Abstract
In this paper, we investigate the general solution and the generalized stability for the quartic, cubic and additive functional equation (briefly, QCA-functional equation)
for any k∈ℤ-{0,±1} in Menger probabilistic normed spaces.
AbstractIn this paper, we obtain Hyers-Ulam stability of the functional equationsf (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w),f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) + 2f (y, w)andf (x + y, z − w) + f (x − y, z + w) = 2f (x, z) − 2f (y, w)in 2-Banach spaces. The quadratic forms ax2 + bxy + cy2, ax2 + by2 and axy are solutions of the above functional equations, respectively.
In this paper, we solve the following additive ?-functional inequalities ||f
(x + y) - f (x) - f (y)|| ? ???(2f (x+y/2) - f(x) + -f (y))??, (1) where ? is
a fixed complex number with |?|<1, and ??2f(x+y/2)-f(x)-
f(y)???||?(f(x+y)-f(x)-f(y))||, (2) where ? is a fixed complex number with
|?|<1/2 , and prove the Hyers-Ulam stability of the additive ?-functional
inequalities (1) and (2) in ?-homogeneous complex Banach spaces and prove the
Hyers-Ulam stability of additive ?-functional equations associated with the
additive ?-functional inequalities (1) and (2) in ?-homogeneous complex
Banach spaces.
In this paper, the Hyers–Ulam–Rassias stabilities of two functional equations,
f
a
x
+
b
y
=
r
f
x
+
s
f
y
and
f
x
+
y
+
z
=
2
f
x
+
y
/
2
+
f
z
, are investigated in the framework of fuzzy normed spaces.
We establish stability results concerning the additive and quadratic functional equations in complete Menger ϕ-normed spaces by using fixed point theory. As particular cases, some theorems regarding the stability of functional equations in β - normed and quasi-normed spaces are obtained.
We obtain the general solution of the generalized mixed additive and quadratic functional equationfx+my+fx−my=2fx−2m2fy+m2f2y,mis even;fx+y+fx−y−2m2−1fy+m2−1f2y,mis odd, for a positive integerm. We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces whenmis an even positive integer orm=3.
Hyers-Ulam stability of quadratic forms in 2-normed spaces