General solution and generalized HyersUlam stability for additive functional equations
Volume 29, Issue 4, pp 343355
https://doi.org/10.22436/jmcs.029.04.04
Publication Date: November 03, 2022
Submission Date: June 12, 2022
Revision Date: July 17, 2022
Accteptance Date: August 13, 2022
Authors
S. S. Santra
 Department of Mathematics, Applied Science Cluster, University of Petroleum and Energy Studies, Dehradun, Uttarakhand  248007, India.
M. Arulselvam
 Government Arts College for Men, Krishnagiri635 001, Tamil Nadu, India.
D. Baleanu
 Department of Mathematics and Computer Science, Faculty of Arts and Sciences, Ankaya University, Ankara, 06790 Etimesgut, Turkey.
 Instiute of Space Sciences, MagureleBucharest, 077125 Magurele, Romania.
 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan, Republic of China.
V. Govindan
 Department of Mathematics, DMI St John The Baptist University Central, Mangochi409, Cental Africa, Malawi.
Kh. M. Khedher
 Department of Civil Engineering, College of Engineering, King Khalid University, Abha 61421, Saudi Arabia.
 Department of Civil Engineering, High Institute of Technological Studies, Mrezgua University Campus, Nabeul 8000, Tunisia.
Abstract
In this paper, we introduce new types of additive functional equations and obtain the solutions to these additive functional equations. Furthermore, we investigate the HyersUlam stability for the additive functional equations in fuzzy normed spaces and random normed spaces using the direct and fixed point approaches. Also, we will present some applications of functional equations in physics. Through these examples, we explain how the functional equations appear in the physical problem, how we use them to solve it, and we talk about solutions that are not used for solving the problem, but which can be of interest. We provide an example to show how functional equations may be used to solve geometry difficulties.
Share and Cite
ISRP Style
S. S. Santra, M. Arulselvam, D. Baleanu, V. Govindan, Kh. M. Khedher, General solution and generalized HyersUlam stability for additive functional equations, Journal of Mathematics and Computer Science, 29 (2023), no. 4, 343355
AMA Style
Santra S. S., Arulselvam M., Baleanu D., Govindan V., Khedher Kh. M., General solution and generalized HyersUlam stability for additive functional equations. J Math Comput SCIJM. (2023); 29(4):343355
Chicago/Turabian Style
Santra, S. S., Arulselvam, M., Baleanu, D., Govindan, V., Khedher, Kh. M.. "General solution and generalized HyersUlam stability for additive functional equations." Journal of Mathematics and Computer Science, 29, no. 4 (2023): 343355
Keywords
 HyersUlam stability
 fuzzy normed space
 random normed space and fixed point
MSC
References

[1]
M. Adam, S. Czerwik, On the stability of the quadratic functional equation in topological spaces, Banach J. Math. Anal., 1 (2007), 245–251

[2]
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66

[3]
T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11 (2003), 687–705

[4]
M. Bohner, T. S. Hassan, T. X. Li, FiteHilleWintnertype oscillation criteria for secondorder halflinear dynamic equation with deviating arguments, Indag. Math. (N.S.), 29 (2018), 548–560

[5]
S. C. Cheng, J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc., 86 (1994), 429–436

[6]
K.S. Chiu, T. X. Li, Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr., 292 (2019), 2153–2164

[7]
P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76–86

[8]
S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59–64

[9]
S. Deepa, A. Ganesh, V. Ibrahimov, S. S. Santra, V. Govindan, K. M. Khedher, S. Noeiaghdam, Fractional Fourier Transform to Stability Analysis of Fractional Differential Equations with Prabhakar Derivatives, Azer. J. Math., 12 (2022), 2218–6816

[10]
J. Dˇzurina, S. R. Grace, I. Jadlovsk´a, T. X. Li, Oscillation criteria for secondorder EmdenFowler delay differential equations with a sublinear neutral term, Math. Nachr., 293 (2020), 910–922

[11]
S. Frassu, T. Li, G. Viglialoro, Improvements and generalizations of results concerning attractionrepulsion chemotaxis models, Math. Meth. Appl. Sci., 2022 (2022), 12 pages

[12]
S. Frassu, G. Viglialorl, Boundedness criteria for a class of indirect (and direct) chemotaxiscomsumption models in high dimensions, Appl. Math. Lett., 132 (2022), 1–7

[13]
P. Gˇavruta, A generalization of the HyersUlamRassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436

[14]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222–224

[15]
K.W. Jun, Y.H. Lee, A generalization of the HyersUlamRassias stability of the Pexiderized quadratic equations. II, Kyungpook Math. J., 47 (2007), 91–103

[16]
P. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York (2009)

[17]
A. K. Katsaras, Fuzzy topological vector spaces. II, Fuzzy Sets and Systems, 12 (1984), 143–154

[18]
M. M. A. Khater, A. E. Ahmed, Strong Langmuir turbulence dynamics through the trigonometric quintic and exponential Bspline schemes, AIMS Math., 6 (2021), 5896–5908

[19]
M. M. A. Khater, A. Bekir, D. C. Lu, R. A. M. Attia, Analytical and semianalytical solutions for timefractional Cahn Allen equation, Math. Methods Appl. Sci., 44 (2021), 2682–2691

[20]
H.M. Kim, On the stability problem for a mixed type of quartic and quadratic functional equation, J. Math. Anal. Appl., 324 (2006), 358–372

[21]
I. Kramosil, J. Mich´alek, Fuzzy metric and statistical metric spaces, Kybernetika (Prague), 11 (1975), 336–344

[22]
Y.H. Lee, On the stability of the monomial functional equation, Bull. Korean Math. Soc., 45 (2008), 397–403

[23]
Y.H. Lee, On the HyersUlamRassias stability of the generalized polynomial function of degree 2, J. Chung Cheong Math. Soc., 2009 (2009), 201–209

[24]
Y.H. Lee, K.W. Jun, On the stability of approximately additive mappings, Proc. Amer. Math. Soc., 128 (2000), 1361– 1369

[25]
T. X. Li, N. Pintus, G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys., 70 (2019), 18 pages

[26]
T.X. Li, Y. V. Rogovchenko, On asymptotic behavior of solutions to higherorder sublinear EmdenFowler delay differential equations, Appl. Math. Lett., 67 (2017), 53–59

[27]
T. X. Li, Y. V. Rogovchenko, Oscillation criteria for second–order superlinear Emden–Fowler neutral differential equations, Monatsh. Math., 184 (2017), 489–500

[28]
T. X. Li, Y. V. Rogovchenko, On the asymptotic behavior of solutions to a class of thirdorder nonlinear neutral differential equations, Appl. Math. Lett., 105 (2020), 7 pages

[29]
T. Li, G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in the attraction dominated regime, Differ. Integ. Equ., 34 (2021), 315–336

[30]
A. K. Mirmostafaee, M. S. Moslehian, Fuzzy versions of HyersUlamRassias theorem, Fuzzy Sets and Systems, 159 (2008), 720–729

[31]
M. Mirzavaziri, M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. (N.S.), 37 (2006), 361–376

[32]
O. Moaaz, A. Muhib, T. Abdeljawad, S. S. Santra, M. Anis, Asymptotic behavior of evenorder noncanonical neutral differential equations, Demonstr. Math., 55 (2022), 28–39

[33]
A. Muhib, I. Dassios, D. Baleanu, S. S. Santra, O. Moaaz, Oddorder differential equations with deviating arguments: asymptomatic behavior and oscillation, Math. Biosci. Eng., 19 (2022), 1411–1425

[34]
A. Najati, A. Rahimi, A fixed point approach to the stability of a generalized cauchy functional equation, Banach J. Math. Anal., 2 (2008), 105–112

[35]
B. Qaraad, O. Moaaz, D. Baleanu, S. S. Santra, R. Ali, E. M. Elabbasy, Thirdorder neutral differential equations of the mixed type: oscillatory and asymptotic behavior, Math. Biosci. Eng., 19 (2022), 1649–1658

[36]
V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91–96

[37]
T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300

[38]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Functional Analysis, 46 (1982), 126–130

[39]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. (2), 108 (1984), 445–446

[40]
T. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 264–284

[41]
P. K. Sahoo, P. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton (2011)

[42]
S. S. Santra, A. Scapellato, Some conditions for the oscillation of secondorder differential equations with several mixed delays, J. Fixed Point Theory Appl., 24 (2022), 11 pages

[43]
F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113–129

[44]
S. M. Ulam, A Collection of Mathematical Problems, Interscience Publisheres, New YorkLondon (1960)

[45]
P.Y. Xiong, H. Jahanshahi, R. Alcaraz, Y.M. Chu, J. F. G´omezAguilar, F. E. Alsaadi, Spectral entropy analysis and synchronization of a multistable fractionalorder chaotic system using a novel neural networkbased chatteringfree sliding mode technique, Chaos Solitons Fractals, 144 (2021), 12 pages