FIXED POINTS AND STABILITY FOR QUARTIC MAPPINGS IN -NORMED LEFT BANACH MODULES ON BANACH ALGEBRAS

doi:10.1016/S0252-9602(13)60067-3

Abstract

Abstract:

The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equation
∑ⁿ_k₌₂(∑^k_i₁₌₂∑^k+1_i2=i1+1…∑ⁿ_in−k_+1=in−k+1)f (∑ⁿ_i_{=1, i6=i1}, …, _in−k+1x_i−∑^n−k+1_r=1x_ir)+ f (∑ⁿ_i₌₁x_i)= 2ⁿ⁻²∑_1≤i<j≤_n(f (x_i+ x_j)+f (x_i− x_j)−2ⁿ⁻⁵(n − 2)∑ⁿ_i₌₁f (2x_i)
(n ∈N, n ≥ 3) in β-Banach modules on Banach algebras.
The concept of Ulam-Hyers-Rassias stability (briefly, HUR-stability) originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

Key words: generalized Hyers-Ulam stability, quartic functional equation, Banach mod-ule, unital Banach algebra, generalized metric space, fixed point method

CLC Number:

39B82

H. Azadi KENARY, A.R. ZOHDI, M. Eshaghi GORDJI. FIXED POINTS AND STABILITY FOR QUARTIC MAPPINGS IN -NORMED LEFT BANACH MODULES ON BANACH ALGEBRAS[J].Acta mathematica scientia,Series B, 2013, 33(4): 1113-1118.

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References

[1] Aczel J, Dhombres J. Functional Equations in Several Variables. Cambridge: Cambridge Univ Press, 1989

[2] Aoki T. On the stability of the linear transformation in Banach spaces. J Math Soc Japan, 1950, 2: 64–66

[3] Balachandran V K. Topological Algebras. New Delhi: Narosa Publishing House, 1999

[4] C?adariu L, Radu V. On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math Ber, 2004, 346: 43–52

[5] Czerwik S. On the stability of the quadratic mapping in normed spaces. Abh Math Sem Univ Hamburg, 1992, 62: 59–64

[6] Gordji M Eshaghi, Khodaei H. Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces. Nonlinear Analysis TMA, 2009, 71: 5629–5643

[7] Gajda Z. On stability of additive mappings. Internat J Math Math Sci, 1991, 14: 431–434

[8] Gˇavruta P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl, 1994, 184: 431–436

[9] Hyers D H. On the stability of the linear functional equation. Proc Natl Acad Sci, 1941, 27: 222–224

[10] Hyers D H, Isac G, Rassias ThM. Stability of Functional Equations in Several Variables. Basel: Birkhauser, 1998

[11] Isac G, Rassias Th M. On the Hyers-Ulam stability of -additive mappings. J Approx Theory, 1993, 72: 131–137

[12] Jung S -M. On the Hyers-Ulam stability of the functional equations that have the quadratic property. J Math Anal Appl, 1998, 222: 126–137

[13] Jung S -M, Sahoo P K. Hyers-Ulam stability of the quadratic equation of Pexider type. J Korean Math Soc, 2001, 38(3): 645–656

[14] Kannappan Pl. Quadratic functional equation and inner product spaces. Results Math, 1995, 27: 368–372

[15] Khodaei H, Rassias Th M. Approximately generalized additive functions in several variables. Int J Non-linear Anal Appl, 2010, 1: 22–41

[16] Margolis B, Diaz J B. A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull Amer Math Soc, 1968, 74: 305–309

[17] Moslehian M S. On the orthogonal stability of the Pexiderized quadratic equation. J Differ Equations Appl, 2005: 999–1004

[18] Najati A, Moghimi M B. Stability of a functional equation deriving from quadratic and additive function in quasi-Banach spaces. J Math Anal Appl, 2008, 337: 399–415

[19] Park C. On the stability of the quadratic mapping in Banach modules. J Math Anal Appl, 2002, 27: 135–144

[20] Park C. On the Hyers-Ulam-Rassias stability of generalized quadratic mappings in Banach modules. J Math Anal Appl, 2004, 291: 214–223

[21] Radu V. The fixed point alternative and the stability of functional equations. Fixed Point Theory, 2003, 4: 91–6

[22] Rassias Th M. On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc, 1978, 72: 297–300

[23] Rassias Th M, Semrl P. On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proc Amer Math Soc, 1992, 114: 989–993

[24] Ulam S M. Problems in Modern Mathematics. Chapter VI. Sci ed. New York: Wiley, 1964