Volume 2011, Article ID 591219,11pages doi:10.1155/2011/591219
Research Article
Multiple Positive Solutions for
m
-Point
Boundary Value Problem on Time Scales
Jie Liu
1, 2and Hong-Rui Sun
21Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China
2School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, China
Correspondence should be addressed to Hong-Rui Sun,[email protected]
Received 29 May 2010; Accepted 6 August 2010
Academic Editor: Feliz Manuel Minh ´os
Copyrightq2011 J. Liu and H.-R. Sun. 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.
The purpose of this article is to establish the existence of multiple positive solutions of the dynamic equation on time scalesφuΔt∇htft, ut, uΔt 0, t ∈ 0, TT, subject to the multi-point boundary conditionuΔ0 0, uT im−12aiuξi, whereφ : R → Ris an increasing homeomorphism and satisfies the relationφxy φxφyforx, y∈R, which generalizes the usuallyp-Laplacian operator. An example applying the result is also presented. The main tool of this paper is a generalization of Leggett-Williams fixed point theorem, and the interesting points are that the nonlinearityf contains the first-order derivative explicitly and the operatorφis not necessarily odd.
1. Introduction
The study of dynamic equations on time scales goes back to its founder Hilger1, and is a new area of still fairly theoretical exploration in mathematics. On one hand, the time scales approach not only unifies calculus and difference equations, but also solves other problems that have a mix of stop-start and continuous behavior. On the other hand, the time scales calculus has tremendous potential for application in biological, phytoremediation of metals, wound healing, stock market and epidemic models2–6.
LetTbe a time scalean arbitrary nonempty closed subset of the real numbersR. For each intervalIofR, we define IT I∩T. For more details on time scales, one can refer to
1–3,5. In this paper we are concerned with the existence of at least triple positive solutions to the followingm-point boundary value problems on time scales
uΔ0 0, uT
m−2
i1
aiuξi, 1.2
whereφ:R → Ris an increasing homeomorphism andφxy φxφyforx, y∈R. Multipoint boundary value problem BVP arise in a variety of different areas of applied mathematics and physics, such as the vibrations of a guy wire of a uniform cross section and composed ofNparts of different densities can be set up as a multipoint boundary value problem 7. Small size bridges are often designed with two supported points, which leads to a standard two-point boundary value condition. And large size bridges are sometimes contrived with multipoint supports, which corresponds to a multipoint boundary value condition8. Especially, if we letutdenotes the displacement of the bridge from the unloaded position, and we emphasize the position of the bridge at supporting points near
t0, we can obtain the multipoint boundary condition1.2. The study of multipoint BVPs for linear second-order ordinary differential equations was initiated by Ilin and Moiseev
9, since then many authors studied more general nonlinear multipoint boundary value problems. We refer readers to8,10–14and the references therein.
Recently, when φ is p-Laplacian operator, that is φu |u|p−2up > 1, and the nonlinear term does not depend on the first-order derivative, the existence problems of positive solutions of boundary value problems have attracted much attention, see10,12,15–
22in the continuous case, see15,23–25in the discrete case and11,13,14,26,27in the general time scale setting. From the process of proving main results in the above references, one can notice that the oddness of thep-Laplacian operator is key to the proof. However in this paper the operatorφis not necessary odd, so it improves and generalizes thep-Laplacian operator. One may note this from Example 3.3 in Section 3. In addition, Bai and Ge 16 generalized the Leggett-Williams fixed point theorems by using fixed point index theory. An application of the theorem is given to prove the existence of three positive solutions to the following second-order BVP:
ut ft, ut, ut0, t∈0,1, 1.3
with Dirichlet boundary condition. They also extended the results to four-point BVP in12. When φu |u|p−2u and the nonlinearity f is not involved with the first-order derivative uΔt, in 27, Sun and Li discussed the existence and multiplicity of positive solutions for problems1.1and1.2. The main tools used are fixed point theorems in cones. Thanks to the above-mentioned research articles16,27, in this paper we consider the existence of multiple positive solutions for the more general dynamic equation on time scales
1.1withm-point boundary condition1.2. An example is also given to illustrate the main results. The obtained results are even new for the special cases of difference equations and differential equations, as well as in the general time scale setting. The main result extends and generalizes the corresponding results of Liu18and Webb21 TR, φu u, T
1, m 3, ft, u, v fu, Sun and Li27 φu |u|p−2u, ft, u, v ft, u. We also emphasize that in this paper the nonlinear term f is involved with the first-order delta derivativeuΔt, the operatorφis not necessary odd and have the more generalized form, and the tool is a generalized Leggett-Williams fixed point theorem16.
final part of this section, we present an example to illustrate the application of the obtained result.
Throughout this paper, the following hypotheses hold:
H10, T ∈ T, 0 < ξ1 < ξ2 < · · · < ξm−2 < ρT, ξi ∈ T, ai ≥ 0 fori 1, . . . , m−2,and
d1−m−2i1 ai >0;
H2η max{t ∈ T : 0 < t ≤ T/2}exists and h ∈ Cld0, TT,0,∞such that 0 < T
0 hs∇s <∞andf:0, TT×0,∞×−∞,∞ → 0,∞is continuous.
2. Preliminaries
In this section, we first present some basic definition, then we define an appropriate Banach space, cone, and integral operator, and finally we list the fixed-point theorem which is needed later.
Definition 2.1. SupposePis a cone in a Banach spaceB. The mapαis said to be a nonnegative continuous concaveconvexfunctional onP provided thatα : P → 0,∞is continuous and
αλx 1−λy≥≤λαx 1−λαy ∀x, y∈P, 0≤λ≤1. 2.1
Let the Banach space B Cld10, TT be endowed with the norm u max{u0,uΔ0}, where
u0t∈max0,T
T
|ut|, uΔ
0t∈sup0,T
T
uΔt, 2.2
and choose the coneP⊂Bas
Pu∈B:ut≥0 fort∈0, TT, uΔ0 0 anduis concave in0, TT. 2.3
Now we define the operatorA:P → Bby
Aut −
T
t
φ−1
−
s
0
hτfτ, uτ, uΔτ∇τΔs
− 1 d
m−2
i1
ai T
ξi
φ−1
−
s
0
hτfτ, uτ, uΔτ∇τΔs.
2.4
From the definition ofAand the assumptions of H1,H2, we can easily obtain that for eachu∈P, Aut≥0 fort∈0, TTandAuΔ0 0. From the fact that
we know thatAuis concave in0, TT. ThusA:P → PandAu0is the maximum value of
Aut. In addition, by direct calculation, we get that each fixed point of the operatorAinP
is a positive solution of1.1and1.2. Similar as the proof of Lemma 2.3 in27, it is easy to see thatA:P → Pis completely continuous.
Supposeαandβare two nonnegative continuous convex functionals satisfying
x ≤L maxαx, βx, x∈P, 2.6
whereLis a positive constant, and
Ω x∈P|αx< r, βx< l/∅, r >0, l >0. 2.7
Letr > a > 0,l > 0 be given,α, βnonnegative continuous convex functionals onP
satisfying the relation2.6and2.7, andγa nonnegative continuous concave functional on
P. We define the following convex sets:
Pα, r;β, lu∈P :αu< r, βu< l,
Pα, r;β, lu∈P :αu≤r, βu≤l,
Pα, r;β, l;γ, au∈P :αu< r, βu< l, γu> a,
Pα, r;β, l;γ, au∈P :αu≤r, βu≤l, γu≥a.
2.8
In order to prove our main results, the following fixed point theorem is important in our argument.
Lemma 2.2see16. LetBbe Banach space,P ⊂Ba cone, andr2 ≥d > b > r1 >0, l2 ≥l1 >0. Assume thatαandβare nonnegative continuous convex functionals satisfying2.6and2.7,γis a nonnegative continuous concave functional onPsuch thatγu≤αufor allu∈Pα, r2;β, l2, and
A:Pα, r2;β, l2 → Pα, r2;β, l2is a completely continuous operator. Suppose
C1{u∈Pα, d;β, l2;γ, b:γu> b}/∅,γAu> bforu∈Pα, d;β, l2;γ, b;
C2αAu< r1,βAu< l1foru∈Pα, r1;β, l1;
C3γAu> bforu∈Pα, r2;β, l2;γ, bwithαAu> d.
ThenAhas at least three fixed pointsu1, u2, u3∈Pα, r2;β, l2with
u1∈P
α, r1;β, l1
, u2∈
Pα, r2;β, l2;γ, b
|γu> b,
u3∈P
α, r2;β, l2
\Pα, r2;β, l2;γ, b
∪Pα, r1;β, l1
.
2.9
3. Main Results
In this section, we impose some growth conditions onfwhich allow us to applyLemma 2.2
1.1and1.2. We note that, from the nonnegativity ofhandf, the solution of1.1and1.2 is nonnegative and concave on0, TT.
First in view of Lemma 2.4 in27, we know that foru∈P, there isut≥T−t/Tu fort∈0, TT.So we get
ut≥ T−η T u ≥
1
2u fort∈
0, ηT. 3.1
Let the nonnegative continuous convex functionals α, β and the nonnegative continuous concave functionalγbe defined on the conePby
αu max
t∈0,TT
|ut|, βu sup
t∈0,TT
uΔt, γu min
t∈0,ηT
ut, u∈P. 3.2
Then, it is easy to see thatumax{αu, βu}and2.6,2.7hold. Now, for convenience we introduce the following notations. Let
Sφ−1
T
0
hτ∇τ
, MT−ηφ−1
η
0
hτ∇τ
,
N
T
0
φ−1
s
0
hτ∇τ
Δs 1 d
m−2
i1
ai T
ξi
φ−1
s
0
hτ∇τ
Δs.
3.3
Theorem 3.1. Assumeft,0,0/≡0fort∈0, TT. If there are positive numbersr2≥2b > b > r1 >
0, l2≥l1 >0withb/M≤min{r2/N, l2/S}, such that the following conditions are satisfied
ift, u, v≤min{−φ−r2/N,−φ−l2/S}fort, u, v∈0, TT×0, r2×−l2,0;
iift, u, v>−φ−b/Mfort, u, v∈0, ηT×b,2b×−l2,0;
iiift, u, v<min{−φ−r1/N,−φ−l1/S}fort, u, v∈0, TT×0, r1×−l1,0.
then the problem1.1,1.2has at least three positive solutionsu1, u2, u3satisfying
max
t∈0,TT
u1t< r1, sup t∈0,TT
uΔ1t< l1,
b < min
t∈0,ηT
u2t≤ max t∈0,TT
u2t≤r2, sup t∈0,TT
uΔ2t≤l2,
max
t∈0,TT
u3t<2b with min t∈0,ηT
u3t< b, sup t∈0,TT
uΔ3t≤l2.
3.4
Proof. By the definition of operatorAand its properties, it suffices to show that the conditions ofLemma 2.2hold with respect to the operatorA.
We first show that if the conditioniis satisfied, then
A:Pα, r2;β, l2
−→Pα, r2;β, l2
In fact, ifu∈Pα, r2;β, l2,then
αu max
t∈0,TT
|ut| ≤r2, βu sup t∈0,TT
uΔt≤l2, 3.6
so assumptioniimplies
ft, ut, uΔt≤min
−φ−r2
N
,−φ
−l2
S
, t∈0, TT. 3.7
On the other hand, foru∈P, there isAu∈P; thenAutis concave in0, TT, andAuΔt≤0 fort∈0, TT, so
αAu max
t∈0,TT
|Aut|Au0
− T 0 φ−1 − s 0
hτfτ, uτ, uΔτ∇τΔs
− 1 d
m−2
i1
ai T ξi φ−1 − s 0
hτfτ, uτ, uΔτ∇τΔs
≤ r2
N T 0 φ−1 s 0 hτ∇τ
Δs 1 d
m−2
i1
ai T ξi φ−1 s 0 hτ∇τ Δs
r2,
βAu sup
t∈0,TT
AuΔt−AuΔT
−φ−1
−
T
0
hτfτ, uτ, uΔτ∇τ
≤ l2
Sφ
−1T 0
hτ∇τ
l2.
3.8
Therefore,3.5holds.
In the same way, ifu∈Pα, r1;β, l1, then conditioniiiimplies
ft, ut, uΔt<min
−φ−r1
N
, φ
−l1
S
fort∈0, TT. 3.9
As in the argument above, we can get thatA:Pα, r1;β, l1 → Pα, r1;β, l1.Thus, condition
C2ofLemma 2.2holds.
Next we show that conditionC1inLemma 2.2holds. We chooseut 2bfort ∈ 0, TT.It is easy to see that
u∈Pα,2b;β, l2;γ, b
and consequently
u∈Pα,2b;β, l2;γ, b
:γu> b/∅. 3.11
Therefore, foru∈Pα,2b;β, l2;γ, b,there are
b≤ut≤2b, uΔt≤l2 fort∈
0, ηT. 3.12
Hence in view of hypothesisii, we have
ft, ut, uΔt>−φ
− b M
fort∈0, ηT. 3.13
So by the definition of the functionalγ, we see that
γAu min
0,ηT
Aut Auη
−
T
η
φ−1
−
s
0
hτfτ, uτ, uΔτ∇τΔs
− 1 d
m−2
i1
ai T
ξi
φ−1
−
s
0
hτfτ, uτ, uΔτ∇τΔs
≥ −
T
η
φ−1
−
s
0
hτfτ, uτ, uΔτ∇τΔs
≥ −
T
η
φ−1
−
η
0
hτfτ, uτ, uΔτ∇τΔs
> b M
T
η
φ−1
η
0
hτ∇τΔs b
MMb.
3.14
Therefore, we getγAu > bforu ∈ Pα,2b;β, l2;γ, b, and conditionC1inLemma 2.2is
fulfilled.
We finally prove thatC3inLemma 2.2holds. In fact, foru∈ Pα, r2;β, l2;γ, bwith
αAu>2b,we have
γAu min
0,ηT
Aut Auη≥ T−η T t∈max0,TT
|Aut| ≥ 1
Thus fromLemma 2.2and the assumption thatft,0,0/≡0 on0, TT, the BVP1.1and1.2 has at least three positive solutionsu1, u2, andu3inPα, r2;β, l2with
u1∈P
α, r1;β, l1
, u2 ∈
Pα, r2;β, l2;γ, b
:γu> b,
u3∈P
α, r2;β, l2
\Pα, r2;β, l2;γ, b
∪Pα, r1;β, l1
.
3.16
The fact that the functionalsαandβonP satisfy an additional relation1/2αu≤γufor
u∈Pimplies that
max
t∈0,TT
u3t<2b. 3.17
The proof is complete.
FromTheorem 3.1, we see that, when assumptions asi,ii, and iiiare imposed appropriately onf, we can establish the existence of an arbitrary odd number of positive solutions of1.1and1.2.
Theorem 3.2. Suppose that there exist constants
0< r1< b1<2b1≤r2 < b2<2b2≤ · · · ≤rn, 0< l1≤l2≤ · · · ≤ln−1, n∈N 3.18
with
bi
M ≤min
ri1
N , li1
S
for1≤i≤n−1 3.19
such that the following conditions hold:
ift, u, v<min{−φ−ri/N,−φ−li/S}fort, u, v∈0, TT×0, ri×−li, li,1≤i≤
n;
iift, u, v>−φ−bi/Mfort, u, v∈0, ηT×bi,2bi×−li1, li1,1≤i≤n.
Then, BVP1.1and1.2has at least2n−1positive solutions.
Proof. Whenn1,it is immediate from conditionithatA:Pα, r1;β, l1 → Pα, r1;β, l1,
which means that Ahas at least one fixed point u1 ∈ Pα, r1;β, l1 by the Schauder fixed
point theorem. Whenn2,it is clear that the hypothesis ofTheorem 3.1holds. Then we can obtain at least three positive solutionsu1, u2,andu3. Following this way, we finish the proof
by induction. The proof is complete.
Example 3.3. LetT 0,1∪ {1 1/2N0}, whereN
0 denote nonnegative integer numbers
set. If we chooseT 2,η 1, m 4, a1 a2 1/3, ξ1 1/2, ξ2 3/2, andht 1 and
consider the following BVP on time scaleT:
φuΔt∇ft, ut, uΔt0, t∈0,2T,
uΔ0 0, u2 1 3u 1 2 1 3u 3 2 , 3.20 where φu ⎧ ⎪ ⎨ ⎪ ⎩
u, u≤0;
u2, u >0.
ft, u, v
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 120t 2 3u 3 v 100 3
, t∈0,2T, u∈−∞,3, v∈−∞,∞;
1
120t18
v
100
3
, t∈0,2T, u∈3,∞, v∈−∞,∞,
3.21
obviously the hypothesesH1,H2hold andft,0,0/≡0 on0,2T. By simple calculations, we have
Sφ−1
2
0
1∇s
√2, Mφ−1
1 0 1∇s 1, N
T 1 d
m−2
i1
aiT−ξi φ−1 T 0 hs∇s
4√2.
3.22 Observe that N T 0 φ−1 s 0 hτ∇τ
Δs 1 d
m−2
i1
ai T ξi φ−1 s 0 hτ∇τ Δs <
T 1 d
m−2
i1
If we chooser2180, b2, r11/3,andl278, l110, thenft, u, vsatisfies
ft, u, v≤ 45 √
2
2 min
−φ
−r2
N
,−φ
−l2
S
<min
−φ−r2
N
,−φ
−l2
S
, t, u, v∈0,2T×0,180×−78,78;
ft, u, v>4 b
M fort, u, v∈0,1T×2,4×−78,78,
ft, u, v< √
2
24 min
−φ
−r1
N
,−φ
−l1
S <min −φ
−r1
N
,−φ
−l1
S
, t, u, v∈0,2T×
0,1
3
×−10,10.
3.24
So all conditions ofTheorem 3.1hold. Thus byTheorem 3.1, the problem3.20has at least three positive solutionsu1, u2, u3such that
max
t∈0,TT u1t<
1
3; t∈sup0,TT
uΔ1t<10,
2< min
t∈0,ηT
u2t≤ max t∈0,TT
u2t≤180; sup t∈0,TT
uΔ2t≤78,
max
t∈0,TT
u3t<4 with min t∈0,ηT
u3t<2, sup t∈0,TT
uΔ3t≤78.
3.25
Acknowledgments
This work was supported by the NNSF of China10801065and NSF of Gansu Province of China0803RJZA096.
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