# Giải gần đúng một số bài toán biên phi tuyến cho phương trình vi phân cấp bốn tt tiếng anh

MINISTRY OF EDUCATION

AND TRAINING

SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY

-----------------------------------

NGUYEN THANH HUONG

SOLVING
SOME NONLINEAR BOUNDARY VALUE PROBLEMS
FOR FOURTH ORDER DIFFERENTIAL EQUATIONS

Major: Applied Mathematics
Code: 9 46 01 12

SUMMARY OF PHD THESIS

Hanoi – 2019

This thesis has been completed:
of Science and Technology

Supervisor 1: Prof. Dr. Dang Quang A
Supervisor 2: Dr. Vu Vinh Quang

Reviewer 1:
Reviewer 2:
Reviewer 3:

The thesis will be defended at the Board of Examiners of Graduate
University of Science and Technology – Vietnam Academy of
Science and Technology at ............................ on..............................

The thesis can be explored at:
- Library of Graduate University of Science and Technology
- National Library of Vietnam

INTRODUCTION

1. Motivation of the thesis
Many phenomena in physics, mechanics and other fields are modeled by
boundary value problems for ordinary differential equations or partial differential
equations with different boundary conditions. The qualitative research as well as
the method of solving these problems are always the topics attracting the attention
of domestic and foreign scientists such as R.P. Agawarl, E. Alves, P. Amster, Z.
Bai, Y. Li, T.F. Ma, H. Feng, F. Minh´os, Y.M. Wang, Dang Quang A, Pham
Ky Anh, Nguyen Dong Anh, Nguyen Huu Cong, Nguyen Van Dao, Le Luong Tai.
The existence, the uniqueness, the positivity of solutions and the iterative method
for solving some boundary value problems for fourth order ordinary differential
equations or partial differential equations have been considered in the works of
Dang Quang A et al. (2006, 2010, 2016-2018). Pham Ky Anh (1982, 1986)
has also some research works on the solvability, the structure of solution sets,

the approximate method of nonlinear periodic boundary value problems. The
existence of solutions, positive solutions of the beam problems are considered in
the works of T.F. Ma (2000, 2003, 2004, 2007, 2010). Theory and numerical
solution of general boundary problems have been mentioned in R.P. Agarwal
(1986), Uri M. Ascher (1995), Herbert B. Keller (1987), M. Ronto (2000).
Among boundary problems, the boundary problem for fourth order ordinary
differential equations and partial differential equations are received great interest
by researchers because they are mathematical models of many problems in mechanics such as the bending of beams and plates. It is possible to classify the
fourth order differential equations into two forms: local fourth order differential
equations and nonlocal ones. A fourth order differential equation containing integral terms is called a nonlocal equation or a Kirchhoff type equation. Otherwise, it
is called a local equation. Below, we will review some typical methods for studying
boundary value problems for fourth order nonlinear differential equations.
The first method is the variational method, a common method of studying the
existence of solutions of nonlinear boundary value problems. With the idea of
reducing the original problem to finding critical points of a suitable functional,
the critical point theorems are used in the study of the existence of these critical
points. There are many works using the variational method (see T.F. Ma (2000,
1

2003, 2004), R. Pei (2010), F. Wang and Y. An (2012), S. Heidarkhani (2016),
John R. Graef (2016), S. Dhar and L. Kong (2018)). However, it must be noted
that, using the variational method, most of authors consider the existence of
solutions, the existence of multiple solutions of the problem (it is possible to
consider the uniqueness of the solution in the case of convex functionals) but there
are no examples of existing solutions, and the method for solving the problem has
not been considered.
The next method is the upper and lower solutions method. The main results of
this method when applying to nonlinear boundary value problems are as follows:
If the problem has upper and the lower solutions, the problem has at least one
solution and this solution is in the range of the upper and the lower solution
sequences with the first approximation being the upper and the lower solution
converge to the maximal and minimal solutions of the problem. In the case of
maximal and minimal solutions coincide, the problem has a unique solution.
We can mention some typical works using the upper and lower solutions method
when studying boundary value problems for nonlinear fourth order differential
equations as follows: J. Ehme (2002), Z. Bai (2004, 2007), Y.M. Wang (2006,
2007), H. Feng (2009), F. Minh´os (2009). From the above works, we find that
the upper and lower solutions method can establish the existence, the uniqueness
of solution, and construct the iterative sequences converging to the solution with
the very important assumption that these solutions exist but the finding of them
is not easy. In addition, they need other assumptions about the right-hand side
function such as the growth at infinity or the Nagumo condition.
Except for the mentioned methods, scientists also use the fixed point methods
in studying nonlinear boundary problems. By using these methods, the original
problem was reduced to the problem of finding fixed points of an operator, then
applying the fixed point theorem to this operator (see R.P. Agarwal (1984), B.
Yang (2005), P. Amster (2008), T.F. Ma (2010), S. Yardimci (2014)).
It should be emphasized that, in the works that apply the fixed point method to
study nonlinear boundary problems, most authors reduce the given problem to the
operator equation for the function to be sought. Using the fixed point theorems
such as ones of Schauder, Leray-Schauder, Krassnosel’skii for this operator, we can
only establish the existence of solutions. Using the Banach fixed point theorem, we
not only establish the existence and uniqueness of solution but also construct an
iterative method which converges with the rate of geometric progression. However,
it must be noted that the selection of the operator and considering this operator
on a suitable space so that the assumptions put on the related functions are simple
and still ensure the conditions to apply the the fixed point theorem in qualitative
research as well as the method of solving nonlinear boundary problems plays very

2

important role.
One of the popular numerical methods used in the approximation of boundary
problems for fourth order ordinary differential equations and partial differential
equations is finite difference method (see T.F. Ma (2003), R.K. Mohanty (2000),
J. Talwar (2012), Y.M. Wang (2007)). By replacing derivatives by difference
formulas, the problem is discretized into algebraic systems of equations. Solving
these systems, we obtain the approximate solution of the problem at grid nodes.
Note that when using finite difference method to study nonlinear boundary value
problems, many works recognize the existence of solutions of the problem and
discrete the problem from the beginning. This approach has a disadvantage that
it is difficult to evaluate the stability, the convergence of the difference scheme
and the error between the exact solution and the approximate one.
When studying nonlinear boundary problems, in addition to the popular methods presented above, we can mention some other methods such as the finite element method, Taylor series method, Fourier series method, Brouwer theory,
Leray-Schauder theory. We can also combine the above methods to get the full
study of both qualitative and quantitative aspects of the problem.
With the continuous development of science, technology, physics, mechanics,
from practical problems in these areas, new boundary problems are posed more
and more complex in both equations and boundary conditions. Authors will
use different methods, approaches and techniques for different problems. Each
proposed method will have its advantages and disadvantages and it is difficult to
confirm that this method is really better than the other method from theory to
experiment. However, our method will study both quantitative and qualitative
aspects of the problems so that the conditions are simple and easy to test. We
also give some numerical examples which illustrate the effectiveness of proposed
method and compare with the results of other authors in some way.
For these reasons, we decide to choose the title ”Solving some nonlinear boundary value problems for fourth order differential equations”.

2. Objectives and scope of the thesis
For some nonlinear boundary problems for fourth order ordinary differential
equations and partial differential equations which are models of problems in bending theory of beams and plates:
- Make qualitative research (the existence, the uniqueness, the positivity of
solutions) by using fixed point theorems and maximum principles without infinite
growth conditions, the Nagumo condition of the right-hand side function.
- Construct iterative methods for solving the problems.
- Give some examples illustrating the applicability of the obtained theoretical
results, including examples showing the advantages of the method in the thesis
3

compared with the methods of some other authors.

3. Research methodology and content of the thesis
- Use the approach of reducing the original nonlinear boundary value problems
to operator equations for the function to be sought or an intermediate function
with the tools of mathematical analysis, functional analysis, theory of differential
equation for studying the existence, the uniqueness and some properties of solutions of some problems for local and nonlocal fourth order differential equations.
- Propose iterative methods for solving these problems and prove the convergence of the iterative processes.
- Give some examples in both cases of known and unknown solution to illustrate the validity of theoretical results and examine the convergence of iterative
methods.

4. The major contributions of the thesis
The thesis proposes a simple but very effective method to study the unique solvability and an iterative method for solving five boundary value problems for nonlinear fourth order ordinary differential equations with different types of boundary
conditions and two boundary value problems for a biharmonic equation and a biharmonic equation of Kirchhoff type by using the reduction of these problems to
the operator equations for the function to be sought or an intermediate function.
Major results:
- Establish the existence, the uniqueness of the solutions of problems under
some easily verified conditions. Consider the positivity of the solution of the
boundary problem for fourth order ordinary differential equations with Dirichlet
boundary condition, combined boundary conditions and the boundary problem
for biharmonic equation.
- Propose iterative methods for solving these problems and prove the convergence with the rate of geometric progression of the iterative processes.
- Give some examples for illustrating the applicability of the obtained theoretical results, including examples showing the advantages of the method in the
thesis compared with the methods of other authors.
- Perform experiments for illustrating the effectiveness of iterative methods.
The thesis is written on the basis of articles [A1]-[A8] in the list of works of
the author related to the thesis.
Besides the introduction, conclusion and references, the contents of the thesis
are presented in three chapters.
The results in the thesis were reported and discussed at:
1. 11th Workshop on Optimization and Scientific Computing, Ba Vi, 24-27/4/2013.
4

2. 4th National Conference on Applied Mathematics, Hanoi, 23-25/12/2015.
3. 14th Workshop on Optimization and Scientific Computing, Ba Vi, 21-23/4/2016.
4. Conference of Applied Mathematics and Informatics, Hanoi University of
Science and Technology, 12-13/11/2016.
5. 10th National Conference on Fundamental and Applied Information Technology Research (FAIR’10), Da Nang, 17-18/8/2017.
6. The second Vietnam International Applied Mathematics Conference (VIAMC 2017), Ho Chi Minh, December 15 to 18, 2017.
7. Scientific Seminar of the Department of Mathematical methods in Information Technology, Institute of Information Technology, Vietnam Academy of
Science and Technology.

5

Chapter 1
Preliminary knowledge
This chapter presents some preparation knowledge needed for subsequent
chapters referenced from the literatures of A.N. Kolmogorov and S.V. Fomin
(1957), E. Zeidler (1986), A.A. Sammarskii (1989, 2001), A. Granas and J.
Dugundji (2003), J. Li (2005), Dang Quang A (2009), R.L. Burden (2011).
• Section 1.1 recalls three fixed point theorems: Brouwer fixed point theorem,
Schauder fixed point theorem, Banach fixed point theorem.
• Section 1.2 presents the definition of the Green function for the boundary
value problem for linear differential equations of order n and some specific
examples of how to define the Green function of boundary problems for
second order and fourth order differential equations with different boundary
conditions.
• Section 1.3 gives some formulas for approximation derivatives and integrals
with second order and fourth order accuracy.
• Section 1.4 presents the formula for approximation of Poisson equation with
fourth order accuracy.
• Section 1.5 mentions the elimination method for three-point equations and
the cyclic reduction method for three-point vector equations.

6

Chapter 2
The existence and uniqueness of a solution and the
iterative method for solving boundary value problems
for nonlinear fourth order ordinary equations
Chapter 2 investigates the unique solvability and an iterative method for solving five boundary value problems for nonlinear fourth order ordinary differential
equations with different types of boundary conditions: simply supported type,
Dirichlet boundary condition, combined boundary conditions, nonlinear boundary conditions. By using the reduction of these problems to the operator equations
for the function to be sought or for an intermediate function, we prove that under
some assumptions, which are easy to verify, the operator is contractive. Then,
the uniqueness of a solution is established, and the iterative method for solving
the problem converges.
This chapter is written on the basis of articles [A2]-[A4], [A6]-[A8] in the list
of works of the author related to the thesis.

2.1.

The boundary value problem for the local nonlinear fourth
order differential equation

2.1.1.

The case of combined boundary conditions

The thesis presents in detail the results of the work [A4] for the problem
u(4) (x) = f (x, u(x), u (x), u (x), u (x)),

0 < x < 1,

u(0) = 0, u (1) = 0, au (0) − bu (0) = 0, cu (1) + du (1) = 0,

(2.1.1)

where a, b, c, d ≥ 0, ρ := ad + bc + ac > 0 and f : [0, 1] × R4 → R is a continuous
function.
2.1.1.1.

The existence and uniqueness of a solution

For function ϕ(x) ∈ C[0, 1], consider the nonlinear operator A : C[0, 1] →
C[0, 1] defined by
(Aϕ)(x) = f (x, u(x), u (x), u (x), u (x)),

7

(2.1.2)

where u(x) is a solution of the problem
u(4) (x) = ϕ(x), 0 < x < 1,
u(0) = 0, u (1) = 0, au (0) − bu (0) = 0, cu (1) + du (1) = 0.

(2.1.3)

Proposition 2.1. A function ϕ(x) is a fixed point of the operator A, i.e., ϕ(x)
is a solution of the operator equation ϕ = Aϕ if and only if the function u(x)
determined from the boundary value problem (2.1.3) satisfiesthe problem (2.1.1).
Set v(x) = u (x), the problem (2.1.3) can be decomposed into two second
problems
v (x) = ϕ(x), 0 < x < 1,
av(0) − bv (0) = 0, cv(1) + dv (1) = 0,

u (x) = v(x), 0 < x < 1,
u(0) = 0, u (1) = 0.

Then
(Aϕ)(x) = f (x, u(x), y(x), v(x), z(x)),

y(x) = u (x),

z(x) = v (x).

For any number M > 0, we define the set
DM = (x, u, y, v, z) | 0 ≤ x ≤ 1, |u| ≤ ρ1 M, |y| ≤ ρ2 M, |v| ≤ ρ3 M, |z| ≤ ρ4 M ,
where

ρ1 =
ρ3 =

1
+
,
24
12ρ

1 a(d + c/2)
2
ρ

2

+

ρ2 =

b(d + c/2)
,
ρ

1
+
,
12

ρ4 =

1 ac
ρ 2

Denote the closed ball in the space C[0, 1] by B[O, M ].
Lemma 2.1. Assume that there exist constants M > 0, K1 , K2 , K3 , K4 ≥ 0 such
that |f (x, u, y, v, z)| ≤ M for all (x, u, y, v, z) ∈ DM . Then, the operator A maps
B[O, M ] into itself. Furthermore, if
|f (x, u2 ,y2 , v2 , z2 ) − f (x, u1 , y1 , v1 , z1 )|
≤ K1 |u2 − u1 | + K2 |y2 − y1 | + K3 |v2 − v1 | + K4 |z2 − z1 |

(2.1.4)

for all (t, ui , yi , vi , zi ) ∈ DM (i = 1, 2) and
q = K1 ρ1 + K2 ρ2 + K3 ρ3 + K4 ρ4 < 1

(2.1.5)

then A is a contraction operator in B[O, M ].
Theorem 2.1. Assume that all the conditions of Lemma 2.1 are satisfied. Then
the problem (2.1.1) has a unique solution u and
u ≤ ρ1 M, u

≤ ρ2 M, u
8

≤ ρ3 M, u

≤ ρ4 M.

Denote
+
DM
= (x, u, y, v, z) | 0 ≤ x ≤ 1, 0 ≤ u ≤ ρ1 M,

0 ≤ y ≤ ρ2 M, −ρ3 M ≤ v ≤ 0, −ρ4 M ≤ z ≤ ρ4 M .
Theorem 2.2. (Positivity of solution)
+
the function f is such that 0 ≤ f (x, u, y, v, z) ≤ M and the
Suppose that in DM
conditions (2.1.4), (2.1.5) of Lemma 2.1 are satisfied. Then the problem (2.1.1)
has a unique nonnegative solution.
2.1.1.2.

Solution method

The iterative method for solving the problem (2.1.1) is proposed as follows:
Iterative method 2.1.1a
i) Given an initial approximation ϕ0 (x), for example, ϕ0 (x) = f (x, 0, 0, 0, 0).
ii) Knowing ϕk (x) (k = 0, 1, 2, ...) solve consecutively two problems
(v k ) (x) = ϕk (x), 0 < x < 1,
av k (0) − b(v k ) (0) = 0, cv k (1) + d(v k ) (1) = 0,

(uk ) (x) = v k (x), 0 < x < 1,
uk (0) = (uk ) (1) = 0.

iii) Update
ϕk+1 (x) = f (x, uk (x), (uk ) (x), v k (x), (v k ) (x)).
qk
Set pk =
ϕ1 − ϕ0 . We have the following result:
1−q
Theorem 2.3. Under the assumptions of Lemma 2.1, Iterative method 2.1.1a
converges and there hold the estimates
uk − u ≤ ρ1 pk , (uk ) − u

≤ ρ2 pk , (uk ) − u

≤ ρ3 pk , (uk ) − u

≤ ρ 4 pk ,

where u is the exact solution of the problem (2.1.1).
Consider the second order boundary value problem
v (x) = g(x), x ∈ (0, 1),
c0 v(0) − c1 v (0) = C, d0 v(1) + d1 v (1) = D,
where c0 , c1 , d0 , d1 ≥ 0, c20 + c21 > 0, d20 + d21 > 0, C, D ∈ R.
Based on the results in the work [A8], we construct a difference scheme of
fourth order accuracy for solving this problem as follows

c1

c
v

(−25v0 + 48v1 − 36v2 + 16v3 − 3v4 ) = F0 ,
0
0

12h
vi−1 − 2vi + vi+1 = Fi , i = 1, 2, ..., N − 1,

 d v + d1 (25v − 48v
0 N
N
N −1 + 36vN −2 − 16vN −3 + 3vN −4 ) = FN ,
12h
h2
h4 2
where F0 = C, FN = D, Fi = h gi + Λgi +
Λ gi , i = 1, 2, ..., N − 1.
12
360
2

9

We introduce the uniform grid ω h = {xi = ih, i = 0, 1, ..., N ; h = 1/N } in
the interval [0, 1]. Denote by V k , U k , Φk the grid functions. For the general grid
function V on ω h we denote Vi = V (xi ) and denote by Vi the first difference
derivative with fourth order accuracy. Consider the following iterative method at
discrete level for solving the problem (2.1.1):
Iterative method 2.1.1b
i) Given
Φ0i = f (xi , 0, 0, 0, 0), i = 0, 1, 2, ..., N.
ii) Knowing Φk (k = 0, 1, 2, ...) solve consecutively two problems

b

aV0k −
(−25V0k + 48V1k − 36V2k + 16V3k − 3U4k ) = 0,

12h

h2
h4 2 k
k
k
k
ΛVi = Φi + ΛΦi +
Λ Φi , i = 1, 2, ..., N − 1,

12
360

 cV k + d (25V k − 48V k + 36V k − 16V k + 3V k ) = 0,
N
N
N −1
N −2
N −3
N −4
12h
 k
U0 = 0,

h4 2 k
h2
k
k
k
Λ Vi , i = 1, 2, ..., N − 1,
ΛUi = Vi + ΛVi +
12
360

k
k
k
k
k

 25UN − 48UN −1 + 36UN −2 − 16UN −3 + 3UN −4 = 0.
12h
iii) Update
Φk+1
= f (xi , Uik , (U k )i , Vik , (V k )i ),
i

i = 0, 1, 2, ..., N.

We give some examples for illustrating the applicability of the obtained theoretical results, including examples of advantages of the method in the thesis
compared to the methods of H. Feng, D. Ji, W. Ge (2009): According to the
proposed method, the problem has a unique solution meanwhile Feng’s method
cannot ensure the existence of a solution.
2.1.2.

The case of Dirichlet boundary condition

The thesis presents in detail the results of the work [A3] for the problem
u(4) (x) = f (x, u(x), u (x), u (x), u (x)),
u(a) = u(b) = 0,
2.1.2.1.

a < x < b,

u (a) = u (b) = 0,

(2.1.6)

The existence and uniqueness of a solution

For function ϕ(x) ∈ C[0, 1], consider the nonlinear problem A : C[a, b] →
C[a, b] defined by
(Aϕ)(x) = f (x, u(x), u (x), u (x), u (x)),
10

(2.1.7)

where u(x) is the solution of the problem
u(4) (x) = ϕ(x),
u(a) = u(b) = 0,

a < x < b,

(2.1.8)

u (a) = u (b) = 0.

Proposition 2.2. If the function ϕ(x) is a fixed point of the operator A, i.e.,
ϕ(x) is a solution of the operator equation
ϕ = Aϕ

(2.1.9)

then the function u(x) determined from the boundary value problem (2.1.8) solves
the problem (2.1.6). Conversely, ifu(x) is a solution of the boundary value problem
(2.1.6) then the function ϕ(x) = f (x, u(x), u (x), u (x), u (x)) is a fixed point of
the operator A defined above by (2.1.7) and (2.1.8).
Thus, the solution of the problem (2.1.6) is reduced to the solution of the
operator equation (2.1.9).
For any number M > 0, we define the set
DM = (x, u, y, v, z) | a ≤ x ≤ b, |u| ≤ C4,0 (b − a)4 M,
|y| ≤ C4,1 (b − a)3 M, |v| ≤ C4,2 (b − a)2 M, |z| ≤ C4,3 (b − a)M ,

where C4,0 = 1/384, C4,1 = 1/72 3, C4,2 = 1/12, C4,3 = 1/2.
By using Schauder fixed point theorem and Bannach fixed point theorem for
the operator A, we establish the existence and uniqueness theorems of the problem
(2.1.6).
Theorem 2.4. Suppose that the function f is continuous and there exists constant
M > 0 such that |f (x, u, y, v, z)| ≤ M for all (x, u, y, v, z) ∈ DM . Then, the
problem (2.1.6) has at least a solution.
Theorem 2.5. Suppose that the assumptions of Theorem 2.4 hold. Additionally,
assume that there exist constants K0 , K1 , K2 , K3 ≥ 0 such that
|f (x, u2 , y2 , v2 , z2 ) − f (x, u1 , y1 , v1 , z1 )| ≤ K0 |u2 − u1 | + K1 |y2 − y1 |
+ K2 |v2 − v1 | + K3 |z2 − z1 |,

(2.1.10)

for all (x, ui , yi , vi , zi ) ∈ DM (i = 1, 2) and
3

Ki C4,k (b − a)4−k < 1.

q=
k=0

Then the problem (2.1.6) has a unique solution u and
u ≤ C4,0 (b − a)4 M,

u

≤ C4,1 (b − a)3 M,

≤ C4,2 (b − a)2 M,

u

≤ C4,3 (b − a)M.

u

11

(2.1.11)

Denote
+
DM
= (x, u, y, v, z) | a ≤ x ≤ b, 0 ≤ u ≤ C4,0 (b − a)4 M,

|y| ≤ C4,1 (b − a)3 M, |v| ≤ C4,2 (b − a)2 M, |z| ≤ C4,3 (b − a)M .
+
the function f is such
Theorem 2.6. (Positivity of solution) Suppose that in DM
that 0 ≤ f (t, x, y, u, z) ≤ M and the conditions (2.1.10), (2.1.11) of Theorem 2.5
are satisfied. The the problem (2.1.6) has a unique nonnegative solution.

2.1.2.2.

Solution method and numerical examples

The iterative method for solving the problem (2.1.6) is proposed as follows:
Iterative method 2.1.2
i) Given ϕ0 (x), for example, ϕ0 (x) = f (x, 0, 0, 0, 0).
b
ii) Knowing ϕk (x), (k = 0, 1, 2, ...) calculate uk (x) = a G(x, t)ϕk (t)dt and the
(m)
derivatives uk (x) of uk (x)
(m)
uk (x)

b

=
a

∂ m G(x, t)
ϕk (t)dt (m = 1, 2, 3).
∂xm

iii) Update
ϕk+1 (x) = f (x, uk (x), uk (x), uk (x), uk (x)).
k
q
Set pk =
ϕ1 − ϕ0 . We have the following result:
1−q
Theorem 2.7. Under the assumptions of Theorem 2.5, Iterative method 2.1.2
converges with the rate of geometric progression and there hold the estimates
uk − u ≤ C4,0 (b − a)4 pk ,

uk − u

≤ C4,1 (b − a)3 pk ,

≤ C4,2 (b − a)2 pk ,

uk − u

≤ C4,3 (b − a)pk ,

uk − u

where u is the exact solution of the problem (2.1.6).
Ch 2.1. Consider the problem
u(4) (x) = f (x, u(x), u (x), u (x), u (x)),
u(a) = A1 ,

u(b) = B1 ,

u (a) = A2 ,

a < x < b,
u (b) = B2 .

(2.1.12)

Set v(x) = u(x) − P (x), where P (x) is the third degree polynomial satisfying the
boundary conditions in this problem and denote
F (x, v(x), v (x), v (x), v (x))
= f (x, v(x) + P (x), (v(x) + P (x)) , (v(x) + P (x)) , (v(x) + P (x)) ).
Then, the problem (2.1.12) becomes
v (4) (x) = F (x, v(x), v (x), v (x), v (x)),
v(a) = v(b) = 0, v (a) = v (b) = 0.

a < x < b,

Therefore, we can apply the results derived above to this problem.
12

Theorem 2.8. Suppose that the function f is continuous and there exists constan
M > 0 such that |f (x, v0 , v1 , v2 , v3 )| ≤ M for all (x, v0 , v1 , v2 , v3 ) ∈ DM , where
DM = (x, v0 , v1 , v2 , v3 ) | a ≤ x ≤ b, |vi | ≤ max |P (i) (x)|
x∈[a,b]

+ C4,i (b − a)4−i M, i = 0, 1, 2, 3 .
Then, the problem (2.1.12) has at least a solution.
We give some examples for illustrating the applicability of the obtained theoretical results, including examples of advantages of the method in the thesis
compared to the methods of R.P. Agarwal (1984): Agarwal can only establish
the existence of a solution of the problem or does not guarantee the existence
of a solution of the problem meanwhile according to the proposed method, the
problem has a unique solution or a unique positive solution.
2.1.3.

The case of nonlinear boundary conditions

The thesis presents in detail the results of the work [A7] for the problem
u(4) (x) = f (x, u, u ),
u(0) = 0,

u(L) = 0,

0 < x < L,
u (0) = g(u (0)),

u (L) = h(u (L)).

(2.1.13)

Set u = v, u = w. Then, the problem (2.1.13) is decomposed to the problems
for w v u

x

 w (x) = f x, v(t)dt, v(x) , 0 < x < L,
u (x) = w(x), 0 < x < L,
0

 w(0) = g(v(0)),

u(0) = 0,

w(L) = h(v(L)),

u(L) = 0.

The solution u(x) from these problems depends on the function v. Consequently, its derivative u also depends on v. Therefore, we can represent this
dependence by an operator T : C[0, L] → C[0, L] defined by T v = u . Combining
with u = v we get the operator equation v = T v, i.e., v is a fixed point of T . To
consider properties of the operator T, we introduce the space
L

S = v ∈ C[0, L],

v(t)dt = 0 .
0

We make the following assumptions on the given functions in the problem
(2.1.13): there exist constants λf , λg , λh ≥ 0 such that
|f (x, u, v) − f (x, u, v)| ≤ λf max |u − u|, |v − v|,
|g(u) − g(u)| ≤ λg |u − u|,

|h(u) − h(u)| ≤ λh |u − u|,

(2.1.14)

for any u, u, v, v. Applying Banach fixed point theorem for T, we establish the
existence and uniqueness of a solution of the problem.
13

Proposition 2.3. With assumption (2.1.14), the problem (2.1.13) has a unique
solution if
L
L3
L
q=
λf max
, 1 + (λg + λh ) < 1.
(2.1.15)
16
2
2
The iterative method for solving the problem (2.1.13) is proposed as follows:
Iterative method 2.1.3
(i) Given an initial approximation v0 (x), for example, v0 (x) = 0.
(ii) Knowing vk (x) (k = 0, 1, 2, ...) solve consecutively two problems

 w (x) = f x, x v (t)dt, v (x) , 0 < x < L,
uk (x) = wk (x), 0 < x < L,
k
k
0 k
 wk (0) = g(vk (0)), wk (L) = h(vk (L)),
uk (0) = uk (L) = 0.
(iii) Update

vk+1 (x) = uk (x).

Theorem 2.9. Under the assumptions (2.1.14), (2.1.15), Iterative method 2.1.3
converges with rate of geometric progression with the quotient q, and there hold
the estimates
uk − u

qk
v1 − v0 ,
1−q

uk − u ≤

L
u −u ,
2 k

where u is the exact solution of the original problem (2.1.13).
For testing the convergence of the method, we perform some experiments for
the case of the known exact solutions and also for the case of the unknown exact
solutions.

2.2.
2.2.1.

The boundary value problem for the nonlocal nonlinear
fourth order differential equation
The case of boundary conditions of simply supported type

The thesis presents in detail the results of the work [A2] for the problem
L
(4)

|u (s)|2 ds u (x)

u (x) − M
0

= f (x, u(x), u (x), u (x), u (x)), 0 < x < L,
u(0) = u(L) = 0, u (0) = u (L) = 0.
2.2.1.1.

(2.2.1)

The existence and uniqueness of a solution

For function ϕ(x) ∈ C[0, L], consider the nonlinear operator A : C[0, L] →
C[0, L] defined by
(Aϕ)(x) = M ( u

2
2 )u

(x) + f (x, u(x), u (x), u (x), u (x)),
14

(2.2.2)

where .

2

is the norm in L2 [0, L], u(x) is a solution of the problem
u(4) (x) = ϕ(x), 0 < x < L,
u(0) = u(L) = 0, u (0) = u (L) = 0.

(2.2.3)

Proposition 2.4. A function ϕ(x) is a fixed point of the operator A, i.e., ϕ(x)
is a solution of the operator equation ϕ = Aϕ if and only if the function u(x)
determined from the boundary value problem (2.2.3) satisfies the problem (2.2.1).
By setting v(x) = u (x), the problem (2.2.3) is decomposed to the problems
v (x) = ϕ(x), 0 < x < L,
v(0) = v(L) = 0,

u (x) = v(x), 0 < x < L,
u(0) = u(L) = 0.

Then the operator A is represented in the form
(Aϕ)(x) := M ( y 22 )v(x) + f (x, u(x), y(x), v(x), z(x)), y(x) = u (x), z(x) = v (x).
For any number R > 0, we define the set
DR := (x, u, y, v, z) | 0 ≤ x ≤ L, |u| ≤

L3 R
L2 R
LR
5L4 R
, |y| ≤
, |v| ≤
, |z| ≤
.
384
24
8
2

Let B[O, R] denote the closed ball in the space C[0, L].
8
Lemma 2.2. If there are constants R > 0, 0 ≤ m ≤ 2 , λM , K1 , K2 , K3 , K4 ≥ 0
L
such that
mL2
,
|M (s)| ≤ m, |f (x, u, y, v, z)| ≤ R 1 −
8
R2 L7
for all (x, u, y, v, z) ∈ DR and 0 ≤ s ≤
, then, the operator A maps B[O, R]
576
|M (s2 ) − M (s1 )| ≤ λM |s2 − s1 |,
|f (x, u2 , y2 , v2 , z2 ) − f (x, u1 , y1 , v1 , z1 )|
≤ K1 |u2 − u1 | + K2 |y2 − y1 | + K3 |v2 − v1 | + K4 |z2 − z1 |,
R 2 L7
(i = 1, 2) and
for all (x, ui , yi , vi , zi ) ∈ DR , 0 ≤ si ≤
576
q = K1

5L4
L3
L2
L mL2 λM R2 L9
+ K2 + K3 + K4 +
+
<1
384
24
8
2
8
2304

then A is a contraction operator in B[O, R].
Theorem 2.10. In conditions of Lemma 2.2, the problem (2.2.1) has a unique
solution u such that
5L4
u ≤
R,
384

u

L3

R,
24

u
15

L2

R,
8

u

L
R.
2

2.2.1.2.

Iterative method and numerical examples

The iterative method for solving the problem (2.2.1) is proposed as follows:
Iterative method 2.2.1
i) Given an initial approximation ϕ0 (x), for example, ϕ0 (x) = f (x, 0, 0, 0, 0).
ii) Knowing ϕk (x) (k = 0, 1, 2, ...) solve successively the problems
uk (x) = vk (x), 0 < x < L,
uk (0) = uk (L) = 0.

vk (x) = ϕk (x), 0 < x < L,
vk (0) = vk (L) = 0,

iii) Update ϕk+1 (x) = M ( uk 22 )uk (x) + f (x, uk (x), uk (x), uk (x), uk (x)).
qk
Set pk =
ϕ1 − ϕ0 . We have the following theorem:
1−q
Theorem 2.11. In conditions of Lemma 2.2, Iterative method 2.2.1 converges to
the exact solution u of the problem (2.2.1) and
5L4
uk − u ≤
pk , uk − u
384

L3

pk , uk − u
24

L2

pk , uk − u
8

L
pk .
2

We give some examples for illustrating the applicability of the obtained theoretical results, including examples of advantages of the method in the thesis
compared to the methods of P. Amster, P.P. C´ardenas Alzate (2008): According to the method proposed, the problem has a unique solution meanwhile these
authors’s method cannot ensure the existence of a solution.
2.2.2.

The case of nonlinear boundary conditions

The thesis presents in detail the results of the work [A6] for the problem
L

|u (s)|2 ds u (x) = f (x, u(x)),

(4)

u (x) − M

0 < x < L,

0

(2.2.4)

L

|u (s)|2 ds u (L) = g(u(L)).

u(0) = u (0) = u (L) = 0, u (L) − M
0

2.2.2.1.

The existence and uniqueness of a solution

By setting v(x) = u (x) − M (||u ||22 )u(x), where .
L2 [0, L], the problem (2.2.4) is reduced to the problems
v (x) = f (x, u(x)),
v(L) = −M
u (x) = M

u
u

2
2

2
2

2

denotes the norm of

0 < x < L,
u(L),

v (L) = g(u(L))

u(x) + v(x),

u(0) = u (0) = 0.
16

0 < x < L,

We can see that u is a solution of the problem (2.2.4) if and only if it is a solution
of the integral equation u(x) = (T u)(x), where
L

(T u)(x) =

G(x, t) M

u

2
2

u(t)

0
L

G(t, s)f (s, u(s))ds + g(u(L))(t − L) − M

+

u

2
2

u(L) dt.

0

Applying Schauder fixed point theorem and Banach fixed point theorem for the
operator T , we establish the existence and uniqueness theorams of the problem
(2.2.4).
Theorem 2.12. Suppose that f, g, M are continuous functions and there exist
constants R, A, B, m > 0 such that
|f (t, u)| ≤ A, ∀(t, u) ∈ [0, L] × [−L2 R, L2 R],
|g(u)| ≤ B, ∀u ∈ [−L2 R, L2 R],
|M (s)| ≤ m,
Then, if

L2
2 A

2

∀s ∈ [0, L3 R ].

+ LB ≤ R(1 − mL2 ), the problem (2.2.4) has at least a solution.

Theorem 2.13. Suppose that the assumptions of Theorem 2.12 hold. Further
assume that there exist constants λf , λg , λM > 0 such that
|f (x, u) − f (x, v)| ≤ λf |u − v|,
|g(u) − g(v)| ≤ λg |u − v|,

∀(x, u), (x, v) ∈ [0, L] × [−L2 R, L2 R],
∀u, v ∈ [−L2 R, L2 R],
2

∀u, v ∈ [0, L3 R ].

|M (u) − M (v)| ≤ λM |u − v|,
2

Then, if q = 4L5 R λM +
unique solution.
2.2.2.2.

L4
2 λf

+ L3 λg + 2mL2 < 1, the problem (2.2.4) has a

Iterative method and numerical examples

The iterative method for solving the problem (2.2.4) is proposed as follows:
Iterative method 2.2.2
i) Given an initial approximation u0 (x), example, u0 (x) = 0, in [0, L].
ii) Knowing uk (x) (k = 0, 1, 2, ...) solve consecutively the final value problem
vk (x) = f (x, uk (x)),
vk (L) = −M

uk

2
2

0 < x < L,
uk (L),

uk+1 (x) = vk (x) + M

uk

uk+1 (0) = uk+1 (0) = 0.
17

2
2

vk (L) = g(uk (L)),
uk (x),

0 < x < L,

Theorem 2.14. Under the assumptions of Theorem 2.13, Iterative method 2.2.2
converges with rate of geometric progression with the quotient q and there hold the
estimates
uk − u

≤ L uk − u

≤ L2 uk − u

≤ L2

qk
u − u0
1−q 1

∞,

where u is the exact solution of the original problem (2.2.4).
We give some examples for illustrating the applicability of the obtained theoretical results, including examples of advantages of the method in the thesis
compared to the methods of T.F. Ma (2003): According to the proposed method,
the problem has a unique solution meanwhile Ma’s method can only establish the
existence of a solution or cannot ensure the existence of a solution.
CONCLUSION OF CHAPTER 2
In this chapter, we investigate the unique solvability and iterative method
for five boundary value problems for local or nonlocal nonlinear fourth order
differential equations with different boundary conditions: The case of boundary
conditions of simply supported type, combined boundary conditions, Dirichlet
boundary condition, nonlinear boundary conditions. By using the reduction of
these problems to the operator equations for the function to be sought or for an
intermediate function, we prove that under some assumptions, which are easy to
verify, the operator is contractive. Then, the uniqueness of a solution is established, and the iterative method for solving the problem converges. We also give
some examples for illustrating the applicability of the obtained theoretical results,
including examples showing the advantages of the method in the thesis compared
with the methods of other authors.

18

Chapter 3
The existence and uniqueness of a solution and the
iterative method for solving boundary value problems
for nonlinear fourth order partial equations
Continuing the development of the techniques in Chapter 2, in Chapter 3, we
also obtain the results of the existence and uniqueness of a solution, and the
convergence of iterative methods for solving two boundary value problems for a
nonlinear biharmonic equation and a nonlinear biharmonic equation of Kirchhoff
type. The results of this chapter are presented in articles [A1], [A4] in the list of
works of the author related to the thesis.

3.1.

The nonlinear boundary value problem for the biharmonic equation

The thesis presents in detail the results of the work [A5] for the problem
∆2 u = f (x, u, ∆u), x ∈ Ω,
u = 0, ∆u = 0, x ∈ Γ,

(3.1.1)

where Ω is a connected bounded domain in R2 with a smooth (or piecewise
smooth) boundary Γ.
3.1.1.

The existence and uniqueness of a solution

For function ϕ(x) ∈ C(Ω), consider the nonlinear operation A : C(Ω) → C(Ω)
defined by
(Aϕ)(x) = f (x, u(x), ∆u(x)),
(3.1.2)
where u(x) is a solution of the problem
∆2 u = ϕ(x), x ∈ Ω,
u = ∆u = 0, x ∈ Γ.

(3.1.3)

Proposition 3.1. A function ϕ(x) is a solution of the operator equation Aϕ = ϕ,
i.e., ϕ(x) is a fixed point of the operator A defined by (3.1.2)-(3.1.3) if and only if
the function u(x) being the solution of the boundary value problem (3.1.3) solves
the problem (3.1.1).
19

Lemma 3.1. Suppose that Ω is a connected bounded domain in RK (K ≥ 2)
with a smooth boundary (or smooth of each piece) Γ. Then, for the solution of
the problem −∆u = f (x), x ∈ Ω, u = 0, x ∈ Γ, there holds the estimate
2
R
u ≤ CΩ f , where u = maxx∈Ω¯ |u(x)|, CΩ =
and R is the radius of the
4
1
circle containing the domain Ω. If Ω is the unit square then u ≤ f .
8
For each positive number M denote
DM = {(x, u, v)| x ∈ Ω, |u| ≤ CΩ2 M, |v| ≤ CΩ M }.
Theorem 3.1. Assume that there exist numbers M, K1 , K2 ≥ 0 such that
|f (x, u, v)| ≤ M,

∀(x, u, v) ∈ DM ,

|f (x, u2 , v2 ) − f (x, u1 , v1 )| ≤ K1 |u2 − u1 | + K2 |v2 − v1 |, ∀(x, ui , vi ) ∈ DM , i = 1, 2.
q := (K2 + CΩ K1 )CΩ < 1.
¯ and u ≤ C 2 M.
Then the problem (3.1.1) has a unique solution u(x) ∈ C(Ω)

Denote
+
= {(x, u, v)| x ∈ Ω, 0 ≤ u ≤ CΩ2 M, −CΩ M ≤ v ≤ 0}.
DM

Theorem 3.2. (Positivity of solution) Assume that there exist numbers M, K1 , K2 ≥
0 such that
+
0 ≤ f (x, u, v) ≤ M, ∀(x, u, v) ∈ DM
.
+
|f (x, u2 , v2 ) − f (x, u1 , v1 )| ≤ K1 |u2 − u1 | + K2 |v2 − v1 |, ∀(x, ui , vi ) ∈ DM
, i = 1, 2,

q := (K2 + CΩ K1 )CΩ < 1.
¯ and
Then the problem (3.1.1) possesses a unique positive solution u(x) ∈ C(Ω)
0 ≤ u(x) ≤ CΩ2 M.
3.1.2.

Solution method and numerical examples

Consider the following iterative process for finding fixed point ϕ of the operator
A and simultaneously for finding the solution u of the original boundary value
problem:
Iterative method 3.1.1
1. Given an initial approximation ϕ0 ∈ B[O, M ], for example,
ϕ0 (x) = f (x, 0, 0), x ∈ Ω.

20

(3.1.4)

2. Knowing ϕk trong Ω (k = 0, 1, ...) solve sequentially two Poisson problems
∆vk

= ϕk ,

vk

= 0,

x ∈ Ω,
x ∈ Γ,

∆uk

= vk ,

uk

= 0,

x ∈ Ω,
x ∈ Γ.

(3.1.5)

3. Update
ϕk+1 = f (x, uk , vk ).

(3.1.6)

Theorem 3.3. Suppose that the assumptions of Theorem 3.1 (or Theorem 3.2)
hold. Then Iterative method 3.1.1 converges and there holds the estimate
qk
2
||uk − u|| ≤ CΩ
ϕ1 − ϕ0 ,
(1 − q)
where u is the exact solution of the problem (3.1.1).
Theorem 3.4. (Monotony) Assume that all the conditions of Theorem 3.1 (or
Theorem 3.2) are satisfied. In addition, we assume that the function f (x, u, v) is
(1)
(2)
increasing in u and decreasing in v for any (x, u, v) ∈ DM . Then, if ϕ0 , ϕ0 ∈
(2)
(1)
B[O, M ] are initial approximations and ϕ0 (x) ≤ ϕ0 (x) for any x ∈ Ω then the
(1)
(2)
sequences {uk }, {uk } generated by the iterative process (3.1.4)-(3.1.6) satisfy the
relation
(2)
(1)
uk (x) ≤ uk (x), k = 0, 1, ...; x ∈ Ω.
Corollary 3.1. Denote ϕmin =

min
(x,u,v)∈DM

f (x, u, v), ϕmax =

max

f (x, u, v).

(x,u,v)∈DM

Under the assumptions of Theorem 3.4, if starting from ϕ0 = ϕmin we obtain the
increasing sequence {uk (x)}, inversely, starting from ϕ0 = ϕmax we obtain the
decreasing sequence {uk (x)}, both of them converge to the exact solution u(x) of
the problem and uk (x) ≤ u(x) ≤ uk (x).
To numerically realize the above iterative method, we use difference schemes
of second and fourth order of accuracy for solving second order boundary value
problems (3.1.5) at each iteration. The numerical examples show the advantage of
the proposed method compared to the methods in Y.M. Wang (2007), Y. An, R.
Liu (2008), S. Hu, L. Wang (2014) on the convergence rate or conclusions about
the uniqueness of the solution.

3.2.

The nonlinear boundary value problem for the biharmonic equation of Kirchhoff type

The thesis presents in detail the results of the work [A1] for the problem
∆2 u = M

|∇u|2 dx ∆u + f (x, u),

u = 0,

∆u = 0,

x ∈ Ω,

(3.2.1)

x ∈ Γ,

where Ω is a connected bounded domain in RK (K ≥ 2) with a smooth (or
piecewise smooth) boundary Γ.
21

3.2.1.

The existence and uniqueness of a solution

For function ϕ(x) ∈ C(Ω), consider the nonlinear operator A : C(Ω) → C(Ω)
defined by
|∇u|2 dx ∆u + f (x, u),

(Aϕ)(x) = M

(3.2.2)

where u(x) is a solution of the problem
∆2 u = ϕ(x), x ∈ Ω,
u = ∆u = 0, x ∈ Γ.

(3.2.3)

Proposition 3.2. A function ϕ(x) is a fixed point of the operator A, i.e., ϕ(x)
is a solution of the operator equationAϕ = ϕ, if and only if the function u(x)
determined from the boundary value problem (3.2.3) satisfies the problem (3.2.1).
For any number R > 0 and the coefficient CΩ defined by Lemma 3.1, we define
the set
DR = (x, u) | x ∈ Ω; |u| ≤ CΩ2 R .
Theorem 3.5. Assume that there exist constants R, λf , m, λM > 0, m ≤ 1/CΩ
such that
|M (s)| ≤ m, |M (s1 ) − M (s2 )| ≤ λM |s1 − s2 |,
|f (x, u)| ≤ R(1 − mCΩ ),

|f (x, u1 ) − f (x, u2 )| ≤ λf |u1 − u2 |,

for all (x, u), (x, ui ) ∈ DR (i = 1, 2); 0 ≤ s, s1 , s2 ≤ CΩ3 R2 SΩ ,
q = λf CΩ2 + mCΩ + 2λM R2 CΩ4 SΩ < 1,
where SΩ is the measure of the domain Ω. Then the problem (3.2.1) has a unique
solution u(x) ∈ C(Ω) which satisfies the estimates u ≤ CΩ2 R, ∆u ≤ CΩ R.
Remark 3.1. As seen from Theorem 3.5 for the existence and uniqueness of
solution of the problem (3.2.1) we require the conditions on the function f (x, u)
and M (s) only in bounded domains. Due to this the assumptions on the growth
of these functions at infinity, which are needed in F. Wang, Y. An (2012) for
the case when M is a function and in other works for the case M = const are
freed. This is an advantage of our result over the results of others. Moreover, the
conditions of Theorem 3.5 are simple and are easy to be verified as will be seen
from the numerical examples.
Remark 3.2. In Theorem 3.5, if M (s) = m = const then λM = 0 and by
q = λf CΩ2 + mCΩ < 1. Thus, the assumptions of the theorem are reduced to the
boundedness and the satisfaction of Lipschitz condition of f (x, u) in the domain
DR . These conditions obviously are not complicated as in Y. An, R. Liu (2008),
R. Pei (2010), S. Hu, L. Wang (2014).
22

Remark 3.3. . In the case if the right-hand side function f = f (u), the conditions
for f in Theorem 3.5 become the boundedness and the Lipschitz condition for f (u)
in the domain DR = u; |u| ≤ CΩ2 R .
3.2.2.

Solution method and numerical examples

The iterative method for solving the problem (3.1.1) is proposed as follows:
Iterative method 3.2.1
i) Given ϕ0 ∈ B[O, R], for example, ϕ0 (x) = f (x, 0), x ∈ Ω.
ii) Knowing ϕk (k = 0, 1, 2, ...) solve successively two second order problems

iii) Update

∆vk

= ϕk ,

vk

= 0,

x ∈ Ω,
x ∈ Γ,

ϕk+1 (x) = M

∆uk

= vk ,

uk

= 0,

2
Ω |∇uk | dx

x ∈ Ω,
x ∈ Γ.

(3.2.4)

vk + f (x, uk ).

Theorem 3.6. Under the assumptions of Theorem 3.5, Iterative method 3.2.1
converges and there holds the estimate
CΩ2 q k
uk − u ≤
ϕ1 − ϕ0 ,
1−q
where u is the exact solution of the problem (3.2.1).
In order to numerically realize the above iterative process we use the difference
scheme of fourth order accuracy for solving (3.2.4), and formulas of fourth order
accuracy for approximating ∇u. In order to test the convergence of the proposed
iterative method we perform some experiments for the case of known exact solutions and also for the case of unknown exact solutions of the problem (3.2.1) in
unit square. Some examples show the advantage of the proposed method compared to the method in F. Wang, Y. An (2012) in the conclusion of the existence
and uniqueness of solution of the problem.
CONCLUSION OF CHAPTER 3
In this chapter, we investigate the unique solvability and iterative method for
two boundary value problems for nonlinear biharmonic equation and nonlinear
biharmonic equation of Kirchhoff type.
- For both problems, under some easily verified conditions, we establish the existence and uniqueness of solution. Especially, for the boundary problem for the
biharmonic equation, we also consider the positive property of the solution.
- We propose iterative methods for solving these problems and prove the convergence of the iterative process. Especially, for the boundary problem for the
biharmonic equation, we also show the monotonicity of the approximation sequences.
23

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