Real Dynamics of Family of Functions for Positive

Abstract: In this paper, the real dynamics of one parameter family of functions, 0 is investigated. It is shown that the real fixed point of h ( x) exists only for 0 1 which is attracting and there is no real fixed point for 1. It is found that the whole real line converges to real attracting fixed point of h ( x) for 0 1. The dynamics of functions in complex plane using real dynamics near to real fixed points have been induced by many researchers. The fixed points play vital role to explore the dynamical behavior of functions. Some related investigation can be found in [1, 3, 4]. The theory of fixed points and dynamics or iteration of transcendental functions can be seen in [2]. A point x is said to be a fixed point of function f (x) if f (x) x. A fixed point x0 is called an attracting, neutral (indifferent) or repelling if |f (x0) | 1, |f (x0) | 1 or |f (x0) | 1respectively. In this paper, we study one parameter family of function which arises from the generalized Bernoulli generating function, Apostol-Bernoulli generating function and Stirling generating function. To study this family of functions, we consider The existence of real fixed point, their nature and real dynamics of the function h H ar