We will show the general solution of the functional equation f(x + ay) + f(x− ay) + 2(a − 1)f(x) = af(x + y) + af(x− y) + 2a(a − 1)f(y) and investigate the Hyers-Ulam stability of the quartic set-valued functional equation.

We present a computational approach for finding all integral solutions of the equation y2 = 1k + 2k + · · ·+xk for even values of k. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we deter...

We present a computational approach for finding all integral solutions of the equation y2 = 1k + 2k + · · ·+xk for even values of k. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we deter...

In this paper, the authors investigate the generalized Ulam-Hyers stability of a Leibniz type additive and quadratic functional equation ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + + − + − + − 3 2 3 3 = ) ( ) ( ) ( z y x f t z y x f t z f t y f t x f ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − + 3 2 3 2 z y x f z y x f in the setting of intuitionistic fuzzy normed spaces using direct and fixed point methods.

in this paper, we consider orthogonal stability of mixed type additive and cubic functional equation of the form $$f(2x+y)+f(2x-y)-f(4x)=2f (x+y)+2f(x-y)-8f(2x) +10f(x)-2f(-x),$$ with $xbot y$, where $bot$  is orthogonality in the sense of ratz.