Geodesic
The Massera-Schäffer problem for a first order linear differential equation
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 2, pp. 477-511
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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 $$\|y\|_{L_p(\mathbb R,\mu )}\le c(p)\|f\|_{L_p(\mathbb R,\theta )}.$$
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 $$\|y\|_{L_p(\mathbb R,\mu )}\le c(p)\|f\|_{L_p(\mathbb R,\theta )}.$$
DOI : 10.21136/CMJ.2021.0548-20
Classification : 34A30
Keywords: admissible space; first order linear differential equation 
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     title = {The {Massera-Sch\"affer} problem for a first order linear differential equation},
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Chernyavskaya, Nina A.; Shuster, Leonid A. The Massera-Schäffer problem for a first order linear differential equation. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 2, pp. 477-511. doi: 10.21136/CMJ.2021.0548-20

[1] Chernyavskaya, N. A.: Conditions for correct solvability of a simplest singular boundary value problem. Math. Nachr. 243 (2002), 5-18. | DOI | MR | JFM

[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 | MR | JFM

[3] Chernyavskaya, N. A., Shuster, L. A.: Spaces admissible for the Sturm-Liouville equation. Commun. Pure Appl. Anal. 17 (2018), 1023-1052. | DOI | MR | JFM

[4] Courant, R.: Differential and Integral Calculus. I. Blackie & Son, Glasgow (1945). | DOI | MR | JFM

[5] Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific, Singapore (2003). | DOI | MR | JFM

[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 | JFM

[7] Massera, J. L., Schäffer, J. J.: Linear Differential Equations and Function Spaces. Pure and Applied Mathematics 21. Academic Press, New York (1966). | MR | JFM

[8] Mynbaev, K. T., Otelbaev, M. O.: Weighted Function Spaces and the Spectrum of Differential Operators. Nauka, Moskva (1988), Russian. | MR | JFM

[9] Titchmarsh, E. C.: The Theory of Functions. Clarendon Press, Oxford (1932). | MR | JFM

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