# Solution of a Cauchy-Jensen stability Ulam type problem

Archivum Mathematicum (2001)

- Volume: 037, Issue: 3, page 161-177
- ISSN: 0044-8753

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topRassias, John Michael. "Solution of a Cauchy-Jensen stability Ulam type problem." Archivum Mathematicum 037.3 (2001): 161-177. <http://eudml.org/doc/248733>.

@article{Rassias2001,

abstract = {In 1978 P. M. Gruber (Trans. Amer. Math. Soc. 245 (1978), 263–277) imposed the following general problem or Ulam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this objects by objects, satisfying the property exactly?" The afore-mentioned problem of P. M. Gruber is more general than the following problem imposed by S. M. Ulam in 1940 (Intersci, Publ., Inc., New York 1960): “Give conditions in order for a linear mapping near an approximately linear mapping to exist". In 1941 D. H. Hyers (Proc. Nat. Acad. Sci., U.S.A. 27 (1941), 411–416) solved a special case of Ulam problem. In 1989 and 1992 we (J. Approx. Th., 57, No. 3 (1989), 268–273; Discuss. Math. 12 (1992), 95–103) solved above Ulam problem. In this paper we introduce the generalized Cauchy-Jensen functional inequality and solve a stability Ulam type problem for this inequality. This problem, according to P. M. Gruber, is of particular interest in probability theory and in the case of functional equations of different types.},

author = {Rassias, John Michael},

journal = {Archivum Mathematicum},

keywords = {Ulam problem; Ulam type problem; stability; Cauchy-Jensen; approximately Cauchy-Jensen; Cauchy-Jensen mapping near an approximately Cauchy-Jensen mapping; Ulam problem; Ulam type problem; stability; Cauchy-Jensen},

language = {eng},

number = {3},

pages = {161-177},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Solution of a Cauchy-Jensen stability Ulam type problem},

url = {http://eudml.org/doc/248733},

volume = {037},

year = {2001},

}

TY - JOUR

AU - Rassias, John Michael

TI - Solution of a Cauchy-Jensen stability Ulam type problem

JO - Archivum Mathematicum

PY - 2001

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 037

IS - 3

SP - 161

EP - 177

AB - In 1978 P. M. Gruber (Trans. Amer. Math. Soc. 245 (1978), 263–277) imposed the following general problem or Ulam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this objects by objects, satisfying the property exactly?" The afore-mentioned problem of P. M. Gruber is more general than the following problem imposed by S. M. Ulam in 1940 (Intersci, Publ., Inc., New York 1960): “Give conditions in order for a linear mapping near an approximately linear mapping to exist". In 1941 D. H. Hyers (Proc. Nat. Acad. Sci., U.S.A. 27 (1941), 411–416) solved a special case of Ulam problem. In 1989 and 1992 we (J. Approx. Th., 57, No. 3 (1989), 268–273; Discuss. Math. 12 (1992), 95–103) solved above Ulam problem. In this paper we introduce the generalized Cauchy-Jensen functional inequality and solve a stability Ulam type problem for this inequality. This problem, according to P. M. Gruber, is of particular interest in probability theory and in the case of functional equations of different types.

LA - eng

KW - Ulam problem; Ulam type problem; stability; Cauchy-Jensen; approximately Cauchy-Jensen; Cauchy-Jensen mapping near an approximately Cauchy-Jensen mapping; Ulam problem; Ulam type problem; stability; Cauchy-Jensen

UR - http://eudml.org/doc/248733

ER -

## References

top- Stability of Isometries, Trans. Amer. Math. Soc. 245 (1978), 263–277. Zbl0393.41020MR0511409
- On the stabililty of the linear functional equation, Proc. Nat. Acad. Sci 27 (1941), 411–416. MR0004076
- Solution of a problem of Ulam, J. Approx. Th. 57 (1989), 268–273. Zbl0672.41027MR0999861
- Solution of a stability problem of Ulam, Discuss. Math. 12 (1992), 95–103. Zbl0878.46032MR1221875
- A collection of mathematical problems, Intersci. Publ., Inc., New York, 1960. Zbl0086.24101MR0120127

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