A FIXED POINT APPROACH TO THE STABILITY OF EULER-LAGRANGE SEXTIC (a, b)-FUNCTIONAL EQUATIONS IN ARCHIMEDEAN AND NON-ARCHIMEDEAN BANACH SPACES

Abstract

In this paper, we present a fixed point method to prove the Hyers-Ulam stability of the system of Euler-Lagrange quadratic-quartic functional equations

{f(ax(1) + bx(2), y) + f(bx(1) + ax(2), y) + ab f(x(1) - x(2), y) = (a(2) + b(2))[f(x(1), y) + f(x(2), y)] + 4ab f(x(1)+x(2)/2, y), f(x, ay(1) + by(2)) + f(x, by(1) + ay(2)) + 1/2ab(a - b)(2) f(x, y(1) - y(2)) = (a(2) -b(2))(2)[f(x, y(1)) + f(x, y(2))] + 8ab f(x, y(1)+y(2)/2) for all numbers a and b with a + b is not an element of {0, +/- 1}, ab + 2 not equal 2(a + b)(2) and ab(a - b)(2) + 4 not equal 4(a + b)(4) in Archimedean and non-Archimedean Banach spaces and we show that the approximation in non-Archimedean Banach spaces is better than the approximation in (Archimedean) Banach spaces.