Găvruţa, P. On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings. (English) Zbl 0993.47002 J. Math. Anal. Appl. 261, No. 2, 543-553 (2001). Let \(\psi: \mathbb{R}_+\to \mathbb{R}_+\) be a mapping. Let \(E_1\), \(E_2\) be normed spaces. A mapping \(f: E_1\to E_2\) is said to be \(\psi\)-additive if there is a \(\theta> 0\) such that \[ \|f(x+ y)- f(x)- f(y)\|\leq\theta(\psi\|x\|+\psi\|y\|) \] for all \(x,y\in E_1\). There are given: an answer to a problem of G. Isac and Th. M. Rassias concerning Hyers-Ulam-Rassias stability of linear mappings and a new characterization of \(\psi\)-additive mappings. Reviewer: D.Przeworska-Rolewicz (Warszawa) Cited in 1 ReviewCited in 24 Documents MSC: 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A30 Norms (inequalities, more than one norm, etc.) of linear operators Keywords:Hyers-Ulam-Rassias stability; characterization of \(\psi\)-additive mappings PDFBibTeX XMLCite \textit{P. Găvruţa}, J. Math. Anal. Appl. 261, No. 2, 543--553 (2001; Zbl 0993.47002) Full Text: DOI Link References: [1] Badea, C., On the Hyers-Ulam stability of mappings: The direct method, (Rassias, Th. M.; Tabor, J., Stability of Mappings of Hyers-Ulam Type (1994), Hadronic Press: Hadronic Press Palm Harbor), 7-13 · Zbl 0845.39011 [2] Gajda, Z., On stability of additive mappings, Internat. J. Math. Math. Sci., 14, 431-434 (1991) · Zbl 0739.39013 [3] Ger, R., The singular case in the stability behaviour of linear mappings, Selected Topics in Functional Equations, Proc. Austrian—Publish. Sem., Graz, 1991. Selected Topics in Functional Equations, Proc. Austrian—Publish. Sem., Graz, 1991, Grazer Math. Ber., 316 (1992), Karl-Franzens-Univ. Graz: Karl-Franzens-Univ. Graz Graz, p. 59-70 · Zbl 0796.39012 [4] Găvruţa, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184, 431-436 (1994) · Zbl 0818.46043 [5] Găvruţa, P., On the Hyers-Ulam-Rassias stability of mappings, (Milovanovic, G. V., Recent Progress in Inequalities (1998), Kluwer Academic: Kluwer Academic Dordrecht), 465-469 · Zbl 0897.39008 [6] Găvruţa, P.; Hossu, M.; Popescu, D.; Căprău, C., On the stability of mappings and an answer to a problem of Th. M. Rassias, Ann. Math. Blaise Pascal, 2, 55-60 (1995) · Zbl 0853.46036 [7] Găvruţa, P.; Cădariu, L., Inequalities for approximately additive mappings, Bul. Ştiint. Univ. Politehn. Timişoara, 43, 24-30 (1998) · Zbl 0974.39019 [8] Gilanyi, A., On the stability of monomial functional equations, Publ. Math. Debrecen, 56, 201-212 (2000) · Zbl 0991.39016 [9] Hyers, D. H., On the stability of linear functional equations, Proc. Natl. Acad. Sci. U.S.A., 27, 222-224 (1941) · Zbl 0061.26403 [10] Hyers, D. H.; Isac, G.; Rassias, Th. M., Stability Functional Equations in Several Variables (1998), Birkhäuser: Birkhäuser Basel · Zbl 0894.39012 [11] Isac, G.; Rassias, Th. M., On the Hyers-Ulam stability of ψ-additive mappings, J. Approx. Theory, 72, 131-137 (1993) · Zbl 0770.41018 [12] Isac, G.; Rassias, Th. M., Functional inequalities for approximately additive mappings, (Rassias, Th. M.; Tabor, J., Stability of Mappings of Hyers-Ulam Type (1994), Hadronic Press: Hadronic Press Palm Harbour), 117-125 · Zbl 0844.39015 [13] Johnson, B. E., Approximative multiplicative maps between Banach algebras, J. London Math. Soc., 37, 294-316 (1998) [14] Jun, K. W.; Shin, D. S.; Kim, B. D., On the Hyers-Ulam-Rassias stability of the Pexider equation, J. Math. Anal. Appl., 239, 20-29 (1999) · Zbl 0940.39021 [15] Jung, S. M., On the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 204, 221-226 (1996) · Zbl 0888.46018 [16] Jung, S. M.; Sahoo, P. K., On the Hyers-Ulam-Rassias stability of an equation of Davison, J. Math. Anal. Appl., 238, 297-304 (1999) · Zbl 0933.39052 [17] Lee, Y. H.; Jun, K. W., On the stability of approximately additive mappings, Proc. Amer. Math. Soc., 128, 1361-1369 (2000) · Zbl 0961.47039 [18] Maligranda, L., Why Hölder’s inequality should be called Roger’s inequality, Math. Inequalities Appl., 1, 69-83 (1998) · Zbl 0889.26001 [19] Rassias, Th. M., On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040 [20] Rassias, Th. M., Remark and problem, report of the 31st Internat. Symp. on Functional Equations, Aequationes Math., 47, 313 (1994) [21] Rassias, Th. M.; Šemrl, P., On the behaviour of mappings which does not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114, 989-993 (1992) · Zbl 0761.47004 [22] Ulam, S. M., Chap. VI, Problems in Modern Mathematics (1960), Wiley: Wiley New York This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.