# Functional equation modeling of adaptive operant-control systems via Matkowski fixed point theory

## TL;DR

This paper introduces a new mathematical approach to model operant-control behavior without initial conditions, using fixed point theory.

## Contribution

A generalized functional equation for operant-control systems is proposed, supported by Matkowski fixed point theory.

## Key findings

- The generalized equation has a unique probabilistic solution, proven via Matkowski fixed point theorem.
- Simulations confirm the validity of the theoretical framework for behavioral modeling.
- Fixed point theory is shown to be effective in analyzing control-based behavioral models.

## Abstract

This paper presents a generalized form of the functional equation used in operant-control models by removing the requirement for initial conditions. The proposed formulation extends earlier studies in mathematical psychology and provides a broader analytical framework for modeling operant-control behavior. Using the Matkowski fixed point theorem, we prove the existence and uniqueness of a probabilistic solution to the generalized equation. Illustrative examples and simulations are included to demonstrate the validity of the theoretical results. This work shows that fixed point theory can effectively support the formulation and analysis of control-based behavioral models.