Studies On The Solvability Of Nonlinear Differential Equations

The results of this research paper can be outlined mainly as the following aspects:1 The existence of entire solutions for the nonlinear elliptic equations in the form ofAu + f(x,u,Vu) = 0is discussed in chapter 1 and chapter 2. First of all, chapter 1 deals with the one-dimentinal case, and twe existing theorems are obtained (theoreml .1-1.2). This result is browden to n-dimentional case in chapter 2 first and an existence theorem (theorem 2.1) for linearly increasing solutions is given. Furthermore, two existing theorems for logarithmic increasing solutions are also obtained in this chapter (theorem 2.2-2.3). All these results are focused in the unbounded solution, which is different from the existing boundary ones, and so our work perfects the concerned results about the existence of solutions to this equation.2 The existence theorems of exponently increasing solutions and decaying solutions for a class of more general nonlinear elliptic equations in the form ofAu - m2u + f(x, u,Vu) = 0are studied in chapter 3 and chapter 4. Three theorems (theorem3.1-3.3) are obtained for exponentlv increasing solutions, and two (theorem4.1-4.2) for decaying ones. These results not only extend the existing ones, but also develop the applications of the sup-sub solution methods and fixed theorems in reseach on the solvability to differential equations.3 Chaper 5 deals with the hyperbolic equation in the form of22utt+A2u-i-M(x, IA" u112 )Au0,which comes from the mathematical descripition of the tension of a extensible beam, and the existence and uniqueness of Cauchy problems for this equation are presented here. An existence and uniqueness theorem (theorem 5.1) of local solvability is proven for this equation, compared with the existing results, our conditions restricted to the nonlinear term of this equation are much general. In fact, the results given here are obtained by breaking all the former restrictions on the nonlinear term.4 The similarity reductions of generalized Burgers equation is treated in chapter 6 by the direct method for finding similarity reductions of patial differential equations, and the corresponding exact special solutions are given also. More over, a group interpretationis provided to the similarity reductions in terms of nonclassical symmetry. Accomplishing with the generalized Burges equation, this work answers partially an open problem presented by Clarckson: that is how to find the similarity reductions of nonlinear PDEs with arbitrary functions by the direct method.5 In the last two chapters, chapter7-8, the global minimized problems of the unconstrained optimization are studied by approching the solutions with the solution curves of a differential equation, which is also called the neural network method. Chapter 7 gives the H-stability of the eqilibrium point set first (theorem7.1), and the other two theorems (theorem7.2-7.3) provide an estimation result for the attractive region of different eqilibrium points. All these results own two essential meanings, the first is that this complete fully the stability of the neural network, the second is that provides a feasible frame for the neural network designs. Just based on these results, two models of neural networks for solving the problems are proposed in chapter 8, and moreober, the feasibility and the validity of our designs are shown by some typical computation examples.