MINISTRY OF EDUCATION

VIETNAM ACADEMY OF

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:

Graduate University of Science and Technology – Vietnam Academy

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

under some additional assumptions. In addition, we can construct two monotone

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 =

2ad + bc + 6bd

1

+

,

24

12ρ

1 a(d + c/2)

2

ρ

2

+

ρ2 =

b(d + c/2)

,

ρ

1

ad + bc + 4bd

+

,

12

4ρ

ρ4 =

1 ac

+ max(ad, bc) .

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

into itself. If, in addition,

|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)

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),

Ω

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)

Ω

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

VIETNAM ACADEMY OF

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:

Graduate University of Science and Technology – Vietnam Academy

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

under some additional assumptions. In addition, we can construct two monotone

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 =

2ad + bc + 6bd

1

+

,

24

12ρ

1 a(d + c/2)

2

ρ

2

+

ρ2 =

b(d + c/2)

,

ρ

1

ad + bc + 4bd

+

,

12

4ρ

ρ4 =

1 ac

+ max(ad, bc) .

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

into itself. If, in addition,

|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)

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),

Ω

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)

Ω

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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