Correct solvability of a general differential equation of the first order in the space $L_p(\mathbb{R})$

N. Chernyavskaya and L.A. Shuster

N. Chernyavskaya, Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105, Israel
L.N. Shuster, Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel


Abstract: We consider the equation \begin{equation} - r(x)y^{\prime }(x)+q(x)y(x)=f(x)\,,\quad x\in \mathbb{R} \end{equation} where $f\in L_p(\mathbb{R}) $, $p\in [1,\infty ]$ ($L_\infty (\mathbb{R}):=C(\mathbb{R})$) and \begin{equation} 0<r\in C^{}(\mathbb{R})\,,\quad 0\le q\in L_1^{}(\mathbb{R})\,. \end{equation} We obtain minimal requirements to the functions $r$ and $q$, in addition to (2), under which equation (1) is correctly solvable in $L_p(\mathbb{R})$, $p\in [1,\infty ]$.

AMSclassification: primary 46E35.

Keywords: correct solvability, differential equation of the first order.

DOI: 10.5817/AM2015-2-87