ON THE STABILITY OF THE GENERAL EULER-LAGRANGE FUNCTIONAL EQUATION

John Michael Rassias
1996 Demonstratio Mathematica  
In 1940 S.M. Ulam [3] imposed at the University of Wisconsin the problem: " Give conditions in order for a linear mapping near an approximately linear mapping to exist". In 1978 P.M. Gruber [1] imposed the general problem: "Suppose that a mathematical object satisfies a certain property approximately. Is it then possible to approximate this object by objects satisfying the property exactly?" In 1989 J.M. Rassias [2] solved the above Ulam problem, or equivalently the Gruber problem for linear
more » ... pings. In this paper the author solves an analogous stability problem for the general 2-dimensional Euler-Lagrange functional inequality (1) WfiaiXí + a 2 x 2 ) + f(a 2 xi -ai® 2 ) -(a* + al)[f(xi) + /(®2)]|| < c, for all 2-dimensional vectors (®i,X2) £ X 2 , with a normed linear space X, a constant c (independent of £1,3:2) > 0, mapping / : X -> Y (where Y is a complete normed linear space), and any fixed reals 01,02 such that 0 < m = a\ + a 2 ^ 0. Besides he introduces the 2-dimensional quadratic weighted means. According to P.M. Gruber [1] the afore-mentioned stability problems are of particular interest in probability theory and in the case of functional equations of different types. DEFINITION 1. For X, Y as above a non-linear mapping Q\ : X -i• Y, such that the functional equation (1)' + a 2 x 2 ) + Qi{a 2 x x -a lX2 ) = (a¡ + a 2 2 )[Q%(x 1) + Qf(x 2 >] holds for all vectors {x\,x 2 ) G X 2 and for any fixed reals 01,02 with m = a 2 + a 2 > 1, is called 2-dimensional quadratic.
doi:10.1515/dema-1996-0411 fatcat:o2zhklm7gbhvlemgf7umikmxgi