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Quadratic-quartic functional equations in RN-spaces. (English) Zbl 1187.39036
Summary: We obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary $$t$$-norms
$f(2x+y)+f(2x - y)=4[f(x+y)+f(x - y)]+2[f(2x) - 4f(x)] - 6f(y).$

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 46B09 Probabilistic methods in Banach space theory
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##### References:
 [1] Ulam SM: Problems in Modern Mathematics. Science edition, John Wiley & Sons, New York, NY, USA; 1964:xvii+150. · Zbl 0137.24201 [2] Hyers, DH, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224, (1941) · Zbl 0061.26403 [3] Rassias, ThM, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, 297-300, (1978) · Zbl 0398.47040 [4] Gajda, Z, On stability of additive mappings, International Journal of Mathematics and Mathematical Sciences, 14, 431-434, (1991) · Zbl 0739.39013 [5] Aczél J, Dhombres J: Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications. Volume 31. Cambridge University Press, Cambridge, UK; 1989:xiv+462. · Zbl 0685.39006 [6] Aoki, T, On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society of Japan, 2, 64-66, (1950) · Zbl 0040.35501 [7] Bourgin, DG, Classes of transformations and bordering transformations, Bulletin of the American Mathematical Society, 57, 223-237, (1951) · Zbl 0043.32902 [8] Găvruţa, P, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications, 184, 431-436, (1994) · Zbl 0818.46043 [9] Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Basel, Switzerland; 1998:vi+313. · Zbl 0907.39025 [10] Isac, G; Rassias, ThM, On the Hyers-Ulam stability of [inlineequation not available: see fulltext.]-additive mappings, Journal of Approximation Theory, 72, 131-137, (1993) · Zbl 0770.41018 [11] Rassias, ThM, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae, 62, 23-130, (2000) · Zbl 0981.39014 [12] Rassias, ThM, On the stability of functional equations in Banach spaces, Journal of Mathematical Analysis and Applications, 251, 264-284, (2000) · Zbl 0964.39026 [13] Kannappan, Pl, Quadratic functional equation and inner product spaces, Results in Mathematics, 27, 368-372, (1995) · Zbl 0836.39006 [14] Skof, F, Proprieta’ locali e approssimazione di operatori, Milan Journal of Mathematics, 53, 113-129, (1983) [15] Cholewa, PW, Remarks on the stability of functional equations, Aequationes Mathematicae, 27, 76-86, (1984) · Zbl 0549.39006 [16] Czerwik, S, On the stability of the quadratic mapping in normed spaces, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 62, 59-64, (1992) · Zbl 0779.39003 [17] Grabiec, A, The generalized Hyers-Ulam stability of a class of functional equations, Publicationes Mathematicae Debrecen, 48, 217-235, (1996) · Zbl 1274.39058 [18] Park, W; Bae, J, On a bi-quadratic functional equation and its stability, Nonlinear Analysis: Theory, Methods & Applications, 62, 643-654, (2005) · Zbl 1076.39027 [19] Chung, JK; Sahoo, PK, On the general solution of a quartic functional equation, Bulletin of the Korean Mathematical Society, 40, 565-576, (2003) · Zbl 1048.39017 [20] Kim, H, On the stability problem for a mixed type of quartic and quadratic functional equation, Journal of Mathematical Analysis and Applications, 324, 358-372, (2006) · Zbl 1106.39027 [21] Miheţ, D, The probabilistic stability for a functional equation in a single variable, Acta Mathematica Hungarica, 123, 249-256, (2009) · Zbl 1212.39036 [22] Miheţ, D, The fixed point method for fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 160, 1663-1667, (2009) · Zbl 1179.39039 [23] Miheţ, D; Radu, V, On the stability of the additive Cauchy functional equation in random normed spaces, Journal of Mathematical Analysis and Applications, 343, 567-572, (2008) · Zbl 1139.39040 [24] Mirmostafaee, AK; Mirzavaziri, M; Moslehian, MS, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 159, 730-738, (2008) · Zbl 1179.46060 [25] Mirmostafaee, AK; Moslehian, MS, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, 159, 720-729, (2008) · Zbl 1178.46075 [26] Mirmostafaee, AK; Moslehian, MS, Fuzzy approximately cubic mappings, Information Sciences, 178, 3791-3798, (2008) · Zbl 1160.46336 [27] Chang SS, Cho YJ, Kang SM: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science, Huntington, NY, USA; 2001:x+338. · Zbl 1080.47054 [28] Schweizer B, Sklar A: Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics. North-Holland, New York, NY, USA; 1983:xvi+275. · Zbl 0546.60010 [29] Sherstnev, AN, On the notion of a random normed space, Doklady Akademii Nauk SSSR, 149, 280-283, (1963) · Zbl 0127.34902 [30] Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and its Applications. Volume 536. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+273. [31] Hadžić, O; Pap, E; Budinčević, M, Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces, Kybernetika, 38, 363-382, (2002) · Zbl 1265.54127 [32] Gordji ME, Rassias JM, Savadkouhi MB: Stability of a mixed type additive and quadratic functional equation in random normed spaces. preprint preprint · Zbl 1265.54127 [33] Gordji, ME; Rassias, JM; Savadkouhi, MB, Approximation of the quadratic and cubic functional equation in RN-spaces, European Journal of Pure and Applied Mathematics, 2, 494-507, (2009) · Zbl 1215.39034
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