Continuous solutions of an iterative equation with multiplication

Abstract

Iterative equation is an equality with an unknown function and its iterates. There were not found a result on iterative equations with multiplication of iterates of the unknown function on R. In this paper we use an exponential function to reduce the equation in conjugation to the well-known form of polynomial-like iterative equation, but we encountered two difficulties: the reduction restricts our discussion of the equation to R + ; the reduced polynomial-like iterative equation is defined on the whole R but known results were given on comapct intervals. We revisit the polynomial-like iterative equation on the whole R and give existence, uniqueness, stability and construction of continuous solutions of our equation on R + . Then we technically extend our solutions from R + to R − .

Differentiable solutions of an equation with product of iterates

“…It is also interesting to discuss Φ of nonlinear combination. In 2007 Zdun and Zhang [16] discussed (1.1) for continuous solutions on the compact space S 1 , the unit circle in C, and recently Gopalakrishna et al [2] investigated (1.1) in the special case Φ(u 1 , u 2 , . .…”

“…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.…”

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In the previous work [2], we considered continuous solutions of an iterative equation involving the multiplication of iterates. In this paper, we continue to investigate this equation for differentiable solutions.

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“…In this paper, we continue to investigate this equation for differentiable solutions. Similar to continuous solutions until [2], there is no obtained result on differentiable solutions of such an equation on non-compact intervals of R.Although our strategy here is to use conjugation to reduce the equation to the well-known polynomial-like iterative equation as in [2], all known results on differentiable solutions of the latter are given on compact intervals. We re-explore polynomial-like iterative equation on the whole of R and prove the existence and uniqueness of differentiable solutions of our equation on R + and R − .…”

“…Although our strategy here is to use conjugation to reduce the equation to the well-known polynomial-like iterative equation as in [2], all known results on differentiable solutions of the latter are given on compact intervals. We re-explore polynomial-like iterative equation on the whole of R and prove the existence and uniqueness of differentiable solutions of our equation on R + and R − .…”