Volume 2007, Article ID 87818,15pages doi:10.1155/2007/87818
Research Article
Positive Solutions of Two-Point Right Focal
Eigenvalue Problems on Time Scales
Yuguo Lin and Minghe Pei
Received 26 May 2007; Revised 20 July 2007; Accepted 21 September 2007
Recommended by Patricia J. Y. Wong
We offer criteria for the existence of positive solutions for two-point right focal eigenvalue problems (−1)n−pyΔn(t)=λ f(t,y(σn−1(t)),yΔ(σn−2(t)),. . .,yΔp−1
(σn−p(t))),t∈[0, 1]∩
T,yΔi(0)=0, 0≤i≤p−1,yΔi(σ(1))=0,p≤i≤n−1, whereλ >0,n≥2, 1≤p≤n−1 are fixed andTis a time scale.
Copyright © 2007 Y. Lin and M. Pei. 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
In this paper, we present results governing the existence of positive solutions to the dif-ferential equation on time scales of the form
(−1)n−pyΔn(t)=λ ft,yσn−1(t),. . .,yΔp−1
σn−p(t), t∈[0, 1]∩T (1.1)
subject to the two-point right focal boundary conditions
yΔi(0)=0, 0≤i≤p−1,
yΔiσ(1)=0, p≤i≤n−1, (1.2)
whereλ >0,p,nare fixed integers satisfyingn≥2, 1≤p≤n−1, 0,1∈T, with 0< σ(1) andρ(σ(1))=1 and f : [0, 1]×Rp→Ris continuous.
We say thaty(t) is a positive solution of BVP (1.1), (1.2) ify(t)∈Cnrd[0, 1] is a solution
BVP (1.1), (1.2) has a positive solution y, thenλis called an eigenvalue and ya corre-sponding eigenfunction of BVP (1.1), (1.2). We let
E=λ >0 : BVP (1.1), (1.2) has at least one positive solution (1.3)
be the set of eigenvalues of BVP (1.1), (1.2).
To understand the notations used in BVP (1.1), (1.2), we recall some standard defini-tions as follows. The reader may refer to [1] for an introduction to the subject.
(a) LetTbe a time scale, that is,Tis a closed subset ofR. We assume thatThas the topology that it inherits from the standard topology onR. Throughout, for any a,b(> a), the interval [a,b] is defined as [a,b]= {t∈T|a≤t≤b}. Analogous notations for open and half-open intervals will also be used in the paper. We also use the notationR[c,d] to denote the real interval{t∈R|c≤t≤d}.
(b) Fort <supTands >infT, the forward jump operatorσand the backward jump operatorρare, respectively, defined by
σ(t)=infτ∈T|τ > t∈T, ρ(s)=supτ∈T|τ < s∈T. (1.4)
We defineσn(t)=σ(σn−1(t)) withσ0(t)=t. Similar definition is used forρn(s). (c) Fixt∈T. Let y:T→R. We define yΔ(t) to be the number (if it exists) with the
property that givenε >0, there is a neighborhoodUoftsuch that for alls∈U,
y
σ(t)−y(s)−yΔ(t)σ(t)−s< εσ(t)−s. (1.5)
We callyΔ(t) the delta derivative ofy(t). DefineyΔn(t) to be the delta derivative ofyΔn−1(t), that is,yΔn(t)=(yΔn−1(t))Δ.
(d) IfFΔ(t)=f(t), then we define the integral
t
a f(τ)Δτ=F(t)−F(a). (1.6)
(e) Ifσ(t)> t, then call the pointtright-scattered; while ifρ(t)< t, then saytis left-scattered. Ifσ(t)=t, then call the pointtright-dense; while ifρ(t)=t, then sayt is left-dense.
Focal boundary value problems have attracted a lot of attention in the recent literature, see [2–7]. Recently, many papers have discussed the existence of nonnegative solution of right focal boundary value problem on time scales, see [8–12]. Motivated by the works mentioned above, the purpose of this article is to present results which guarantee the existence of one or more positive solutions to BVP (1.1), (1.2).
2. Preliminary
Lemma2.2 [1]. For nonnegative integern,
hn(t,s)=(−1)ngn(s,t), t∈T, s∈Tk
)k.Further, the functions satisfy the inequalities
hn(t,s)≥0, gn(t,s)≥0 ∀t≥s. (2.6)
Lemma2.3 [9]. Green’s function of the boundary value problem
Lemma2.4. Letk(t,s)be Green’s function of the equation
(−1)n−pyΔn−p+1(t)=0, t∈0,σn−p+1(1) (2.9)
subject to the boundary conditions
yΔi(0)=0, 0≤i≤p−1,
yΔiσ(1)=0, p≤i≤n−1. (2.10)
Then
L(t)·gn−pσ(s), 0)≤k(t,s)≤gn−pσ(s), 0, (t,s)∈0,σn−p+1(1)×[0, 1], (2.11)
where
L(t)= t
σn−p+1(1)≤1, t∈
0,σn−p+1(1). (2.12)
Proof. It is clear that
k(t,s)=KΔtp−1(t,s)
= ⎧ ⎪ ⎨ ⎪ ⎩
(−1)n−phn−p
0,σ(s)−hn−p
t,σ(s), t≤σ(s),
(−1)n−phn−p
0,σ(s), t≥σ(s),
= ⎧ ⎪ ⎨ ⎪ ⎩
gn−p
σ(s), 0−gn−p
σ(s),t, t≤σ(s),
gn−p
σ(s), 0, t≥σ(s),
(2.13)
wheret∈[0,σn−p+1(1)] ands∈[0, 1].
Obviously,
L(t)gn−p
σ(s), 0≤gn−p
σ(s), 0, t≥σ(s). (2.14)
Next, we will prove by induction that fork=1, 2,. . ., andt≤σ(s),
L(t)gkσ(s), 0≤gkσ(s), 0−gkσ(s),t≤gkσ(s), 0. (2.15)
Fork=1, we have
g1
σ(s), 0−g1
σ(s),t=σ(s)−σ(s)−t
=t≥σn−pt+1(1)·σ(s)=L(t)g1
σ(s), 0. (2.16)
Fort≤σ(s),
continuous operator such that either
(i)Tu ≤ u,u∈C∩∂Ω1; Tu ≥ u,u∈C∩∂Ω2;or
(ii)Tu ≥ u,u∈C∩∂Ω1; Tu ≤ u,u∈C∩∂Ω2.
Then,Thas a fixed point inC∩(Ω2\Ω1).
3. Main results
In this section, by usingLemma 2.6, we offer criteria for the existence of positive solution of BVP (1.1), (1.2).
(A2) There exists constanta >0 such that
lim
up→0+(t,u1,u2,...,up−min1)∈[0,1]×R[0,a]p−1
ft,u1,u2,. . .,up
up = ∞
. (3.2)
(A3) f(t,u1,u2,. . .,up) is nondecreasing in uj for each fixed (t,u1,u2,. . .,uj−1,uj+1,
. . .,up)
Definition 3.1. Define f ∈Crd(T:R) to be right-dense continuous if for all t∈T,
lims→t+f(s)= f(t) at every right-dense pointt∈T, lims→t−f(s) exists and is finite at
every left-dense pointt∈T.
LetCrdn([0, 1]) denote the space of functions:
Cn rd
[0, 1]=y:y∈C0,σn(1),. . .,yΔn−1
∈C0,σ(1),yΔn∈Crd
[0, 1]. (3.3)
LetB= {y∈Cn
rd([0, 1]) :yΔ
i
(0)=0, 0≤i≤p−2}be a Banach space with the norm
y =supt∈[0,σn−p+1(1)]|yΔ p−1
(t)|, and let
C=y∈B:yΔp−1(t)≥L(t)y,t∈0,σn−p+1(1), (3.4)
whereL(t) is given inLemma 2.4.
It is obvious thatCis a cone inB. FromyΔi(0)=0, 0≤i≤p−2, it follows that for all y∈C,
hp−i(t, 0)
σn−p+1(1)·y ≤yΔ
i
(t)≤δy, i=1, 2,. . .,p−1, (3.5)
where
δ:=σn(1)p−1. (3.6)
Remark 3.2. If u,v∈C and uΔp−1(t)≥vΔp−1(t), t∈[0,σn−p+1(1)], it follows from
uΔi(0)=vΔi(0)=0, 0≤i≤p−2 thatuΔi(t)≥vΔi(t),t∈[0,σn−i(1)], 0≤i≤p−1.
Let the operatorS:C→Bbe defined by
(Sy)(t)=λ σ(1)
0 K(t,s)f
s,yσn−1(s),. . .,yΔp−1σn−p(s)Δs, t∈0,σn(1),
(Sy)Δp−1(t)=λ σ(1)
0 k(t,s)f
s,yσn−1(s),. . .,yΔp−1
σn−p(s)Δs, t∈0,σn−p+1(1).
(3.7)
To obtain a positive solution of BVP (1.1), (1.2), we seek a fixed point of the operator Sin the coneC.
Proof. FromLemma 2.4, we know that fort∈[0,σn−p+1(1)],
Lemma3.4. The operatorS:C→Cis completely continuous.
Proof. First we will prove that the operatorSis continuous. Let{ym},y∈Cbe such that
Then, it is easy to see that
Next, to show complete continuity, we will apply Arzela-Ascoli theorem. LetΩbe a bounded subset ofCand lety∈Ω. Now there existsL >0 such that for ally∈Ω,
supyΔp−1≤L, supyΔi≤δL, i=0, 1,. . .,p−2, (3.14)
whereδis given in (3.6). Let
M= sup
(s,u1,u2,...,up)∈[0,1]×R[0,δL]p−1×R[0,L] f
s,u1,u2,. . .,up. (3.15)
Clearly, we have fort∈[0,σn(1)],
(Sy)(t)≤λM σ(1)
0 K(t,s)Δs≤λMt∈[0,supσn(1)]
σ(1)
0 K(t,s)Δs, (3.16)
and fort,t∈[0,σn(1)],
(Sy)(t)−(Sy)(t)≤λM σ(1) 0
K(t,s)−K(t,s)Δs. (3.17)
The Arzela-Ascoli theorem guarantees thatSΩis relatively compact, soS:C→Cis
com-pletely continuous.
For anyL >0, define
rL=ML L
gn−p+1
σ(1), 0−1, (3.18)
where
ML= sup
(t,u1,u2,...,up)∈[0,1]×R[0,δL]p−1×R[0,L]
ft,u1,u2,. . .,up
, (3.19)
andδis given in (3.6).
Theorem3.5. Let(A1)hold. For anyλ∈R(0,rL], BVP (1.1), (1.2) has at least one positive
solutionysuch thaty ≥L.
Proof. LetL >0 be given and letλ∈R(0,rL] be fixed. We separate the proof into the following two steps.
Step 1. Let
Ω1=
y∈B:y< L. (3.20)
It follows fromLemma 2.4that for ally∈∂Ω1∩C,
(Sy)Δp−1(t)=λ σ(1)
0 k(t,s)f
s,yσn−1(s),. . .,yΔp−1
σn−p(s)Δs
≤λML σ(1)
0 gn−p
σ(s), 0Δs
=λML·gn−p+1
σ(1), 0≤L, t∈0,σn−p+1(1).
Hence
Sy ≤ y, y∈∂Ω1∩C. (3.22)
Step 2. From (A1), we know that there existsη > L(ηcan be chosen arbitrarily large) such
that for all (u1,u2,. . .,up)∈R[σ1η,∞)×R[σ2η,∞)× ··· ×R[σpη,∞),
min t∈[ε,1]
ft,u1,u2,. . .,up
up ≥
σ(1)
ε gn−pσ(s), 0Δs −1
λσp , (3.23)
where
σi=
hp−i+1(ε, 0)
σn−p+1(1), i=1, 2,. . .,p. (3.24)
So,
ft,u1,u2,. . .,up≥
σ(1)
ε gn−pσ(s), 0Δs −1
η
λ (3.25)
on [ε, 1]×R[σ1η,∞)×R[σ2η,∞)× ··· ×R[σpη,∞).
UsingLemma 2.4, we know that
(Sy)Δp−1σn−p+1(1)=λ σ(1)
0 k
σn−p+1(1),sfs,yσn−1(s),. . .,yΔp−1σn−p(s)Δs
≥λ σ(1)
ε gn−p
σ(s), 0Δs· σ(1)
ε gn−pσ(s), 0Δs −1
η
λ =η.
(3.26)
By lettingΩ2= {y∈B:y< η}, we have
Sy ≥ y, y∈∂Ω2∩C. (3.27)
Therefore, it follows fromLemma 2.6that BVP (1.1), (1.2) has a solutiony∈Csuch
thaty ≥L.
Theorem3.6. Let(A2)hold. For anyλ∈R(0,rL] (L∈R(0,a]), BVP (1.1), (1.2) has at
least one positive solutionysuch that0<y ≤L.
Proof. LetL∈R(0,a] be given and letλ∈R(0,rL] be fixed. Let
Ω3=
y∈B:y< L. (3.28)
Then fory∈C∩∂Ω3, we have fromLemma 2.4that fort∈[0,σn−p+1(1)],
(Sy)Δp−1(t)=λ σ(1)
0 k(t,s)f
s,yσn−1(s),. . .,yΔp−1
σn−p(s)Δs
≤λML σ(1)
0 gn−p
σ(s), 0Δs=λML·gn−p+1
σ(1), 0≤L.
Therefore,
Sy ≤ y, y∈C∩∂Ω3. (3.30)
From (A2), there existsη,r0whereλη
σ(1)
0 gn−p(σ(s), 0)L(s)Δs >1 withr0< Lsuch that
ft,u1,u2,. . .,up
≥ηup, (3.31)
on [0, 1]×R[0,δr0]p−1×R[0,r0], whereδis given in (3.6).
Fory∈Candy =r0, we have fromLemma 2.4that
(Sy)Δp−1σn−p+1(1)=λ σ(1)
0 k
σn−p+1(1),sfs,yσn−1(s),. . .,yΔp−1
σn−p(s)Δs
≥λ σ(1)
0 gn−p
σ(s), 0ηyΔp−1σn−p(s)Δs
≥λ σ(1)
0 gn−p
σ(s), 0L(s)ηyΔs >y =r0.
(3.32)
By lettingΩ4= {y∈B:y< r0}, we have
Sy ≥ y, y∈C∩∂Ω4. (3.33)
Therefore, it follows fromLemma 2.6that BVP (1.1), (1.2) has a solutiony∈Csuch that
0< r0≤ y ≤L.
Theorem3.7. Let(A2)and(A3)hold. Suppose thatλ0∈E. ThenR(0,λ0]⊆E.
Proof. Let y0 be the eigenfunction corresponding to the eigenvalue λ0. Then for t∈
[0,σn−p+1(1)],
yΔ0p−1(t)=λ0 σ(1)
0 k(t,s)f
s,y0
σn−1(s),. . .,yΔ0p−1
σn−p(s)Δs. (3.34)
Fromy0∈C, we have
t
σn−p+1(1)·y0≤yΔ
p−1
0 (t)≤y0, t∈
0,σn−p+1(1). (3.35)
We will consider two cases.
Case 1. f(t, 0, 0,. . ., 0)≡0,t∈[0, 1].Define
K=y∈C: 0≤yΔp−1(t)≤yΔ0p−1(t),t∈
For y∈K andλ∈R(0,λ0), fromLemma 3.3, (A3) andRemark 3.2, we have that for
For y∈K and λ∈R(λ∗,λ0), from Remark 3.2 and (A3), we have that for t∈[0,
Hence,SmapsK intoK. Schauder’s fixed point theorem guarantees that Shas a fixed point inK. ThusR(rL,λ0)⊆R(λ∗,λ0)⊆E.
Therefore,R(0,λ0]⊂E.
From Theorems3.5,3.6, and3.7, we can easily get the following results.
It follows that
1≥λ σ(1)
0 gn−p
σ(s), 0NL(s)Δs. (3.47)
Let
λ∗=
N σ(1)
0 gn−p
σ(s), 0L(s)Δs −1
. (3.48)
Therefore, BVP (1.1), (1.2) has no solution forλ > λ∗.
Acknowledgment
The authors thank the referee for valuable suggestions which led to improvement of the original manuscript.
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Yuguo Lin: Department of Mathematics, Bei Hua University, Jilin City 132013, China Email address:[email protected]
