The inverse and direct problem for equation of the first kind of convolution on the half-line

In this paper we study the integral equation first kind of convolution on the semi-infinite interval. The next two tasks are:

• Task is reconstruction of history. From the integral equation it is required to find two functions $u(t)$ for $t>0$ and $f(t)$ for $0<t<b$ for given values of the right side of the equation $f(t)$ for $t>b$ ($u(t)$ — solution of integral equation).
• The problem of inversion of the integral operator.

Uniqueness theorems are proved, necessary and sufficient conditions for solvability are found, explicit formulas for solutions are received.