Dhombres Type Functional Equations with Non-Trivial Solutions

Abstract

In such equations the function φ is given and one looks for solution functions, f ; that is, f is the “unknown”. Interesting is the case when all functions are continuous. Several cases are well known; for example, if φ is an increasing homeomorphism of an interval J ⊆ (0,∞) then the range Rf ⊆ J of any solution is an interval with the end-points fixed by φ, which contains no fixed point 6= 1. There is a characterization of these φ that allow only monotone solutions, and characterization of the monotone solutions; they form a “parametric family”where parameter is an initial monotone function defined on a compact subinterval of R+, see [1]. Also characterization of the continuous solutions in this case is known [2]. On the other hand, if φ is a decreasing homeomorphism then there can be no nonconstant solutions at all. The only known example of such solution is for the function φ : y 7→ α/y, with α ∈ (0, 1). In this case Rf consists of periodic points of period 2, except for the point √ α which is fixed [3]. A general question is then this: