Examining the approximation of the second-order functional equations developed by using fuzzy p-normed space

Subject Areas : Analyze

ehsan movahednia ¹ , parvaneh lolo ^{2 *}

1 - Department of Mathematics, Behbahan Khatam Alanbia University of Technology- Iran
2 - Department of Mathematics, Behbahan Khatam Alanbia University of Technology- Iran

Received: 2023-04-25 Accepted : 2023-07-02 Published : 2024-10-17

Keywords: , &quot, &quot, قضیه نقطه ثابت&quot, پایداری معادلات تابعی&quot, فضای فازی&quot,

Abstract :

در ﭘﺎﺋﻴﺰ ﺳﺎل ٠۴٩١، اوﻻمدر سخنرانی معروف خود در باﺷﮕﺎه رﻳﺎضیات داﻧﺸﮕﺎه وﻳﺴﻨﺴﻴﻦ ﺗﻌﺪادی از ﻣﺴﺎﺋﻞ ﺣﻞ ﻧﺸﺪه را اراﺋﻪ ﻧﻤﻮد. اﻳﻦ ﺳﺨﻨﺮاﻧﯽ ﻧﻘﻄﻪ آﻏﺎزی ﺑﺮ ﻧﻈﺮﻳﻪ ی ﭘﺎﻳﺪاری ﻣﻌﺎدﻻت ﺗﺎﺑﻌﯽ ﮔﺮدﻳﺪ. اوﻻم ﻣﺴﺄﻟﻪ ی ﭘﺎﻳﺪاری را در ﺣﺎﻟﺘﯽ ﻛﻠﯽ ﺑیان ﻧﻤﻮد و در ﺣﺎﻟﺘﯽ ﺧﺎص، ﻣﺴﺄﻟﻪ ی ﭘﺎﻳﺪاری ﻣﻌﺎدﻻت ﺗﺎﺑﻌﯽ را اﻳﻦ ﮔﻮﻧﻪﺑیانﻛﺮد: “... ﻣﻔﻬﻮم ﭘﺎﻳﺪاری ﻗﻀﺎﻳﺎی رﻳﺎﺿﯽ از ﻧﻈﺮﮔﺎﻫﯽ ﻛﻠﯽ اﻳﻦ ﮔﻮﻧﻪ ﻣﻄﺮح ﻣﯽ ﺷﻮد: اﮔﺮ ﻓﺮض ﻳﻚ ﻗﻀﻴﻪ ﺗﻘﺮﻳﺒﺎ ﺑﺮﻗﺮار ﺑﺎﺷﺪ، آﻳﺎ ﻣﯽ ﺗﻮان ادﻋﺎ ﻧﻤﻮد ﻛﻪ ﺣﮑﻢ ﻗﻀﻴﻪ ﻧﻴﺰ ﺗﻘﺮﻳﺒﺎ ﺑﺮﻗﺮار ﺧﻮاﻫﺪ ﻣﺎﻧدبرای ﻳﻚ ﻣﻌﺎدﻟﻪ ی ﺗﺎﺑﻌﯽ ﻣﺴﺄﻟﻪ ی ﭘﺎﻳﺪاری اﻳﻨﮕﻮﻧﻪ ﻣﻄﺮح ﻣﯽ ﮔﺮدد: اﮔﺮ ﻣﻌﺎدﻟﻪ ی ﺗﺎﺑﻌﯽ ﻣﻔﺮوض دارای ﺟﻮاب ﺗﻘﺮﻳﺒﯽ دﻟﺨﻮاﻫﯽ ﺑﺎﺷﺪ، آﻳﺎ اﻳﻦ ﺟﻮاب ﺗﻘﺮﻳﺒﯽ ﺑﻪ اﻧﺪازه ﻛﺎﻓﯽ ﺑﻪ ﻳﻚ ﺟﻮاب دﻗﻴﻖ از ﻣﻌﺎدﻟﻪ ی ﺗﺎﺑﻌﯽ ﻣﻔﺮوض ﻧﺰدﻳﻚ ﺧﻮاﻫﺪ ﺑﻮد؟ ... اﮔﺮ ﺟﻮاب اﻳﻦ ﺳﺆال ﻣﺜﺒﺖ ﺑﺎﺷﺪ، ﻣﻌﺎدﻟﻪ ی ﺗﺎﺑعی ﻣﻔﺮوض را ﭘﺎﻳﺪار ﮔﻮﻳﻴﻢ. ٢ E ﺗﻘﺮﯾﺒﺎ ﺟﻤﻌﯽ ٢ E و ١ E : f ﺑﺮ روی ﻓﻀﺎﻫﺎی ﺑﺎﻧﺎخ ١ → E ٢ ٠ > ϵ دﻟﺨﻮاه، اﮔﺮ ﻧﮕﺎﺷﺖ ﺑﺎﺷﺪ، ﯾﻌﻨﯽ راﺑﻄﻪ ∥f(x + y) − f(x) − f(y)∥ ≤ ϵ E : A ﭼﻨﺎن ﻣﻮﺟﻮد اﺳﺖ ﮐﻪ راﺑﻄﻪ ی ١ → E ٢ ﺑﺮﻗﺮار ﺑﺎﺷﺪ، آﻧﮕﺎه ٠ > k و ﻧﮕﺎﺷﺖ ﺟﻤﻌﯽ ﯾﮑﺘﺎی ∥A(x) − f(x)∥ ≤ ϵ ﺑﺮﻗﺮار ﺧﻮاﻫﺪ ﺑﻮد. روﺷﯽ ﮐﻪ ﻫﺎﯾﺮز در اﺛﺒﺎت ﭘﺎﯾﺪاری ﻣﻌﺎدﻟﻪ ی ﺗﺎﺑﻌﯽ ﺟﻤﻌﯽ اﺳﺘﻔﺎده ﻧﻤﻮد را روش ﻣﺴﺘﻘﯿﻢ ﻧﺎﻣﯿﺪه اﻧﺪ. اﮔﺮ ﭘﺎﯾﺪاری ﯾﮏ ﻣﻌﺎدﻟﻪ ی ﺗﺎﺑﻌﯽ ﻣﺸﺎﺑﻪ ﺑﺎ ﻗﻀﯿﻪ ﻫﺎﯾﺮز و ﺑﺎ اﺳﺘﻔﺎده از ϵ ﺛﺎﺑ.ﺖ .ﮔﺮدد، آن را ﭘﺎﯾﺪاری ﻫﺎﯾﺮز- اوﻻم ﻣﯽ ﻧﺎﻣﻨﺪ

References:

[1] Ulam. S. M. (2004). Problems in modern mathematics. Courier Corporation.

[2] Hyers. H. D. (1941) On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224.

[3] Rassias. T. M. (1978) On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, 297-300.

[4] Mihet. D and Radu. V. (2008) On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl,343,567-572.

[5] Mirmostafaee. A. K. , Mirzavaziri. M. and Moslehian. M. S. (2008) Fuzzy stability of the jensen functional equation, Fuzzy Sets and Systems, 159, 730-738.

[6] A.K. Mirmostafaee, A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces, Fuzzy Sets Syst. 160 (2009) 1653-1662..

[7] Modarres Mosadegh. S. M. S. and Movahednia. E. (2018) Stability of preserving lattice cubic functional equation in Menger probabilistic normed Riesz spaces, Journal of Fixed Point Theory and Applications, 20, 34-43.

[8] Movahednia. E. and Mursaleen. M. (2016) Stability of a generalized quadratic functional equation in intuitionistic fuzzy 2-normed space, Filomat, 30, 449-457.

[9] Movahednia. E., Cho. Y, Park. C and Paokanta. S. (2020), On approximate solution of lattice functional equations in Banach f-algebras, AIMS Mathematics, 5, 5458-5469.

[10] Park. C, Eshaghi Gorgji. M and Saadati. R. (2012) Random homomorphisms and random derivations in random normed algebras via fixed point method, J. Inequal. Appl, 13, 1-4.

[11] Salehi. N and Modarres. S. M. S (2020) A fixed point method for stability of involutions on multi-Banach algebra, Journal of Fixed Point Theory and Applications, 22.

[12] Movahednia. E. (2011) Fuzzy stability of quadratic functional equations in general cases, International Scholarly Research Notices, 1--9.

[13] Gordji. M. E., Alizadeh. Z., Khodaei. H. and Park. C. (2012) On approximate homomorphisms: a fixed point approach, Mathematical Sciences, 6.1, 1-8.

[14] Cadariu. L and Radu. V. (2003) Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math,4, Art. ID 4.

[15] Bag. T. and Samanta .S. K. (2003) Finite dimensional fuzzy normed linear spaces, Journal of Fuzzy Mathematics, 11, 687-–705.

[16] Diaz. J and Margolis. B. (1968) A fixed point theorem of the alternative for contractions on a generalized complete metrice space, Bull. Amer. Math. Soc, 74, 305-309.

[17] Katsaras. A. K. (1984) Fuzzy topological vector spaces 2, Fuzzy Sets and Systems, 12, 143-–154.

[18] Kramosil. I. and Michalek. J. (1975) Fuzzy metric and statistical metric spaces, Kybernetika,11, 326-–334.

[19] Lo′lo′. P, Movahednia. E, and De la Sen M. (2022) Hyers Stability and Multi Fuzzy Banach Algebra. Mathematics. 10(1):106. https://doi.org/10.3390/math10010106.