This paper calculates the exact quadratic variation in space and quartic variation in time for the solutions to a one dimensional stochastic heat equation driven by a multiplicative space-time white noise.

In this paper, we obtain the general solution and investigate the generalized Hyers-Ulam stability of the new generalized mixed type additive and quadratic functional equation in fuzzy normed space. Mathematics Subject Classification 39B55, 39B52, 39B82

The stability problem of functional equations originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms. Let G1, · be a group and let G2, ∗, d be a metric group with the metric d ·, · . Given > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d h x · y , h x ∗ h y < δ for all x, y ∈ G1, then there exists a homomorphism H...

In the paper we establish the general solution of the function equation f(2x+y)+f(2x-y) = f(x+y)+f(x-y)+2f(2x)-2f(x) and investigate the Hyers-Ulam-Rassias stability of this equation in 2-Banach spaces.

In this paper, we solve the quadratic $alpha$-functional equations $2f(x) + 2f(y) = f(x + y) + alpha^{-2}f(alpha(x-y)); (0.1)$ where $alpha$ is a fixed non-Archimedean number with $alpha^{-2}neq 3$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the quadratic $alpha$-functional equation (0.1) in non-Archimedean Banach spaces.