## On the asymptoticity aspect of Hyers-Ulam stability of mappings.(English)Zbl 0894.39012

Let $$E_1$$ be a real normed vector space and $$E_2$$ a real Banach space, let $$\varepsilon>0$$. Answering a problem posed by S. M. Ulam, in 1941 D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] proved that if $$f:E_1\to E_2$$ satisfies $$\| f(x+y)-f(x)-f(y)\| \leq \varepsilon$$ for all $$x,y \in E_1$$, then there exists a unique additive mapping $$T:E_1\to E_2$$ such that $$\| f(x)-T(x)\| \leq\varepsilon$$ for all $$x \in E_1$$.
The assumptions were weakened by Th. M. Rassias [Proc. Am. Math. Soc. 72, 297-300 (1978; Zbl 0398.47040)], who proved that if $$f$$ satisfies, for some $$0< p < 1$$, $$\| f(x+y)-f(x)-f(y)\| \leq \varepsilon (\| x\| ^p+\| y\| ^p)$$ for all $$x,y \in E_1$$, then there is a unique additive mapping $$T:E_1\to E_2$$ with $$\| f(x)-T(x)\| \leq \varepsilon \beta(p) \| x\| ^p$$ for all $$x\in E_1$$, where $$\beta(p)=2/(2-2^p)$$.
The present paper generalizes this result further. The authors show that if $$f$$ satisfies $$\| f(x+y)-f(x)-f(y)\| \leq\varepsilon (\| x\| ^p+\| y\| ^p)$$ whenever $$\| x\| ^p+\| y\| ^p>M^p$$ for some $$M>0$$, then there is an additive mapping $$T:E_1 \to E_2$$ with $$\| f(x)-T(x)\| \leq \varepsilon \beta(p) \| x\| ^p$$ for all $$\| x\| >M/2^{1/p}$$.

### MSC:

 39B72 Systems of functional equations and inequalities 47J20 Variational and other types of inequalities involving nonlinear operators (general) 39B52 Functional equations for functions with more general domains and/or ranges

### Citations:

Zbl 0061.26403; Zbl 0398.47040
Full Text:

### References:

 [1] J. Ralph Alexander, Charles E. Blair, and Lee A. Rubel, Approximate versions of Cauchy’s functional equation, Illinois J. Math. 39 (1995), no. 2, 278 – 287. · Zbl 0824.39007 [2] Herbert Amann, Fixed points of asymptotically linear maps in ordered Banach spaces, J. Functional Analysis 14 (1973), 162 – 171. · Zbl 0263.47043 [3] Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620 – 709. · Zbl 0345.47044 [4] Nguyên Phuong Các and Juan A. Gatica, Fixed point theorems for mappings in ordered Banach spaces, J. Math. Anal. Appl. 71 (1979), no. 2, 547 – 557. · Zbl 0448.47035 [5] Jacek Chmieliński, On the stability of the generalized orthogonality equation, Stability of mappings of Hyers-Ulam type, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1994, pp. 43 – 57. · Zbl 0844.39014 [6] P. D. T. A. Elliott, Cauchy’s functional equation in the mean, Adv. in Math. 51 (1984), no. 3, 253 – 257. · Zbl 0541.39003 [7] Roman Ger, On functional inequalities stemming from stability questions, General inequalities, 6 (Oberwolfach, 1990) Internat. Ser. Numer. Math., vol. 103, Birkhäuser, Basel, 1992, pp. 227 – 240. · Zbl 0770.39007 [8] R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Polish Acad. Sci. Math. 43 (1995), 143-151. CMP 97:03 · Zbl 0833.39007 [9] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. · Zbl 0061.26403 [10] Donald H. Hyers and Themistocles M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125 – 153. · Zbl 0806.47056 [11] George Isac, Opérateurs asymptotiquement linéaires sur des espaces localement convexes, Colloq. Math. 46 (1982), no. 1, 67 – 72 (French). · Zbl 0498.47023 [12] George Isac and Themistocles M. Rassias, On the Hyers-Ulam stability of \?-additive mappings, J. Approx. Theory 72 (1993), no. 2, 131 – 137. · Zbl 0770.41018 [13] George Isac and Themistocles M. Rassias, Stability of \Psi -additive mappings: applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), no. 2, 219 – 228. · Zbl 0843.47036 [14] M. A. Krasnosel$$^{\prime}$$skiĭ, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. [15] Themistocles M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297 – 300. · Zbl 0398.47040 [16] Themistocles M. Rassias and Peter Šemrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), no. 2, 325 – 338. · Zbl 0789.46037 [17] Fulvia Skof, On the approximation of locally \?-additive mappings, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117 (1983), no. 4-6, 377 – 389 (1986) (Italian, with English summary). · Zbl 0794.39008 [18] Fulvia Skof, On the stability of functional equations on a restricted domain and a related topic, Stability of mappings of Hyers-Ulam type, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1994, pp. 141 – 151. · Zbl 0844.39006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.