On Solutions of Some Functional Equations of Radical Type

Abstract

Many authors have studied various functional equations of forms patterned on the equation expressed as ga2+b2=g(a)+g(b), which can be considered, e.g., for real functions. Such equations are usually referred to as radical functional equations or of the radical type. Authors mainly study the so-called Ulam stability of such equations, i.e., they investigate how much the mappings satisfying the equations approximately (in a sense) differ from the exact solutions of these equations. Quite often, information about the solutions of these equations is also provided, but unfortunately, sometimes, such information is given in a misleading or incomplete way. It seems, therefore, that there is a need for a publication containing simple descriptions of such solutions (with appropriate examples), which would help in easy correction of such information and avoidance of similar problems for future authors. This is the main motivation for this expository paper. We present a general approach to the topic and consider two general forms of such equations. Moreover, the results presented in this paper show significant symmetry between the solutions of numerous functional equations and the solutions of equations of the radical type that correspond to them. To make this publication accessible to a wider audience, we omit various related information, avoid advanced generalizations, and present several simple examples.