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Let D'L∞(R2n) and A'L∞(R2n) be the spaces of bounded distributions and bounded hyperfunctions respectively. In this paper we consider the Ulam's stability of the sine-cosine functional equation u ...T - u ⊗ v + v ⊗ u ∈ D'L∞(R2n) [resp. A'L∞(R2n)], where u, v are Gelfand hyperfunctions, T : R2n → Rn such that T(x, y) = x - y for all x, y ∈ Rn, and ... ⊗ denote pullback and tensor product of generalized functions respectively. [ABSTRACT FROM AUTHOR]

Let $G$ be a commutative group and $\mathbb{C}$ the field of complex numbers, $\mathbb{R}^{+}$ the set of positive real numbers and $f,g,h,k:G\times \mathbb{R}^{+}\rightarrow \mathbb{C}$. In this paper, we first consider the Levi-Civitá functional inequality $$\begin{eqnarray}\displaystyle f(x+y,t+s)-g(x,t)h(y,s)-k(y,s)\leq {\rm\Phi}(t,s),\quad x,y\in G,t,s>0, & & \displaystyle \nonumber\end{eqnarray}$$ where ${\rm\Phi}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ is a symmetric decreasing function in the sense that ${\rm\Phi}(t_{2},s_{2})\leq {\rm\Phi}(t_{1},s_{1})$ for all $0