Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 705740, 10 pages
doi:10.1155/2011/705740
Research Article
Existence and Iteration of Monotone Positive
Solution of BVP for an Elastic Beam Equation
Jian-Ping Sun and Xian-Qiang Wang
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
Correspondence should be addressed to Jian-Ping Sun, jpsun@lut.cn
Received 5 October 2010; Accepted 28 January 2011
Academic Editor: Jitao Sun
Copyright q 2011 J.-P. Sun and X.-Q. Wang. 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.
This paper is concerned with the existence of monotone positive solution of boundary value
problem for an elastic beam equation. By applying iterative techniques, we not only obtain the
existence of monotone positive solution but also establish iterative scheme for approximating the
solution. It is worth mentioning that the iterative scheme starts off with zero function, which is
very useful and feasible for computational purpose. An example is also included to illustrate the
main results.
1. Introduction
It is well-known that beam is one of the basic structures in architecture. The deformations of
an elastic beam in equilibrium state can be described by the following equation of deflection
curve:
d2
dx2
d2 u
EIz 2
dx
qx,
1.1
where E is Yang’s modulus constant, Iz is moment of inertia with respect to z axes, and qx
is loading at x. If the loading of beam considered is in relation to deflection and rate of change
of deflection, we need to study more general equation
u4 x f x, ux, u x .
1.2
According to different forms of supporting, various boundary value problems BVPs should
be considered.
2
Mathematical Problems in Engineering
Owing to its importance in engineering, physics, and material mechanics, BVPs for
elastic beam equations have attracted much attention from many authors see, for example,
1–15 and the references therein. However, almost all of the papers we mentioned focused
their attention on the existence of solutions or positive solutions. In the existing literature,
there are few papers concerned with the computational methods of solutions or positive
solutions. It is worth mentioning that Zhang 16 obtained the existence of positive solutions
and established iterative schemes for approximating the solutions for an elastic beam
equation with a corner. The main tools used were monotone iterative techniques. For
monotone iterative methods, one can refer 17–19 and the references therein.
Motivated greatly by the above-mentioned excellent works, in this paper we
investigate the existence and iteration of monotone positive solution for the following elastic
beam equation BVP
u4 t ft, ut, u t,
t ∈ 0, 1,
u0 u 1 u 0 u 1 0.
1.3
In material mechanics, the equation in 1.3 describes the deflection or deformation of an
elastic beam under a certain force. The boundary conditions in 1.3 mean that the elastic
beam is simply fixed at the end t 0 and fastened with a sliding clamp at the end t 1. By
applying iterative techniques, we not only obtain the existence of monotone positive solution
but also establish iterative scheme for approximating the solution. It is worth mentioning
that the iterative scheme starts off with zero function, which is very useful and feasible for
computational purpose. An example is also included to illustrate the main results.
Throughout this paper, we always assume that the following condition is satisfied:
A f ∈ C0, 1 × 0, ∞ × 0, ∞, 0, ∞.
2. Preliminary
In order to obtain the main results of this paper, we first present several fundamental lemmas
in this section.
Lemma 2.1. Let y ∈ C0, 1. Then, the BVP
u4 t yt,
t ∈ 0, 1,
u0 u 1 u 0 u 1 0,
2.1
has a unique solution
ut 1
Gt, sysds,
t ∈ 0, 1,
2.2
0
where
⎧ 2
2
1 ⎨s 6t − 3t − s ,
Gt, s 6 ⎩t6s − 3s2 − t2 ,
0 ≤ s ≤ t ≤ 1,
0 ≤ t ≤ s ≤ 1.
2.3
Mathematical Problems in Engineering
3
Lemma 2.2. For any t, s ∈ 0, 1 × 0, 1, we have
0≤
∂Gt, s
≤ s1 − t,
∂t
1 2
1
t s ≤ Gt, s ≤
2t − t2 s.
3
2
2.4
Proof. For any fixed s ∈ 0, 1, it is easy to know that
⎧
⎪
⎨s1 − t,
∂Gt, s
⎪
∂t
⎩s1 − t − 1 s − t2 ,
2
0 ≤ s ≤ t ≤ 1,
0 ≤ t ≤ s ≤ 1,
2.5
which shows that
0≤
∂Gt, s
≤ s1 − t,
∂t
for t, s ∈ 0, 1 × 0, 1,
2.6
and so,
Gt, s t
0
∂Gτ, s
dτ ≤
∂τ
t
s1 − τdτ 0
1
2t − t2 s,
2
for t, s ∈ 0, 1 × 0, 1.
2.7
On the other hand, it follows from the expression of Gt, s that
1 1
1 s 6t − 3t2 − s2 ≥ s 6t − 4t2 ≥ t2 s,
6
6
3
0 ≤ s ≤ t ≤ 1,
1 1
1 Gt, s t 6s − 3s2 − t2 ≥ t 6s − 4s2 ≥ t2 s,
6
6
3
0 ≤ t ≤ s ≤ 1.
Gt, s 2.8
Let E C1 0, 1 be equipped with the norm u max{u∞ , u ∞ }. Then, E is a
Banach space. Denote
K
2
u ∈ E : ut ≥ t2 u∞ , u t ≥ 0, for t ∈ 0, 1 .
3
2.9
Then, it is easy to verify that K is a cone in E. Note that this induces an order relation º in E
by defining u º v if and only if v − u ∈ K. Now, we define an operator T on K by
Tut 1
Gt, sf s, us, u s ds,
t ∈ 0, 1.
0
Obviously, fixed points of T are monotone and nonnegative solutions of the BVP 1.3.
2.10
4
Mathematical Problems in Engineering
Lemma 2.3. T : K → K is completely continuous.
Proof. First, we prove TK ⊂ K. Suppose that u ∈ K. In view of Lemma 2.2, on the one hand,
0 ≤ Tut
1
Gt, sf s, us, u s ds
0
≤
≤
1
2t − t2
2
1
2
1
1
2.11
sf s, us, u s ds
0
sf s, us, u s ds,
t ∈ 0, 1,
0
which shows that
Tu∞ ≤
1
2
1
sf s, us, u s ds.
2.12
0
On the other hand,
Tut 1
Gt, sf s, us, u s ds
0
1
≥ t2
3
1
sf s, us, u s ds,
2.13
t ∈ 0, 1,
0
which together with 2.12 implies that
Tut ≥
2 2
t Tu∞ ,
3
2.14
t ∈ 0, 1.
Again, by Lemma 2.2, we have
Tu t 1
0
∂Gt, s f s, us, u s ds ≥ 0,
∂t
t ∈ 0, 1.
2.15
So, it follows from 2.14, 2.15 and the definition of K that TK ⊂ K.
Next, we show that T is a compact operator. Let D ⊂ K be a bounded set. Then, there
exists M1 > 0 such that u ≤ M1 for any u ∈ D. For any {yk }∞
k1 ⊂ TD, there exist
∞
{xk }k1 ⊂ D such that yk Txk . Denote
M2 sup f t, x, y : t, x, y ∈ 0, 1 × 0, M1 × 0, M1 .
2.16
Mathematical Problems in Engineering
5
Then, for any positive integer k, it follows from Lemma 2.2 that
yk t |Txk t|
1
Gt, sf s, xk s, xk s ds
0
1
2t − t2
≤
2
≤
M2
,
4
1
0
sf s, xk s, xk s ds
2.17
t ∈ 0, 1,
which indicates that {yk }∞
k1 is uniformly bounded. Similarly, we have
y t Txk t
k
1 ∂Gt, s f s, xk s, xk s ds
0
∂t
≤ 1 − t
1
0
≤
M2
,
2
sf s, xk s, xk s ds
2.18
t ∈ 0, 1.
This shows that {yk }∞
is uniformly bounded, which implies that {yk }∞
k1 is equicontinuous.
k1
By Arzela-Ascoli theorem, we know that {yk }∞
has
a
convergent
subsequence
in C0, 1.
k1
converges
in
C0,
1.
Without loss of generality, we may assume that {yk }∞
k1
On the other hand, for any > 0, by the uniform continuity of ∂Gt, s/∂t, we know
that there exists a δ > 0 such that for any t1 , t2 ∈ 0, 1 with |t1 − t2 | < δ,
∂Gt1 , s ∂Gt2 , s <
−
M 1,
∂t
∂t
2
s ∈ 0, 1.
2.19
So, for any positive integer k, t1 , t2 ∈ 0, 1 with |t1 − t2 | < δ, we get
y t1 − y t2 Txk t1 − Txk t2 k
k
1
∂Gt1 , s ∂Gt2 , s f s, xk s, x s ds
−
k
∂t
∂t
0
M2 <
M2 1
< ,
2.20
6
Mathematical Problems in Engineering
is equicontinuous. Again, it follows from Arzela-Ascoli theorem
which shows that {yk }∞
k1
that {yk }∞
has
a
convergent
subsequence in C0, 1. Therefore, {yk }∞
k1 has a convergent
k1
subsequence in K.
Finally, we prove that T is continuous. Suppose that um , u ∈ K and um − u →
0 m → ∞. Then, there exists M3 > 0 such that for any positive integer m, um ≤ M3 .
Denote
M4 sup f t, x, y : t, x, y ∈ 0, 1 × 0, M3 × 0, M3 .
2.21
Then, for any positive integer m and t ∈ 0, 1, by Lemma 2.2, we have
Gt, sfs, um s, um s ≤
M4
M4 s,
2t − t2 s ≤
2
2
∂Gt, s
fs, um s, um s ≤ M4 1 − ts ≤ M4 s,
∂t
s ∈ 0, 1,
2.22
s ∈ 0, 1.
So, it follows from Lebesgue dominated convergence theorem that
1
lim Tum t lim
m→∞
m→∞
1
0
Gt, sf s, um s, um s ds
Gt, sf s, us, u s ds
0
Tut,
lim Tum t lim
m→∞
1
m→∞
1
0
0
t ∈ 0, 1,
∂Gt, s f s, um s, um s ds
∂t
2.23
∂Gt, s f s, us, u s ds
∂t
Tu t,
t ∈ 0, 1,
which implies that T is continuous. Therefore, T : K → K is completely continuous.
3. Main Results
Theorem 3.1. Assume that ft, 0, 0 /
≡ 0 for t ∈ 0, 1, and there exists a constant a > 0 such that
ft, u1 , v1 ≤ ft, u2 , v2 ≤ 2a,
0 ≤ t ≤ 1, 0 ≤ u1 ≤ u2 ≤ a, 0 ≤ v1 ≤ v2 ≤ a.
3.1
Mathematical Problems in Engineering
7
If we construct a iterative sequence vn1 Tvn , n 0, 1, 2, . . ., where v0 t ≡ 0 for t ∈ 0, 1, then
∗
1
{vn }∞
n1 converges to v in C 0, 1, which is a monotone positive solution of the BVP 1.3 and satisfy
0 ≤ v∗ t ≤ a for t ∈ 0, 1.
0 < v∗ t ≤ a for t ∈ 0, 1,
3.2
Proof. Let Ka {u ∈ K : u ≤ a}. We assert that T : Ka → Ka . In fact, if u ∈ Ka , then
0 ≤ us ≤ max us ≤ u ≤ a,
0≤s≤1
0 ≤ u s ≤ max u s ≤ u ≤ a,
0≤s≤1
for s ∈ 0, 1,
3.3
which together with the condition 3.1 and Lemma 2.2 implies that
0 ≤ Tut 1
Gt, sf s, us, u s ds < a,
t ∈ 0, 1,
0
1
0 ≤ Tu t 0
∂Gt, s f s, us, u s ds ≤ a,
∂t
3.4
t ∈ 0, 1.
Hence, we have shown that T : Ka → Ka .
∗
1
Now, we assert that {vn }∞
n1 converges to v in C 0, 1, which is a monotone positive
solution of the BVP 1.3 and satisfies
0 < v∗ t ≤ a
0 ≤ v∗ t ≤ a for t ∈ 0, 1.
for t ∈ 0, 1,
3.5
Indeed, in view of v0 ∈ Ka and T : Ka → Ka , we have vn ∈ Ka , n 1, 2, . . .. Since
∞
the set {vn }∞
n0 is bounded and T is completely continuous, we know that the set {vn }n1 is
relatively compact.
In what follows, we prove that {vn }∞
n0 is monotone by induction. First, by Lemma 2.2,
we have
1 2
t
3
1
sfs, 0, 0ds
0
≤ v1 t − v0 t Tv0 t 1
Gt, sfs, 0, 0ds ≤
0
1
2
1
sfs, 0, 0ds,
t ∈ 0, 1,
0
3.6
which implies that
v1 t − v0 t ≥
2 2
t v1 − v0 ∞ ,
3
3.7
t ∈ 0, 1.
At the same time, it is obvious that
v1 t
−
v0 t
v1 t
Tv0 t 1
0
∂Gt, s
fs, 0, 0ds ≥ 0,
∂t
t ∈ 0, 1.
3.8
8
Mathematical Problems in Engineering
It follows from 3.7 and 3.8 that v1 − v0 ∈ K, which shows that v0
that vk−1 º vk . Then, in view of Lemma 2.2 and 3.1, we have
vk1 t − vk t 1
0
≤
vk1 t − vk t 1
2
Gt, s f s, vk s, vk s − f s, vk−1 s, vk−1
s ds
1
0
1
0
º v1 . Next, we assume
s f s, vk s, vk s − f s, vk−1 s, vk−1
s ds,
t ∈ 0, 1,
3.9
Gt, s f s, vk s, vk s − f s, vk−1 s, vk−1
s ds
1
≥ t2
3
1
0
s f s, vk s, vk s − f s, vk−1 s, vk−1
s ds,
t ∈ 0, 1,
which implies that
vk1 t − vk t ≥
2 2
t vk1 − vk ∞ ,
3
t ∈ 0, 1.
3.10
At the same time, by Lemma 2.2 and 3.1, we also have
vk1
t − vk t 1
0
∂Gt, s f s, vk s, vk s − f s, vk−1 s, vk−1
s ds ≥ 0,
∂t
t ∈ 0, 1.
3.11
It follows from 3.10 and 3.11 that vk1 − vk ∈ K, which indicates that vk º vk1 . Thus, we
have shown that vn º vn1 , n 0, 1, 2 . . ..
∗
Since {vn }∞
n1 is relatively compact and monotone, there exists a v ∈ Ka such that
vn − v∗ → 0 n → ∞, which together with the continuity of T and the fact that vn1 Tvn
implies that v∗ Tv∗ . Moreover, in view of ft, 0, 0 /
≡ 0 for t ∈ 0, 1, we know that the zero
function is not a solution of the BVP 1.3. Thus, v∗ ∞ > 0. So, it follows from v∗ ∈ Ka that
0 < v∗ t ≤ a
0 ≤ v∗ t ≤ a for t ∈ 0, 1.
for t ∈ 0, 1,
3.12
4. An Example
Example 4.1. Consider the BVP
u4 t 2
1
1
tut u t 1,
2
8
t ∈ 0, 1,
u0 u 1 u 0 u 1 0.
4.1
Mathematical Problems in Engineering
9
If we let ft, u, v 1/2tu 1/8v2 1 for t, u, v ∈ 0, 1 × 0, ∞ × 0, ∞, then
all the hypotheses of Theorem 3.1 are fulfilled with a 1. It follows from Theorem 3.1 that
the BVP 4.1 has a monotone positive solution v∗ satisfying
0 < v∗ t ≤ 1
for t ∈ 0, 1,
0 ≤ v∗ t ≤ 1 for t ∈ 0, 1.
4.2
Moreover, the iterative scheme is
v0 t ≡ 0,
1
vn1 t 6
t ∈ 0, 1,
t 1
2
1 2
2
svn s vn s 1 ds
s 6t − 3t − s
2
8
0
1
6
1 1
2
1 2
2
svn s vn s 1 ds,
t 6s − 3s − t
2
8
t
t ∈ 0, 1, n 0, 1, 2, . . . .
4.3
The first, second, third, and fourth terms of the scheme are as follows:
v0 t ≡ 0,
v1 t 1
1
1
t − t3 t4 ,
3
6
24
v2 t 5 2 79 3 1 4
25 5
23 6
1 7
47 8
11 9
1 10
73
t
t −
t t t −
t −
t t −
t t ,
216
144
432
64
1152
8640
192
11520
8640
6912
v3 t 6935 2 404629 3
60361 4
71689 5
110633 6
22181 7
378577
t
t −
t t t t −
t
1119744
186624
2239488
8957952
2985984
14929920
1492992
373759 8
2467627 9
2449943 10
800341 11
16256149 12
t t −
t −
t t
119439360
477757440
716636160
1592524800
11466178560
−
3556357 13
655267 14
70313 15
25471
363911 17
t −
t t −
t16 −
t
7166361600
4777574400
477757440
2388787200
9555148800
7781
25973
8917
31
25
t18 −
t19 t20 −
t21 t22 .
298598400
2866544640
4777574400
143327232
2293235712
4.4
Acknowledgment
This work is supported by the National Natural Science Foundation of China 10801068.
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