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We consider the equation $$-{\left(r\left(x\right)y^{\text{'}}\left(x\right)\right)}^{\text{'}}+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ,$$ where $f\in L_p\left(ℝ\right)$, $p\in (1,\infty )$ and $$r>0,\frac{1}{r}\in L_1^{\mathrm{loc}}\left(ℝ\right),q\in L_1^{\mathrm{loc}}\left(ℝ\right).$$ For particular equations of this form, we suggest some methods for the study of the question on requirements to the functions $r$ and $q$ under which the above equation is correctly solvable in the space $L_p\left(ℝ\right),$ $p\in (1,\infty ).$

We consider the equation $$-{\left(r\left(x\right)y^{\text{'}}\left(x\right)\right)}^{\text{'}}+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ(*)$$ where $f\in L_p\left(ℝ\right)$, $p\in (1,\infty )$ and $$\begin{array}{c}r>0,q\ge 0,\frac{1}{r}\in L_1^{\mathrm{loc}}\left(ℝ\right),q\in L_1^{\mathrm{loc}}\left(ℝ\right),\\ \underset{\left|d\right|\to \infty }{lim}{\int }_{x-d}^x\frac{\mathrm{d}t}{r\left(t\right)}·{\int }_{x-d}^{x}q\left(t\right)\mathrm{d}t=\infty .\end{array}$$ In an earlier paper, we obtained a criterion for correct solvability of ($*$) in $L_p\left(ℝ\right),$ $p\in (1,\infty ).$ In this criterion, we use values of some auxiliary implicit functions in the coefficients $r$ and $q$ of equation ($*$). Unfortunately, it is usually impossible to compute values of these functions. In the present paper we obtain sharp by order, two-sided estimates (an estimate of a function $f\left(x\right)$ for $x\in (a,b)$ through a function $g\left(x\right)$ is sharp by order if $c^{-1}\left|g\left(x\right)\right|\le \left|f\left(x\right)\right|\le c\left|g\left(x\right)\right|,$ $x\in (a,b),$...

We consider the Massera-Schäffer problem for the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ,$$ where $f\in L_p^{\mathrm{loc}}\left(ℝ\right),$ $p\in [1,\infty )$ and $0\le q\in L_1^{\mathrm{loc}}\left(ℝ\right).$ By a solution of the problem we mean any function $y,$ absolutely continuous and satisfying the above equation almost everywhere in $ℝ.$ Let positive and continuous functions $\mu \left(x\right)$ and $\theta \left(x\right)$ for $x\in ℝ$ be given. Let us introduce the spaces $$\begin{array}{cc}\hfill L_p(ℝ,\mu )& =\left\{f\in L_p^{\mathrm{loc}}{\left(ℝ\right):\parallel f\parallel }_{L_p(ℝ,\mu )}^p={\int }_{-\infty }^{\infty }{\left|\mu \left(x\right)f\left(x\right)\right|}^p\mathrm{d}x<\infty \right\},\\ \hfill L_p(ℝ,\theta )& =\left\{f\in L_p^{\mathrm{loc}}{\left(ℝ\right):\parallel f\parallel }_{L_p(ℝ,\theta )}^p={\int }_{-\infty }^{\infty }{\left|\theta \left(x\right)f\left(x\right)\right|}^p\mathrm{d}x<\infty \right\}.\end{array}$$ We obtain requirements to the functions $\mu$, $\theta$ and $q$ under which (1) for every function $f\in L_p(ℝ,\theta )$ there exists a unique solution $y\in L_p(ℝ,\mu )$ of the above equation; (2) there is an absolute constant $c\left(p\right)\in (0,\infty )$ such...

We consider the equation $$-r\left(x\right)y^{\text{'}}\left(x\right)+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ$$ where $f\in L_p\left(ℝ\right)$, $p\in [1,\infty ]$ ($L_{\infty }\left(ℝ\right):=C\left(ℝ\right)$) and $$0<r\in C^{}\left(ℝ\right),0\le q\in L_1^{}\left(ℝ\right).$$ We obtain minimal requirements to the functions $r$ and $q$, in addition to (), under which equation () is correctly solvable in $L_p\left(ℝ\right)$, $p\in [1,\infty ]$.

We consider the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y(x-\varphi \left(x\right))=f\left(x\right),x\in ℝ,$$ where $\varphi$ and $q$ ($q\ge 1$) are positive continuous functions for all $x\in ℝ$ and $f\in C\left(ℝ\right)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $ℝ$, which satisfies the equation for all $x\in ℝ$. We show that under certain additional conditions on the functions $\varphi$ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$\parallel y^{\text{'}}{\parallel }_{C\left(ℝ\right)}+{\parallel qy\parallel }_{C\left(ℝ\right)}\le c{\parallel f\parallel }_{C\left(ℝ\right)},$$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.