Abstract
We study continuous solutions of the 3rd-order iterative equation of the linear dependence for hyperbolic cases and generalize the results to the n th-order polynomial-like iterative equation, in some cases recursively. This paper partly answers the question raised in Remark 7 in (Yang and Zhang in Aequ. Math. 67:80-105, 2004) and the open problem raised by Matkowski in (Aequ. Math. 37:119, 1989) in the 26th international symposium on functional equations. MSC:39B12, 37E05.
Highlights
Consider a real function f : R → R
While attention was paid to the case of nonlinear F and further generalized forms [, ], the case of linear F, that is, f n(x) + λn– f n– (x) + · · · + λ f (x) + λ x = c, ( . )
In what follows we prove some cases, listed in Tables and, which can be generalized to the nth-order polynomial-like iterative equation recursively
Summary
Consider a real function f : R → R. In order to prove the case (i), let f be a C strictly increasing solution with fixed points. ) satisfying f (x) < max{r x, r x}, constructed by Theorem of [ ], are C strictly increasing solutions of
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