tailieunhanh - Handbook of mathematics for engineers and scienteists part 137

Handbook of mathematics for engineers and scienteists part 137. 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. | 920 Difference Equations and Other Functional Equations Remark. System and therefore the original functional equation may have several even infinitely many solutions or no solutions at all. Example. Consider the nonlinear equation y2 x f x y a - x . Replacing x by a - x in we get y2 a - x f a - x y x . Eliminating y a - x from - we obtain two solutions of the original equation y x f2 x f a - x 1 3 and y x 0. 2 . Consider a reciprocal equation of the form F x y x y a x 0. Replacing x by a x we obtain a similar equation with the unknown function having the same arguments F a x y a x y x 0. Eliminating y a x from this and the original equation we come to the usual algebraic or transcendental equation of the form x y x 0. In other words solutions of the original functional equation y y x are defined in a parametric manner by means of a system of two algebraic or transcendental equations F x y t 0 F a x t y 0 where t is a parameter. . Reciprocal cyclic functional equations of general form. Reciprocal functional equations have the form F x y x y x y 2 x . x. 1 x 0. Here we use the notation Q n x Q Q n_1 x and Q x is a cyclic function satisfying the condition n x x. The value n is called the order of a cyclic reciprocal equation. Successively replacing n times the argument x by Q x in the functional equation we obtain the following system the original equation coincides with the first equation of this system F x yo yi . yn-1 0 F z yi y2 . yo 0 17426 . F o n_1 x yn-i yo . yra_ 0 where we have set y0 y x y1 y Q x . yn-1 y Q n-1 x condition . 2. 5 implies that yn y0 Eliminating y1 y2 . yn-1 from the system of nonlinear algebraic or transcendental equations . we obtain a solution of the functional equation . 2. 4 in implicit form x y0 0 where y0 y x . . Nonlinear Difference and Functional Equations with a Single Variable 921 . Nonlinear Functional Equations .