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On the new Hyers-Ulam-Rassias stability of the generalized cubic set-valued mapping in the incomplete normed spaces. (English) Zbl 1482.39042

Summary: We present a novel generalization of the Hyers-Ulam-Rassias stability definition to study a generalized cubic set-valued mapping in normed spaces. In order to achieve our goals, we have applied a brand new fixed point alternative. Meanwhile, we have obtained a practicable example demonstrating the stability of a cubic mapping that is not defined as stable according to the previously applied methods and procedures.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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[1] K.J. Arrow, G.A.C. Debereu,Existence of an equilibrium for a competitive economy, Econometrica,22:265-290, 1954,https://doi.org/10.2307/1907353. · Zbl 0055.38007
[2] R.J. Aumann, Integrals of set-valued functions,J. Math. Anal. Appl.,12:1-12, 1965,https: //doi.org/10.1016/0022-247X(65)90049-1. · Zbl 0163.06301
[3] H. Baghani, M. Eshaghi, M. Ramezani, Orthogonal sets: The axiom of choice and proof of a fixed point theorem,J. Fixed Point Theory Appl.,18:465-477, 2016,https://doi.org/ 10.1007/s11784-016-0297-9. · Zbl 1454.54029
[4] H. Baghani, M. Ramezani, A fixed point theorem for a new class of set-valued mappings in R-complete (not necessarily complete) metric spaces,Filomat,31:3875-3884, 2017,https: //doi.org/10.2298/FIL1712875B. · Zbl 1478.54043
[5] H. Baghani, M. Ramezani,Coincidence and fixed points for multivalued mappings in incomplete metric spaces with application,Filomat,33:13-26, 2019,https://doi.org/ 10.2298/FIL1901013B. · Zbl 1499.54160
[6] J. Brzde¸k, K. Ciepli´nski, A fixed point theorem inn-Banach spaces and Ulam stability,J. Math. Anal. Appl.,470(1):632-646, 2019,https://doi.org/10.1016/j.jmaa.2018.10. 028. · Zbl 1441.39025
[7] H.-Y. Chu, D.S. Kang, On the stability of ann-dimensional cubic functional equation,J. Math. Anal. Appl.,325:595-607, 2007,https://doi.org/10.1016/j.jmaa.2006.02. 003. · Zbl 1106.39025
[8] H.-Y. Chu, A. Kim, S.K. Yoo, On the stability of the generalized cubic set-valued functional equation,Appl. Math. Lett.,37:7-14, 2014,https://doi.org/10.1016/j.aml. 2014.05.008. · Zbl 1314.39031
[9] G. Debreu, Integration of correspondences, inProceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part I, Univ. California Press, Oakland, CA, 1966, pp. 351-372.
[10] M. Eshaghi, M. Ramezani, D. la Sen, Y.J. Cho, On orthogonal sets and Banach fixed point theorem,Fixed Point Theory,18:569-578, 2017,https://doi.org/10.24193/fptro.2017.2.45. · Zbl 1443.47051
[11] G.L. Forti, Hyers-Ulam stability of functional equations in several variables,Aequationes Math.,50:143-190, 1995,https://doi.org/10.1007/BF01831117. · Zbl 0836.39007
[12] R. Fukutaka, M. Onitsuka, Best constant in Hyers-Ulam stability of first-order homogeneous linear differential equations with a periodic coefficient,J. Math. Anal. Appl.,473(2):1432- 1446, 2019,https://doi.org/10.1016/j.jmaa.2019.01.030. · Zbl 1447.34019
[13] Z. Gajda, On isometric mappings,Int. J. Math. Math. Sci., to appear. · Zbl 0739.39013
[14] P. Gávruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,J. Math. Anal. Appl.,184:431-436, 1994,https://doi.org/1999.6546. · Zbl 0818.46043
[15] M. Eshaghi Gordji, H. Habibi, Existence and uniqueness of solutions to a first-order differential equation via fixed point theorem in orthogonal metric space,Facta Univ., Ser. Math. Inf.,34(1): 123-135, 2019,https://doi.org/10.22190/FUMI1901123G. · Zbl 1488.54134
[16] M. Eshaghi Gordji, H. Habibi, Fixed point theory in-connected orthogonal metric space, Sahand Commun. Math. Anal.,16:35-46, 2019,https://doi.org/10.22130/scma. 2018.72368.289. · Zbl 1449.54059
[17] M. Eshaghi Gordji, H. Habibi, M.B. Sahabi,Orthogonal sets; orthogonal contractions,Asian-Eur. J. Math.,12(3):1950034, 2019,https://doi.org/10.1142/ S1793557119500347. · Zbl 1489.54115
[18] C. Hess, Set-valued integration and set-valued probability theory: An overview, in E. Pap (Ed.),Handbook of Measure Theory, North-Holland, Amsterdam, 2002,https://doi. org/10.1016/B978-044450263-6/50015-4. · Zbl 1022.60011
[19] D.H. Hyers, On the stability of the linear functional equation,Proc. Natl. Acad. Sci. USA, 27:222-224, 1941,https://doi.org/10.1073/pnas.27.4.222. · JFM 67.0424.01
[20] G. Isac, T.M. Rassias,Stability ofψ-additive mappings: Applications to nonlinear analysis,Int. J. Math. Math. Sci.,19:219-228, 1996,https://doi.org/10.1155/ S0161171296000324. · Zbl 0843.47036
[21] S.Y. Jang, C. Park, Y. Cho, Hyers-Ulam stability of a generalized additive set-valued functional equation,J. Inequal. Appl.,101, 2013,https://doi.org/10.1186/1029-242X2013-101. · Zbl 1282.39030
[22] K.-W. Jun, H.-M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,J. Math. Anal. Appl.,274:867-878, 2002,https://doi.org/10.1016/ S0022-247X(02)00415-8. · Zbl 1021.39014
[23] S.-M. Jung,Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011,https://doi.org/10.1007/978-1-4419-9637-4. · Zbl 1221.39038
[24] S.-M. Jung, D. Popa, Th.M. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups,J. Glob. Optim.,59:165-171, 2014,https: //doi.org/10.1007/s10898-013-0083-9. · Zbl 1295.33004
[25] Y.-S. Jung, I.-S. Chang, The stability of a cubic type functional equation with the fixed point alternative,J. Math. Anal. Appl.,306:752-760, 2005,https://doi.org/10.1016/j. jmaa.2004.10.017. · Zbl 1077.39026
[26] H.A. Kenary, H. Rezaei, Y. Gheisari, C. Park, On the stability of set-valued functional equations with the fixed point alternative,Fixed Point Theory Appl.,81, 2012,https://doi.org/ 10.1186/1687-1812-2012-81. · Zbl 1283.39010
[27] S. Khalehoghli, H. Rahimi, M. Eshaghi Gordji,R-topological spaces and SR-topological spaces with their applications,Math. Sci.,14:249-255, 2020,https://doi.org/10. 1007/s40096-020-00338-5. · Zbl 1475.54002
[28] G. Lu, C. Park, Hyers-Ulam stability of additive set-valued functional euqtions,Appl. Math. Lett.,24:1312-1316, 2011,https://doi.org/10.1016/j.aml.2011.02.024. · Zbl 1220.39030
[29] L.W. McKenzie,On the existence of general equilibrium for a competitive market, Econometrica,27:54-71, 1959,https://doi.org/10.2307/1907777. · Zbl 0095.34302
[30] K. Nikodem, On quadratic set-valued functions,Publ. Math.,30:297-301, 1983,https: //doi.org/10.1007/BF02591511. · Zbl 0537.39002
[31] K. Nikodem, On Jensen’s functional equation for set-valued functions,Rad. Mat.,3:23-33, 1987,https://doi.org/10.1007/s00025-017-0679-3. · Zbl 0628.39013
[32] K. Nikodem, Set-valued solutions of the Pexider functional equation,Funkc. Ekvacioj, Ser. Int.,31(2):227-231, 1988,https://doi.org/10.1007/s00025-017-0679-3. · Zbl 0698.39007
[33] H.K. Pathak, N. Shahzad, A generalization of Nadler’s fixed point theorem and its application to nonconvex integral inclusions,Topol. Methods Nonlinear Anal.,41:207-227, 2013. · Zbl 1326.47055
[34] M. Ramezani, Orthogonal metric space and convex contractions,Int. J. Nonlinear Anal. Appl., 6:127-132, 2015,https://doi.org/10.22075/IJNAA.2015.261. · Zbl 1321.54096
[35] M. Ramezani, H. Baghani, The Meir-Keeler fixed point theorem in incomplete modular spaces with application,J. Fixed Point Theory Appl.,19:2369-2382, 2017,https://doi.org/ 10.1007/s11784-017-0440-2. · Zbl 1493.47071
[36] T.M. Rassias, On the stability of the linear mapping in Banach spaces,Proc. Am. Math. Soc.,27(2):297-300, 1978,https://doi.org/10.1090/S0002-9939-19780507327-1. · Zbl 0398.47040
[37] T.M. Rassias, P. Šemrl,On the behavior of mappings which do not satisfy Hyers-Ulam stability,Proc. Am. Math. Soc.,114(4):989-993, 1992,https://doi.org/10.2307/ 2159617. · Zbl 0761.47004
[38] S.M. Ulam,Problems in Modern Mathematics, Wiley, New York, 1940,https://doi. org/10.4236/ojpp.2012.21010 · Zbl 0137.24201
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