The Convergence Of Solutions For Nonlinear Singular Differential Systems

In this paper, we study the convergence of solutions for nonlinear singular differential systems by using the theory of linear singular systems, matrix theory, comparison principle,upper-lower solution method, monotone iterative technique, quasilinearization method and so on. This paper consists of seven chapters.In Chapter 1, we give the significance of singular systems, research status as well as the main contents of this paper. In Chapter 2, by weakening the restrictions of concavity and convexity on the right hand side function, we investigate the higher order of convergence of approximate solutions of the initial value problem for nonlinear singular differential systems under weaker conditions. In Chapter 3, we study the quadratic convergence of approximate solutions of the initial value problem for nonlinear singular differential systems with “maxima” by assuming the concavity or convexity on the right hand side function. By further extend this result,we study the higher order of convergence of its approximate solutions under the assumptions of hyperconcavity or hyperconvexity on the right hand side function, an numerical example is given to verify the main results. In Chapter 4, we devote to discuss the quadratic convergence of approximate solutions of the periodic boundary value problem for nonlinear singular differential systems with “maxima” by assuming suitable conditions on the right hand side function.Furthermore, by assuming weaker conditions on the right hand side function, we discuss the quadratic convergence of its approximate solutions. In Chapter 5, the nonlinear singular differential systems with “maxima” be extended to the nonlinear singular difference systems with“maxima”, we study the quadratic convergence of approximate solutions of the initial value problem for nonlinear singular difference systems with “maxima” under the assumptions of concavity or convexity on the right hand side function. In Chapter 6, an example is given to illustrate the application of generalized quasilinearization method in practical fields, we discuss the higher order of convergence of approximate solutions for telegraph systems by weakening the restrictions of concavity and convexity on the right hand side function. In Chapter 7, the main contents of this paper are summarized.