# Article

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Keywords:
correct solvability; differential equation of the first order
Summary:
We consider the equation $$- r(x)y^{\prime }(x)+q(x)y(x)=f(x)\,,\quad x\in \mathbb{R}$$ where $f\in L_p(\mathbb{R})$, $p\in [1,\infty ]$ ($L_\infty (\mathbb{R}):=C(\mathbb{R})$) and $$0<r\in C^{}(\mathbb{R})\,,\quad 0\le q\in L_1^{}(\mathbb{R})\,.$$ We obtain minimal requirements to the functions $r$ and $q$, in addition to (), under which equation () is correctly solvable in $L_p(\mathbb{R})$, $p\in [1,\infty ]$.
References:
[1] Chernyavskaya, N.: Conditions for correct solvability of a simplest singular boundary value problem. Math. Nachr. 243 (2002), 5–18. DOI 10.1002/1522-2616(200209)243:1<5::AID-MANA5>3.0.CO;2-B | MR 1923831 | Zbl 1028.34018
[2] Chernyavskaya, N., Shuster, L.: Conditions for correct solvability of a simplest singular boundary value problem of general form. I. Z. Anal. Anwendungen 25 (2006), 205–235. DOI 10.4171/ZAA/1285 | MR 2229446 | Zbl 1122.34021
[3] Chernyavskaya, N., Shuster, L.: Conditions for correct solvability of a simplest singular boundary value problem of general form. II. Z. Anal. Anwendungen 26 (2007), 439–458. DOI 10.4171/ZAA/1334 | MR 2341766 | Zbl 1139.34010
[4] Kantorovich, L.W., Akilov, G.P.: Functional Analysis. Nauka, Moscow, 1977. MR 0511615
[5] Lukachev, M., Shuster, L.: On uniqueness of soltuion of a linear differential equation without boundary conditions. Funct. Differ. Equ. 14 (2007), 337–346. MR 2323215
[6] Mynbaev, K., Otelbaev, M.: Weighted Function Spaces and the Spectrum of Differential Operators. Nauka, Moscow, 1988. MR 0950172
[7] Opic, B., Kufner, A.: Hardy Type Inequalities. Pitman Research Notes in Mathematics Series, vol. 219, Harlow, Longman, 1990. MR 1069756 | Zbl 0698.26007
[8] Otelbaev, M.: Estimates of the Spectrum of the Sturm-Liouville Operator. Alma-Ata, Gilim, 1990, in Russian. Zbl 0747.47029

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