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
$0<t_{1}\leq t_{2}$
and
$0<s_{1}\leq s_{2}$
. As an application, we solve the Hyers–Ulam stability problem of the Levi-Civitá functional equation
$$\begin{eqnarray}\displaystyle u\circ S-v\otimes w-k\circ {\rm\Pi}\in {\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})\quad [\text{respectively}\;{\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})] & & \displaystyle \nonumber\end{eqnarray}$$
in the space of Gelfand hyperfunctions, where
$u,v,w,k$
are Gelfand hyperfunctions,
$S(x,y)=x+y,{\rm\Pi}(x,y)=y,x,y\in \mathbb{R}^{n}$
, and
$\circ$
,
$\otimes$
,
${\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$
and
${\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$
denote pullback, tensor product and the spaces of bounded distributions and bounded hyperfunctions, respectively.