Special solutions of a general iterative functional equation

Continuous solutions of an iterative equation with multiplication

“…, u n ) = n k=1 u α k k , where G is given and g is unknown. Unlike those [6,9,11,13,17,18] on compact intervals, our work to (1.2) is concentrated to solving (1.1) on the whole R. Our strategy is to restrict our discussion of (1.2) on R + := (0, +∞) and use an exponential function to reduce in conjugation to the well-known form of polynomial-like iterative equation (1.1) on the whole R. Note that all found results on the polynomial-like iterative equation are given either on a compact interval or near a fixed point, none of which can be applied to our case. In this paper we generally discuss a polynomial-like iterative equation on the whole R and use obtained result to give solutions of equation (1.2) on R + and R − := (−∞, 0).…”

“…Although there can be found several papers [6,9,11] on the general Lipschitzian Φ, more efforts were still made to the basic form…”

Differentiable solutions of an equation with product of iterates

“…In this paper, we continue to inverstige (1.3) considering its differentiable solutions. Unlike those [6,11,13,17,18] on compact intervals, our work to (1.3) is focused on investigating (1.2) on the whole R as done in [2]. Our approach is to restrict the discussion of (1.3) on R + and use an exponential function to reduce in conjugation to the well-known form of polynomiallike iterative equation (1.2) on the whole R. Note that all found results on (1.2) are given on a compact interval, none of which are applicable to our case.…”

“…Although there are plentiful results (see [6,9,11] for example) on the solutions of (1.1) when Φ is a Lipschitzian, the basic form…”

Hyers-Ulam Stability of Iterative Equation in the Class of Lipschitz Functions

“…In 1986, Zhang [10] constructed an interesting operator called "structural operator" for (1) and used the fixed point theory in Banach space to get the solutions of (1). From then on (1) and other types of equations were discussed extensively by employing this idea (see [8,9,[11][12][13][14][15][16] and references therein). In 2002, by means of a modification of Zhang's method applied in [10], Kulczycki and Tabor [11] investigated the existence of Lipschitzian solutions of the iterative equation…”