# On Solutions and Stability of Stochastic Functional Equations Emerging in Psychological Theory of Learning

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## Abstract

**:**

## 1. Introduction and Preliminaries

What if an animal or human does not move for any prey or response and sticks to its original position?

## 2. Auxiliary Information and Results

- $\left(1\right)$
- $d(x,y)=0$ if and only if $x=y$;
- $\left(2\right)$
- $d(x,y)=d(y,x)$;
- $\left(3\right)$
- $d(x,z)\le d(x,y)+d(y,z)$.

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Some Preliminary Remarks

## 4. Main Results

**Hypothesis**

**1**

**Hypothesis**

**2**

**Hypothesis**

**3**

- -
- ${x}_{0}$ is a fixed point of ${L}_{1}$ and ${L}_{2}$, ${a}_{1}\left({x}_{0}\right)={A}_{1}$, ${a}_{2}\left({x}_{0}\right)={A}_{2}$;
- -
- ${x}_{0}$ is a fixed point of ${L}_{3}$ and ${L}_{4}$, ${a}_{1}\left({x}_{0}\right)={a}_{2}\left({x}_{0}\right)=0$;
- -
- ${x}_{0}$ is a fixed point of ${L}_{1}$ and ${L}_{4}$, ${a}_{1}\left({x}_{0}\right)={A}_{1}$, ${a}_{2}\left({x}_{0}\right)=0$.

**Theorem**

**2.**

- (a)
- There exist points ${u}_{1},{u}_{2}\in X$ such that$$\begin{array}{c}\hfill {L}_{1}\left({u}_{1}\right)={L}_{3}\left({u}_{1}\right),\phantom{\rule{2.em}{0ex}}{L}_{2}\left({u}_{2}\right)={L}_{4}\left({u}_{2}\right)\end{array}$$and$$\begin{array}{c}\hfill \lambda :={\lambda}_{0}+\delta \left(X\right)(\parallel {a}_{1}{\parallel}_{e}({\kappa}_{1}+{\kappa}_{3})+\parallel {a}_{2}{\parallel}_{e}({\kappa}_{2}+{\kappa}_{4}))<1.\end{array}$$
- (b)
- There exist ${\kappa}_{5},{\kappa}_{6}\in [0,\infty )$ such that$$\begin{array}{c}\hfill d\left({L}_{1}\left(x\right),{L}_{3}\left(y\right)\right)\le {\kappa}_{5}d(x,y),\phantom{\rule{2.em}{0ex}}d\left({L}_{2}\left(x\right),{L}_{4}\left(y\right)\right)\le {\kappa}_{6}d(x,y)\end{array}$$for all $x,y\in X$ with $x\ne y$, and$$\begin{array}{c}\hfill \lambda :={\lambda}_{0}+\delta \left(X\right)(\parallel {a}_{1}{\parallel}_{e}({\kappa}_{1}+{\kappa}_{5})+\parallel {a}_{2}{\parallel}_{e}({\kappa}_{2}+{\kappa}_{6}))<1.\end{array}$$
- (c)
- There exist ${\gamma}_{1},{\gamma}_{2}\in [0,\infty )$ with$$\begin{array}{c}\hfill d\left({L}_{1}\left(x\right),{L}_{3}\left(x\right)\right)\le {\gamma}_{1},\phantom{\rule{2.em}{0ex}}d\left({L}_{2}\left(x\right),{L}_{4}\left(x\right)\right)\le {\gamma}_{2},\phantom{\rule{2.em}{0ex}}x\in X,\end{array}$$and$$\begin{array}{c}\hfill \lambda :={\lambda}_{0}+\parallel {a}_{1}{\parallel}_{e}{\gamma}_{1}+{\parallel {a}_{2}\parallel}_{e}\phantom{\rule{0.166667em}{0ex}}{\gamma}_{2}<1.\end{array}$$

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 5. Remarks on Ulam Stability

**Definition**

**1.**

**Corollary**

**1.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Estes, W.K.; Straughan, J.H. Analysis of a verbal conditioning situation in terms of statistical learning theory. J. Exp. Psych.
**1954**, 47, 225–234. [Google Scholar] [CrossRef] [PubMed] - Bush, A.A.; Wilson, T.R. Two-choice behavior of paradise fish. J. Exp. Psych.
**1956**, 51, 315–322. [Google Scholar] [CrossRef] - Bush, R.; Mosteller, F. Stochastic Models for Learning; Wiley: New York, NY, USA, 1955. [Google Scholar]
- Epstein, B. On a difference equation arising in a learning-theory model. Israel J. Math.
**1966**, 4, 145–152. [Google Scholar] [CrossRef] - Istrăţescu, V.I. On a functional equation. J. Math. Anal. Appl.
**1976**, 56, 133–136. [Google Scholar] [CrossRef] - Turab, A.; Sintunavarat, W. On the solution of the traumatic avoidance learning model approached by the Banach fixed point theorem. J. Fixed Point Theory Appl.
**2020**, 22, 50. [Google Scholar] [CrossRef] - Grant, D.A.; Hake, H.W.; Hornseth, J.P. Acquisition and extinction of a verbal conditioned response with differing percentages of reinforcement. J. Exp. Psych.
**1951**, 42, 1–5. [Google Scholar] [CrossRef] [PubMed] - Turab, A.; Sintunavarat, W. On analytic model for two-choice behavior of the paradise fish based on the fixed point method. J. Fixed Point Theory Appl.
**2019**, 21, 56. [Google Scholar] [CrossRef] - Humphreys, L.G. Acquisition and extinction of verbal expectations in a situation analogous to conditioning. J. Exp. Psych.
**1939**, 25, 294–301. [Google Scholar] [CrossRef] - Jarvik, M.E. Probability learning and a negative recency effect in the serial anticipation of alternative symbols. J. Exp. Psych.
**1951**, 41, 291–297. [Google Scholar] [CrossRef] - Schein, E.H. The effect of reward on adult imitative behavior. J. Abnorm. Soc. Psych.
**1954**, 49, 389–395. [Google Scholar] [CrossRef] - Miller, N.E.; Dollard, J. Social Learning and Imitation; Yale University Press: New Haven, CT, USA, 1941. [Google Scholar]
- Schwartz, N. An Experimental Study of Imitation. The Effects of Reward and Age. Ph.D. Thesis, Radcliffe College, Cambridge, MA, USA, 1953. [Google Scholar]
- Neimark, E.D. Effects of Type of Non-reinforcement and Number of Alternative Responses in Two Verbal Conditioning Situation. Ph.D. Thesis, Indiana University, Bloomington, IN, USA, 1953. [Google Scholar]
- Aydi, H.; Karapinar, E.; Rakocevic, V. Nonunique fixed point theorems on b-metric spaces via simulation functions. Jordan J. Math. Stat.
**2019**, 12, 265–288. [Google Scholar] - Karapinar, E. Recent advances on the results for nonunique fixed in various spaces. Axioms
**2019**, 8, 72. [Google Scholar] [CrossRef] [Green Version] - Alsulami, H.H.; Karapinar, E.; Rakocevic, V. Ciric type nonunique fixed point theorems on b-metric spaces. Filomat
**2017**, 31, 3147–3156. [Google Scholar] [CrossRef] - Ethier, S.N.; Kurtz, T.G. Markov Processes, Characterization and Convergence; Wiley: New York, NY, USA, 1986. [Google Scholar]
- Kouritzin, M.A. On tightness for probability measures on Skorokhod spaces. Trans. Am. Math. Soc.
**2016**, 368, 5675–5700. [Google Scholar] [CrossRef] - Diaz, J.B.; Margolis, B. A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc.
**1968**, 74, 305–309. [Google Scholar] [CrossRef] [Green Version] - Berinde, V.; Khan, A.R. On a functional equation arising in mathematical biology and theory of learning. Creat. Math. Inform.
**2015**, 24, 9–16. [Google Scholar] [CrossRef] - Hyers, D.H. On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA
**1941**, 27, 222–224. [Google Scholar] [CrossRef] [Green Version] - Rassias, T.M. On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc.
**1978**, 72, 297–300. [Google Scholar] [CrossRef] - Aoki, T. On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn.
**1950**, 2, 64–66. [Google Scholar] [CrossRef] - Brzdęk, J.; Popa, D.; Raşa, I.; Xu, B. Ulam Stability of Operators; Academic Press: Oxford, UK, 2018. [Google Scholar]
- Hyers, D.H.; Isac, G.; Rassias, T.M. Stability of Functional Equations in Several Variables; Birkhauser: Basel, Switzerland, 1998. [Google Scholar]
- Jung, S.-M. Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis; Springer: New York, NY, USA, 2011. [Google Scholar]
- Ulam, S.M. A Collection of the Mathematical Problems; Interscience Publ.: New York, NY, USA, 1960. [Google Scholar]
- Brzdęk, J.; Ciepliński, K.; Leśniak, Z. On Ulam’s type stability of the linear equation and related issues. Discrete Dyn. Nat. Soc.
**2014**, 2014, 536791. [Google Scholar] [CrossRef] - Agarwal, R.P.; Xu, B.; Zhang, W. Stability of functional equations in single variable. J. Math. Anal. Appl.
**2003**, 288, 852–869. [Google Scholar] [CrossRef] [Green Version] - Xu, B.; Brzdęk, J.; Zhang, W. Fixed point results and the Hyers-Ulam stability of linear equations of higher orders. Pac. J. Math.
**2015**, 273, 483–498. [Google Scholar] [CrossRef] - Brzdęk, J.; Popa, D.; Xu, B. On approximate solutions of the linear functional equation of higher order. J. Math. Anal. Appl.
**2011**, 373, 680–689. [Google Scholar] [CrossRef] [Green Version]

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**MDPI and ACS Style**

Turab, A.; Brzdęk, J.; Ali, W.
On Solutions and Stability of Stochastic Functional Equations Emerging in Psychological Theory of Learning. *Axioms* **2022**, *11*, 143.
https://doi.org/10.3390/axioms11030143

**AMA Style**

Turab A, Brzdęk J, Ali W.
On Solutions and Stability of Stochastic Functional Equations Emerging in Psychological Theory of Learning. *Axioms*. 2022; 11(3):143.
https://doi.org/10.3390/axioms11030143

**Chicago/Turabian Style**

Turab, Ali, Janusz Brzdęk, and Wajahat Ali.
2022. "On Solutions and Stability of Stochastic Functional Equations Emerging in Psychological Theory of Learning" *Axioms* 11, no. 3: 143.
https://doi.org/10.3390/axioms11030143