Institute of Mathematics

Abstract:  We consider the Massera-Schäffer problem for the equation $-y'(x)+q(x)y(x)=f(x)$, $x\in\mathbb R$, 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 $L_p(\mathbb R,\mu) =\biggl\{ f\in L_p^{\rm loc}(\mathbb R) \colon\|f\|_{L_p(\mathbb R,\mu)}^p=\int_{-\infty}^\infty|\mu(x)f(x)|^p {\rm d} x<\infty\biggr\}$, $L_p(\mathbb R,\theta) =\biggl\{f\in L_p^{\rm loc}(\mathbb R) \colon\|f\|_{L_p(\mathbb R,\theta)}^p=\int_{-\infty}^\infty|\theta(x)f(x)|^p {\rm d} x<\infty\biggr\}$. 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 $\|y\|_{L_p(\mathbb R,\mu)}\le c(p)\|f\|_{L_p(\mathbb R,\theta)}$.