## 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
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### References:

  Aoki, T., On the stability of the linear transformation in Banach spaces, J. math. soc. Japan, 2, 64-66, (1950) · Zbl 0040.35501  Batko, B., On the stability of an alternative functional equation, Math. inequal. appl., 8, 4, 685-691, (2005) · Zbl 1096.39026  Batko, B.; Tabor, J., Stability of an alternative Cauchy equation on a restricted domain, Aequationes math., 57, 221-232, (1999) · Zbl 0935.39011  Bourgin, D.G., Classes of transformations and bordering transformations, Bull. amer. math. soc. (N.S.), 57, 223-237, (1951) · Zbl 0043.32902  Forti, G.L., An existence and stability theorem for a class of functional equations, Stochastics, 4, 23-30, (1980) · Zbl 0442.39005  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  Forti, G.L., Hyers – ulam stability of functional equations in several variables, Aequationes math., 50, 143-190, (1995) · Zbl 0836.39007  Gajda, Z., On stability of additive mappings, Int. J. math. math. sci., 14, 431-434, (1991) · Zbl 0739.39013  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  Gavruta, P., An answer to a question of John M. Rassias concerning the stability of Cauchy equation, (), 67-71  Ger, R., A survey of recent results on stability of functional equations, (), 5-36  Ger, R., On functional inequalities stemming from stability questions, General inequalities, vol. 6, (1992), Birkhäuser Basel · Zbl 0770.39007  Ger, R., The singular case in the stability behaviour of linear mappings, Grazer math. ber., 316, 59-70, (1991) · Zbl 0796.39012  Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403  Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, (1998), Birkhäuser Boston · Zbl 0894.39012  Isac, G.; Rassias, Th.M., On the hyers – ulam stability of ψ-additive mappings, J. approx. theory, 72, 131-137, (1993) · Zbl 0770.41018  Jung, S., Hyers – ulam – rassias stability of functional equations in mathematical analysis, (2001), Hadronic Press, Inc. Palm Harbor · Zbl 0980.39024  Kuczma, M., On some alternative functional equations, Aequationes math., 17, 182-198, (1978) · Zbl 0398.39007  Rassias, J.M., Complete solution of the multi-dimensional problem of Ulam, Discuss. math., 14, 101-107, (1994) · Zbl 0819.39012  Rassias, J.M., On approximation of approximately linear mappings by linear mappings, J. funct. anal., 46, 126-130, (1982) · Zbl 0482.47033  Rassias, J.M., On approximation of approximately linear mappings by linear mappings, Bull. sci. math., 108, 445-446, (1984) · Zbl 0599.47106  Rassias, J.M., Solution of a problem of Ulam, J. approx. theory, 57, 268-273, (1989) · Zbl 0672.41027  Rassias, J.M., Solution of a stability problem of Ulam, Discuss. math., 12, 95-103, (1992) · Zbl 0779.47005  Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040  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  Rassias, Th.M.; Šemrl, P., On the hyers – ulam stability of linear mappings, J. math. anal. appl., 173, 325-338, (1993) · Zbl 0789.46037  Ulam, S.M., A collection of mathematical problems, (1960), Interscience Publ. New York · Zbl 0086.24101
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