions which can be found in
11–13and the references therein.
Definition 1.1. For t < supT and r > infT, 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, tis said to be right scattered, and ifρr< r, ris said to be left scattered. Ifσt t, t is said to be right dense, and ifρr r, r is said to be left dense. If Thas a right-scattered minimumm,defineTκT− {m}; otherwise setTκT.IfThas a left-scattered maximumM, defineTκT− {M}; otherwise setTκT.
Definition 1.2. For x : T→R and t ∈ Tκ, define the deltaderivative of xt, xΔt, to be the numberwhen it exists, with the property that, for anyε > 0, there is a neighborhoodUoft such that
x
σt−xs−xΔtσt−s< εσt−s ∀s∈U. 1.4
Forx : T→Randt∈ Tκ,define the nabla derivative ofxt, x∇t,to be the numberwhen it exists, with the property that, for anyε >0, there is a neighborhoodV oftsuch that
x
Definition 1.3. IfFΔt ft, then define the delta integral by atfsΔsFt−Fa.IfΦ∇t ft, then define the nabla integral byatfs∇s Φt−Φa.
The following lemma is crucial to prove our main results.
4 Advances in Difference Equations
It follows from2.3that the following lemma holds.
Lemma 2.1. LetFλbe defined by2.3. Ifx∈P, then
iFλP⊂P.
iiFλ :P→Pis completely continuous.
The proof ofLemma 2.1can be found in15.
We need to define further subsets of0, Twith respect to the delayμ. Set
Y1:
t∈0, T:μt<0; Y2:
t∈0, T:μt≥0. 2.5
Throughout this paper, we assumeY1/∅andφq
Y1ar∇r>0.
Lemma 2.2. Suppose that (H1)–(H5) hold. Then there exists aλ∗ > 0such that the operatorFλ has a
fixed pointx∗∈P \ {θ}atλ∗, whereθis the zero element of the Banach spaceE.
Proof. Set
et B0
φq
T
0
ar∇r
t
0
φq
T
s ar∇r
Δs, t∈0, T. 2.6
We know thate∈P.Letλ∗M−fe1,where
Mfe rmax∈ 0,Tf
er, eμr≥c >0,
Fλ∗xt B0
φq
T
0
λ∗arfxr, xμr∇r
t
0
φq
T
s λ
∗arfxr, xμr∇rΔs, t∈0, T.
2.7
From above, we have
et≥Fλ∗et. 2.8
Letx0t etandxnt Fλ∗xn−1t, n1,2, . . . , t∈0, T.Then
x0t≥x1t≥ · · · ≥xnt≥ · · · ≥
cλ∗q−1et.
2.9
By the Lebesgue dominated convergence theorem16together withH3, it follows that
{xn}∞n0 {Fλn∗x0}n∞0 decreases to a fixed pointx∗ ∈ P \ {θ}of the operatorFλ∗.The proof is
complete.
Lemma 2.3. Suppose that (H1)–(H6) hold and thatI ⊂ b,∞for someb > 0. Then there exists a constantCI >0such that for allλ∈Iand all possible fixed pointsxofFλatλ, one has||x||< CI.
Proof. Set
We need to prove that there exists a constant CI > 0 such that x < CI for all x ∈ S.If the
number of elements ofSis finite, then the result is obvious. If not, without loss of generality, we assume that there exists a sequence{xn}∞n0such that limn→∞xn ∞, wherexn ∈P is
which is a contradiction. The proof is complete.
Lemma 2.4. Suppose that (H1)–(H5) hold and that the operatorFλ has a positive fixed pointxinP at
6 Advances in Difference Equations
0 decreases to a fixed point
x∗∈P\ {θ}of the operatorFλ∗, andx∗< x.The proof is complete.
which is a contradiction. The proof is complete.
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 toxλ∗ ∈P.Note that
xnt B0
φq
T
0
λnarfxnr, xnμr∇r
t
0
φq
T
s λnarf
xnr, xnμr∇r
Δs.
2.21
Taking the limitation n→∞ to both sides of 2.21, and using the Lebesgue dominated convergence theorem16, we have
xλ∗ B0
φq
T
0
λ∗arfxλ∗r, xλ∗μr∇r
t
0
φq
T
s λ
∗arfx
λ∗r, xλ∗μr∇r
Δs,
2.22
which shows thatFλhas a positive fixed pointxλ∗atλλ∗.The proof is complete.
Theorem 2.7. Suppose that (H1)–(H6) hold. Then there exists aλ∗>0such that1.1has at least two, one, and no positive solutions for0< λ < λ∗, λλ∗andλ > λ∗,respectively.
Proof. Assume thatH1–H5hold. Then there exists aλ∗ > 0 such thatFλ has a fixed point
xλ∗ ∈P \ {θ}atλ λ∗.In view ofLemma 2.4,Fλ also has a fixed pointxλ < xλ∗, xλ ∈P\ {θ} and 0 < λ < λ∗.Note thatf is continuous onR2. For 0 < λ < λ∗,there exists aδ0 > 0 such
that
fxλ∗rδ, xλ∗μrδ−fxλ∗r, xλ∗μr≤f0,0 λ∗
λ −1
forr ∈0, T, 0< δ≤δ0.
2.23
Hence,
λarfxλ∗r δ, xλ∗μrδ−λ∗arfxλ∗r, xλ∗μr
λarfxλ∗r δ, xλ∗μrδ−fxλ∗r, xλ∗μr
−λ∗−λarfxλ∗r, xλ∗μr
≤λ∗−λarf0,0−λ∗−λfxλ∗r, xλ∗μr
λ∗−λarf0,0−fxλ∗r, xλ∗μr
≤0, ∀r ∈0, T.
2.24
From above, we have
8 Advances in Difference Equations
SetR1 ||xλ∗t δ||fort∈0, TandPR1 {x∈P :||x||< R1}. We haveFλx /xforx∈∂R1. ByLemma 2.1,iFλ, PR1, P 1.In view ofH6, we can chooseL > R1>0 such that
fx, ψs≥Jxp−1,
Jλq−1δ2
Tβ φq
Y1
ar∇r
>1 forx > L, s∈−τ,0.
2.26
Set
R2 TδβL1, PR2
x∈P :x< R2
. 2.27
Similar toLemma 2.3, it is easy to obtain that Fλx
FλxT
≥δφq
T
0
λarfxr, xμr∇r
≥δφq
Y1
λarfxr, ψμr∇r
> δJλq−1min
t∈Y1
xtφq
Y1
ar∇r
≥ Jλq− 1δ2
T β xφq
Y1
ar∇r
>x for x∈∂PR2.
2.28
In view ofLemma 2.1,iFλ, PR2, P 0.By the additivity of fixed point index,
iFλ, PR2 \PR1, P
iFλ, PR2, P
−iFλ, PR1, P
−1. 2.29
So,Fλhas 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.
References
1R. Avery and J. Henderson, “Existence of three positive pseudo-symmetric solutions for a one-dimensionalp-Laplacian,”Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 395–404, 2003.
3A. 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.
4F.-H. Wong, “Existence of positive solutions for m-Laplacian boundary value problems,” Applied Mathematics Letters, vol. 12, no. 3, pp. 11–17, 1999.
5C. Song, “Positive solutions of functional difference equations withp-Laplacian operator,”Advances in Difference Equations, vol. 2006, Article ID 82784, 9 pages, 2006.
6H.-R. Sun and W.-T. Li, “Positive solutions for nonlinear three-point boundary value problems on time scales,”Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 508–524, 2004.
7E. 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.
8H. Luo and Q. Ma, “Positive solutions to a generalized second-order three-point boundary-value problem on time scales,”Electronic Journal of Differential Equations, vol. 2005, no. 17, pp. 1–14, 2005.
9Z. He, “Double positive solutions of three-point boundary value problems forp-Laplacian dynamic equations on time scales,”Journal of Computational and Applied Mathematics, vol. 182, no. 2, pp. 304–315, 2005.
10W.-T. Li and X.-L. Liu, “Eigenvalue problems for second-order nonlinear dynamic equations on time scales,”Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 578–592, 2006.
11M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Application, Birkh¨auser, Boston, Mass, USA, 2001.
12R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,”Results in Mathematics, vol. 35, no. 1-2, pp. 3–22, 1999.
13S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
14D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 ofNotes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.
15C. Song, “Existence of solutions for p-Laplacian functional dynamic equations on time scales,”
Electronic Journal of Differential Equations, vol. 2006, no. 113, pp. 1–8, 2006.