Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1982. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 16)

II. TANULMÁNYOK A TERMÉSZETTUDOMÁNYOK KÖRÉBÖL - Dr. Szepessy Bálint: Megjegyzések a valós függvények iterálásához II

REMARKS ON ITERATION OF REAL FUNCTION By Bálint Szepessy (Summary) Let f (x) be a real valued function defined on interval [a, b]. If f (x) satisfies the conditions (i) f (x) is a continuous function at every inside points of the interval [a, b]

; furthermore f (x) is continuous on the right and on the left at point a and b respectively; (ii) f (x) map? the interval [a, b] onto itself; (iii) there is riD subinterval of the interval [a. b] where f (x) is a constant function; then f (x) is called iterational basic function. Let us define the iterated functions of f (x) by f

u (x) = x. fi (x) = f (x) and f

n (x) = f (f

n_i /x/) for n > 1. If f (c) = c for some real c then c is called fix point of f (x) of order one, and if f

n (c) 4= c for n = 1, 2, . . ., r— 1 but f

r (c) = c. then c is called fix point of f (x) of order r. In [9] we studied the iterational basic functions f (x) having fix points and we gave sufficient conditions for f (x) which have fix points of order r > k. where k is an arbitrary integer. In this paper conditions are studied for f (x) satisfying restrictions (i), (ii) and (iii) and having fix points of order r = 2. Among others we prove the following theorem: Let d be a real number with a < d < b. If the function f (x) satisfies the conditions f (a) = d, f (d) = b, f (b) = d; f (x) is a monotonically decreasing function on interval [d, b], and in case a < x < d we have x < f (x) < b then there are fix points only of order one or of order two on interval [a, b]. 565