  In this paper the convolution integrals $\int_0^t(t-s)^{\lambda -1}b(s)ds$
with hyper-singular kernels are considered, where $\lambda\le 0$ and $b$ is a
smooth or $b$ is in $L^1(\mathbb{R}_+)$. For such $\lambda$ these integrals
diverge classically even for smooth $b$. These convolution integrals are
defined in this paper for $\lambda\le 0$, $\lambda\neq 0,-1,-2,...$. Integral
equations and inequalities are considered with the hyper-singular kernels
$(t-s)^{\lambda -1}_+$ for $\lambda\le 0$, where $t^\lambda_+:=0$ for $t<0$. In
particular, one is interested in the value $\lambda=-\frac 14$ because it is
important for the Navier-Stokes problem (NSP). Integral equations of the type
$b(t)=b_0(t)+ \int_0^t(t-s)^{\lambda-1}b(s)ds$, $\lambda\le 0$, are studied.
The solution of these equations is investigated, existence and uniqueness of
the solution is proved for $\lambda=-\frac 1 4$. This special value of
$\lambda$ is of basic importance for a study of the Navier-Stokes problem
(NSP). The above results are applied to the analysis of the NSP in the space
$\mathbb{R}^3$ without boundaries. It is proved that the NSP is contradictory
in the following sense: even if one assumes that the initial data
$v_0(x):=v(x,0)\not\equiv 0$, $\nabla \cdot v_0(x)=0$ one proves that the
solution $v(x,t)$ to the NSP has the property $v(x,0)=0$. This paradox shows
that the NSP is not a correct description of the fluid mechanics problem and it
proves that the NSP does not have a solution.
