Geodesic
Methods of analysis of the condition for correct solvability in $L_p (\mathbb R)$ of general Sturm-Liouville equations
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 1067-1098 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We consider the equation $$\label {1} - (r(x)y'(x))'+q(x)y(x)=f(x),\quad x\in \mathbb R \eqno {(*)} $$ where $f\in L_p(\mathbb R)$, $p\in (1,\infty )$ and \begin {gather} r>0,\quad q\ge 0,\quad \frac {1}{r}\in L_1^{\rm loc}(\mathbb R),\quad q\in L_1^{\rm loc}(\mathbb R), \nonumber \\ \lim _{|d|\to \infty }\int _{x-d}^x \frac {{\rm d} t}{r(t)}\cdot \int _{x-d}^x q(t) {\rm d} t=\infty . \nonumber \end {gather} In an earlier paper, we obtained a criterion for correct solvability of ($*$) in $L_p(\mathbb R),$ $p\in (1,\infty ).$ 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 $f(x)$ for $x\in (a,b)$ through a function $g(x)$ is sharp by order if $c^{-1}|g(x)|\le |f(x)|\le c|g(x)|,$ $x\in (a,b),$ $c=\rm const$) of auxiliary functions, which guarantee efficient study of the problem of correct solvability of ($*$) in $L_p(\mathbb R),$ $p\in (1,\infty ).$
We consider the equation $$\label {1} - (r(x)y'(x))'+q(x)y(x)=f(x),\quad x\in \mathbb R \eqno {(*)} $$ where $f\in L_p(\mathbb R)$, $p\in (1,\infty )$ and \begin {gather} r>0,\quad q\ge 0,\quad \frac {1}{r}\in L_1^{\rm loc}(\mathbb R),\quad q\in L_1^{\rm loc}(\mathbb R), \nonumber \\ \lim _{|d|\to \infty }\int _{x-d}^x \frac {{\rm d} t}{r(t)}\cdot \int _{x-d}^x q(t) {\rm d} t=\infty . \nonumber \end {gather} In an earlier paper, we obtained a criterion for correct solvability of ($*$) in $L_p(\mathbb R),$ $p\in (1,\infty ).$ 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 $f(x)$ for $x\in (a,b)$ through a function $g(x)$ is sharp by order if $c^{-1}|g(x)|\le |f(x)|\le c|g(x)|,$ $x\in (a,b),$ $c=\rm const$) of auxiliary functions, which guarantee efficient study of the problem of correct solvability of ($*$) in $L_p(\mathbb R),$ $p\in (1,\infty ).$
DOI : 10.1007/s10587-014-0154-1
Classification : 34B24
Keywords: correct solvability; Sturm-Liouville equation 
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     title = {Methods of analysis of the condition for correct solvability in $L_p (\mathbb R)$ of general {Sturm-Liouville} equations},
     journal = {Czechoslovak Mathematical Journal},
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Chernyavskaya, Nina A.; Shuster, Leonid A. Methods of analysis of the condition for correct solvability in $L_p (\mathbb R)$ of general Sturm-Liouville equations. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 1067-1098. doi: 10.1007/s10587-014-0154-1

[1] Chernyavskaya, N. A., El-Natanov, N., Shuster, L. A.: Weighted estimates for solutions of a Sturm-Liouville equation in the space $L_1(\mathbb R)$. Proc. R. Soc. Edinb., Sect. A, Math. 141 (2011), 1175-1206. | MR

[2] Chernyavskaya, N., Shuster, L.: A criterion for correct solvability in $L_p(\mathbb R)$ of a general Sturm-Liouville equation. J. Lond. Math. Soc., II. Ser. 80 (2009), 99-120. | DOI | MR

[3] Chernyavskaya, N., Shuster, L.: A criterion for correct solvability of the Sturm-Liouville equation in the space $L_p(R)$. Proc. Am. Math. Soc. 130 (2002), 1043-1054. | DOI | MR | Zbl

[4] Chernyavskaya, N., Shuster, L.: Regularity of the inversion problem for a Sturm-Liouville equation in $L_p(\mathbb R)$. Methods Appl. Anal. 7 (2000), 65-84. | MR

[5] Chernyavskaya, N., Shuster, L.: Estimates for the Green function of a general Sturm-Liouville operator and their applications. Proc. Am. Math. Soc. 127 (1999), 1413-1426. | DOI | MR | Zbl

[6] Chernyavskaya, N., Shuster, L.: Solvability in $L_p$ of the Neumann problem for a singular non-homogeneous Sturm-Liouville equation. Mathematika 46 (1999), 453-470. | DOI | MR

[7] Chernyavskaya, N., Shuster, L.: Solvability in $L_p$ of the Dirichlet problem for a singular nonhomogeneous Sturm-Liouville equation. Methods Appl. Anal. 5 (1998), 259-272. | MR | Zbl

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

[9] Titchmarsh, E. C.: The Theory of Functions. (2. ed.). University Press, Oxford (1939). | MR

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