Refined ULAM Stability for Euler-Lagrange Type Mappings in Hilbert Spaces

We have introduced the Euler-Lagrange quadratic mappings, motivated from the pertinent algebraic equation vertical bar a(1)x(1) + a(2)x(2)vertical bar(2) + vertical bar a(2)x(1) - a(1)x(2)vertical bar(2) = (a(1)(2) + a(2)(2))[vertical bar x(1)vertical bar(2) +vertical bar x(2)vertical bar(2)], for the special case alpha(i) = 1 (i = 1,2). a i Thus we introduced and investigated the relative functional equation in our publications. In addition we generalized the above algebraic equation to the following equation m(1)m(2) vertical bar a(1)x(1) + a(2)x(2)vertical bar(2) + vertical bar m(2)a(2)x(1) - m(1)a(1)x(2)vertical bar(2) = (m(1)vertical bar a(1)vertical bar(2)+ m(2)vertical bar a(2)vertical bar(2))[m(2)vertical bar x(1)vertical bar(2) + m(1)vertical bar x(2)vertical bar(2)] Therefore we introduced and investigated the general pertinent Euler-Lagrange quadratic mappings via our paper. Analogous quadratic mappings were introduced and investigated through our following publications. Before 1992 these mappings and equations were not known at all in functional equations and inequalities. However, a completely different kind of Euler-Lagrange partial differential equations is known in calculus of variations. We are now introducing the general Euler-Lagrange type equation parallel to ax + by parallel to(2) + parallel to bx +ay parallel to(2) + ax -by parallel to(2) + parallel to bx -ay parallel to(2) = 2(a(2) +b(2))[parallel to x parallel to(2) + parallel to y parallel to(2)] which holds in real Hilbert spaces and establish the refined Ulam stability for these mappings.