APPROXIMATE SOLUTION OF A p-th ROOT FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN (2,β)-BANACH SPACES
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摘要
In this paper, using the Brzdek's fixed point theorem [9,Theorem 1] in non-Archimedean(2,β)-Banach spaces, we prove some stability and hyperstability results for an p-th root functional equation ■where p∈{1, …, 5}, a_1,…, a_k are fixed nonzero reals when p ∈ {1,3,5} and are fixed positive reals when p ∈{2,4}.
In this paper, using the Brzdek's fixed point theorem [9,Theorem 1] in non-Archimedean(2,β)-Banach spaces, we prove some stability and hyperstability results for an p-th root functional equation ■where p∈{1, …, 5}, a_1,…, a_k are fixed nonzero reals when p ∈ {1,3,5} and are fixed positive reals when p ∈{2,4}.
In this paper, using the Brzdek's fixed point theorem [9,Theorem 1] in non-Archimedean(2,β)-Banach spaces, we prove some stability and hyperstability results for an p-th root functional equation ■where p∈{1, …, 5}, a_1,…, a_k are fixed nonzero reals when p ∈ {1,3,5} and are fixed positive reals when p ∈{2,4}.
引文
[1]Agarwal R P, Xu B, Zhang W. Stability of functional equations in single variable. J Math Anal Appl, 2003,288:852-869
[2]Aiemsomboon L, Sintunavarat W. On a new type of stability of a radical quadratic functional equation using Brzdek's fixed point theorem. Acta Math Hungar, 2017, 151(1):35-46
[3]Alizadeh Z, Ghazanfari A G. On the stability of a radical cubic functional equation in quasi-β-spaces. J Fixed Point Theory Appl, 2016, 18:843-853
[4]Aoki T. On the stability of the linear transformation in Banach spaces. J Math Soc Japan, 1950, 2:64-66
[5]Bourgin D G. Approximately isometric and multiplicative transformations on continuous function rings.Duke Math J, 1949, 16:385-397
[6]Bourgin D G. Classes of transformations and bordering transformations. Bull Amer Math Soc, 1951, 57:223-237
[7]Brzdek J. Hyperstability of the Cauchy equation on restricted domains. Acta Math Hungar, 2013, 141(1/2):58-67
[8]Brzdek J. Remarks on solutions to the functional equations of the radical type. Adv Theory Nonlinear Anal Appl, 2017, 1(2):125-135
[9]Brzdek J, Cieplinski K. A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal, 2011, 74:6861-6867
[10]Brzdek J, Cieplinski K. Hyperstability and superstability. Abstr Appl Anal, 2003, 2013:Article ID 401756
[11]Brzdek J, Cieplinski K. On a fixed point theorem in 2-Banach spaces and some of its applications. Acta Math Sci, 2018, 38B(2):377-390
[12]Brzdek J, Fechner W, Moslehian M S, Sikorska J. Recent developments of the conditional stability of the homomorphism equation. Banach J Math Anal, 2015, 9:278-327
[13]Dung N V, Hang V T L. The generalized hyperstability of general linear equations in quasi-Banach spaces.J Math Anal Appl, 2018, 462:131-147
[14]EL-Fassi Iz. On a new type of hyperstability for radical cubic functional equation in non-Archimedean metric spaces. Results Math, 2017, 72:991-1005
[15]EL-Fassi Iz. Approximate solution of radical quartic functional equation related to additive mapping in2-Banach spaces. J Math Anal Appl, 2017, 455:2001-2013
[16]EL-Fassi Iz. A new type of approximation for the radical quintic functional equation in non-Archimedean(2,β)-Banach spaces. J Math Anal Appl, 2018, 457:322-335
[17]EL-Fassi Iz. On the general solution and hyperstability of the general radical quintic functional equation in quasi-β-Banach spaces. J Math Anal Appl, 2018, 466:733-748
[18]EL-Fassi Iz. Solution and approximation of radical quintic functional equation related to quintic mapping in quasi-β-Banach spaces. RACSAM, 2018, https://doi.org/10.1007/s13398-018-0506-z
[19]EL-Fassi Iz, Brzdek J, Chahbi A, Kabbaj S. On hyperstability of the biadditive functional equation. Acta Math Sci, 2017, 37B(6):1727-1739
[20]El-Fassi Iz, Kabbaj S. On the generalized orthogonal stability of the Pexiderized quadratic functional equation in modular space. Math Slovaca, 2017,67(1):165-178
[21]Gahler S. Lineare 2-normierte Raume. Math Nachr, 1964, 28:1-43(in German)
[22]Gahler S. Uber 2-Banach-Raume. Math Nachr, 1969, 42:335-347(in German)
[23]Gajda Z. On stability of additive mappings. Int J Math Math Sci, 1991, 14:431-434
[24]Gavruta P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl, 1994, 184:431-436
[25]Gordji M E, Khodaei H, Rassias Th M. A Functional Equation Having Monomials and Its Stability.Optimization and Its Applications, 96. Springer, 2014
[26]Hensel K. Uber eine neue Begriindung der Theorie der algebraischen Zahlen. Jahresber Dtsch Math-Ver,1897, 6:83-88
[27]Hyers D H. On the stability of the linear functional equation. Proc Nat Acad Sci, USA, 1941, 27:222-224
[28]Katsaras A K, Beoyiannis A. Tensor products of non-Archimedean weighted spaces of continuous functions.Georgian Math J, 1999, 6:33-44
[29]Khodaei H, Eshaghi Gordji M, Kim S S, Cho Y J. Approximation of radical functional equations related to quadratic and quartic mappings. J Math Anal Appl, 2012, 395:284-297
[30]Khrennikov A. Non-Archimedean Analysis:Quantum Paradoxes, Dynamical Systems and Biological Models. Mathematics and Its Applications, 427. Dordrecht:Kluwer Academic, 1997
[31]Maksa G, Pales Z. Hyperstability of a class of linear functional equations. Acta Math Acad Paedagog Nyhaazi(NS), 2001, 17:107-112
[32]Moslehian M S, Rassias Th M. Stability of functional equations in non-Archimedean spaces. Appl Anal Discrete Math, 2007, 1:325-334
[33]Nyikos P J. On some non-Archimedean spaces of Alexandrof and Urysohn. Topol Appl, 1999, 91:1-23
[34]Rassias Th M. On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc, 1978, 72:297-300
[35]Rassias Th M. On a modified Hyers-Ulam sequence. J Math Anal Appl, 1991, 158:106-113
[36]Rassias Th M, Semrl P. On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proc Amer Math Soc, 1992, 114:989-993
[37]Ulam S M. A Collection of Mathematical Problems. New York:Interscience Publishers, 1960; Reprinted as:Problems in Modern Mathematics. New York:John Wiley&Sons, Inc, 1964
[38]White A. 2-Banach spaces. Math Nachr, 1969, 42:43-60
[39]Xu B, Brzdek J, Zhang W. Fixed point results and the Hyers-Ulam stability of linear equations of higher orders. Pacific J Math, 2015, 273(2):483-498
[2]Aiemsomboon L, Sintunavarat W. On a new type of stability of a radical quadratic functional equation using Brzdek's fixed point theorem. Acta Math Hungar, 2017, 151(1):35-46
[3]Alizadeh Z, Ghazanfari A G. On the stability of a radical cubic functional equation in quasi-β-spaces. J Fixed Point Theory Appl, 2016, 18:843-853
[4]Aoki T. On the stability of the linear transformation in Banach spaces. J Math Soc Japan, 1950, 2:64-66
[5]Bourgin D G. Approximately isometric and multiplicative transformations on continuous function rings.Duke Math J, 1949, 16:385-397
[6]Bourgin D G. Classes of transformations and bordering transformations. Bull Amer Math Soc, 1951, 57:223-237
[7]Brzdek J. Hyperstability of the Cauchy equation on restricted domains. Acta Math Hungar, 2013, 141(1/2):58-67
[8]Brzdek J. Remarks on solutions to the functional equations of the radical type. Adv Theory Nonlinear Anal Appl, 2017, 1(2):125-135
[9]Brzdek J, Cieplinski K. A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal, 2011, 74:6861-6867
[10]Brzdek J, Cieplinski K. Hyperstability and superstability. Abstr Appl Anal, 2003, 2013:Article ID 401756
[11]Brzdek J, Cieplinski K. On a fixed point theorem in 2-Banach spaces and some of its applications. Acta Math Sci, 2018, 38B(2):377-390
[12]Brzdek J, Fechner W, Moslehian M S, Sikorska J. Recent developments of the conditional stability of the homomorphism equation. Banach J Math Anal, 2015, 9:278-327
[13]Dung N V, Hang V T L. The generalized hyperstability of general linear equations in quasi-Banach spaces.J Math Anal Appl, 2018, 462:131-147
[14]EL-Fassi Iz. On a new type of hyperstability for radical cubic functional equation in non-Archimedean metric spaces. Results Math, 2017, 72:991-1005
[15]EL-Fassi Iz. Approximate solution of radical quartic functional equation related to additive mapping in2-Banach spaces. J Math Anal Appl, 2017, 455:2001-2013
[16]EL-Fassi Iz. A new type of approximation for the radical quintic functional equation in non-Archimedean(2,β)-Banach spaces. J Math Anal Appl, 2018, 457:322-335
[17]EL-Fassi Iz. On the general solution and hyperstability of the general radical quintic functional equation in quasi-β-Banach spaces. J Math Anal Appl, 2018, 466:733-748
[18]EL-Fassi Iz. Solution and approximation of radical quintic functional equation related to quintic mapping in quasi-β-Banach spaces. RACSAM, 2018, https://doi.org/10.1007/s13398-018-0506-z
[19]EL-Fassi Iz, Brzdek J, Chahbi A, Kabbaj S. On hyperstability of the biadditive functional equation. Acta Math Sci, 2017, 37B(6):1727-1739
[20]El-Fassi Iz, Kabbaj S. On the generalized orthogonal stability of the Pexiderized quadratic functional equation in modular space. Math Slovaca, 2017,67(1):165-178
[21]Gahler S. Lineare 2-normierte Raume. Math Nachr, 1964, 28:1-43(in German)
[22]Gahler S. Uber 2-Banach-Raume. Math Nachr, 1969, 42:335-347(in German)
[23]Gajda Z. On stability of additive mappings. Int J Math Math Sci, 1991, 14:431-434
[24]Gavruta P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl, 1994, 184:431-436
[25]Gordji M E, Khodaei H, Rassias Th M. A Functional Equation Having Monomials and Its Stability.Optimization and Its Applications, 96. Springer, 2014
[26]Hensel K. Uber eine neue Begriindung der Theorie der algebraischen Zahlen. Jahresber Dtsch Math-Ver,1897, 6:83-88
[27]Hyers D H. On the stability of the linear functional equation. Proc Nat Acad Sci, USA, 1941, 27:222-224
[28]Katsaras A K, Beoyiannis A. Tensor products of non-Archimedean weighted spaces of continuous functions.Georgian Math J, 1999, 6:33-44
[29]Khodaei H, Eshaghi Gordji M, Kim S S, Cho Y J. Approximation of radical functional equations related to quadratic and quartic mappings. J Math Anal Appl, 2012, 395:284-297
[30]Khrennikov A. Non-Archimedean Analysis:Quantum Paradoxes, Dynamical Systems and Biological Models. Mathematics and Its Applications, 427. Dordrecht:Kluwer Academic, 1997
[31]Maksa G, Pales Z. Hyperstability of a class of linear functional equations. Acta Math Acad Paedagog Nyhaazi(NS), 2001, 17:107-112
[32]Moslehian M S, Rassias Th M. Stability of functional equations in non-Archimedean spaces. Appl Anal Discrete Math, 2007, 1:325-334
[33]Nyikos P J. On some non-Archimedean spaces of Alexandrof and Urysohn. Topol Appl, 1999, 91:1-23
[34]Rassias Th M. On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc, 1978, 72:297-300
[35]Rassias Th M. On a modified Hyers-Ulam sequence. J Math Anal Appl, 1991, 158:106-113
[36]Rassias Th M, Semrl P. On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proc Amer Math Soc, 1992, 114:989-993
[37]Ulam S M. A Collection of Mathematical Problems. New York:Interscience Publishers, 1960; Reprinted as:Problems in Modern Mathematics. New York:John Wiley&Sons, Inc, 1964
[38]White A. 2-Banach spaces. Math Nachr, 1969, 42:43-60
[39]Xu B, Brzdek J, Zhang W. Fixed point results and the Hyers-Ulam stability of linear equations of higher orders. Pacific J Math, 2015, 273(2):483-498