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Stability of a pexiderial functional equation in random normed spaces. (English) Zbl 1231.39010

Summary: The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias stability theorem [Proc. Am. Math. Soc. 72, 297–300 (1978; Zbl 0398.47040)]. Recently, the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation \(f(x+y)+f(x-y)=2f(x)+2f(y)\) proved in the earlier work. In this paper, using direct method we prove the generalized Hyers-Ulam stability of the following Pexiderial functional equation \(f(x+y)+f(x-y)=2g(x)+2g(y)\) in random normed space.

MSC:

39B22 Functional equations for real functions
39B52 Functional equations for functions with more general domains and/or ranges

Citations:

Zbl 0398.47040
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References:

[1] Azadi Kenary, H., Cho, Y.J.: Stability of mixed additive-quadratic Jensen type functional equation in various spaces. Comput. Math. Appl. (2011). doi: 10.1016/j.camwa.2011.03.024 · Zbl 1235.39024
[2] Cholewa, P.W.: Remarks on the stability of functional equations. Aequ. Math. 27(1–2), 76–86 (1984) · Zbl 0549.39006 · doi:10.1007/BF02192660
[3] Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 62, 59–64 (1992) · Zbl 0779.39003 · doi:10.1007/BF02941618
[4] Gavruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184(3), 431–436 (1994) · Zbl 0818.46043 · doi:10.1006/jmaa.1994.1211
[5] Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941) · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[6] Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of functional equations in several variables. In: Progress in Nonlinear Differential Equations and their Applications, vol. 34. Birkhäuser, Basel (1998) · Zbl 0907.39025
[7] Hyers, D.H., Isac, G., Rassias, Th.M.: On the asymptoticity aspect of Hyers-Ulam stability of mappings. Proc. Am. Math. Soc. 126(2), 425–430 (1998) · Zbl 0894.39012 · doi:10.1090/S0002-9939-98-04060-X
[8] Hyers, D.H., Rassias, Th.M.: Approximate homomorphisms. Aequ. Math. 44(2–3), 125–153 (1992) · Zbl 0806.47056 · doi:10.1007/BF01830975
[9] Jordan, P., von Neumann, J.: On inner products in linear metric spaces. Ann. Math. 36(3), 719–723 (1935) · JFM 61.0435.05 · doi:10.2307/1968653
[10] Jung, S.-M.: On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl. 222(1), 126–137 (1998) · Zbl 0928.39013 · doi:10.1006/jmaa.1998.5916
[11] Jung, S.-M.: On the Hyers-Ulam-Rassias stability of a quadratic functional equation. J. Math. Anal. Appl. 232(2), 384–393 (1999) · Zbl 0926.39013 · doi:10.1006/jmaa.1999.6282
[12] Jung, S.-M.: Stability of the quadratic equation of Pexider type. Abh. Math. Semin. Univ. Hamb. 70, 175–190 (2000) · Zbl 0991.39018 · doi:10.1007/BF02940912
[13] Kannappan, P.: Quadratic functional equation and inner product spaces. Results Math. 27(3–4), 368–372 (1995) · Zbl 0836.39006 · doi:10.1007/BF03322841
[14] Najati, A., Park, C.: On the stability of a cubic functional equation. Acta Math. Sin. Engl. Ser. (2011, to appear) · Zbl 1159.39014
[15] Park, C.: Universal Jensen’s equations in Banach modules over a C algebra and its unitary group. Acta Math. Sin. Engl. Ser. 20(6), 1047–1056 (2004) · Zbl 1087.39514 · doi:10.1007/s10114-004-0409-0
[16] Park, C., Hou, J., Oh, S.: Homomorphisms between JC algebras and Lie C algebras. Acta Math. Sin. Engl. Ser. 21(6), 1391–1398 (2005) · Zbl 1121.39030 · doi:10.1007/s10114-005-0629-y
[17] Park, C., Rassias, Th.M.: The N-isometric isomorphisms in linear N-normed C algebras. Acta Math. Sin. Engl. Ser. 22(6), 1863–1890 (2006) · Zbl 1125.39028 · doi:10.1007/s10114-005-0878-9
[18] Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72(2), 297–300 (1978) · Zbl 0398.47040 · doi:10.1090/S0002-9939-1978-0507327-1
[19] Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251(1), 264–284 (2000) · Zbl 0964.39026 · doi:10.1006/jmaa.2000.7046
[20] Saadati, R., Vaezpour, M., Cho, Y.J.: A note to paper ”On the stability of cubic mappings and quartic mappings in random normed spaces”. J. Inequal. Appl. 2009, 214530 (2009) · Zbl 1176.39024 · doi:10.1155/2009/214530
[21] Schewizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland, New York (1983)
[22] Skof, F.: Local properties and approximation of operators. Rend. Semin. Mat. Fis. Milano 53, 113–129 (1983) · Zbl 0599.39007 · doi:10.1007/BF02924890
[23] Ulam, S.M.: A collection of mathematical problems. Intersci. Tracts Pure Appl. Math. (8) (1960) · Zbl 0086.24101
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