tailieunhanh - Handbook of mathematics for engineers and scienteists part 136

Handbook of mathematics for engineers and scienteists part 136. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | . Linear Functional Equations 913 homogeneous equation with F x 0 the following three cases are possible the notation used here is in agreement with that of Paragraph i Equation on I either has a one-parameter family of continuous solutions or no solutions at all. If has a continuous solution y0 x on I then the general continuous solution on I is given by z z a GW where a is an arbitrary constant and G x is given by . ii Equation on I has a continuous solution depending on an arbitrary function or has no continuous solutions on I . iii Equation on I either has a continuous solution or no solutions at all. 2 . Let x g I g I and f x e R0 I . Suppose that g x and F x are continuous functions g x 0 for x and g 1. Then equation has a unique continuous solution that can be represented by the series m - t FGf n D n Gn 1 x n 0 where the function Gn x is defined by . . Equations of special form with g x const. 1 . Consider the equation y f x y x F x which is a special case of equation with g x -1. A Let g I f x g R0 I and suppose that F x is a continuous function. If there exists a continuous solution of equation it can be represented by the power series y x 2 F -1 n F f n x - F . B Suppose that the assumptions of Item A hold and moreover there exist positive constants A k and C such that F x - F C x - K x e - A 8 n I and is a strongly attractive fixed point of f x . Then equation has a continuous solution on I. 2 . Consider the functional equation y f x - Ay x F x A 0. Let x g I a fi . Assume that on the interval I the function f x is continuous strongly increasing and satisfies the condition a f x x and F x is a function of bounded variation. Depending on the value of the parameter A the following cases are possible. 914 Difference Equations and Other Functional Equations Case A 1. There is a unique solution m - E . n 0 which coincides