In mathematical psychology, the model of decision practice represents the development of moral judgment that deals with the time to decide the meaning of the various choices and selecting one of them for use. Most animal behavior research classifies such situations as two distinct phenomena. On the other hand, reward plays a big part in this kind of study since, based on the selected side and food location, such circumstances may be classified into four categories. This paper intends to investigate such types of behavior and establish a general functional equation for it. The proposed functional equation can be used to describe several psychological and learning theory models in the existing literature. By using the fixed point theory tools, we obtain the results related to the existence, uniqueness, and stability of a solution to the proposed functional equation. Finally, we give two examples to support our main results.

Recently Berinde and Păcurar [Approximating fixed points of enriched contractions in Banach spaces. {\em J. Fixed Point Theory Appl.} {\bf 22} (2020), no. 2., 1--10], first introduced the idea of enriched contraction mappings and proved the existence of a fixed point of an enriched contraction mapping using the well-known fact that any fixed point of {the averaged mapping $T_\lambda$, where $\lambda\in (0,1]$, is also a fixed point of the initial mapping $T$}. In this work, we introduce the idea of weak enriched contraction mappings, and a new generalization of an averaged mapping called double averaged mapping. The first attempt is to prove the existence and uniqueness of the fixed point of a double averaged mapping associated with a weak enriched contraction mapping. Based on this result on Banach spaces, we give some sufficient conditions for the equality of all fixed points of a double averaged mapping and the set {of all fixed points of a weak enriched contraction mapping.} Moreover, our results show that an appropriate Kirk's iterative algorithm can be used to approximate a fixed point of a weak enriched contraction mapping. An illustrative example for showing the efficiency of our results is given.