Summary of Mathematics doctoral thesis: Iterative method for solving two point boundary value problems for fourth order differential equations and systems

The objective of the thesis is to develop the iterative method and combining it with other methods to study qualitative and especially the method of solving some two-point boundary problems for the fourth-order differential equations and systems, arising in beam bending theory without using condition of growth rate at infinity, Nagumo condition, etc. of the right-hand side function.
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Summary of Mathematics doctoral thesis: Iterative method for solving two point boundary value problems for fourth order differential equations and systems MINISTRY OF EDUCATION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND ECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ……..….***………… NGÔ THỊ KIM QUY ITERATIVE METHOD FOR SOLVING TWO-POINT BOUNDARY VALUE PROBLEMS FOR FOURTH ORDER DIFFERENTIAL EQUATIONS AND SYSTEMS Major : Applied Mathematics Code: 62 46 01 12 SUMMARY OF MATHEMATICS DOCTORAL THESIS Hanoi – 2017 This thesis was completed at: Graduate University of Science and Technology Vietnam Academy of Science and Technology Supervisor 1: Prof. Dr. Dang Quang A Supervisor 2: Assoc. Prof. Dr. Ha Tien Ngoan Reviewer 1: … Reviewer 2: … Reviewer 3: …. The Dissertation will be officially presented in front of the Doctoral Dissertation Grading Committee, meating at: Graduate University of Science and Technology Vietnam Academy of Science and Technology At ................Date..................Month...................Year 201… The Dissertation is avaiable at: 1. Library of Graduate University of Science and Technology 2. National Library of Vietnam INTRODUCTION 1. Motivation of the thesis Many problems in physics, mechanics and some other fields are described by differential equations or systems of differential equations with different bound- ary conditions. It is possible to classify the fourth-order differential equations into two forms: fully fourth-order differential equations and non-fully fourth- order ones. A fourth-order differential equation whose right-hand side function contains an unknown function and its derivatives of all order (from first to third order) is called a fully fourth-order differential equation. Otherwise, the equation is called a non-fully fourth-order differential equation. The boundary value problems for differential equations have attracted the attention of scientists such as Alve, Amster, Bai, Li, Ma, Feng, Minh´os, etc. Some Vietnamese mathematicians and mechanics, namely, Dang Quang A, Pham Ky Anh, Nguyen Van Dao, Nguyen Dong Anh, Le Xuan Can, Nguyen Huu Cong, Le Luong Tai, etc. also studied methods for solving the boundary value problems for differential equations. Among the differential equations, the nonlinear fourth-order differential equation has been of great interest recently as it is the mathematical model of many problems in mechanics. Here we take a look at some of the boundary value problems for the nonlinear fourth-order differential equations. Firstly, consider the problem of elastic beams as described by the nonlinear fourth-order differential equation u(4) (x) = f (x, u(x), u00 (x)) (0.0.2) or u(4) (x) = f (x, u(x), u0 (x)) (0.0.3) where u is the deflection of the beam, 0 ≤ x ≤ L. The conditions at two ends of beams are given in dependence of the constraints of the problems. There have been many research results on the qualitative aspects of the problems such as existence, uniqueness and positivity of solutions. Noteworthy is the works of Alves et al. (2009), Amster et al. (2008), Bai (2004), Li (2010), Ma et al. (1997), ..., where the upper and lower solution method, the variational method, the methods of fixed point theorems are used. In these works the conditions of the boundedness of the right-hand function or of its growth rate at infinity is indispensable. In the articles mentioned above, the fourth-order differential equation does not contain third-order derivative. For the last ten years, the fully fourth-order 1 differential equations, namely the equation u(4) (x) = f (x, u(x), u0 (x), u00 (x), u000 (x)) (0.0.6) has attracted the interest of many authors (Ehme et al. (2002), Feng et al. (2009), Li et al. (2013), Li (2016), Minh et al. (2009), Pei et al. (2011), ...). The main results in the these papers are the study of the existence, uniqueness and positivity of the solution. The tools used are Leray-Schauder’s degree theory (see Pei et al. (2011)), the Schauder fixed point theorem based on the monotone method in the present of lower and the upper solutions (see Bai (2007) ), Ehme et al. (2002), Feng et al. (2009), Minh´os et al. (2009)) or Fourier analysis (see Li et al. (2013)). However, in all of the articles mentioned above, the authors need a very important assumption that the function f : [0, 1]× R4 → R satisfies the Nagumo condition and some other conditions of monotonicity and growth at infinity. It should be emphasized that in the monotone method the assumption of the presence of lower and upper solutions is always needed and the finding of them is not easy. The system of fourth-order differential equations have not been studied much, such as Kang et al. (2012), L¨ u et al. (2005), Zhu et al. (2010), in which the authors considered the equations containing only even-order deriva- tives associated with the simply supported boundary conditions. Under very complicated conditions, by using a fixed point index theorem in cones, the au- thors obtained the existence of positive solutions. But it should be emphasized that the obtained results are of pure theoretical character because no examples of existing solutions are shown. Minh´os and Coxe (2017, 2018) for the first time considered the system of coupled fully fourth-order of differential equations. The authors have provided sufficient conditions for solving the system by using the lower and upper solu- tions method and the Schauder fixed point theorem. Demonstrating this result is very cumbersome and complicated an ...