Quasilinearization for functional differential equations with retardation and anticipation

The rapid convergence for nonlinear singular differential systems with "maxima''

“…This method originated in the dynamic programming theory and was initially applied by Bellman and Kalaba [8]. A systematic development of the method to ordinary differential equations has been provided by Lakshmikantham and Vatsala [18], and there are some generalized results of the method to various types of differential systems, we can refer to the monographs [16,17], for functional differential equations [4,12], for impulsive equations [3,6], for partial differential equations [5,9,14,24], for others [21,22,27] and references cited therein. However, there were few applicable results of the method to singular differential systems [1,13,19,23].…”

A Bernstein polynomial approach for solution of nonlinear integral equations

“…Some systematic studies of this property have been given in [1, 5-7, 11, 13-15, 18]. This method is also a powerful tool to obtain the approximate solution of nonlinear problems included such as differential equations [1,3,5,9,14,17], functional differential equations [2,7], integral equations [13,15] and integro-differential equations [4,18].…”

Quasilinearization And Numerical Solution Of Nonlinear Volterra Integro-differential Equations

“…This technique works fruitfully for the problems that their nonlinear parts involve convex or concave functions and gives two sequences of linear problems that their solutions are upper and lower solutions to the nonlinear problem and are converging monotonically and quadratically to the unique solution of the given nonlinear problem. Recently, this method is applied to a variety of problems [2][3][4][5] and in the continuation the convexity assumption was relaxed and the method was generalized and extended in various directions to make it applicable to a large class of problems [6][7][8][9][10][11][12][13][14][15].…”