On a Class of Functional Equations over the Real and over the Complex Fields

Abstract

In the present review-paper, we start by recalling some of our earlier results on the construction of a nontrivial function defined implicitly by the equation (1.1), without using the implicit function theorem. This is the first aim of the paper. Here the function  is given, satisfying some conditions. All these considerations work in the real case, for functions and a class of operators. The second aim is to consider the complex case, proving the analyticity of the function defined implicitly, under the hypothesis that is analytic and verifies natural conditions, related to the real case. Some consequences are deduced.  Finally, one illustrates the preceding results by an application to a concrete functional and respectively operatorial equation. Related examples are given, some of them pointing out elementary functions for which equation (1.1) leads to nontrivial solutions which can be expressed by means of elementary functions.