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On the stability of set-valued functional equations with the fixed point alternative. (English) Zbl 1283.39010
In the paper, several functional equations (Cauchy functional equations, Jensen functional equation, quadratic functional equations) for set-valued mappings are considered. The authors discuss the problem of stability of such equations in the sense of Ulam-Hyers and Rassias. To prove the stability results, they apply an interesting theorem on the existence of fixed points of a contractive mapping in a generalized metric space (called a fixed point alternative).

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 54H25 Fixed-point and coincidence theorems (topological aspects) 54C60 Set-valued maps in general topology
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##### References:
 [1] Aumann, RJ, Integrals of set-valued functions, J Math Anal Appl, 12, 1-12, (1965) · Zbl 0163.06301 [2] Debreu, G, Integration of correspondences, (1966) · Zbl 0211.52803 [3] Arrow, KJ; Debreu, v, Existence of an equilibrium for a competitive economy, Econometrica, 22, 265-290, (1954) · Zbl 0055.38007 [4] McKenzie, LW, On the existence of general equilibrium for a competitive market, Econometrica, 27, 54-71, (1959) · Zbl 0095.34302 [5] Hindenbrand W: Core and Equilibria of a Large Economy. Princeton University Press, Princeton; 1974. [6] Aubin JP, Frankow H: Set-Valued Analysis. Birkhäuser, Boston; 1990. · Zbl 0713.49021 [7] Castaing, C; Valadier, M, Convex analysis and measurable multifunctions, vol. 580, (1977), Berlin · Zbl 0346.46038 [8] Klein E, Thompson A: Theory of Correspondence. Wiley, New York; 1984. [9] Hess, C, Set-valued integration and set-valued probability theory: an overview, (2002), Amsterdam · Zbl 1022.60011 [10] Cascales, T; Rodrigeuz, J, Birkhoff integral for multi-valued functions, J Math Anal Appl, 297, 540-560, (2004) · Zbl 1066.46037 [11] Brzdëk, T; Popa, D; Xu, B, Seletions of set-valued maps satisfying a linear inclusion in a single variable, Nonlinear Anal TMA, 74, 324-330, (2011) · Zbl 1205.39025 [12] Cardinali, T; Nikodem, K; Papalini, F, Some results on stability and characterization of $$K$$-convexity of set-valued functions, Ann Polon Math, 58, 185-192, (1993) · Zbl 0786.26016 [13] Nikodem, K, On quadratic set-valued functions, Publ Math Debrecen, 30, 297-301, (1984) · Zbl 0537.39002 [14] Nikodem, K, On Jensen’s functional equation for set-valued functions, Radovi Mat, 3, 23-33, (1987) · Zbl 0628.39013 [15] Nikodem, K, Set-valued solutions of the Pexider functional equation, Funkcialaj Ekvacioj, 31, 227-231, (1988) · Zbl 0698.39007 [16] Nikodem, K, $$K$$-convex and $$K$$-concave set-valued functions, No. 559, (1989) [17] Piao, YJ, The existence and uniqueness of additive selection for (α, β)-(β, α) type subadditive set-valued maps, J Northeast Normal Univ, 41, 38-40, (2009) [18] Popa, D, Additive selections of (α, β)-subadditive set-valued maps, Glas Mat Ser III, 36, 11-16, (2001) · Zbl 1039.28013 [19] Ulam SM: Problems in Modern Mathematics, chapter VI, Science ed. Wiley, New York; 1940. [20] Hyers, DH, On the stability of the linear functional equation, Proc Natl Acad Sci USA, 27, 222-224, (1941) · Zbl 0061.26403 [21] Aoki, T, On the stability of the linear transformation in Banach spaces, J Math Soc Japan, 2, 64-66, (1950) · Zbl 0040.35501 [22] Rassias, ThM, On the stability of the linear mapping in Banach spaces, Proc Am Math Soc, 72, 297-300, (1978) · Zbl 0398.47040 [23] Gǎvruta, P, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J Math Anal Appl, 184, 431-436, (1994) · Zbl 0818.46043 [24] Skof, F, Proprietà locali e approssimazione di operatori, Rend Sem Mat Fis Milano, 53, 113-129, (1983) · Zbl 0599.39007 [25] Cholewa, PW, Remarks on the stability of functional equations, Aequationes Math, 27, 76-86, (1984) · Zbl 0549.39006 [26] Czerwik, S, On the stability of the quadratic mapping in normed spaces, Abh Math Sem Univ Hamburg, 62, 59-64, (1992) · Zbl 0779.39003 [27] Aczel J, Dhombres J: Functional Equations in Several Variables. Cambridge Univ Press, Cambridge; 1989. · Zbl 0685.39006 [28] Azadi Kenary, H; Park, C, Direct and fixed point methods approach to the generalized Hyers-Ulam stability for a functional equation having monomials as solutions, Iranian J Sci Tech Trans A, A4, 301-307, (2011) [29] Azadi Kenary, H; Rezaei, H; Talebzadeh, S; Park, C, Stability for the Jensen equation in C*- algebras: a fixed point alternative approach, Adv Diff Eq, 2012, 17, (2012) · Zbl 1278.39039 [30] Azadi Kenary, H; Jang, SY; Park, C, A fixed point approach to the Hyers-Ulam stability of a functional equation in various normed spaces, Fixed Point Theory Appl, 2011, 67, (2011) · Zbl 1278.39038 [31] Cădariu, L; Radu, V, Fixed points and the stability of Jensen’s functional equation, J Inequal Pure Appl Math, 4, 4, (2003) · Zbl 1043.39010 [32] Cădariu, L; Radu, V, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math Ber, 346, 43-52, (2004) · Zbl 1060.39028 [33] Cădariu, L; Radu, V, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl, 2008, 1-5, (2008) [34] Cho, YJ; Saadati, R, Lattice non-Archimedean random stability of ACQ functional equation, Adv Diff Equ, 2011, 31, (2011) · Zbl 1273.39024 [35] Ebadian, A; Ghobadipour, N; Gordji, ME, A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C*-ternary algebras, J Math Phys, 51, 10, (2010) · Zbl 1314.46063 [36] Eshaghi Gordji M, Alizadeh Z, Cho YJ, Khodaei H: On approximate\bfC*-ternary\bfm-Homomorphisms: A fixed point approach.Fixed Point Theory Appl 2011.,2011(14): (Article ID FPTA,454093) · Zbl 1216.39036 [37] Eshaghi Gordji, M; Ghaemi, MB; Kaboli Gharetapeh, S; Shams, S; Ebadian, A, On the stability of J*-derivations, J Geom Phys, 60, 454-459, (2010) · Zbl 1188.39029 [38] Eshaghi Gordji, M; Khodaei, H, The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces, No. 15, (2010) · Zbl 1221.39036 [39] Eshaghi Gordji, M; Khodaei, H; Rassias, JM, Fixed points and stability for quadratic mappings in β-normed left Banach modules on Banach algebras, (2011) · Zbl 1276.39016 [40] Eshaghi Gordji, M; Najati, A, Approximately J*-homomorphisms: a fixed point approach, J Geom Phys, 60, 809-814, (2010) · Zbl 1192.39020 [41] Gordji, ME; Park, C; Savadkouhi, MB, The stability of a quartic type functional equation with the fixed point alternative, Fixed Point Theory, 11, 265-272, (2010) · Zbl 1208.39036 [42] Eshaghi Gordji, M; Azadi Kenary, H; Rezaei, H; Lee, YW; Kim, GH, Solution and Hyers-Ulam-Rassias stability of generalized mixed type additive-quadratic functional equations in fuzzy Banach spaces, No. 22, (2012) · Zbl 1242.39032 [43] Gordji, ME; Savadkouhi, MB, Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces, Appl Math Lett, 23, 1198-1202, (2010) · Zbl 1204.39028 [44] Găvruta, P; Găvruta, L, A new method for the generalized Hyers-Ulam-Rassias stability, Int J Nonlinear Anal Appl, 1, 11-18, (2010) · Zbl 1281.39038 [45] Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998. · Zbl 0907.39025 [46] Isac, G; Rassias, ThM, On the Hyers-Ulam stability of $$ψ$$-additive mappings, J Approx Theory, 72, 131-137, (1993) · Zbl 0770.41018 [47] Isac, G; Rassias, ThM, Stability of $$ψ$$-additive mappings: applications to nonlinear analysis, Int J Math Math Sci, 19, 219-228, (1996) · Zbl 0843.47036 [48] Jun, K; Cho, Y, Stability of the generalized Jensen type quadratic functional equations, J. Chungcheong Math. Soc, 20, 515-523, (2007) [49] Kenary, HA; Cho, YJ, Stability of mixed additive-quadratic Jensen type functional equation in various spaces, Comput Math Appl, 61, 2704-2724, (2011) · Zbl 1235.39024 [50] Khodaei, H; Rassias, ThM, Approximately generalized additive functions in several variables, Int J Nonlinear Anal Appl, 1, 22-41, (2010) · Zbl 1281.39041 [51] Mohammadi, M; Cho, YJ; Park, C; Vetro, P; Saadati, R, Random stability of an additive-quadratic-quartic functional equation, No. 18, (2010) · Zbl 1187.39045 [52] Najati, A, Stability of homomorphisms on JB* triples associated to a Cauchy-Jensen type functional equation, J Math Inequal, 1, 83-103, (2007) · Zbl 1155.39307 [53] Najati, A; Cho, YJ, Generalized Hyers-Ulam stability of the pexiderized Cauchy functional equation in non-Archimedean spaces, No. 11, (2011) · Zbl 1213.39018 [54] Najati, A; Cho, YJ, Generalized Hyers-Ulam stability of the pexiderized Cauchy functional equation in non-Archimedean spaces, No. 11, (2011) · Zbl 1213.39018 [55] Najati A, Kang JI, Cho YJ: Local stability of the pexiderized Cauchy and Jensen’s equations in fuzzy spaces.J Inequal Appl 2011.,78(2011): doi:10.1186/1029-242X-2011-78 doi:10.1186/1029-242X-2011-78 [56] Park, C, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, No. 9, (2008) [57] Park, C, Fuzzy stability of a functional equation associated with inner product spaces, Fuzzy Set Syst, 160, 1632-1642, (2009) · Zbl 1182.39023 [58] Radu, V, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4, 91-96, (2003) · Zbl 1051.39031 [59] Rassias ThM (ed) (Ed): Functional Equations and Inequalities. Kluwer Academic, Dordrecht; 2000. · Zbl 0163.06301 [60] Rassias, ThM, On the stability of functional equations and a problem of Ulam, Acta Math Appl, 62, 23-130, (2000) · Zbl 0981.39014 [61] Saadati, R; Cho, YJ; Vahidi, J, The stability of the quartic functional equation in various spaces, Comput Math Appl, 60, 1994-2002, (2010) · Zbl 1205.39029 [62] Kim, H; Ko, H; Son, J, On the stability of a modified Jensen type cubic mapping, J Chungcheong Math Soc, 21, 129-138, (2008) [63] Diaz, J; Margolis, B, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull Am Math Soc, 74, 305-309, (1968) · Zbl 0157.29904 [64] Park, C, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, No. 2007, (2007) · Zbl 1167.39018
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