On some recent developments in Ulam’s type stability. (English) Zbl 1259.39019

Brillouët-Belluot, Nicole; Brzdȩk, Janusz; Ciepliński, Krzysztof

Abstr. Appl. Anal. 2012, Article ID 716936, 41 p. (2012).

Summary: We present a survey of some selected recent developments (results and methods) in the theory of Ulam’s type stability. In particular, we provide some information on hyperstability and the fixed point methods.

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
39B52	Functional equations for functions with more general domains and/or ranges

Keywords:

Ulam stability; hyperstability; fixed point method

References:

[1]	R. P. Agarwal, B. Xu, and W. Zhang, “Stability of functional equations in single variable,” Journal of Mathematical Analysis and Applications, vol. 288, no. 2, pp. 852-869, 2003. · Zbl 1053.39042
[2]	S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing, River Edge, NJ, USA, 2002. · Zbl 1011.39019
[3]	G.-L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143-190, 1995. · Zbl 0836.39007
[4]	D. H. Hyers and Th. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125-153, 1992. · Zbl 0806.47056
[5]	D. H. Hyers, G. Isac, and Th. M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific Publishing, River Edge, NJ, USA, 1997. · Zbl 0878.47040
[6]	D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser Boston, Boston, Mass, USA, 1998. · Zbl 0907.39025
[7]	S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001. · Zbl 0980.39024
[8]	S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, NY, USA, 2011. · Zbl 1221.39038
[9]	Z. Moszner, “Sur les définitions différentes de la stabilité des équations fonctionnelles,” Aequationes Mathematicae, vol. 68, no. 3, pp. 260-274, 2004. · Zbl 1060.39031
[10]	Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23-130, 2000. · Zbl 0981.39014
[11]	L. Székelyhidi, “Ulam’s problem, Hyers’s solution-and to where they led,” in Functional Equations and Inequalities, vol. 518, pp. 259-285, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. · Zbl 0976.39028
[12]	G. Pólya and G. Szeg\Ho, Aufgaben und Lehrsätze aus der Analysis I, Julius Springer, Berlin, Germany, 1925. · JFM 51.0173.01
[13]	S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, NY, USA, 1960, Reprinted as: Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964. · Zbl 0086.24101
[14]	D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941. · Zbl 0061.26403
[15]	T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64-66, 1950. · Zbl 0040.35501
[16]	J. Brzd\cek, “A note on stability of additive mappings,” in Stability of Mappings of Hyers-Ulam Type, Th. M. Rassias and J. Tabor, Eds., pp. 19-22, Hadronic Press, Palm Harbor, Fla, USA, 1994. · Zbl 0844.39012
[17]	L. Maligranda, “A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions-a question of priority,” Aequationes Mathematicae, vol. 75, no. 3, pp. 289-296, 2008. · Zbl 1158.39019
[18]	Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040
[19]	Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431-434, 1991. · Zbl 0739.39013
[20]	Th. M. Rassias, “On a modified Hyers-Ulam sequence,” Journal of Mathematical Analysis and Applications, vol. 158, no. 1, pp. 106-113, 1991. · Zbl 0746.46038
[21]	Th. M. Rassias and P. \vSemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,” Proceedings of the American Mathematical Society, vol. 114, no. 4, pp. 989-993, 1992. · Zbl 0761.47004
[22]	D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223-237, 1951. · Zbl 0043.32902
[23]	J.-Y. Chung, “A remark on some stability theorems,” Aequationes Mathematicae, vol. 75, no. 3, pp. 271-275, 2008. · Zbl 1148.39022
[24]	P. G\uavru\cta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431-436, 1994. · Zbl 0818.46043
[25]	J. Brzd\cek and A. Pietrzyk, “A note on stability of the general linear equation,” Aequationes Mathematicae, vol. 75, no. 3, pp. 267-270, 2008. · Zbl 1149.39018
[26]	J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126-130, 1982. · Zbl 0482.47033
[27]	J. M. Rassias, “On a new approximation of approximately linear mappings by linear mappings,” Discussiones Mathematicae, vol. 7, pp. 193-196, 1985. · Zbl 0592.46004
[28]	J. Brzd\cek and J. Sikorska, “A conditional exponential functional equation and its stability,” Nonlinear Analysis A, vol. 72, no. 6, pp. 2923-2934, 2010. · Zbl 1187.39032
[29]	J. Sikorska, “Generalized stability of the Cauchy and the Jensen functional equations on spheres,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 650-660, 2008. · Zbl 1157.39019
[30]	J. Sikorska, “On a Pexiderized conditional exponential functional equation,” Acta Mathematica Hungarica, vol. 125, no. 3, pp. 287-299, 2009. · Zbl 1212.39039
[31]	F. Skof, “On the stability of functional equations on a restricted domain and a related topic,” in Stability of Mappings of Hyers-Ulam Type, Th. M. Rassias and J. Tabor, Eds., pp. 141-151, Hadronic Press, Palm Harbor, Fla, USA, 1994. · Zbl 0844.39006
[32]	M. Kuczma, Functional Equations in a Single Variable, Państwowe Wydawnictwo Naukowe, Warsaw, Poland, 1968. · Zbl 0196.16403
[33]	M. Kuczma, B. Choczewski, and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0703.39005
[34]	J. Brzd\cek, J. Chudziak, and Z. Páles, “A fixed point approach to stability of functional equations,” Nonlinear Analysis A, vol. 74, no. 17, pp. 6728-6732, 2011. · Zbl 1237.39022
[35]	J. Brzd\cek, D. Popa, and B. Xu, “Hyers-Ulam stability for linear equations of higher orders,” Acta Mathematica Hungarica, vol. 120, no. 1-2, pp. 1-8, 2008. · Zbl 1174.39012
[36]	J. Brzd\cek and S.-M. Jung, “A note on stability of a linear functional equation of second order connected with the Fibonacci numbers and Lucas sequences,” Journal of Inequalities and Applications, Article ID 793947, 10 pages, 2010. · Zbl 1198.39036
[37]	J. Brzd\cek and S.-M. Jung, “A note on stability of an operator linear equation of the second order,” Abstract and Applied Analysis, vol. 2011, Article ID 602713, 15 pages, 2011. · Zbl 1273.39023
[38]	J. Brzd\cek, D. Popa, and B. Xu, “On approximate solutions of the linear functional equation of higher order,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 680-689, 2011. · Zbl 1204.65147
[39]	J. A. Baker, “The stability of certain functional equations,” Proceedings of the American Mathematical Society, vol. 112, no. 3, pp. 729-732, 1991. · Zbl 0735.39004
[40]	D. Brydak, “On the stability of the functional equation \varphi [f(x)]=g(x)\varphi (x)+F(x),” Proceedings of the American Mathematical Society, vol. 26, pp. 455-460, 1970. · Zbl 0271.39009
[41]	J. Brzd\cek, D. Popa, and B. Xu, “The Hyers-Ulam stability of nonlinear recurrences,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 443-449, 2007. · Zbl 1123.39022
[42]	J. Brzd\?k, D. Popa, and B. Xu, “Remarks on stability of the linear functional equation in single variable,” in Nonlinear Analysis: Stability, Approximation, and Inequalities, P. Pardalos, H. M. Srivastava, and P. Georgiev, Eds., pp. 81-119, Springer, New York, NY, USA, 2012. · Zbl 1248.39022
[43]	G.-L. Forti, “Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 127-133, 2004. · Zbl 1052.39031
[44]	S.-M. Jung, “On a general Hyers-Ulam stability of gamma functional equation,” Bulletin of the Korean Mathematical Society, vol. 34, no. 3, pp. 437-446, 1997. · Zbl 0965.39023
[45]	D. Popa, “Hyers-Ulam-Rassias stability of the general linear equation,” Nonlinear Functional Analysis and Applications, vol. 7, no. 4, pp. 581-588, 2002. · Zbl 1031.39021
[46]	D. Popa, “Hyers-Ulam-Rassias stability of a linear recurrence,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 591-597, 2005. · Zbl 1079.39027
[47]	D. Popa, “Hyers-Ulam stability of the linear recurrence with constant coefficients,” Advances in Difference Equations, no. 2, pp. 101-107, 2005. · Zbl 1095.39024
[48]	T. Trif, “On the stability of a general gamma-type functional equation,” Publicationes Mathematicae Debrecen, vol. 60, no. 1-2, pp. 47-61, 2002. · Zbl 1004.39023
[49]	T. Trif, “Hyers-Ulam-Rassias stability of a linear functional equation with constant coefficients,” Nonlinear Functional Analysis and Applications, vol. 11, no. 5, pp. 881-889, 2006. · Zbl 1115.39028
[50]	E. Turdza, “On the stability of the functional equation \varphi [f(x)]=g(x)\varphi (x)+F(x),” Proceedings of the American Mathematical Society, vol. 30, pp. 484-486, 1971. · Zbl 0224.39005
[51]	B. Xu and W. Zhang, “Construction of continuous solutions and stability for the polynomial-like iterative equation,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1160-1170, 2007. · Zbl 1111.39020
[52]	R. Badora and J. Brzd\?k, “A note on a fixed point theorem and the Hyers-Ulam stability,” Journal of Difference Equations and Applications, vol. 18, pp. 1115-1119, 2012. · Zbl 1254.39012
[53]	J. Brzd\cek and K. Ciepliński, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces,” Nonlinear Analysis A, vol. 74, no. 18, pp. 6861-6867, 2011. · Zbl 1237.39022
[54]	Th. M. Rassias, “Problem,” Aequationes Mathematicae, vol. 39, p. 309, 1990.
[55]	A. Bahyrycz, “Forti’s example of an unstable homomorphism equation,” Aequationes Mathematicae, vol. 74, no. 3, pp. 310-313, 2007. · Zbl 1165.39021
[56]	G.-L. Forti, “The stability of homomorphisms and amenability, with applications to functional equations,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 57, pp. 215-226, 1987. · Zbl 0619.39012
[57]	Z. Moszner, “On the stability of functional equations,” Aequationes Mathematicae, vol. 77, no. 1-2, pp. 33-88, 2009. · Zbl 1207.39044
[58]	J. Brzd\cek, D. Popa, and B. Xu, “Note on nonstability of the linear recurrence,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 76, pp. 183-189, 2006. · Zbl 1126.39019
[59]	J. Brzd\cek, D. Popa, and B. Xu, “Remarks on stability of linear recurrence of higher order,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1459-1463, 2010. · Zbl 1206.39019
[60]	J. Brzd\cek, D. Popa, and B. Xu, “On nonstability of the linear recurrence of order one,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 146-153, 2010. · Zbl 1193.39006
[61]	J. Brzd\cek, D. Popa, and B. Xu, “Note on nonstability of the linear functional equation of higher order,” Computers & Mathematics with Applications, vol. 62, no. 6, pp. 2648-2657, 2011. · Zbl 1231.39012
[62]	J. Brzd\cek, D. Popa, and B. Xu, “A note on stability of the linear functional equation of higher order and fixed points of an operator,” Fixed Point Theory. In press.
[63]	G.-L. Forti and J. Schwaiger, “Stability of homomorphisms and completeness,” Comptes Rendus Mathématiques, vol. 11, no. 6, pp. 215-220, 1989. · Zbl 0697.39013
[64]	Z. Moszner, “Stability of the equation of homomorphism and completeness of the underlying space,” Opuscula Mathematica, vol. 28, no. 1, pp. 83-92, 2008. · Zbl 1156.39020
[65]	W. Jabłoński and J. Schwaiger, “Stability of the homogeneity and completeness,” Österreichische Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche Klasse, vol. 214, pp. 111-132, 2005. · Zbl 1142.39022
[66]	Z. Moszner, “On stability of some functional equations and topology of their target spaces,” Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, vol. 11, pp. 69-94, 2012. · Zbl 1292.39027
[67]	A. Najati, “On the completeness of normed spaces,” Applied Mathematics Letters, vol. 23, no. 8, pp. 880-882, 2010. · Zbl 1200.39011
[68]	J. Schwaiger, “Some remarks on the stability of the multi-Jensen equation,” Central European Journal of Mathematics. In press. · Zbl 1273.39026
[69]	G.-L. Forti, “Elementary remarks on Ulam-Hyers stability of linear functional equations,” Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 109-118, 2007. · Zbl 1111.39026
[70]	J. Brzd\cek, “On a method of proving the Hyers-Ulam stability of functional equations on restricted domains,” The Australian Journal of Mathematical Analysis and Applications, vol. 6, no. 1, Article 4, 10 pages, 2009. · Zbl 1175.39014
[71]	J. Brzd\cek, “A note on stability of the Popoviciu functional equation on restricted domain,” Demonstratio Mathematica, vol. 43, no. 3, pp. 635-641, 2010. · Zbl 1221.39035
[72]	J. Sikorska, “On a direct method for proving the Hyers-Ulam stability of functional equations,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 99-109, 2010. · Zbl 1198.39039
[73]	G. H. Kim, “On the stability of functional equations with square-symmetric operation,” Mathematical Inequalities & Applications, vol. 4, no. 2, pp. 257-266, 2001. · Zbl 0990.39028
[74]	G. H. Kim, “Addendum to “On the stability of functional equations on square-symmetric groupoid”,” Nonlinear Analysis A, vol. 62, no. 2, pp. 365-381, 2005. · Zbl 1112.39022
[75]	Z. Páles, “Hyers-Ulam stability of the Cauchy functional equation on square-symmetric groupoids,” Publicationes Mathematicae Debrecen, vol. 58, no. 4, pp. 651-666, 2001. · Zbl 0980.39022
[76]	Z. Páles, P. Volkmann, and R. D. Luce, “Hyers-Ulam stability of functional equations with a square-symmetric operation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 95, no. 22, pp. 12772-12775, 1998. · Zbl 0930.39020
[77]	D. Popa, “Selections of set-valued maps satisfying functional inclusions on square-symmetric grupoids,” in Functional Equations in Mathematical Analysis, Th. M. Rasssias and J. Brzd\?k, Eds., vol. 52 of Springer Optimization and its Applications, pp. 261-272, Springer, New York, NY, USA, 2012. · Zbl 1248.39042
[78]	J. Rätz, “On approximately additive mappings,” in General Inequalities 2, vol. 47 of International Series of Numerical Mathematics, pp. 233-251, Birkhäuser, Boston, Mass, USA, 1980. · Zbl 0433.39014
[79]	J. Tabor and J. Tabor, “Stability of the Cauchy functional equation in metric groupoids,” Aequationes Mathematicae, vol. 76, no. 1-2, pp. 92-104, 2008. · Zbl 1160.39010
[80]	G.-L. Forti, “Continuous increasing weakly bisymmetric groupoids and quasi-groups in \Bbb R,” Mathematica Pannonica, vol. 8, no. 1, pp. 49-71, 1997. · Zbl 0880.39017
[81]	V. Radu, “The fixed point alternative and the stability of functional equations,” Fixed Point Theory, vol. 4, no. 1, pp. 91-96, 2003. · Zbl 1051.39031
[82]	M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Cauchy’s Equation and Jensen’s Inequality, Birkhäuser, Basel, Switzerland, 2009. · Zbl 1221.39041
[83]	W. Prager and J. Schwaiger, “Multi-affine and multi-Jensen functions and their connection with generalized polynomials,” Aequationes Mathematicae, vol. 69, no. 1-2, pp. 41-57, 2005. · Zbl 1072.39025
[84]	K. Ciepliński, “On multi-Jensen functions and Jensen difference,” Bulletin of the Korean Mathematical Society, vol. 45, no. 4, pp. 729-737, 2008. · Zbl 1172.39033
[85]	K. Ciepliński, “Stability of the multi-Jensen equation,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 249-254, 2010. · Zbl 1211.39017
[86]	K. Ciepliński, “Generalized stability of multi-additive mappings,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1291-1294, 2010. · Zbl 1204.39026
[87]	K. Ciepliński, “Stability of multi-additive mappings in non-Archimedean normed spaces,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 376-383, 2011. · Zbl 1204.39027
[88]	K. Ciepliński, “On the generalized Hyers-Ulam stability of multi-quadratic mappings,” Computers & Mathematics with Applications, vol. 62, no. 9, pp. 3418-3426, 2011. · Zbl 1236.39025
[89]	K. Ciepliński, “Stability of multi-Jensen mappings in non-Archimedean normed spaces,” in Functional Equations in Mathematical Analysis, Th. M. Rasssias and J. Brzd\?k, Eds., vol. 52 of Springer Optimization and its Applications, pp. 79-86, Springer, New York, NY, USA, 2012. · Zbl 1248.39026
[90]	K. Ciepliński, “Stability of multi-additive mappings in \beta -Banach spaces,” Nonlinear Analysis A, vol. 75, no. 11, pp. 4205-4212, 2012. · Zbl 1243.39022
[91]	W. Prager and J. Schwaiger, “Stability of the multi-Jensen equation,” Bulletin of the Korean Mathematical Society, vol. 45, no. 1, pp. 133-142, 2008. · Zbl 1151.39023
[92]	T. Z. Xu, “On the stability of multi-Jensen mappings in \beta -normed spaces,” Applied Mathematics Letters, vol. 25, pp. 1866-1870, 2012. · Zbl 1253.39035
[93]	T. Z. Xu, “Stability of multi-Jensen mappings in non-Archimedean normed spaces,” Journal of Mathematical Physics, vol. 53, no. 2, Article 023507, 9 pages, 2012. · Zbl 1275.47135
[94]	L. G\uavru\cta, “Matkowski contractions and Hyers-Ulam stability,” Buletinul \cStiin\ctific al Universit\ua\ctii Politehnica din Timi\csoara, vol. 53, no. 2, pp. 32-35, 2008. · Zbl 1240.39059
[95]	M. Akkouchi, “Stability of certain functional equations via a fixed point of Ćirić,” Filomat, vol. 25, no. 2, pp. 121-127, 2011. · Zbl 1265.39045
[96]	D. Mihe\ct, “The Hyers-Ulam stability for two functional equations in a single variable,” Banach Journal of Mathematical Analysis, vol. 2, no. 1, pp. 48-52, 2008. · Zbl 1143.39015
[97]	L. C\uadariu, L. G\uavru\cta, and P. G\uavru\cta, “Fixed points and generalized Hyers-Ulam stability,” Abstract and Applied Analysis, vol. 2012, Article ID 712743, 10 pages, 2012. · Zbl 1252.39030
[98]	F. Bojor, “Note on the stability of first order linear differential equations,” Opuscula Mathematica, vol. 32, no. 1, pp. 67-74, 2012. · Zbl 1253.34022
[99]	S.-M. Jung, “A fixed point approach to the stability of differential equations y’=F(x,y),” Bulletin of the Malaysian Mathematical Sciences Society, vol. 33, no. 1, pp. 47-56, 2010. · Zbl 1184.26012
[100]	L. P. Castro and A. Ramos, “Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations,” Banach Journal of Mathematical Analysis, vol. 3, no. 1, pp. 36-43, 2009. · Zbl 1177.45010
[101]	P. G\uavru\cta and L. G\uavru\cta, “A new method for the generalized Hyers-Ulam-Rassias stability,” International Journal of Nonlinear Analysis and Applications, vol. 1, pp. 11-18, 2010.
[102]	S.-M. Jung, “A fixed point approach to the stability of a Volterra integral equation,” Fixed Point Theory and Applications, vol. 2007, Article ID 57064, 9 pages, 2007. · Zbl 1155.45005
[103]	K. Ciepliński, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey,” Annals of Functional Analysis, vol. 3, no. 1, pp. 151-164, 2012. · Zbl 1252.39032
[104]	A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. · Zbl 0920.11087
[105]	L. M. Arriola and W. A. Beyer, “Stability of the Cauchy functional equation over p-adic fields,” Real Analysis Exchange, vol. 31, no. 1, pp. 125-132, 2005/06. · Zbl 1099.39019
[106]	J. Schwaiger, “Functional equations for homogeneous polynomials arising from multilinear mappings and their stability,” Annales Mathematicae Silesianae, no. 8, pp. 157-171, 1994. · Zbl 0820.39011
[107]	M. S. Moslehian and Th. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 325-334, 2007. · Zbl 1257.39019
[108]	K. Baron, A. Simon, and P. Volkmann, “On functions having Cauchy differences in some prescribed sets,” Aequationes Mathematicae, vol. 52, no. 3, pp. 254-259, 1996. · Zbl 0861.39019
[109]	J. A. Baker, “On some mathematical characters,” Glasnik Matemati\vcki III, vol. 25, no. 2, pp. 319-328, 1990. · Zbl 0782.39007
[110]	D. Cenzer, “The stability problem for transformations of the circle,” Proceedings of the Royal Society of Edinburgh, vol. 84, no. 3-4, pp. 279-281, 1979. · Zbl 0439.39004
[111]	D. Cenzer, “The stability problem: new results and counterexamples,” Letters in Mathematical Physics, vol. 10, no. 2-3, pp. 155-160, 1985. · Zbl 0595.39010
[112]	J. Brzd\cek, “On approximately additive functions,” Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 299-307, 2011. · Zbl 1235.39017
[113]	J. Brzd\cek and J. Tabor, “Stability of the Cauchy congruence on restricted domain,” Archiv der Mathematik, vol. 82, no. 6, pp. 546-550, 2004. · Zbl 1059.39018
[114]	E. Manstavi\vcius, “Value concentration of additive functions on random permutations,” Acta Applicandae Mathematicae, vol. 79, pp. 1-8, 2003. · Zbl 1030.60014
[115]	K. Baron, F. Halter-Koch, and P. Volkmann, “On orthogonally exponential functions,” Archiv der Mathematik, vol. 64, no. 5, pp. 410-414, 1995. · Zbl 0821.39006
[116]	K. Baron and G.-L. Forti, “Orthogonality and additivity modulo \Bbb Z,” Results in Mathematics, vol. 26, no. 3-4, pp. 205-210, 1994. · Zbl 0828.39010
[117]	K. Baron and P. Kannappan, “On the Cauchy difference,” Aequationes Mathematicae, vol. 46, no. 1-2, pp. 112-118, 1993. · Zbl 0796.39011
[118]	K. Baron and P. Volkmann, “On the Cauchy equation modulo Z,” Fundamenta Mathematicae, vol. 131, no. 2, pp. 143-148, 1988. · Zbl 0662.39005
[119]	K. Baron and P. Volkmann, “On a theorem of van der Corput,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 61, pp. 189-195, 1991. · Zbl 0753.39002
[120]	J. Brzd\cek, “On functionals which are orthogonally additive modulo \Bbb Z,” Results in Mathematics, vol. 30, no. 1-2, pp. 25-38, 1996. · Zbl 0862.39012
[121]	J. Brzd\cek, “On the Cauchy difference on normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 66, pp. 143-150, 1996. · Zbl 0864.39009
[122]	J. Brzd\cek, “On orthogonally exponential and orthogonally additive mappings,” Proceedings of the American Mathematical Society, vol. 125, no. 7, pp. 2127-2132, 1997. · Zbl 0870.39011
[123]	J. Brzd\cek, “On orthogonally exponential functionals,” Pacific Journal of Mathematics, vol. 181, no. 2, pp. 247-267, 1997. · Zbl 1010.39011
[124]	J. Brzd\cek, “On measurable orthogonally exponential functions,” Archiv der Mathematik, vol. 72, no. 3, pp. 185-191, 1999. · Zbl 0924.39018
[125]	J. Brzd\cek, “On the isosceles orthogonally exponential mappings,” Acta Mathematica Hungarica, vol. 87, no. 1-2, pp. 147-152, 2000. · Zbl 0963.46015
[126]	J. Brzd\cek, “The Cauchy and Jensen differences on semigroups,” Publicationes Mathematicae Debrecen, vol. 48, no. 1-2, pp. 117-136, 1996. · Zbl 0862.39011
[127]	T. Kochanek and W. Wyrobek, “Measurable orthogonally additive functions modulo a discrete subgroup,” Acta Mathematica Hungarica, vol. 123, no. 3, pp. 239-248, 2009. · Zbl 1212.39038
[128]	W. Wyrobek, “Orthogonally additive functions modulo a discrete subgroup,” Aequationes Mathematicae, vol. 78, no. 1-2, pp. 63-69, 2009. · Zbl 1212.39038
[129]	J. Brzd\cek, “Subgroups of the Clifford group,” Aequationes Mathematicae, vol. 41, no. 2-3, pp. 123-135, 1991. · Zbl 0737.20021
[130]	P. Fischer and Z. Słodkowski, “Christensen zero sets and measurable convex functions,” Proceedings of the American Mathematical Society, vol. 79, no. 3, pp. 449-453, 1980. · Zbl 0444.46010
[131]	J. P. R. Christensen, “On sets of Haar measure zero in abelian Polish groups,” Israel Journal of Mathematics, vol. 13, pp. 255-260, 1972. · Zbl 0249.43002
[132]	J. P. R. Christensen, “Borel structures in groups and semigroups,” Mathematica Scandinavica, vol. 28, pp. 124-128, 1971. · Zbl 0217.08502
[133]	Z. Kominek and M. Kuczma, “Theorems of Bernstein-Doetsch, Piccard and Mehdi and semilinear topology,” Archiv der Mathematik, vol. 52, no. 6, pp. 595-602, 1989. · Zbl 0683.46006
[134]	D. G. Bourgin, “Approximately isometric and multiplicative transformations on continuous function rings,” Duke Mathematical Journal, vol. 16, pp. 385-397, 1949. · Zbl 0033.37702
[135]	G. Maksa and Z. Páles, “Hyperstability of a class of linear functional equations,” Acta Mathematica, vol. 17, no. 2, pp. 107-112, 2001. · Zbl 1004.39022
[136]	Gy. Szabó, “A conditional Cauchy equation on normed spaces,” Publicationes Mathematicae Debrecen, vol. 42, no. 3-4, pp. 265-271, 1993. · Zbl 0807.39010
[137]	C. Borelli Forti, “Solutions of a nonhomogeneous Cauchy equation,” Radovi Matemati\vcki, vol. 5, no. 2, pp. 213-222, 1989. · Zbl 0697.39009
[138]	J. Brzd\cek, “On a generalization of the Cauchy functional equation,” Aequationes Mathematicae, vol. 46, no. 1-2, pp. 56-75, 1993. · Zbl 0788.39004
[139]	T. M. K. Davison and B. Ebanks, “Cocycles on cancellative semigroups,” Publicationes Mathematicae Debrecen, vol. 46, no. 1-2, pp. 137-147, 1995. · Zbl 0860.39037
[140]	B. Ebanks, “Generalized Cauchy difference functional equations,” Aequationes Mathematicae, vol. 70, no. 1-2, pp. 154-176, 2005. · Zbl 1079.39017
[141]	B. Ebanks, “Generalized Cauchy difference equations. II,” Proceedings of the American Mathematical Society, vol. 136, no. 11, pp. 3911-3919, 2008. · Zbl 1206.39022
[142]	B. Ebanks, P. Kannappan, and P. K. Sahoo, “Cauchy differences that depend on the product of arguments,” Glasnik Matemati\vcki III, vol. 27, no. 47, pp. 251-261, 1992. · Zbl 0780.39007
[143]	B. Ebanks, P. K. Sahoo, and W. Sander, Characterizations of Information Measures, World Scientific Publishing, River Edge, NJ, USA, 1998. · Zbl 0923.94002
[144]	B. Ebanks and H. Stetkær, “Continuous cocycles on locally compact groups,” Aequationes Mathematicae, vol. 78, no. 1-2, pp. 123-145, 2009. · Zbl 1207.39033
[145]	B. Ebanks and H. Stetkær, “Continuous cocycles on locally compact groups: II,” Aequationes Mathematicae, vol. 80, no. 1-2, pp. 57-83, 2010. · Zbl 1216.39031
[146]	J. Erd\Hos, “A remark on the paper “On some functional equations” by S. Kurepa,” Glasnik Matematicki, vol. 14, pp. 3-5, 1959.
[147]	I. Feny\Ho and G.-L. Forti, “On the inhomogeneous Cauchy functional equation,” Stochastica, vol. 5, no. 2, pp. 71-77, 1981. · Zbl 0492.39002
[148]	A. Járai, G. Maksa, and Z. Páles, “On Cauchy-differences that are also quasisums,” Publicationes Mathematicae Debrecen, vol. 65, no. 3-4, pp. 381-398, 2004. · Zbl 1071.39026
[149]	B. Jessen, J. Karpf, and A. Thorup, “Some functional equations in groups and rings,” Mathematica Scandinavica, vol. 22, pp. 257-265, 1968. · Zbl 0183.04004
[150]	J. Brzd\?k, “Stability of the equation of the p-Wright affine functions,” Aequationes Mathematicae. In press. · Zbl 1272.39015
[151]	E. Gselmann, “Hyperstability of a functional equation,” Acta Mathematica Hungarica, vol. 124, no. 1-2, pp. 179-188, 2009. · Zbl 1212.39044
[152]	Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of the generalized polynomial function of degree 2,” Journal of Chungcheong Mathematical Society, vol. 22, pp. 201-209, 2009.
[153]	G. Maksa, “The stability of the entropy of degree alpha,” Journal of Mathematical Analysis and Applications, vol. 346, no. 1, pp. 17-21, 2008. · Zbl 1149.39024
[154]	E. Jabłońska, “On solutions of some generalizations of the Goł\cab-Schinzel equation,” in Functional Equations in Mathematical Analysis, Th. M. Rasssias and J. Brzd\?k, Eds., vol. 52 of Springer Optimization and its Applications, pp. 509-521, Springer, New York, NY, USA, 2012. · Zbl 1248.39018
[155]	J. Chudziak, “On a functional inequality related to the stability problem for the Goł\cab-Schinzel equation,” Publicationes Mathematicae Debrecen, vol. 67, no. 1-2, pp. 199-208, 2005. · Zbl 1084.39023
[156]	J. Chudziak and J. Tabor, “On the stability of the Goł\cab-Schinzel functional equation,” Journal of Mathematical Analysis and Applications, vol. 302, no. 1, pp. 196-200, 2005. · Zbl 1064.39027
[157]	J. Chudziak, “Stability of the generalized Goł\cab-Schinzel equation,” Acta Mathematica Hungarica, vol. 113, no. 1-2, pp. 133-144, 2006. · Zbl 1121.39022
[158]	J. Chudziak, “Approximate solutions of the generalized Goł\cab-Schinzel equation,” Journal of Inequalities and Applications, vol. 2006, Article ID 89402, 8 pages, 2006. · Zbl 1134.39022
[159]	R. Ger and P. \vSemrl, “The stability of the exponential equation,” Proceedings of the American Mathematical Society, vol. 124, no. 3, pp. 779-787, 1996. · Zbl 0846.39013
[160]	J. Chudziak, “Approximate solutions of the Goł\cab-Schinzel equation,” Journal of Approximation Theory, vol. 136, no. 1, pp. 21-25, 2005. · Zbl 1083.39025
[161]	J. Chudziak, “Stability problem for the Goł\cab-Schinzel type functional equations,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 454-460, 2008. · Zbl 1132.39021
[162]	J. Brzd\cek, “On the quotient stability of a family of functional equations,” Nonlinear Analysis A, vol. 71, no. 10, pp. 4396-4404, 2009. · Zbl 1172.39032
[163]	N. Brillouët-Belluot, J. Brzd\cek, and J. Chudziak, “On solutions of Aczél’s equation and some related equations,” Aequationes Mathematicae, vol. 80, no. 1-2, pp. 27-44, 2010. · Zbl 1215.39023
[164]	J. Brzd\cek, “On stability of a family of functional equations,” Acta Mathematica Hungarica, vol. 128, no. 1-2, pp. 139-149, 2010. · Zbl 1274.39052
[165]	A. Charifi, B. Bouikhalene, S. Kabbaj, and J. M. Rassias, “On the stability of a Pexiderized Goł\cab-Schinzel equation,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3193-3202, 2010. · Zbl 1193.39016
[166]	W. Fechner, “Stability of a composite functional equation related to idempotent mappings,” Journal of Approximation Theory, vol. 163, no. 3, pp. 328-335, 2011. · Zbl 1211.39019
[167]	A. Najdecki, “On stability of a functional equation connected with the Reynolds operator,” Journal of Inequalities and Applications, vol. 2007, Article ID 79816, 3 pages, 2007. · Zbl 1133.39031
[168]	A. Mach, Algebraic Constructions of Solutions for the Translation Equation, vol. 11 of Lecture Notes in Nonlinear Analysis, Juliusz Schauder Center for Nonlinear Studies, Toruń, Poland, 2010. · Zbl 1221.39033
[169]	Z. Moszner, “The translation equation and its application,” Demonstratio Mathematica, vol. 6, pp. 309-327, 1973. · Zbl 0282.39009
[170]	Z. Moszner, “General theory of the translation equation,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 17-37, 1995. · Zbl 0876.39007
[171]	G. Targoński, Topics in Iteration Theory, vol. 6, Vandenhoeck & Ruprecht, Göttingen, Germany, 1981.
[172]	J. Chudziak, “Approximate dynamical systems on interval,” Applied Mathematics Letters, vol. 25, no. 3, pp. 532-537, 2012. · Zbl 1244.39019
[173]	W. Jabłoński and L. Reich, “Stability of the translation equation in rings of formal power series and partial extensibility of one-parameter groups of truncated formal power series,” Österreichische Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche Klasse, vol. 215, pp. 127-137, 2006. · Zbl 1159.39316
[174]	A. Mach, “On some functional equations involving Babbage equation,” Results in Mathematics, vol. 51, no. 1-2, pp. 97-106, 2007. · Zbl 1144.39022
[175]	A. Mach and Z. Moszner, “On stability of the translation equation in some classes of functions,” Aequationes Mathematicae, vol. 72, no. 1-2, pp. 191-197, 2006. · Zbl 1104.39024
[176]	B. Przebieracz, “On the stability of the translation equation,” Publicationes Mathematicae Debrecen, vol. 75, no. 1-2, pp. 285-298, 2009. · Zbl 1212.26009
[177]	B. Przebieracz, “On the stability of the translation equation and dynamical systems,” Nonlinear Analysis A, vol. 75, no. 4, pp. 1980-1988, 2012. · Zbl 1242.39044
[178]	J. Brzd\cek, “On the Baxter functional equation,” Aequationes Mathematicae, vol. 52, no. 1-2, pp. 105-111, 1996. · Zbl 0857.39011
[179]	N. Brillouët-Belluot and B. Ebanks, “Localizable composable measures of fuzziness. II,” Aequationes Mathematicae, vol. 60, no. 3, pp. 233-242, 2000. · Zbl 0974.39016
[180]	J. Brzd\cek, D. Popa, and B. Xu, “Selections of set-valued maps satisfying a linear inclusion in a single variable,” Nonlinear Analysis A, vol. 74, no. 1, pp. 324-330, 2011. · Zbl 1205.39025
[181]	K. Nikodem and D. Popa, “On selections of general linear inclusions,” Publicationes Mathematicae Debrecen, vol. 75, no. 1-2, pp. 239-249, 2009. · Zbl 1212.39041
[182]	M. Piszczek, “The properties of functional inclusions and Hyers-Ulam stability,” Aequationes Mathematicae. In press. · Zbl 1271.39031
[183]	M. Piszczek, “On selections of set-valued inclusions in a single variable with applications to several variables,” Results in Mathematics. In press. · Zbl 1277.39032
[184]	D. Popa, “A stability result for a general linear inclusion,” Nonlinear Functional Analysis and Applications, vol. 9, no. 3, pp. 405-414, 2004. · Zbl 1067.39043
[185]	D. Popa, “Functional inclusions on square-symmetric groupoids and Hyers-Ulam stability,” Mathematical Inequalities & Applications, vol. 7, no. 3, pp. 419-428, 2004. · Zbl 1058.39026
[186]	D. Popa, “A property of a functional inclusion connected with Hyers-Ulam stability,” Journal of Mathematical Inequalities, vol. 3, no. 4, pp. 591-598, 2009. · Zbl 1189.39032
[187]	B. Paneah, “A new approach to the stability of linear functional operators,” Aequationes Mathematicae, vol. 78, no. 1-2, pp. 45-61, 2009. · Zbl 1207.39046
[188]	J. Tabor and J. Tabor, “General stability of functional equations of linear type,” Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 192-200, 2007. · Zbl 1111.39027
[189]	I. G. Cho and H. J. Koh, “Several stability problems of a quadratic functional equation,” Communications of the Korean Mathematical Society, vol. 26, no. 1, pp. 99-113, 2011. · Zbl 1214.39012
[190]	H.-Y. Chu, G. Han, and D. S. Kang, “On stability problems with shadowing property and its application,” Bulletin of the Korean Mathematical Society, vol. 48, no. 4, pp. 637-688, 2011. · Zbl 1241.39011
[191]	S.-H. Lee, H. Koh, and S.-H. Ku, “Investigation of the stability via shadowing property,” Journal of Inequalities and Applications, vol. 2009, Article ID 156167, 12 pages, 2009. · Zbl 1185.39022
[192]	J. Tabor and J. Tabor, “Restricted stability and shadowing,” Publicationes Mathematicae Debrecen, vol. 73, no. 1-2, pp. 49-58, 2008. · Zbl 1199.39069
[193]	B. Batko, “Stability of Dhombres’ equation,” Bulletin of the Australian Mathematical Society, vol. 70, no. 3, pp. 499-505, 2004. · Zbl 1066.39027
[194]	B. Batko, “On the stability of Mikusiński’s equation,” Publicationes Mathematicae Debrecen, vol. 66, no. 1-2, pp. 17-24, 2005. · Zbl 1119.39304
[195]	B. Batko, “On the stability of an alternative functional equation,” Mathematical Inequalities & Applications, vol. 8, no. 4, pp. 685-691, 2005. · Zbl 1096.39026
[196]	B. Batko, “Stability of an alternative functional equation,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 303-311, 2008. · Zbl 1133.39025
[197]	B. Batko, “Note on superstability of Mikusiński’s functional equation,” in Functional Equations in Mathematical Analysis, Th. M. Rasssias and J. Brzd\?k, Eds., vol. 52 of Springer Optimization and its Applications, pp. 15-17, Springer, New York, NY, USA, 2012. · Zbl 1248.39021
[198]	T. Kochanek and M. Lewicki, “Stability problem for number-theoretically multiplicative functions,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2591-2597, 2007. · Zbl 1117.39018
[199]	T. Kochanek, “Stability aspects of arithmetic functions,” Acta Arithmetica, vol. 132, no. 1, pp. 87-98, 2008. · Zbl 1155.11049
[200]	T. Kochanek, “Corrigendum to “Stability aspects of arithmetic functions II” (Acta Arith.139 (2009), 131-146),” Acta Arithmetica, vol. 149, pp. 83-98, 2011. · Zbl 1222.11101
[201]	Z. Boros and E. Gselmann, “Hyers-Ulam stability of derivations and linear functions,” Aequationes Mathematicae, vol. 80, no. 1-2, pp. 13-25, 2010. · Zbl 1215.39033
[202]	A. Fo\vsner, “On the generalized Hyers-Ulam stability of module left (m,n)-derivations,” Aequationes Mathematicae, vol. 84, no. 1-2, pp. 91-98, 2012. · Zbl 1267.39022
[203]	D. S. Cimpean and D. Popa, “Hyers-Ulam stability of Euler’s equation,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1539-1543, 2011. · Zbl 1225.35051
[204]	D. S. Cîmpean and D. Popa, “On the stability of the linear differential equation of higher order with constant coefficients,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 4141-4146, 2010. · Zbl 1211.34065
[205]	S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order,” Applied Mathematics Letters, vol. 17, no. 10, pp. 1135-1140, 2004. · Zbl 1061.34039
[206]	S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. III,” Journal of Mathematical Analysis and Applications, vol. 311, no. 1, pp. 139-146, 2005. · Zbl 1087.34534
[207]	S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. II,” Applied Mathematics Letters, vol. 19, no. 9, pp. 854-858, 2006. · Zbl 1125.34328
[208]	S.-M. Jung, “Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 549-561, 2006. · Zbl 1106.34032
[209]	S.-M. Jung, “Hyers-Ulam stability of linear partial differential equations of first order,” Applied Mathematics Letters, vol. 22, no. 1, pp. 70-74, 2009. · Zbl 1163.39308
[210]	S.-M. Jung and J. Brzd\cek, “Hyers-Ulam stability of the delay equation y’(t)=\lambda y(t-\tau ),” Abstract and Applied Analysis, vol. 2010, Article ID 372176, 10 pages, 2010. · Zbl 1210.34108
[211]	S.-M. Jung and K.-S. Lee, “Hyers-Ulam stability of first order linear partial differential equations with constant coefficients,” Mathematical Inequalities & Applications, vol. 10, no. 2, pp. 261-266, 2007. · Zbl 1120.26011
[212]	S.-M. Jung and S. Min, “On approximate Euler differential equations,” Abstract and Applied Analysis, vol. 2009, Article ID 537963, 8 pages, 2009. · Zbl 1179.35241
[213]	S.-M. Jung and Th. M. Rassias, “Ulam’s problem for approximate homomorphisms in connection with Bernoulli’s differential equation,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 223-227, 2007. · Zbl 1118.39014
[214]	S.-M. Jung and Th. M. Rassias, “Generalized Hyers-Ulam stability of Riccati differential equation,” Mathematical Inequalities & Applications, vol. 11, no. 4, pp. 777-782, 2008. · Zbl 1200.39010
[215]	Y. Li and Y. Shen, “Hyers-Ulam stability of nonhomogeneous linear differential equations of second order,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 576852, 7 pages, 2009. · Zbl 1181.34014
[216]	N. Lungu and D. Popa, “Hyers-Ulam stability of a first order partial differential equation,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 86-91, 2012. · Zbl 1236.39030
[217]	T. Miura and G. Hirasawa, “The Hyers-Ulam and Ger type stabilities of the first order linear differential equations,” in Functional Equations in Mathematical Analysis, Th. M. Rasssias and J. Brzd\?k, Eds., vol. 52 of Springer Optimization and its Applications, pp. 191-200, Springer, New York, NY, USA, 2012. · Zbl 1250.34003
[218]	T. Miura, S.-M. Jung, and S.-E. Takahasi, “Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations y’=\lambda y,” Journal of the Korean Mathematical Society, vol. 41, no. 6, pp. 995-1005, 2004. · Zbl 1069.34079
[219]	T. Miura, S. Miyajima, and S.-E. Takahasi, “Hyers-Ulam stability of linear differential operator with constant coefficients,” Mathematische Nachrichten, vol. 258, pp. 90-96, 2003. · Zbl 1039.34054
[220]	T. Miura, S. Miyajima, and S.-E. Takahasi, “A characterization of Hyers-Ulam stability of first order linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 136-146, 2003. · Zbl 1045.47037
[221]	D. Popa and I. Ra\csa, “On the Hyers-Ulam stability of the linear differential equation,” Journal of Mathematical Analysis and Applications, vol. 381, no. 2, pp. 530-537, 2011. · Zbl 1222.34069
[222]	D. Popa and I. Ra\csa, “The Fréchet functional equation with application to the stability of certain operators,” Journal of Approximation Theory, vol. 164, no. 1, pp. 138-144, 2012. · Zbl 1238.39008
[223]	D. Popa and I. Ra\csa, “Hyers-Ulam stability of the linear differential operator with nonconstant coefficients,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1562-1568, 2012. · Zbl 1368.34075
[224]	S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation y’=\lambda y,” Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309-315, 2002. · Zbl 1011.34046
[225]	G. Wang, M. Zhou, and L. Sun, “Hyers-Ulam stability of linear differential equations of first order,” Applied Mathematics Letters, vol. 21, no. 10, pp. 1024-1028, 2008. · Zbl 1159.34041
[226]	T. Miura, G. Hirasawa, S.-E. Takahasi, and T. Hayata, “A note on the stability of an integral equation,” in Functional Equations in Mathematical Analysis, Th. M. Rasssias and J. Brzd\?k, Eds., vol. 52 of Springer Optimization and its Applications, pp. 207-222, Springer, New York, NY, USA, 2012. · Zbl 1248.45008