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

Chernyavskaya, Nina A., and Shuster, Leonid A.. "The Massera-Schäffer problem for a first order linear differential equation." Czechoslovak Mathematical Journal 72.2 (2022): 477-511. <http://eudml.org/doc/298310>.

@article{Chernyavskaya2022,
abstract = {We consider the Massera-Schäffer problem for the equation \[ -y^\{\prime \}(x)+q(x)y(x)=f(x),\quad x\in \mathbb \{R\}, \] 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 \lbrace f\in L\_p^\{\rm loc\}(\mathbb \{R\}) \colon \Vert f\Vert \_\{L\_p(\mathbb \{R\},\mu )\}^p=\int \_\{-\infty \}^\infty |\mu (x)f(x)|^p \{\rm d\} x<\infty \biggr \rbrace ,\\ L\_p(\mathbb \{R\},\theta )&=\biggl \lbrace f\in L\_p^\{\rm loc\}(\mathbb \{R\}) \colon \Vert f\Vert \_\{L\_p(\mathbb \{R\},\theta )\}^p=\int \_\{-\infty \}^\infty |\theta (x)f(x)|^p \{\rm d\} x<\infty \biggr \rbrace . \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 \[\Vert y\Vert \_\{L\_p(\mathbb \{R\},\mu )\}\le c(p)\Vert f\Vert \_\{L\_p(\mathbb \{R\},\theta )\}.\]},
author = {Chernyavskaya, Nina A., Shuster, Leonid A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {admissible space; first order linear differential equation},
language = {eng},
number = {2},
pages = {477-511},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Massera-Schäffer problem for a first order linear differential equation},
url = {http://eudml.org/doc/298310},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Chernyavskaya, Nina A.
AU - Shuster, Leonid A.
TI - The Massera-Schäffer problem for a first order linear differential equation
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 2
SP - 477
EP - 511
AB - We consider the Massera-Schäffer problem for the equation \[ -y^{\prime }(x)+q(x)y(x)=f(x),\quad x\in \mathbb {R}, \] 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 \lbrace f\in L_p^{\rm loc}(\mathbb {R}) \colon \Vert f\Vert _{L_p(\mathbb {R},\mu )}^p=\int _{-\infty }^\infty |\mu (x)f(x)|^p {\rm d} x<\infty \biggr \rbrace ,\\ L_p(\mathbb {R},\theta )&=\biggl \lbrace f\in L_p^{\rm loc}(\mathbb {R}) \colon \Vert f\Vert _{L_p(\mathbb {R},\theta )}^p=\int _{-\infty }^\infty |\theta (x)f(x)|^p {\rm d} x<\infty \biggr \rbrace . \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 \[\Vert y\Vert _{L_p(\mathbb {R},\mu )}\le c(p)\Vert f\Vert _{L_p(\mathbb {R},\theta )}.\]
LA - eng
KW - admissible space; first order linear differential equation
UR - http://eudml.org/doc/298310
ER -