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Keywords:
admissible space; first order linear differential equation
admissible space; first order linear differential equation
Summary:
We consider the Massera-Schäffer problem for the equation $$ -y'(x)+q(x)y(x)=f(x),\quad x\in \mathbb R, $$ where $f\in L_p^{\rm loc}(\mathbb R),$ $p\in [1,\infty )$ and $0\le q\in L_1^{\rm loc}(\mathbb R).$ By a solution of the problem we mean any function $y,$ absolutely continuous and satisfying the above equation almost everywhere in $\mathbb R.$ Let positive and continuous functions $\mu (x)$ and $\theta (x)$ for $x\in \mathbb R$ be given. Let us introduce the spaces \begin {eqnarray*} L_p(\mathbb R,\mu )&=\biggl \{ f\in L_p^{\rm loc}(\mathbb R) \colon \|f\|_{L_p(\mathbb R,\mu )}^p=\int _{-\infty }^\infty |\mu (x)f(x)|^p {\rm d} x<\infty \biggr \},\\ L_p(\mathbb R,\theta )&=\biggl \{f\in L_p^{\rm loc}(\mathbb R) \colon \|f\|_{L_p(\mathbb R,\theta )}^p=\int _{-\infty }^\infty |\theta (x)f(x)|^p {\rm d} x<\infty \biggr \}. \end {eqnarray*} We obtain requirements to the functions $\mu $, $\theta $ and $q$ under which (1) for every function $f\in L_p(\mathbb R,\theta )$ there exists a unique solution $y\in L_p(\mathbb R,\mu )$ of the above equation; (2) there is an absolute constant $c(p)\in (0,\infty )$ such that regardless of the choice of a function $f\in L_p(\mathbb R,\theta )$ the solution of the above equation satisfies the inequality $$\|y\|_{L_p(\mathbb R,\mu )}\le c(p)\|f\|_{L_p(\mathbb R,\theta )}.$$
References:
[1] Chernyavskaya, N. A.: 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. A., Shuster, L. A.: Conditions for correct solvability of a simplest singular boundary value problem of general form. I. Z. Anal. Anwend. 25 (2006), 205-235. DOI 10.4171/ZAA/1285 | MR 2229446 | Zbl 1122.34021
[3] Chernyavskaya, N. A., Shuster, L. A.: Spaces admissible for the Sturm-Liouville equation. Commun. Pure Appl. Anal. 17 (2018), 1023-1052. DOI 10.3934/cpaa.2018050 | MR 3809112 | Zbl 1397.34052
[4] Courant, R.: Differential and Integral Calculus. I. Blackie & Son, Glasgow (1945). DOI 10.1002/9781118033241 | MR 1009558 | Zbl 0018.30001
[5] Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific, Singapore (2003). DOI 10.1142/5129 | MR 1982932 | Zbl 1065.26018
[6] Lukachev, M., Shuster, L.: On uniqueness of the solution of a linear differential equation without boundary conditions. Funct. Differ. Equ. 14 (2007), 337-346. MR 2323215 | Zbl 1148.34303
[7] Massera, J. L., Schäffer, J. J.: Linear Differential Equations and Function Spaces. Pure and Applied Mathematics 21. Academic Press, New York (1966). MR 0212324 | Zbl 0243.34107
[8] Mynbaev, K. T., Otelbaev, M. O.: Weighted Function Spaces and the Spectrum of Differential Operators. Nauka, Moskva (1988), Russian. MR 0950172 | Zbl 0651.46037
[9] Titchmarsh, E. C.: The Theory of Functions. Clarendon Press, Oxford (1932). MR 3728294 | Zbl 0005.21004
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