## Stability of an alternative functional equation.(English)Zbl 1133.39025

The author considers the alternative functional equation
$f(x+y)+f(x)+f(y) \neq 0 \implies f(x+y)=f(x)+f(y),$
where $$f:S \to X$$, $$(S,+)$$ is an abelian semigroup and $$(X,\|\cdot\|)$$ a Banach space. The problem to be solved is the following: assume that $$f:S \to X$$ satisfies
$\| f(x+y)+f(x)+f(y)\|>\Phi_1(x,y) \implies \| f(x+y)-f(x)-f(y)\| \leq \Phi_2(x,y),$
where $$\Phi_1, \Phi_2:S \times S \to \mathbb R_+$$ are given functions. Does there exist an additive function $$a:S \to X$$ such that $\| f(x)-a(x)\| \leq \Psi(x),$ where $$\Psi:S \to \mathbb R_+$$ is a function which an be explicitly computed starting from $$\Phi_1$$ and $$\Phi_2$$? The answer is positive if the functions $$\Phi_1$$ and $$\Phi_2$$ satisfy one of the following conditions.
1. Each of the series $\sum_k^{\infty} 2^{-k}\Phi_i(2^kjx,2^kx),$ where $$j\in \{1,2,3\}$$, $$i\in \{1,2\}$$, converges for every $$x \in S$$ and
$\lim_{k \to +\infty} 2^{-k}\Phi_i(2^kx,2^ky)=0$
for $$x,y \in S$$;
2. $$S$$ is uniquely $$2$$-divisible, each of the series $\sum_k^{\infty} 2^k \Phi_i(2^{-k}jx,2^{-k}x),$ where $$j\in \{1,2,3\}$$, $$i\in \{1,2\}$$, converges for every $$x \in S$$ and
$\lim_{k \to +\infty} 2^k \Phi_i(2^{-k}x,2^{-k}y)=0$ for $$x,y \in S$$.

### MSC:

 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 39B55 Orthogonal additivity and other conditional functional equations
Full Text:

### References:

 [1] Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, 64-66 (1950) · Zbl 0040.35501 [2] Batko, B., On the stability of an alternative functional equation, Math. Inequal. Appl., 8, 4, 685-691 (2005) · Zbl 1096.39026 [3] Batko, B.; Tabor, J., Stability of an alternative Cauchy equation on a restricted domain, Aequationes Math., 57, 221-232 (1999) · Zbl 0935.39011 [4] Bourgin, D. G., Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. (N.S.), 57, 223-237 (1951) · Zbl 0043.32902 [5] Forti, G. L., An existence and stability theorem for a class of functional equations, Stochastics, 4, 23-30 (1980) · Zbl 0442.39005 [6] Forti, G. L., Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl., 295, 127-133 (2004) · Zbl 1052.39031 [7] Forti, G. L., Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50, 143-190 (1995) · Zbl 0836.39007 [8] Gajda, Z., On stability of additive mappings, Int. J. Math. Math. Sci., 14, 431-434 (1991) · Zbl 0739.39013 [9] Gavruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184, 431-436 (1994) · Zbl 0818.46043 [10] Gavruta, P., An answer to a question of John M. Rassias concerning the stability of Cauchy equation, (Advances in Equations and Inequalities. Advances in Equations and Inequalities, Hadronic Math. Ser. (1999)), 67-71 [11] Ger, R., A survey of recent results on stability of functional equations, (Proceedings of the 4th International Conference on Functional Equations and Inequalities (1994), Pedagogical University in Cracow), 5-36 [12] Ger, R., On Functional Inequalities Stemming from Stability Questions, General Inequalities, vol. 6 (1992), Birkhäuser: Birkhäuser Basel · Zbl 0770.39007 [13] Ger, R., The singular case in the stability behaviour of linear mappings, Grazer Math. Ber., 316, 59-70 (1991) · Zbl 0796.39012 [14] Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27, 222-224 (1941) · Zbl 0061.26403 [15] Hyers, D. H.; Isac, G.; Rassias, Th. M., Stability of Functional Equations in Several Variables (1998), Birkhäuser: Birkhäuser Boston · Zbl 0894.39012 [16] Isac, G.; Rassias, Th. M., On the Hyers-Ulam stability of $$Ψ$$-additive mappings, J. Approx. Theory, 72, 131-137 (1993) · Zbl 0770.41018 [17] Jung, S., Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis (2001), Hadronic Press, Inc.: Hadronic Press, Inc. Palm Harbor · Zbl 0980.39024 [18] Kuczma, M., On some alternative functional equations, Aequationes Math., 17, 182-198 (1978) · Zbl 0398.39007 [19] Rassias, J. M., Complete solution of the multi-dimensional problem of Ulam, Discuss. Math., 14, 101-107 (1994) · Zbl 0819.39012 [20] Rassias, J. M., On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46, 126-130 (1982) · Zbl 0482.47033 [21] Rassias, J. M., On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math., 108, 445-446 (1984) · Zbl 0599.47106 [22] Rassias, J. M., Solution of a problem of Ulam, J. Approx. Theory, 57, 268-273 (1989) · Zbl 0672.41027 [23] Rassias, J. M., Solution of a stability problem of Ulam, Discuss. Math., 12, 95-103 (1992) · Zbl 0779.47005 [24] Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040 [25] Rassias, Th. M.; Šemrl, P., On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114, 989-993 (1992) · Zbl 0761.47004 [26] Rassias, Th. M.; Šemrl, P., On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl., 173, 325-338 (1993) · Zbl 0789.46037 [27] Ulam, S. M., A Collection of Mathematical Problems (1960), Interscience Publ.: Interscience Publ. New York · Zbl 0086.24101
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.