Existence, Uniqueness, and Stability Analysis of the Probabilistic Functional Equation Emerging in Mathematical Biology and the Theory of Learning

Probabilistic functional equations have been used to analyze various models in computational biology and learning theory. It is worth noting that they are linked to the symmetry of a system of functional equations’ transformation. Our objective is to propose a generic probabilistic functional equation that can cover most of the mathematical models addressed in the existing literature. The notable fixed-point tools are utilized to examine the existence, uniqueness, and stability of the suggested equation’s solution. Two examples are also given to emphasize the significance of our findings.

[1]  W. Sintunavarat,et al.  On a solution of the probabilistic predator–prey model approached by the fixed point methods , 2020 .

[2]  Yilun Shang,et al.  L 1 group consensus of multi-agent systems with stochastic inputs under directed interaction topology , 2013, Int. J. Control.

[3]  O. Baghani,et al.  Hyers-Ulam Stability of Nonlinear Integral Equation , 2010 .

[4]  W. Sintunavarat,et al.  On the solutions of the two preys and one predator type model approached by the fixed point theory , 2020 .

[5]  W. Sintunavarat,et al.  On analytic model for two-choice behavior of the paradise fish based on the fixed point method , 2019, Journal of Fixed Point Theory and Applications.

[6]  W. Estes,et al.  Analysis of a verbal conditioning situation in terms of statistical learning theory , 1954 .

[7]  W. Sintunavarat,et al.  On the solution of the traumatic avoidance learning model approached by the Banach fixed point theorem , 2020 .

[8]  T. Rassias.  On the stability of the linear mapping in Banach spaces , 1978 .

[9]  Dirk Ifenthaler,et al.  Stochastic Models of Learning , 2012 .

[10]  T. R. Wilson,et al.  Two-choice behavior of paradise fish. , 1956, Journal of experimental psychology.

[11]  J. Bae,et al.  A Fixed Point Approach to the Stability of a Cauchy-Jensen Functional Equation , 2012 .

[12]  L. Humphreys.  Acquisition and extinction of verbal expectations in a situation analogous to conditioning. , 1939 .

[13]  E. Rojas,et al.  HYERS-ULAM AND HYERS-ULAM-RASSIAS STABILITY OF NONLINEAR INTEGRAL EQUATIONS WITH DELAY , 2011 .

[14]  D. A. Grant,et al.  Acquisition and extinction of a verbal conditioned response with differing percentages of reinforcement. , 1951, Journal of experimental psychology.

[15]  S. Banach.  Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales , 1922 .

[16]  M. Jarvik,et al.  Probability learning and a negative recency effect in the serial anticipation of alternative symbols. , 1951, Journal of experimental psychology.