doi:10.1155/2009/123565
Research Article
Existence of Solutions for Nonlinear
Four-Point
p
-Laplacian Boundary Value
Problems on Time Scales
S. Gulsan Topal, O. Batit Ozen, and Erbil Cetin
Department of Mathematics, Ege University, Bornova, 35100 Izmir, Turkey
Correspondence should be addressed to S. Gulsan Topal,f.serap.topal@ege.edu.tr
Received 16 March 2009; Accepted 20 July 2009
Recommended by Alberto Cabada
We are concerned with proving the existence of positive solutions of a nonlinear second-order four-point boundary value problem with ap-Laplacian operator on time scales. The proofs are based on the fixed point theorems concerning cones in a Banach space. Existence result forp-Laplacian boundary value problem is also given by the monotone method.
Copyrightq2009 S. Gulsan Topal et al. 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 any time scale such that0,1be subset ofT. The concept of dynamic equations on time scales can build bridges between differential and difference equations. This concept not only gives us unified approach to study the boundary value problems on discrete intervals with uniform step size and real intervals but also gives an extended approach to study on discrete case with non uniform step size or combination of real and discrete intervals. Some basic definitions and theorems on time scales can be found in1,2.
In this paper, we study the existence of positive solutions for the following nonlinear four-point boundary value problem with ap-Laplacian operator:
φp
xΔ∇t htft, xt 0, t∈0,1, 1.1
αφp
xρ0 −Ψφp
xΔξ 0, γφpxσ1 δφpxΔ
η 0, 1.2
whereφpsis an operator, that is,φps |s|p−2sforp >1,φp−1s φqs,where 1/p
H1the functionf∈ C0,1×0,∞,0,∞,
H2the function h ∈ CldT,0,∞ and does not vanish identically on any closed subinterval ofρ0, σ1and 0<ρσ01ht∇t <∞,
H3 Ψ : R → R is continuous and satisfies that there exist B2 ≥ B1 > 0 such that
B1s≤Ψs≤B2s,fors∈0,∞.
In recent years, the existence of positive solutions for nonlinear boundary value problems withp-Laplacians has received wide attention, since it has led to several important mathematical and physical applications3,4. In particular, forp 2 orφps sis linear,
the existence of positive solutions for nonlinear singular boundary value problems has been obtained5,6.p-Laplacian problems with two-, three-, andm-point boundary conditions for ordinary differential equations and difference equations have been studied in7–9and the references therein. Recently, there is much attention paid to question of positive solutions of boundary value problems for second-order dynamic equations on time scales, see10–13. In particular, we would like to mention some results of Agarwal and O’Regan14, Chyan and Henderson5, Song and Weng15, Sun and Li16, and Liu17, which motivate us to consider thep-Laplacian boundary value problem on time scales.
The aim of this paper is to establish some simple criterions for the existence of positive solutions of thep-Laplacian BVP1.1-1.2. This paper is organized as follows. InSection 2 we first present the solution and some properties of the solution of the linearp-Laplacian BVP corresponding to1.1-1.2. Consequently we define the Banach space, cone and the integral operator to prove the existence of the solution of1.1-1.2. InSection 3, we state the fixed point theorems in order to prove the main results and we get the existence of at least one and two positive solutions for nonlinearp-Laplacian BVP1.1-1.2. Finally, using the monotone method, we prove the existence of solutions forp-Laplacian BVP inSection 4.
2. Preliminaries and Lemmas
In this section, we will give several fixed point theorems to prove existence of positive solutions of nonlinearp-Laplacian BVP1.1-1.2. Also, to state the main results in this paper, we employ the following lemmas. These lemmas are based on the linear dynamic equation:
φp
xΔ∇t yt 0. 2.1
Lemma 2.1. Suppose condition (H2) holds, then there exists a constantθ∈ρ0,σ1−ρ0/2
that satisfies
0<
σ1−θ
θ
ht∇t <∞. 2.2
Furthermore, the function
At
t
θ
φq
t
s
hu∇u Δs
σ1−θ
t
φq
s
t
hu∇u
is a positive continuous function, therefore,Athas a minimum onθ, σ1−θ, hence one supposes that there existsL >0such thatAt≥Lfort∈θ, σ1−θ.
Proof. It is easily seen thatAtis continuous onθ, σ1−θ. Let
A1t t
θ
φq
t
s
hu∇u Δs, A2t σ1−θ
t
φq
s
t
hu∇u
Δs. 2.4
Then, from conditionH2, we have that the functionA1tis strictly monoton nondecreasing
on θ, σ1 −θ and A1θ 0, the function A2t is strictly monoton nonincreasing on θ, σ1−θandA2σ1−θ 0, which impliesL mint∈θ,σ1−θAt>0.
Throughout this paper, letE C0,1, thenEis a Banach space with the normx
supt∈0,1|xt|. Let
K {x∈E:xt≥0, xtconcave function on0,1}. 2.5
Lemma 2.2. Letxt∈Kandθbe as inLemma 2.1, then
xt≥ θ
σ1−ρ0x, ∀t∈θ, σ1−θ. 2.6
Proof. Supposeτ inf{ς ∈ ρ0, σ1 : supt∈ρ0,σ1xt xς}.We have three different cases.
iτ ∈ ρ0, θ. It follows from the concavity of xt that each point on the chard betweenτ, xτandσ1, xσ1is below the graph ofxt, thus
xt≥xτ xσ1−xτ
σ1−τ t−τ, t∈θ, σ1−θ, 2.7
then
xt≥ min
t∈θ,σ1−θ
xτ xσ1−xτ σ1−τ t−τ
xτ xσ1−xτ
σ1−τ σ1−θ−τ σ1−θ−τ
σ1−τ xσ1
θ
σ1−τxτ
≥ θ
σ1−ρ0xτ,
2.8
iiτ ∈θ, σ1−θ. Ift∈θ, τ, similarly, we have
xt≥xτ xτ−x
ρ0
τ−ρ0 t−τ
≥xτ xτ−x
ρ0
τ−ρ0 θ−τ
θ−ρ0
τ−ρ0xτ
τ−θ τ−ρ0x
ρ0
≥ θ−ρ0
σ1−ρ0xτ≥
θ
σ1−ρ0xτ.
2.9
Ift∈τ, σ1−θ, similarly, we have
xt≥xτ xσ1−xτ σ1−τ t−τ
≥ min
t∈θ,σ1−θ
xτ xσ1−xτ σ1−τ t−τ
θ σ1−τxτ
σ1−τ−θ
σ1−τ xσ1
≥ θ
σ1−ρ0xτ,
2.10
this meansxt≥θ/σ1−ρ0xfort∈θ, σ1−θ. iiiτ ∈σ1−θ, σ1. Similarly we have
xt≥xτ xτ−x
ρ0
τ−ρ0 t−τ, t∈θ, σ1−θ, 2.11
then
xt≥ min
t∈θ,σ1−θ
xτ xτ−x
ρ0
τ−ρ0 t−τ
θ−ρ0
τ−ρ0xτ
τ−θ τ−ρ0x
ρ0
≥ θ
σ1−ρ0xτ,
2.12
know
whereτis a solution of the following equation
V1t V2t, t∈
Integrating2.1onτ, t,we get
Using the second boundary condition and the formula2.18fort η, we have
xσ1 φq
Lemma 2.4. Suppose that the conditions inLemma 2.3hold. Then there exists a constantAsuch that
the solutionxtofp-Laplacian BVP2.1-1.2satisfies
max
t∈ρ0,σ1|xt| ≤At∈ρmax0,σ1
Now, we define a mappingT :K → Egiven by
Lemma 2.5. Suppose that the conditions (H1)–(H3) hold.T:K → Kis completely continuous.
Proof. SupposeP ⊂ Kis a bounded set. LetM > 0 be such thatx ≤ M,x ∈ P. For any
By the Arzela-Ascoli theorem, we can easily see that T is completely continuous operator.
For convenience, we set
In order to follow the main results of this paper easily, now we state the fixed point theorems which we applied to prove Theorems3.1–3.4.
Theorem 2.6see18 Krasnoselskii fixed point theorem. LetEbe a Banach space, and let
K⊂Ebe a cone. AssumeΩ1andΩ2are open, bounded subsets ofEwith0∈Ω1, Ω1 ⊂Ω2, and let
A:K∩Ω2\Ω1
−→K 2.31
be a completely continuous operator such that either
iAu ≤ uforu∈K∩∂Ω1, Au ≥ uforu∈K∩∂Ω2; iiAu ≥ uforu∈K∩∂Ω1, Au ≤ uforu∈K∩∂Ω2
hold. ThenAhas a fixed point inK∩Ω2\Ω1.
Theorem 2.7see19 Schauder fixed point theorem. LetEbe a Banach space, and letA :
E → Ebe a completely continuous operator. AssumeK ⊂Eis a bounded, closed, and convex set. If
AK⊂K, thenAhas a fixed point inK.
Theorem 2.8see20 Avery-Henderson fixed point theorem. LetPbe a cone in a real Banach
spaceE. Set
Pφ, r u∈ P:φu< r. 2.32
Ifμandφare increasing, nonnegative, continuous functionals onP, letθbe a nonnegative continuous functional onPwithθ0 0such that for some positive constantsrandM,
φu≤θu≤μu, u ≤Mφu 2.33
for allu∈ Pφ, r. Suppose that there exist positive numbersp < q < rsuch thatθλu≤λθufor all0≤λ≤1andu∈∂Pθ, q.
IfA:Pφ, r → Pis a completely continuous operator satisfying iφAu> rfor allu∈∂Pφ, r,
iiθAu< qfor allu∈∂Pθ, q,
iiiPμ, q/∅andμAu> pfor allu∈∂Pμ, p,
thenAhas at least two fixed pointsu1andu2such that
p < μu1 withθu1< q, q < θu2 withφu2< r. 2.34
3. Main Results
Theorem 3.1. Assume that (H1)–(H3) are satisfied. In addition, suppose thatfsatisfies
A1ft, x≥φpmk1forθk1/σ1−ρ0≤x≤k1, A2ft, x≤φpMk2for0≤x≤k2,
wherem∈R1,∞andM∈0, R2. Then thep-Laplacian BVP1.1-1.2has a positive solution
xtsuch thatk1≤ x ≤k2.
Proof. Without loss of generality, we supposek1< k2. For anyx∈K, byLemma 2.2, we have
xt≥ θ
σ1−ρ0x, ∀t∈θ, σ1−θ. 3.1
We define two open subsetsΩ1 andΩ2ofEsuch thatΩ1 {x∈ K :x < k1}and Ω2 {x∈K:x< k2}.
Forx∈∂Ω1, by3.1, we have
k1 x ≥xt≥ θ
σ1−ρ0x ≥
θ
σ1−ρ0k1, t∈θ, σ1−θ. 3.2
Fort∈θ, σ1−θ, ifA1holds, we will discuss it from three perspectives.
iIfτ ∈θ, σ1−θ, thus forx∈∂Ω1, byA1andLemma 2.1, we have
2Tx 2Txτ
≥ τ
ρ0
φq
τ
s
hrfr, xr∇r
Δs
σ1
τ
φq
s
τ
hrfr, xr∇r
Δs
≥mk1 τ
θ
φq
τ
s
hr∇r
Δsmk1 σ1−θ
τ
φq
s
τ
hr∇r
Δs
≥mk1Aτ≥mk1L≥R1k1L 2k1 2x.
3.3
iiIfτ ∈σ1−θ, σ1, thus forx∈∂Ω1, byA1andLemma 2.1, we have
Tx Txτ
≥ τ
ρ0φq
τ
s
hrfr, xr∇r
Δs
≥mk1 σ1−θ
θ
φq
σ1−θ
s
hr∇r Δs
≥mk1Aσ1−θ≥mk1L≥2k1 > k1 x.
iiiIfτ ∈ρ0, θ, thus forx∈∂Ω1, byA1andLemma 2.1, we have
Existence of at least one positive solution is also proved using Schauder fixed point theoremTheorem 2.7. Then we have the following result.
Theorem 3.2. Assume that (H1)–(H3) are satisfied. IfRsatisfies
Q
R2 ≤R,
whereQsatisfies
φpQ≥max x≤R
ft, xt fort∈0,1, 3.8
then thep-Laplacian BVP1.1-1.2has at least one positive solution.
Proof. LetKR : {x∈K:x ≤R}. Note thatKRis closed, bounded, and convex subset ofE
to which the Schauder fixed point theorem is applicable. DefineT :KR → Eas in2.27for
t∈ρ0, σ1. It can be shown thatT :KR → Eis continuous. Claim thatT :KR → KR. Let
x∈KR. By using the similar methods used in the proof ofTheorem 3.1, we have
Tx Txτ
φq
1
αΨ
τ
ξ
hrfr, xr∇r
τ
ρ0φq
τ
s
hrfr, xr∇r
Δs
≤φq
B2
α
σ1
ρ0hrfr, xr∇r
σ1
ρ0φq
σ1
ρ0hrfr, xr∇r Δs
≤Q
φq
B2
α
σ1−ρ0
φq σ1
ρ0hr∇r
Q 1 R2 ≤R,
3.9
which impliesTx ∈ KR. The compactness of the operatorT : KR → KR follows from the
Arzela-Ascoli theorem. HenceT has a fixed point inKR.
Corollary 3.3. Iffis continuous and bounded on0,1×R, then thep-Laplacian BVP1.1-1.2
has a positive solution.
Now we will give the sufficient conditions to have at least two positive solutions for
p-Laplacian BVP1.1-1.2. Set
Pt: φq
t
θ
hr∇r φq
σ1−θ
t
hr∇r , t∈θ, σ1−θ. 3.10
The functionPtis positive and continuous onθ, σ1−θ. Therefore,Pthas a minimum onθ, σ1−θ. Hence we suppose there existsN >0 such thatPt≥N.
Also, we define the nonnegative, increasing continuous functionsΥ,Φ,andΓby
Υx 1
2xθ xσ1−θ,
Φx max
t∈ρ0,θ∪σ1−θ,σ1xt,
Γx max
t∈ρ0,σ1xt.
We observe here that, for everyx ∈ K,Υx ≤ Φx ≤ Γx and fromLemma 2.2,x ≤
σ1−ρ0/θΥx. Also, for 0≤λ≤1,Φλx λΦx.
Theorem 3.4. Assume that (H1)–(H3) are satisfied. Suppose that there exist positive numbersa <
b < csuch that the function f satisfies the following conditions: ift, x≥φpmaforx∈0, a,
iift, x≤φpMbforx∈0,σ1−ρ0/θb,
iiift, x≥φp2/θncforx∈θ/σ1−ρ0c,σ1−ρ0/θc,
for positive constantsm∈R1,∞,M∈0, R2,andn∈0, N. Then thep-Laplacian BVP1.1 1.2has at least two positive solutionsx1, x2such that
a < max
t∈ρ0,σ1x1t with t∈ρ0,θmax∪σ1−θ,σ1x1t< b,
b < max
t∈ρ0,θ∪σ1−θ,σ1x2t with
1
2x2θ x2σ1−θ< c.
3.12
Proof. Define the cone as in 2.5. From Lemmas 2.2and 2.3 and the conditionsH1and H2, we can obtainTK⊂K. Also fromLemma 2.5, we see thatT :K → Kis completely continuous.
We now show that the conditions ofTheorem 2.8are satisfied.
To fulfill propertyiofTheorem 2.8, we choosex∈∂PΥ, c, thusΥx 1/2xθ xσ1−θ c. Recalling thatx ≤σ1−ρ0/θΥx σ1−ρ0/θc, we have
θ
σ1−ρ0x ≤xt≤
σ1−ρ0
θ c. 3.13
Then assumptioniiiimpliesft, x> φp2/θncfort∈θ, σ1−θ. We have three
different cases.
aIfτ∈σ1−θ, σ1, we have
ΥTx 1
2Txθ Txσ1−θ
≥Txθ φq
1
αΨ
τ
ξ
hrfr, xr∇r
θ
ρ0
φq
τ
s
hrfr, xr∇r
Δs
≥ θ
ρ0
φq
τ
s
hrfr, xr∇r
Δs≥
θ
ρ0
φq
σ1−θ
θ
hrfr, xr∇r Δs
≥ θ
ρ0
φq
σ1−θ
θ
hrφp
2
θnc
∇r Δs 2 θncφq
σ1−θ
θ
hr∇r θ−ρ0
≥ 2
θncPθθ≥
2
NcPθ≥2c.
Theniiyieldsft, x≤φpMbfort∈ρ0, σ1.
So conditioniiofTheorem 2.8holds.
bIfτ∈θ, σ1−θ, we have
4. Monotone Method
In this section, we will prove the existence of solution ofp-Laplacian BVP1.1-1.2by using upper and lower solution method. We define the set
D:
x:φp
xΔ∇ is continuous on0,1
. 4.1
Definition 4.1. A real-valued functionut ∈ Donρ0, σ1is a lower solution for1.1 1.2if
φp
uΔ∇t htft, ut>0, t∈0,1,
αφp
uρ0 −Ψφp
uΔξ ≤0, γφpuσ1 δφpuΔ
η≤0.
4.2
Similarly, a real-valued functionvt∈Donρ0, σ1is an upper solution for1.1 1.2if
φp
vΔ∇t htft, vt<0, t∈0,1,
αφp
vρ0 −Ψφp
vΔξ ≥0, γφpvσ1 δφpvΔ
η≥0.
4.3
We will prove when the lower and the upper solutions are given in the well order, that is,u≤v, thep-Laplacian BVP1.1-1.2admits a solution lying between both functions.
Theorem 4.2. Assume that (H1)–(H3) are satisfied anduandvare, respectively, lower and upper
solutions for thep-Laplacian BVP1.1-1.2such thatu≤vonρ0, σ1. Then thep-Laplacian BVP1.1-1.2has a solutionxt∈ut, vtonρ0, σ1.
Proof. Consider thep-Laplacian BVP:
φp
xΔ∇t htFt, xt 0, t∈0,1,
αφp
xρ0 −Ψφp
xΔξ 0, γφpxσ1 δφpxΔ
η 0,
4.4
where
Ft, xt
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ft, vt, xt> vt,
ft, xt, ut≤xt≤vt,
ft, ut, xt< ut,
4.5
Clearly, the functionF is bounded fort ∈ 0,1 and satisfies conditionH1. Thus byTheorem 3.2, there exists a solutionxtof thep-Laplacian BVP4.4. We first show that
xt≤vtonρ0, σ1. Setzt xt−vt. Ifxt≤vtonρ0, σ1is not true, then there exists at0 ∈ρ0, σ1such thatzt0 maxt∈ρ0,σ1{xt−vt} >0 has a positive
maximum. Consequently, we know thatzΔt0 ≤ 0 and there existst1 ∈ ρ0, t0such that
zΔt ≥ 0 ont1, t0. On the other hand by the continuity ofztatt0,we know there exists
t2∈ρ0, t0such thatzt>0 ont2, t0.Lett max{t1, t2}, then we havezΔt≥0 ont, t0.
Thus we get
zΔt≥0 ⇒φp
xΔt≥φp
vΔt,
zΔt0≤0 ⇒φp
xΔt0
≤φp
vΔt0
.
4.6
Therefore,
0≥φp
xΔt0
−φp
vΔt0
−φp
xΔt−φp
vΔt
t0
t
φp
xΔ−φp
vΔ∇t∇t
t0
t
φp
xΔ∇t−φp
vΔ∇t
∇t
>
t0
t
−htft, vt htft, vt∇t 0,
4.7
which is a contradiction and thust0cannot be an element ofρ0, σ1.
Ift0 ρ0, from the boundary conditions, we have
αφp
xρ0≤B2φp
xΔξ ⇒φp
α1−qxρ0≤φ p
B1−2 qxΔξ,
αφp
vρ0≥B1φp
vΔξ ⇒φp
α1−qvρ0≥φp
B1−1 qvΔξ.
4.8
Thus we get
α1−qxρ0≤B21−qxΔξ, α1−qvρ0≥B11−qvΔξ. 4.9
From this inequalities, we have
α1−qx−vρ0≤B21−qxΔξ−B11−qvΔξ≤B11−qx−vΔξ,
α1−qzρ0≤B1−1 qzΔξ≤0,
4.10
Ift0 σ1, from the boundary conditions, we have
γφpxσ1 −δφp
xΔη ⇒φp
γ1−qxσ1
−φp
δ1−qxΔη φp
−δ1−qxΔη,
γφpvσ1≥ −δφp
vΔη ⇒φp
γ1−qvσ1
≥ −φp
δ1−qvΔη φp
−δ1−qvΔη.
4.11
Thus we get
γ1−qxσ1 δ1−qxΔη, γ1−qvσ1≥ −δ1−qvΔη. 4.12
From this inequalities, we have
γ1−qx−vσ1≤ −δ1−qx−vΔη,
γ1−qzσ1≤ −δ1−qzΔη≤0,
4.13
which is a contradiction. Thus we havext≤vtonρ0, σ1.
Similarly, we can getut≤xtonρ0, σ1. Thusxtis a solution ofp-Laplacian BVP1.1-1.2which lies betweenuandv.
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