The Massera-Schäffer problem for a first order linear differential equation

Chernyavskaya, Nina A. ; Shuster, Leonid A.

Czechoslovak Mathematical Journal, Tome 72 (2022) no. 2, pp. 477-511

Résumé

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 $$\|y\|_{L_p(\mathbb R,\mu )}\le c(p)\|f\|_{L_p(\mathbb R,\theta )}.$$

Détail

DOI : 10.21136/CMJ.2021.0548-20

Classification : 34A30
Keywords: admissible space; first order linear differential equation