ON THE ULAM-GAVRUTA-RASSIAS STABILITY OF THE ORTHOGONALLY EULER-LAGRANGE TYPE FUNCTIONAL EQUATION

Abstract

In 1940 S. M. Ulam proposed at the Mathematics club of the University of Wisconsin the problem: “Given conditions in order for a linear mappings near an approximately linear mappings to exists.” In 1941 D. H. Hyers solved the Ulam problem for linear mappings. In 1951 D. G. Bourgin solved the Ulam problem for additive mappings. In 1982-2004 J. M. Rassias, M. J. Rassias established the Hyers-Ulam stability for the Ulam problem for different mappings. In 1992-2000, J. M. Rassias investigated the Ulam stability for Euler-Lagrange mappings. In 2005 J. M. Rassias solved the Ulam problem for Euler-Lagrange type quadratic functional equations.On the other hand, the orthogonal Cauchy functional equation with an abstract orthogonality relation, was first investigated by S. Gudder and D. Strawther. J. Ratz introduced a new definition of orthogonality by using more restrictive axioms than of S. Gudder and D. Strawther. Moreover, he investigated the structure of orthogonally additive mappings. J. Ratz and Gy. Szabo investigated the problem in a rather more general framework. The orthogonally standard quadratic equation was first investigated by F. Vajzovic when in a Hilbert space, is the scalar field. Many more mathematicians like H. Drljevic, M. Fochi, M. S. Moslehian, Gy. Szabo generalized this result. In this paper the authors wish is to prove the Ulam-Gavruta-Rassias stability for the orthogonally Euler-Lagrange type functional equation.