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

Abstract

Octav Olteanu

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), 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) leads to nontrivial solutions ?????? that can be expressed by means of
elementary functions.

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