Solution of a quadratic stability Ulam type problem

Rassias, John Michael. "Solution of a quadratic stability Ulam type problem." Archivum Mathematicum 040.1 (2004): 1-16. <http://eudml.org/doc/249322>.

@article{Rassias2004,
abstract = {In 1940 S. M. Ulam (Intersci. Publ., Inc., New York 1960) imposed at the University of Wisconsin the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist”. According to P. M. Gruber (Trans. Amer. Math. Soc. 245 (1978), 263–277) the afore-mentioned problem of S. M. Ulam belongs to 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?” In 1941 D. H. Hyers (Proc. Nat. Acad. Sci. 27 (1941), 411–416) established the stability Ulam problem with Cauchy inequality involving a non-negative constant. Then in 1989 we (J. Approx. Theory, 57 (1989), 268–273) solved Ulam problem with Cauchy functional inequality, involving a product of powers of norms. Finally we (Discuss. Math. 12 (1992), 95–103) established the general version of this stability problem. In this paper we solve a stability Ulam type problem for a general quadratic functional inequality. Moreover, we introduce an approximate eveness on approximately quadratic mappings of this problem. These problems, according to P. M. Gruber (1978), are of particular interest in probability theory and in the case of functional equations of different types. Today there are applications in actuarial and financial mathematics, sociology and psychology, as well as in algebra and geometry.},
author = {Rassias, John Michael},
journal = {Archivum Mathematicum},
keywords = {Ulam problem; Ulam type problem; stability; quadratic; approximate eveness; approximately quadratic; quadratic mapping near an approximately quadratic mapping; quadratic functional equation; approximate evenness; approximately quadratic; Ulam stability; normed real linear spaces},
language = {eng},
number = {1},
pages = {1-16},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Solution of a quadratic stability Ulam type problem},
url = {http://eudml.org/doc/249322},
volume = {040},
year = {2004},
}

TY - JOUR
AU - Rassias, John Michael
TI - Solution of a quadratic stability Ulam type problem
JO - Archivum Mathematicum
PY - 2004
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 040
IS - 1
SP - 1
EP - 16
AB - In 1940 S. M. Ulam (Intersci. Publ., Inc., New York 1960) imposed at the University of Wisconsin the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist”. According to P. M. Gruber (Trans. Amer. Math. Soc. 245 (1978), 263–277) the afore-mentioned problem of S. M. Ulam belongs to 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?” In 1941 D. H. Hyers (Proc. Nat. Acad. Sci. 27 (1941), 411–416) established the stability Ulam problem with Cauchy inequality involving a non-negative constant. Then in 1989 we (J. Approx. Theory, 57 (1989), 268–273) solved Ulam problem with Cauchy functional inequality, involving a product of powers of norms. Finally we (Discuss. Math. 12 (1992), 95–103) established the general version of this stability problem. In this paper we solve a stability Ulam type problem for a general quadratic functional inequality. Moreover, we introduce an approximate eveness on approximately quadratic mappings of this problem. These problems, according to P. M. Gruber (1978), are of particular interest in probability theory and in the case of functional equations of different types. Today there are applications in actuarial and financial mathematics, sociology and psychology, as well as in algebra and geometry.
LA - eng
KW - Ulam problem; Ulam type problem; stability; quadratic; approximate eveness; approximately quadratic; quadratic mapping near an approximately quadratic mapping; quadratic functional equation; approximate evenness; approximately quadratic; Ulam stability; normed real linear spaces
UR - http://eudml.org/doc/249322
ER -