Quasi graph-additive functions with a prescribed branch

TL;DR

This paper characterizes solutions to a specific functional equation with prescribed values on the non-positive real line, providing explicit formulas for solutions generated by continuous, strictly monotone functions.

Contribution

It introduces a method to extend prescribed functions on the non-positive half-line to solutions of the equation, with a complete characterization for continuous, strictly monotone generators.

Findings

Any function negative on the negative half-line and greater than the identity can generate a solution.

Solutions generated by continuous, strictly monotone functions are explicitly characterized.

A closed-form expression for these solutions is established.

Abstract

Here, we investigate the solutions to equation \[f(f(-x)+x)=f(-f(x))+f(x),\qquad x\in\mathbb{R}\] that are prescribed on the non-positive half-line. We will refer to this prescribed function as the generator of the corresponding solution. We show that any function taking negative values on the negative half-line and being strictly greater than the identity can be extended to a solution. Nevertheless, the solutions generated by continuous, strictly monotone functions can be well characterized. As our main result, we establish a closed-form expression for these functions.