tailieunhanh - Handbook of mathematics for engineers and scienteists part 138

Handbook of mathematics for engineers and scienteists part 138. 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. | . Functional Equations with Several Variables 927 Remark. For the sake of analysis it is sometimes convenient to choose suitable values of the parameter a in order to simplify system . Example. Consider the equation f x y f x - y 2f x cos y. This equation holds as identity for y 0 and any f x . Substituting into we get f t f -t 2C1 cos t f t 2a f t 2f t a cos a f t 2a f -t 2C2 cos t a where Ci f 0 C2 f a . System becomes much simpler for a n 2. In this case cos a 0 and the function f t a is dropped from the equations summing up the first two equations term by term and subtracting the third equation from the resulting relation we immediately find a solution of the functional equation in the form f t Ci cos t C2 sin t. Verification shows that the function is indeed a solution of the functional equation . 4 . Now consider a functional equation more general than f x f y f x y f x - y x y 0. Letting y 0 we get f x a f x f x x 0 0 where a f 0 . If the left-hand side of this relation does not vanish identically for all f x then it can be resolved with respect to f x . Then inserting an admissible solution f x into the original equation we find possible values of the parameter a there are cases in which the equation has no solutions . 5 . If the left-hand side of identically vanishes for all f x and a the following approach can be used. In we consecutively take x 0 y t x t y 2t x 2t y t x t y t. We get a f t f t f -t 0 t 0 f t f 2t f 3t f -t t 2t 0 f 2t f t f 3t f t 2t t 0 f t f t f 2t a t t 0 where a f 0 . Eliminating f -t f 2t and f 3t from system we come to an equation for f t . The solution obtained in this manner should be inserted into the original equation . 928 Difference Equations and Other Functional Equations . Method of Argument Elimination by Test Functions . .