Recent results in the theory of functional equations in a single variable

Abstract

In 2001 Aequationes Mathematicae published the survey paper Re- cent results on functional equations in a single variable, perspectives and open prob- lems written jointly with Witold Jarczyk in which we pay attention on papers pub- lished in the last decade. I continue this review and discuss mainly papers published or written in 2000 or later. Contents 1. General remarks 2. Iterative roots and equations with superpositions of the unknown function 3. Some linear equations 4. Simultaneous equations 5. Schroder's and Bottcher's equations 6. Miscellaneous results 1. General remarks. Iterative functional equations methods are connected mainly with real, complex and functional analysis. But recently some linear func- tional equations in a single variable have been solved with the aid of probabilistic tools. Among recent papers developing this new method are: Derfel (1989), Ger and Sablik (1998), Pittenger and Ryff (1999), Deliu and Spruill (2000), Baron and Jarczyk (M). Some results from these papers are presented in the survey paper Baron and Jarczyk (2001). Later on I will present also some results of Kapica (A). Until quite lately the Hyers-Ulam stability was considered almost exclusively for functional equations in several variables. It seems that this changes: a lot of recent papers deal with this type of stability also for iterative functional equations. I mention here Lee and Jun (1998), where the Hyers-Ulam stability of φ(x + α) = aφ(x)