Measure of quality and certainty approximation of functional inequalities

Zahra Eidinejad; Reza Saadati; Donal O'Regan; Fehaid Salem Alshammari; Zahra Eidinejad; Reza Saadati; Donal O'Regan; Fehaid Salem Alshammari

doi:10.3934/math.2024100

AIMS Mathematics

2024, Volume 9, Issue 1: 2022-2031. doi: 10.3934/math.2024100

Previous Article Next Article

Research article

Measure of quality and certainty approximation of functional inequalities

1.
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
2.
School of Mathematical and Statistical Sciences, University of Galway, Galway, University Road, H91 TK33 Galway, Ireland
3.
Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh 11432, Saudi Arabia

Received: 18 October 2023 Revised: 09 November 2023 Accepted: 12 November 2023 Published: 19 December 2023
MSC : 46L05, 47B47, 47H10, 46L57, 39B62

To make a decision to select a suitable approximation for the solution of a functional inequality, we need reliable information. Two useful information ideas are quality and certainty, and the measure of quality and certainty approximation of the solution of a functional inequality helps us to find the optimum approximation. To measure quality and certainty, we used the idea of the Z-number (Z-N) and we introduced the generalized Z-N (GZ-N) as a diagonal matrix of the form $ diag(X, Y, X\ast Y) $, where $ X $ is a fuzzy set time-stamped, $ Y $ is the probability distribution function and the third part is the fuzzy-random trace of the first and the second subjects. This kind of diagonal matrix allowed us to define a new model of control functions to stabilize our problem. Using stability analysis, we obtained the most suitable approximation for functional inequalities.
- fuzzy sets,
- probability distribution function,
- Z-numbers,
- functional inequality,
- Jordan-von Neumann functional equation,
- Gauss hypergeometric function,
- Mittag Leffler function
Citation: Zahra Eidinejad, Reza Saadati, Donal O'Regan, Fehaid Salem Alshammari. Measure of quality and certainty approximation of functional inequalities[J]. AIMS Mathematics, 2024, 9(1): 2022-2031. doi: 10.3934/math.2024100

Related Papers:

Abstract

To make a decision to select a suitable approximation for the solution of a functional inequality, we need reliable information. Two useful information ideas are quality and certainty, and the measure of quality and certainty approximation of the solution of a functional inequality helps us to find the optimum approximation. To measure quality and certainty, we used the idea of the Z-number (Z-N) and we introduced the generalized Z-N (GZ-N) as a diagonal matrix of the form $ diag(X, Y, X\ast Y) $, where $ X $ is a fuzzy set time-stamped, $ Y $ is the probability distribution function and the third part is the fuzzy-random trace of the first and the second subjects. This kind of diagonal matrix allowed us to define a new model of control functions to stabilize our problem. Using stability analysis, we obtained the most suitable approximation for functional inequalities.

References

[1]	A. Gilányi, Eine zur Parallelogrammgleichung äquivalente Ungleichung, Aequationes Math., 62 (2001), 303–309. https://doi.org/10.1007/pl00000156 doi: 10.1007/pl00000156
[2]	S. M. Ulam, A collection of the mathematical problems, New York: Interscience Publ., 1960.
[3]	W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math., 71 (2006), 149–161. https://doi.org/10.1007/s00010-005-2775-9 doi: 10.1007/s00010-005-2775-9
[4]	A. Gilányi, On a problem by K. Nikodem, Math. Inequal. Appl., 5 (2002), 707–710. https://dx.doi.org/10.7153/mia-05-71 doi: 10.7153/mia-05-71
[5]	L. A. Zadeh, A note on $Z$-numbers, Inform. Sci., 181 (2011), 2923–2932. https://doi.org/10.1016/j.ins.2011.02.022 doi: 10.1016/j.ins.2011.02.022
[6]	S. Pirmuhammadi, T. Allahviranloo, M. Keshavarz, The parametric form of $Z$-number and its application in $Z$-number initial value problem, Int. J. Intell. Syst., 32 (2017) 1030–1061. https://doi.org/10.1002/int.21883 doi: 10.1002/int.21883
[7]	S. Ezadi, T. Allahviranloo, Numerical solution of linear regression based on $Z$-numbers by improved neural network, Intell. Autom. Soft Comput., 2017, 1–11. https://doi.org/10.1080/10798587.2017.1328812 doi: 10.1080/10798587.2017.1328812
[8]	R. A. Aliev, A. V. Alizadeh, O. H. Huseynov, The arithmetic of discrete $Z$-numbers, Inform. Sci., 290 (2015), 134–155. https://doi.org/10.1016/j.ins.2014.08.024 doi: 10.1016/j.ins.2014.08.024
[9]	A. Ahadi, R. Saadati, Generalized Z-number approximation for the fractional two-point iterative equation with a boundary condition, Inform. Sci., 649 (2023), 119673. https://doi.org/10.1016/j.ins.2023.119673 doi: 10.1016/j.ins.2023.119673
[10]	B. Kang, D. Wei, Y. Li, Y. Deng, Decision making using $Z$-numbers under uncertain environment, J. Comput. Inform. Syst., 8 (2012), 2807–2814.
[11]	R. R. Yager, On Z-valuations using Zadeh's Z-numbers, Int. J. Intell. Syst., 27 (2012), 259–278. https://doi.org/10.1002/int.21521 doi: 10.1002/int.21521
[12]	W. A. J. Luxemburg, On the convergence of successive approximations in the theory of ordinary differential equations, Indagat. Math., 61 (1958), 540–546. https://doi.org/10.1016/s1385-7258(58)50077-8 doi: 10.1016/s1385-7258(58)50077-8
[13]	Z. Eidinejad, R. Saadati, R. Mesiar, Optimum approximation for $\varsigma$-Lie homomorphisms and jordan $\varsigma$-Lie homomorphisms in $\varsigma$-Lie algebras by aggregation control functions, Mathematics, 10 (2022), 1704. https://doi.org/10.3390/math10101704 doi: 10.3390/math10101704
[14]	R. Saadati, Nonlinear contraction and fuzzy compact operator in fuzzy Banach algebras, Fixed Point Theory, 20 (2019), 289–297. https://doi.org/10.24193/fpt-ro.2019.1.19 doi: 10.24193/fpt-ro.2019.1.19
[15]	M. Demma, R. Saadati, P. Vetro, Fixed point results on $b$-metric space via Picard sequences and $b$-simulation functions, Iran. J. Math. Sci. Inform., 11 (2016), 123–136. https://doi.org/10.7508/ijmsi.2016.01.011 doi: 10.7508/ijmsi.2016.01.011
[16]	R. Saadati, A note on "Some results on the IF-normed spaces", Chaos Soliton. Fract., 41 (2009), 206–213. https://doi.org/10.1016/j.chaos.2007.11.027 doi: 10.1016/j.chaos.2007.11.027
[17]	S. Shakeri, L. Ciric, R. Saadati, Common fixed point theorem in partially ordered $L$-fuzzy metric spaces, Fixed Point Theory Appl., 2010, 1–13. http://dx.doi.org/10.1155/2010/125082 doi: 10.1155/2010/125082
[18]	C. Park, D. O'Regan, R. Saadati, Stability of some set-valued functional equations, Appl. Math. Lett., 24 (2011), 1910–1914. https://doi.org/10.1016/j.aml.2011.05.017 doi: 10.1016/j.aml.2011.05.017
[19]	T. Rasham, M. S. Shabbir, P. Agarwal, S. Momani, On a pair of fuzzy dominated mappings on closed ball in the multiplicative metric space with applications, Fuzzy Set. Syst., 437 (2022), 81–96. https://doi.org/10.1016/j.fss.2021.09.002 doi: 10.1016/j.fss.2021.09.002
[20]	T. Rasham, A. Asif, H. Aydi, M. Sen, On pairs of fuzzy dominated mappings and applications, Adv. Differ. Equ., 2021, 1–12. https://doi.org/10.1186/s13662-021-03569-5 doi: 10.1186/s13662-021-03569-5
[21]	T. Rasham, G. Marino, A. Shahzad, C. Park, A. Shoaib, Fixed point results for a pair of fuzzy mappings and related applications in $b$-metric like spaces, Adv. Differ. Equ., 2021,259. https://doi.org/10.1186/s13662-021-03418-5 doi: 10.1186/s13662-021-03418-5
[22]	D. Mihet, V. Radu, On the stability of the additive cauchy functional equation in random normed spaces, J. Math. Anal. Appl., 343 (2008), 567–572. https://doi.org/10.1016/j.jmaa.2008.01.100 doi: 10.1016/j.jmaa.2008.01.100
[23]	J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305–309. https://doi.org/10.1090/s0002-9904-1968-11933-0 doi: 10.1090/s0002-9904-1968-11933-0
[24]	L. Cădariu, V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math., 4 (2003).
[25]	J. S. An, On an additive functional inequality in normed modules over a $C^$-algebra, Pure Appl. Math.*, 15 (2008), 393–400.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)