Summary of Phd thesis: Solving some nonlinear boundary value problems for fourth order differential equations

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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.

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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 me- chanics 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 inte- gral 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 meth- ods presented above, we can mention some other methods such as the finite el- ement 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 bound- ary 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 bend- ing 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 solu- tions of some problems for local and nonlocal fourth order differential equations. - Propose iterative methods for solving these problems and prove the conver- gence of the iterative processes. - Give some examples in both cases of known and unknown solution to illus- trate 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 solv- ability and an iterative method for solving five boundary value problems for non- linear fourth order ordinary differential equations with different types of boundary conditions and two boundary value problems for a biharmonic equation and a bi- harmonic 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 conver- gence with the rate of geometric progression of the iterative processes. - Give some examples for illustrating the applicability of the obtained theo- retical 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 Tech- nology Research (FAIR’10), Da Nang, 17-18/8/2017. 6. The second Vietnam International Applied Mathematics Conference (VI- AMC 2017), Ho Chi Minh, December 15 to 18, 2017. 7. Scientific Seminar of the Department of Mathematical methods in Informa- tion 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 solv- ing 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 bound- ary 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), u0 (x), u00 (x), u000 (x)), 0 < x < 1, (2.1.1) u(0) = 0, u0 (1) = 0, au00 (0) − bu000 (0) = 0, cu00 (1) + du000 (1) = 0, 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), u0 (x), u00 (x), u000 (x)), (2.1.2) 7
where u(x) is a solution of the problem u(4) (x) = ϕ(x), 0 < x < 1, (2.1.3) u(0) = 0, u0 (1) = 0, au00 (0) − bu000 (0) = 0, cu00 (1) + du000 (1) = 0. 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) = u00 (x), the problem (2.1.3) can be decomposed into two second problems 00 00 v (x) = ϕ(x), 0 < x < 1, u (x) = v(x), 0 < x < 1, av(0) − bv 0 (0) = 0, cv(1) + dv 0 (1) = 0, u(0) = 0, u0 (1) = 0. Then (Aϕ)(x) = f (x, u(x), y(x), v(x), z(x)), y(x) = u0 (x), z(x) = v 0 (x). For any number M > 0, we define the set n o DM = (x, u, y, v, z) | 0 ≤ x ≤ 1, |u| ≤ ρ1 M, |y| ≤ ρ2 M, |v| ≤ ρ3 M, |z| ≤ ρ4 M , 1 2ad + bc + 6bd 1 ad + bc + 4bd where ρ1 = + , ρ2 = + , 24 12ρ 12 4ρ 1 a(d + c/2) 2 b(d + c/2) 1 ac ρ3 = + , ρ4 = + max(ad, bc) . 2 ρ ρ 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 )| (2.1.4) ≤ K1 |u2 − u1 | + K2 |y2 − y1 | + K3 |v2 − v1 | + K4 |z2 − z1 | 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 kuk ≤ ρ1 M, ku0 k ≤ ρ2 M, ku00 k ≤ ρ3 M, ku000 k ≤ ρ4 M. 8
Denote n + DM = (x, u, y, v, z) | 0 ≤ x ≤ 1, 0 ≤ u ≤ ρ1 M, o 0 ≤ y ≤ ρ2 M, −ρ3 M ≤ v ≤ 0, −ρ4 M ≤ z ≤ ρ4 M . Theorem 2.2. (Positivity of solution) + Suppose that in DM the function f is such that 0 ≤ f (x, u, y, v, z) ≤ M and the 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 k 00 k 00 (v ) (x) = ϕk (x), 0 < x < 1, (u ) (x) = v k (x), 0 < x < 1, av k (0) − b(v k )0 (0) = 0, cv k (1) + d(v k )0 (1) = 0, uk (0) = (uk )0 (1) = 0. iii) Update ϕk+1 (x) = f (x, uk (x), (uk )0 (x), v k (x), (v k )0 (x)). qk Set pk = kϕ1 − ϕ0 k. 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 kuk − uk ≤ ρ1 pk , k(uk )0 − u0 k ≤ ρ2 pk , k(uk )00 − u00 k ≤ ρ3 pk , k(uk )000 − u000 k ≤ ρ4 pk , where u is the exact solution of the problem (2.1.1). Consider the second order boundary value problem 00 v (x) = g(x), x ∈ (0, 1), c0 v(0) − c1 v 0 (0) = C, d0 v(1) + d1 v 0 (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 0 v0 − (−25v0 + 48v1 − 36v2 + 16v3 − 3v4 ) = F0 ,   12h vi−1 − 2vi + vi+1 = Fi , i = 1, 2, ..., N − 1,  d v + d1 (25v − 48v  N −1 + 36vN −2 − 16vN −3 + 3vN −4 ) = FN ,  0 N N 12h 2 h2 h4 2 where F0 = C, FN = D, Fi = h gi + Λgi + Λ gi , i = 1, 2, ..., N − 1. 12 360 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 Vi0 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, h2 h4 2 k    k k k ΛUi = Vi + ΛVi + Λ Vi , i = 1, 2, ..., N − 1, k k 12 360 k k k  25UN − 48UN −1 + 36UN −2 − 16UN −3 + 3UN −4 = 0.    12h iii) Update Φk+1 i = f (xi , Uik , (U k )0i , Vik , (V k )0i ), i = 0, 1, 2, ..., N. We give some examples for illustrating the applicability of the obtained the- oretical 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), u0 (x), u00 (x), u000 (x)), a < x < b, (2.1.6) u(a) = u(b) = 0, u0 (a) = u0 (b) = 0, 2.1.2.1. 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), u0 (x), u00 (x), u000 (x)), (2.1.7) 10
where u(x) is the solution of the problem u(4) (x) = ϕ(x), a < x < b, (2.1.8) u(a) = u(b) = 0, u0 (a) = u0 (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), u0 (x), u00 (x), u000 (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 n DM = (x, u, y, v, z) | a ≤ x ≤ b, |u| ≤ C4,0 (b − a)4 M, o 3 2 |y| ≤ C4,1 (b − a) M, |v| ≤ C4,2 (b − a) 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 | (2.1.10) + K2 |v2 − v1 | + K3 |z2 − z1 |, for all (x, ui , yi , vi , zi ) ∈ DM (i = 1, 2) and 3 X q= Ki C4,k (b − a)4−k < 1. (2.1.11) k=0 Then the problem (2.1.6) has a unique solution u and kuk ≤ C4,0 (b − a)4 M, ku0 k ≤ C4,1 (b − a)3 M, ku00 k ≤ C4,2 (b − a)2 M, ku000 k ≤ C4,3 (b − a)M. 11
Denote n + DM = (x, u, y, v, z) | a ≤ x ≤ b, 0 ≤ u ≤ C4,0 (b − a)4 M, o 3 2 |y| ≤ C4,1 (b − a) M, |v| ≤ C4,2 (b − a) M, |z| ≤ C4,3 (b − a)M . + Theorem 2.6. (Positivity of solution) Suppose that in DM the function f is such 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). Rb 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) Z b m (m) ∂ G(x, t) uk (x) = ϕk (t)dt (m = 1, 2, 3). a ∂xm iii) Update ϕk+1 (x) = f (x, uk (x), u0k (x), u00k (x), u000 k (x)). k q Set pk = kϕ1 − ϕ0 k. 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 kuk − uk ≤ C4,0 (b − a)4 pk , ku0k − u0 k ≤ C4,1 (b − a)3 pk , ku00k − u00 k ≤ C4,2 (b − a)2 pk , ku000 000 k − u k ≤ C4,3 (b − a)pk , 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), u0 (x), u00 (x), u000 (x)), a < x < b, (2.1.12) u(a) = A1 , u(b) = B1 , u0 (a) = A2 , u0 (b) = B2 . 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 0 (x), v 00 (x), v 000 (x)) = f (x, v(x) + P (x), (v(x) + P (x))0 , (v(x) + P (x))00 , (v(x) + P (x))000 ). Then, the problem (2.1.12) becomes v (x) = F (x, v(x), v 0 (x), v 00 (x), v 000 (x)), (4) a < x < b, v(a) = v(b) = 0, v 0 (a) = v 0 (b) = 0. 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 n DM = (x, v0 , v1 , v2 , v3 ) | a ≤ x ≤ b, |vi | ≤ max |P (i) (x)| x∈[a,b] o 4−i + C4,i (b − a) 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 the- oretical 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, u0 ), 0 < x < L, (2.1.13) u(0) = 0, u(L) = 0, u00 (0) = g(u0 (0)), u00 (L) = h(u0 (L)). Set u0 = v, u00 = w. Then, the problem (2.1.13) is decomposed to the problems for w v u  x (  w00 (x) = f x, R v(t)dt, v(x) , 0 < x < L,  u00 (x) = w(x), 0 < x < L, 0  w(0) = g(v(0)),  w(L) = h(v(L)), u(0) = 0, u(L) = 0. The solution u(x) from these problems depends on the function v. Conse- quently, its derivative u0 also depends on v. Therefore, we can represent this dependence by an operator T : C[0, L] → C[0, L] defined by T v = u0 . Combining with u0 = 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 n ZL o 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|, (2.1.14) |g(u) − g(u)| ≤ λg |u − u|, |h(u) − h(u)| ≤ λh |u − u|, 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 L3 L 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   w00 (x) = f x, R x v (t)dt, v (x) , 0 < x < L, ( k 0 k k u00k (x) = wk (x), 0 < x < L,  wk (0) = g(vk (0)), wk (L) = h(vk (L)), uk (0) = uk (L) = 0. (iii) Update vk+1 (x) = u0k (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 qk L 0 ku0k − u0 k ≤ kv1 − v0 k, kuk − uk ≤ kuk − u0 k, 1−q 2 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. The boundary value problem for the nonlocal nonlinear fourth order differential equation 2.2.1. The case of boundary conditions of simply supported type The thesis presents in detail the results of the work [A2] for the problem Z L (4) u (x) − M c |u (s)| ds u00 (x) 0 2 0 (2.2.1) = f (x, u(x), u0 (x), u00 (x), u000 (x)), 0 < x < L, u(0) = u(L) = 0, u00 (0) = u00 (L) = 0. 2.2.1.1. 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 c(ku0 k2 )u00 (x) + f (x, u(x), u0 (x), u00 (x), u000 (x)), (Aϕ)(x) = M (2.2.2) 2 14
where k.k2 is the norm in L2 [0, L], u(x) is a solution of the problem u(4) (x) = ϕ(x), 0 < x < L, (2.2.3) u(0) = u(L) = 0, u00 (0) = u00 (L) = 0. 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) = u00 (x), the problem (2.2.3) is decomposed to the problems 00 00 v (x) = ϕ(x), 0 < x < L, u (x) = v(x), 0 < x < L, v(0) = v(L) = 0, u(0) = u(L) = 0. Then the operator A is represented in the form c(kyk2 )v(x) + f (x, u(x), y(x), v(x), z(x)), y(x) = u0 (x), z(x) = v 0 (x). (Aϕ)(x) := M 2 For any number R > 0, we define the set n 5L4 R L3 R L2 R LR o DR := (x, u, y, v, z) | 0 ≤ x ≤ L, |u| ≤ , |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 , λMc, K1 , K2 , K3 , K4 ≥ 0 L such that 2 mL |Mc(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 c(s2 ) − M c(s1 )| ≤ λ c|s2 − s1 |, M |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 for all (x, ui , yi , vi , zi ) ∈ DR , 0 ≤ si ≤ (i = 1, 2) and 576 2 9 5L4 L3 L2 L mL2 λM cR L q = K1 + K2 + K3 + K4 + +
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 00 00 vk (x) = ϕk (x), 0 < x < L, uk (x) = vk (x), 0 < x < L, vk (0) = vk (L) = 0, uk (0) = uk (L) = 0. c(ku0 k2 )u00 (x) + f (x, uk (x), u0 (x), u00 (x), u000 (x)). iii) Update ϕk+1 (x) = M k 2 k k k k k q Set pk = kϕ1 − ϕ0 k. 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 0 0 L3 00 00 L2 L kuk − uk ≤ pk , kuk − u k ≤ pk , kuk − u k ≤ pk , ku000 000 k −u k≤ pk . 384 24 8 2 We give some examples for illustrating the applicability of the obtained the- oretical results, including examples of advantages of the method in the thesis compared to the methods of P. Amster, P.P. C´ardenas Alzate (2008): Accord- ing 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 Z L (4) u (x) − M c |u (s)| ds u00 (x) = f (x, u(x)), 0 < x < L, 0 2 0 Z L (2.2.4) u(0) = u0 (0) = u00 (L) = 0, u000 (L) − M c |u0 (s)|2 ds u0 (L) = g(u(L)). 0 2.2.2.1. The existence and uniqueness of a solution By setting v(x) = u00 (x) − M c(||u0 ||2 )u(x), where k.k2 denotes the norm of 2 L2 [0, L], the problem (2.2.4) is reduced to the problems v 00 (x) = f (x, u(x)), 0 < x < L, v(L) = −M ku k2 u(L), v 0 (L) = g(u(L)) 0 2 00 0 2 u (x) = M ku k2 u(x) + v(x), c 0 < x < L, u(0) = u0 (0) = 0. 16
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 Z L h 0 2 (T u)(x) = G(x, t) M ku k2 u(t) c 0 Z L i 0 2 + G(t, s)f (s, u(s))ds + g(u(L))(t − L) − M ku k2 u(L) dt. c 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 c 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], c(s)| ≤ m, ∀s ∈ [0, L3 R2 ]. |M L2 Then, if 2 A + 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 c > 0 such that |f (x, u) − f (x, v)| ≤ λf |u − v|, ∀(x, u), (x, v) ∈ [0, L] × [−L2 R, L2 R], |g(u) − g(v)| ≤ λg |u − v|, ∀u, v ∈ [−L2 R, L2 R], |M c(u) − M c(v)| ≤ λ c|u − v|, ∀u, v ∈ [0, L3 R2 ]. M 2 L4 Then, if q = 4L5 R λM c + 2 λf + L3 λg + 2mL2 < 1, the problem (2.2.4) has a unique solution. 2.2.2.2. 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 vk00 (x) = f (x, uk (x)), 0 < x < L, vk (L) = −M kuk k2 uk (L), vk0 (L) = g(uk (L)), c 0 2 u00k+1 (x) 0 2 = vk (x) + M kuk k2 uk (x), c 0 < x < L, uk+1 (0) = u0k+1 (0) = 0. 17
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 qk kuk − uk∞ ≤ Lku0k − u0 k∞ ≤ L2 ku00k − u00 k∞ ≤ L2 ku001 − u000 k∞ , 1−q where u is the exact solution of the original problem (2.2.4). We give some examples for illustrating the applicability of the obtained the- oretical 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 estab- lished, 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