Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 618413, 13 pages
doi:10.1155/2009/618413
Research Article
Successive Iteration and Positive Solutions
for Nonlinear m-Point Boundary Value Problems
on Time Scales
Yanbin Sang
Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China
Correspondence should be addressed to Yanbin Sang, sangyanbin@126.com
Received 3 June 2009; Accepted 16 September 2009
Recommended by Yong Zhou
We study the existence of positive solutions for a class of m-point boundary value problems on
time scales. Our approach is based on the monotone iterative technique and the cone expansion
and compression fixed point theorem of norm type. Without the assumption of the existence of
lower and upper solutions, we do not only obtain the existence of positive solutions of the problem,
but also establish the iterative schemes for approximating the solutions.
Copyright q 2009 Yanbin Sang. 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
The purpose of this paper is to consider the existence of positive solutions and establish the
corresponding iterative schemes for the following m-point boundary value problems BVP
on time scales:
uΔ∇ t ft, ut 0,
βu0 − γuΔ 0 0,
u1 t ∈ 0, 1T ,
m−2
αi uξi ,
m ≥ 3.
1.1
i1
The study of dynamic equations on time scales goes back to its founder Hilger 1,
and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject
is the notion that dynamic equations on time scales can build bridges between continuous
and discrete mathematics. Some preliminary definitions and theorems on time scales can be
found in 2–5 which are good references for the calculus of time scales.
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Discrete Dynamics in Nature and Society
In recent years, by applying fixed point theorems, the method of lower and upper
solutions and critical point theory, many authors have studied the existence of positive
solutions for two-point and multipoint boundary value problems on time scales, for details,
see 2, 3, 6–18 and references therein. However, to the best of our knowledge, there are few
papers which are concerned with the computational methods of the multipoint boundary
value problems on time scales. We would like to mention some results of Sun and Li 16,
Aykut Hamel and Yoruk 12, Anderson and Wong 10, Wang et al. 18, and Jankowski
13, which motivated us to consider the BVP 1.1.
In 16, Sun and Li considered the existence of positive solutions of the following
dynamic equations on time scales:
uΔ∇ t atft, ut 0,
βu0 − γuΔ 0 0,
t ∈ 0, T ,
1.2
αu η uT ,
where β, γ ≥ 0, βγ > 0, η ∈ 0, ρT , 0 < α < T/η. They obtained the existence of single and
multiple positive solutions of 1.2 by using a fixed point theorem and the Leggett-Williams
fixed point theorem, respectively.
Very recently, in 12, Aykut Hamel and Yoruk discussed the following dynamic
equation on time scales:
uΔ∇ t ft, ut 0,
βu0 − γuΔ 0 0,
u1 t ∈ 0, 1 ⊂ T,
m−2
αi uξi ,
m ≥ 3.
1.3
i1
They obtained some results for the existence of at least two and three positive solutions
to the BVP 1.3 by using fixed point theorems in a cone and the associated Green’s
function.
In related paper, in 10, Anderson and Wong studied the second-order time
scale semipositone boundary value problem with Sturm-Liouville boundary conditions or
multipoint conditions as in
puΔ
∇
t λft, ut 0,
αua − β puΔ a 0,
n
φi puΔ ti ,
αua − β puΔ a i1
t ∈ a, bT ,
γuσ b δ puΔ b 0,
or
n
ψi puΔ ti .
γuσ b δ puΔ b i1
1.4
Discrete Dynamics in Nature and Society
3
On the other hand, the method of lower and upper solutions has been effectively
used for proving the existence results for dynamic equations on time scales. In 18, Wang
et al. considered a method of generalized quasilinearization, with even-order k k ≥ 2
convergence, for the BVP
∇
− ptxΔ qtxσ ft, xσ gt, xσ ,
τ1 x ρa − τ2 xΔ ρa 0,
t ∈ a, bT ,
xσb − τ3 x η 0.
1.5
The main contribution in 18 relaxed the monotone conditions on f i t, x, g i t, x 1 < i <
k including a more general concept of upper and lower solution in mathematical biology, so
that the high-order convergence of the iterations was ensured for a larger class of nonlinear
functions on time scales.
In 13, Jankowski investigated second-order differential equations with deviating
arguments on time scales of the form
−xΔΔ t ft, xt, xαt ≡ Fxt,
x0 k1 ∈ R,
t ∈ J,
xT k2 ∈ R.
1.6
They formulated sufficient conditions, under which such problems had a minimal and a
maximal solution in a corresponding region bounded by upper-lower solutions.
We would also like to mention the result of Yao 19. In 19, Yao considered
the positive solutions to the following two classes of nonlinear second-order three-point
boundary value problems:
u t f t, ut, u t 0,
u t ft, ut 0,
u0 0,
0 ≤ t ≤ 1,
0 ≤ t ≤ 1,
1.7
αu η u1,
where both η and α are given constants satisfying 0 < η < 1, 0 < α < 1/η. By improving
the classical monotone iterative technique of Amann 20, two successive iterative schemes
were established for the BVP 1.7. It was worth stating that the first terms of the iterative
schemes were constant functions or simple functions. We note that Ma et al. 21 and Sun et al.
22, 23 have also applied the similar methods to p-laplacian boundary value problems with
T R.
In this paper, we will investigate the iterative and existence of positive solutions
for the BVP 1.1, by considering the “heights” of the nonlinear term f on some bounded
sets and applying monotone iterative techniques on a Banach space, we do not only
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Discrete Dynamics in Nature and Society
obtain the existence of positive solutions for the BVP 1.1, but also give the iterative
schemes for approximating the solutions. We should point out that the monotone condition
imposed on the nonlinear term f will play crucial role in obtaining the iterative schemes
for approximating the solutions. In essence, we combine the method of lower and upper
solutions with the cone expansion and compression fixed point theorem of norm type. The
idea of this paper comes from Yao 19, 24, 25.
Let T be a time scale which has the subspace topology inherited from the standard
topology on R. For each interval I of R, we define IT I ∩ T.
For the remainder of this article, we denote the set of continuous functions from 0, 1T
to R by C0, 1T , R. Let C0, 1T , R be endowed with the ordering x ≤ y if xt ≤ yt for all
t ∈ 0, 1T , and u
maxt∈0,1T |ut| is defined as usual by maximum norm. The C0, 1T , R
is a Banach space.
Throughout this paper, we will assume that the following assumptions are
satisfied:
H1 β, γ ≥ 0, 0 < β γ ≤ 1, ξi ∈ 0, ρ1T for i 1, 2, . . . , m − 2 with 0 < ξ1 < ξ2 < · · · <
ξm−2 < ρ1;
H2 m−2
α ∈ 0, 1 with αi ∈ 0, ∞ for i 1, 2, . . . , m − 2 and d β1 −
i
γ1 − m−2
i1 αi > 0;
i1
m−2
i1
αi ξi H3 f : 0, 1T × 0, ∞ → 0, ∞ is continuous.
2. Preliminaries and Several Lemmas
To prove the main results in this paper, we will employ several lemmas. These lemmas are
based on the linear boundary value problem
uΔ∇ t ht 0,
βu0 − γuΔ 0 0,
t ∈ 0, 1T ,
u1 m−2
αi uξi ,
m ≥ 3.
2.1
i1
Lemma 2.1 see 12. It holds d β1 −
for the BVP
m−2
i1
αi ξi γ1 −
−uΔ∇ t 0,
βu0 − γuΔ 0 0,
m−2
i1
αi /
0; then the Green’s function
t ∈ 0, 1T ,
u1 m−2
αi uξi ,
i1
m ≥ 3,
2.2
Discrete Dynamics in Nature and Society
5
is given by
Gt, s ⎤
⎡
⎧
m−2
⎪
⎪
⎪
⎪ βs γ ⎣1 − t −
aj ξj − t ⎦,
⎪
⎪
⎪
j1
⎪
⎪
⎪
⎪
⎪
⎪
⎪ if 0 ≤ t ≤ 1, 0 ≤ s ≤ ξ1 , s ≤ t;
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
m−2
i−1
⎪
⎪
⎪
⎪
βs
γ
ξ
α
−
t
βs
γ
αj βξj γ t − s,
−
t
−
1
j j
⎪
⎪
⎪
⎪
ji
j1
⎪
⎪
⎪
⎪
⎪
⎪
1 ⎨ if ξr−1 ≤ t ≤ ξr , 2 ≤ r ≤ m − 1, ξi−1 ≤ s ≤ ξi , 2 ≤ i ≤ r, s ≤ t;
d⎪
⎤
⎡
⎪
⎪
⎪
m−2
⎪
⎪
⎪
⎪
αj ξj − s ⎦,
βt γ ⎣1 − s −
⎪
⎪
⎪
⎪
ji
⎪
⎪
⎪
⎪
⎪
⎪
⎪
if ξr−1 ≤ t ≤ ξr , 2 ≤ r ≤ m − 2, ξi−1 ≤ s ≤ ξi , r ≤ i ≤ m − 2, t ≤ s;
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
βt γ 1 − s,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
if 0 ≤ t ≤ 1, ξm−2 ≤ s ≤ 1, t ≤ s.
Here for the sake of convenience, one writes
m2
im1
2.3
hi 0 for m2 < m1 .
Lemma 2.2 see 12. Assume that conditions H1 –H3 are satisfied. Then
i Gt, s ≥ 0 for t, s ∈ 0, 1T ;
ii there exist a number Ψ ∈ 0, 1 and a continuous function θ : 0, 1T → R such that
Gt, s ≤ θs for t, s ∈ 0, 1T ,
Gt, s ≥ Ψθs for t ∈ ξ1 , 1T , s ∈ 0, 1T ,
2.4
where
θs max 1,
Ψ
m−2
i1
ξ1
αi
βs γ 1 − s
,
d
1
max 1, m−2
i1 αi /ξ1
⎧
⎨
2.5
⎫
s−1
m−2
⎬
αj 1 − ξj , αj βξj γ αj 1 − ξj
βξ1 γ 1 − ξm−2 ,
× min
.
⎭
2≤s≤m−2⎩
js
j1
j1
m−2
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Discrete Dynamics in Nature and Society
Let B C0, 1T , R. It is easy to see that the BVP 1.1 has a solution u ut if and
only if u is a fixed point of the operator equation:
T ut 1
2.6
Gt, sfs, us∇s.
0
Denote
K
u ∈ B : u is nonnegative, concave, and min ut ≥ Ψ
u
,
t∈ξ1 ,1T
2.7
where Ψ is the same as in Lemma 2.2. By 12, Lemma 3.1, we can obtain that T K ⊂ K and
T : K → K is completely continuous.
3. Successive Iteration and One Positive Solution for 1.1
For notational convenience, we denote
A
−1
1
max
t∈0,1T
Gt, s∇s
,
B
0
−1
1
max
t∈0,1T
Gt, s∇s
.
3.1
ξ1
Constants A, B are not easy to compute explicitly. For convenience, we can replace A by A ,
B by B , where
A 1
−1
θs∇s
,
1
−1
B Ψ θs∇s .
0
3.2
ξ1
Obviously, 0 < A < A < B < B .
Theorem 3.1. Assume H1 –H3 hold, and there exist two positive numbers a, b with b < a such
that
C1 max{ft, a : t ∈ 0, 1T } ≤ aA, min{ft, Ψb : t ∈ ξ1 , 1T } ≥ bB;
C2 ft, u1 ≤ ft, u2 for any t ∈ 0, 1T , 0 ≤ u1 ≤ u2 ≤ a.
Then the BVP 1.1 has at least one positive solution u∗ such that b ≤ u∗ ≤ a and limn → ∞ T n u
u∗ ,
n
∗
converges uniformly to u in 0, 1T , where u
t ≡ a, t ∈ 0, 1T .
that is, T u
Remark 3.2. The iterative scheme in Theorem 3.1 is u1 T u
, un1 T un , n 1, 2, . . . . It starts
off with constant function u
t ≡ a, t ∈ 0, 1T .
Proof of Theorem 3.1. Denote Kb, a {u ∈ K : b ≤ u
≤ a}. If u ∈ Kb, a, then
max ut ≤ a,
t∈0,1T
min ut ≥ Ψ
u
≥ bΨ.
t∈ξ1 ,1T
3.3
Discrete Dynamics in Nature and Society
7
By Assumptions C1 and C2 , we have
ft, ut ≤ ft, a ≤ aA,
t ∈ 0, 1T ;
ft, ut ≥ ft, bΨ ≥ bB,
t ∈ ξ1 , 1T .
3.4
It follows that
1
T u
max Gt, sfs, us∇s
t∈0,1T 0
≤ aA max
1
t∈0,1T
T u
≥ max
t∈0,1T
Gt, s∇s a;
0
3.5
1
Gt, sfs, us∇s
ξi
≥ bB max
1
t∈0,1T
Gt, s∇s b.
ξi
Thus, we assert that T : Kb, a → Kb, a.
Let u
t ≡ a, t ∈ 0, 1T , then u
∈ Kb, a. Let u1 T u
, then u1 ∈ Kb, a. Denote un1 T un , n 1, 2, . . . . Since T Kb, a ⊂ Kb, a, we have un ∈ T Kb, a ⊂ Kb, a, n 1, 2, . . . .
Since T is completely continuous, we assert that {un }∞
n1 has a convergent subsequence
∗
∗
and
there
exists
u
∈
Kb,
a,
such
that
u
→
u
.
{unk }∞
nk
k1
Now, since u1 ∈ Kb, a, we have
u1 t ≤ u1 ≤ a u
t,
t ∈ 0, 1T .
3.6
By Assumption C2 ,
u2 t T u1 t
1
Gt, sfs, u1 s∇s
0
≤
1
3.7
Gt, sfs, u
s∇s
0
Tu
t u1 t.
By the induction, then
un1 t ≤ un t,
t ∈ 0, 1T , n 1, 2, . . . .
3.8
Hence, T n u
un → u∗ . Applying the continuity of T and un1 T un , we get T u∗ u∗ . Since
∗
u ≥ b > 0 and u∗ is a nonnegative concave function, we conclude that u∗ is a positive
solution of the BVP 1.1.
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Discrete Dynamics in Nature and Society
Corollary 3.3. Assume that H1 –H3 hold, and the following conditions are satisfied:
C1 liml → 0 mint∈ξ1 ,1T ft, l/l > Ψ−1 B, liml → ∞ maxt∈0,1T ft, l/l < A (particularly,
liml → 0 mint∈ξ1 ,1T ft, l/l ∞, liml → ∞ maxt∈0,1T ft, l/l 0);
C2 ft, u1 ≤ ft, u2 for any t ∈ 0, 1T , u1 ≤ u2 , u1 , u2 ∈ 0, ∞.
Then the BVP 1.1 has at least one positive solution u∗ ∈ K and there exists a positive number a such
u∗ , that is,
that limn → ∞ T n u
t − u∗ t| 0,
lim sup |T n u
n→∞
t∈0,1T
3.9
where u
t ≡ a, t ∈ 0, 1T .
Theorem 3.4. Assume H1 –H3 hold, and the following conditions are satisfied:
D1 there exists a > 0 such that ft, · : 0, a → 0, ∞ is nondecreasing for any t ∈ 0, 1T
and max{ft, a : t ∈ 0, 1T } ≤ aA;
D2 ft, 0 > 0, for any t ∈ 0, 1T .
Then the BVP 1.1 has one positive solution u∗ such that 0 < u∗ ≤ a and limn → ∞ T n 0 u∗ , that
is, T n 0 converges uniformly to u∗ in 0, 1T . Furthermore, if there exists 0 < ω < 1 such that
ft, l2 − ft, l1 ≤ ωA|l2 − l1 |,
t ∈ 0, 1T , 0 ≤ l1 , l2 ≤ a.
3.10
Then T n1 0 − u∗ ≤ ωn /1 − ω
T 0
.
Proof. Denote K0, a {u ∈ K : u
≤ a}. Similarly to the proof of Theorem 3.1, we can know
1 ∈ K0, a. Denote u
n1 t T u
n , n 1, 2, . . . .
that T : K0, a → K0, a. Let u
1 T 0, then u
Copying the corresponding proof of Theorem 3.1, we can prove that
u
n1 t ≥ u
n t,
t ∈ 0, 1T , n 1, 2, . . . .
3.11
Since T is completely continuous, we can get that there exists u∗ ∈ K0, a such that
n1 t T u
n , we can obtain that T u∗ u∗ . We
u
n → u∗ . Applying the continuity of T and u
note that ft, 0 > 0, for all t ∈ 0, 1T , it implies that the zero function is not the solution of
the problem 1.1. Therefore, u∗ is a positive solution of 1.1.
Now, since
ft, l2 − ft, l1 ≤ ωA|l2 − l1 |,
t ∈ 0, 1T , 0 ≤ l1 , l2 ≤ a.
3.12
Discrete Dynamics in Nature and Society
9
If u1 , u2 ∈ K0, a and u2 t ≥ u1 t, t ∈ 0, 1T , then
1
T u2 − T u1 max Gt, s fs, u2 s − fs, u1 s ∇s
t∈0,1T 0
≤ ωA max
1
t∈0,1T
Gt, s|u2 s − u1 s|∇s
0
3.13
≤ ωA
u2 − u1 A−1
ω
u2 − u1 .
Hence, we can deduce that
n1 T u
n ≤ ωn T 0 − 0
ωn T 0
,
un2 − u
n1 − T u
n1 ≤ ω
unk2 − u
nk
ω
nk−1
3.14
ωn
T 0
.
1−ω
3.15
··· ω
n
ωn
T 0
.
T 0
<
1−ω
It implies that
T n1 0 − u∗ ≤
The proof is complete.
4. Existence of n positive solutions
Theorem 4.1. Assume H1 –H3 hold, and there exist 2n positive numbers a1 , . . . , an , b1 , . . . , bn
with b1 < a1 < b2 < a2 < · · · < bn < an such that
E1 max{ft, ai : t ∈ 0, 1T } ≤ ai A, min{ft, Ψbi : t ∈ ξ1 , 1T } ≥ bi B, i 1, 2, . . . , n;
E2 ft, u1 ≤ ft, u2 for any t ∈ 0, 1T , 0 ≤ u1 ≤ u2 ≤ an .
Then the BVP 1.1 has n positive solutions u∗i , i 1, 2, . . . , n such that bi ≤ u∗i ≤ ai and
i u∗i , that is,
limn → ∞ T n u
lim sup T n u
i t − u∗i t 0,
n→∞
t∈0,1T
4.1
where u
i t ≡ ai , t ∈ 0, 1T , i 1, 2, . . . , n.
Corollary 4.2. Assume that H1 –H3 and C1 –C2 hold, and the following condition is satisfied
E there exist 2n − 1 positive numbers a1 < b2 < a2 < · · · < bn−1 < an−1 < bn such that
!
"
max ft, ai : t ∈ 0, 1T < ai A, i 1, . . . , n − 1,
!
"
min ft, Ψbi : t ∈ ξi , 1T > bi B, i 2, . . . , n.
4.2
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Discrete Dynamics in Nature and Society
Then the BVP 1.1 has n positive solutions u∗i , i 1, 2, . . . , n, and there exists a positive number
i u∗i , where u
i t ≡ ai , t ∈ 0, 1T , i 1, 2, . . . , n.
an with an > bn such that limn → ∞ T n u
5. Examples
#
Example 5.1. Let T 0, 1/3 1/2, 1. Considering the following BVP:
uΔ∇ t ft, u 0,
u0 0,
t ∈ 0, 1T ,
$ %
$ %
1
1
1
1
u1 u
u
,
8
3
6
2
5.1
where ft, u 200/109u2 1, it is easy to check that ft, 0 1 > 0, for any t ∈ 0, 1T .
Further calculations give us
d
A 7
,
8
1
−1
θs∇s
0
1
−1
8
s1 − s∇s
7 0
5.2
1/3
−1
1/2
1
8
s1 − sds s1 − s∇s s1 − sds
7 0
ρ1/2
1/2
567
.
109
Choose a 1, it is easy to check that ft, · : 0, 1T → 0, ∞ is nondecreasing for any
t ∈ 0, 1T and
max ft, 1 t∈0,1T
567
200
1≤1·
.
109
109
5.3
Let u
0 t ≡ 0, for n 0, 1, 2, . . . , we have
%
$
%
200
8 1
200
u
n s 1 ∇s t 1 − s
u
n s 1 ∇s
u
n1 t − t − s
109
7 0
109
0
1/3 $
1/2 $
%$
%$
%
% 200
200
8 1
1
1
1
−s
u
n s 1 ∇s −s
u
n s 1 ∇s .
− t
7 8 0
3
109
6 0
2
109
5.4
t
$
Discrete Dynamics in Nature and Society
11
By Theorem 3.4, the BVP 5.1 has one positive solution u∗ such that 0 < u∗ ≤ 1 and T n 0 →
u∗ . On the other hand, for any 0 ≤ u1 , u2 ≤ 1, we have
fu1 − fu2 200 u2 − u2 2
109 1
≤
400
567 400
·
|u1 − u2 | |u1 − u2 |
109
109 567
400 A |u1 − u2 |.
567
5.5
Then,
T n1 0 − u∗ ≤
$
%
567 400 n
400/567n
0
T 0
.
T
1 − 400/567
167 567
5.6
The first and second terms of this scheme are as follows:
u
0 t 0,
⎧ 2
199t
t
⎪
⎪
⎪
⎨− 2 378 ,
u
1 t ⎪
⎪
1
t2 199t
⎪
⎩− ,
2
378 72
&
'
1
t ∈ 0, ,
3
&
'
1
t∈
,1 .
2
5.7
Now, we compute the third term of this scheme.
For t ∈ 0, 1/3,
u
2 t 175 4 9950 3 t2
t −
t − 327
61803
2
$
%
6720175
199
t.
70084602 378
5.8
$
%
6720175
100097
t.
70084602 185409
5.9
For t ∈ 1/2, 1,
u
2 t 175 4 9950 3 503t2
t −
t −
327
61803
981
#
Example 5.2. Let T {0} {1/3n : n ∈ N0 }. Considering the BVP on T,
uΔ∇ (
uΔ 0 0,
ut 0,
t ∈ 0, 1T ,
$ %
$ %
1
1
1
1
u
.
u1 u
3
9
9
3
5.10
By direct computation, we can get
d
5
,
9
A 9
,
16
B 81
,
8
Ψ
5
.
54
5.11
12
Discrete Dynamics in Nature and Society
Choose a 100, b 1/1875, it is easy to see that the nonlinear term ft, u fu possesses the following properties
)
ut
a f : 0, 1T × 0, ∞ → 0, ∞ is continuous;
b ft, u1 ≤ ft, u2 for any t ∈ 0, 1T and 0 ≤ u1 ≤ u2 ≤ 100;
√
c )
max{ft, 100 : t ∈ 0, 1T } 100 ≤ aA 100 × 9/16, min{ft, Ψb : t ∈ ξ1 , 1T } 5/54 × 1/1875 > bB 1/1875 × 81/8.
By Theorem 3.1, the BVP 5.10 has one positive solution u∗ such that 1/1875 ≤ u∗ ≤ 100
u∗ , where u
t ≡ 100, t ∈ 0, 1T . Let u0 t ≡ 100, t ∈ 0, 1T . For n and limn → ∞ T n u
0, 1, 2, . . . , we have
un1 1
Gt, sfs, un s∇s
0
(
(
9 1
− t − s un s∇s 1 − s un s∇s
5 0
0
$
$
%(
%(
1 1/3 1
3 1/9 1
−s
−s
un s∇s −
un s∇s.
−
5 0
9
5 0
3
t
5.12
Remark 5.3. By Theorems 3.1, 3.2, and 3.3 in 12, 16, 17, the existence of positive solutions
for the BVP 5.1 can be obtained, however, we cannot give a way to find the solutions which
will be useful from an application viewpoint. Therefore, Theorem 3.1 improves and extends
the main results of 12, 16, 17. On the other hand, in Example 5.2, since f0 0, we cannot
obtain the above mentioned results by use of Theorem 3.4, thus, Theorems 3.1 and 3.4 do not
contain each other.
Acknowledgment
This project supported by the Youth Science Foundation of Shanxi Province 2009021001-2.
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