American Institute of Mathematical Sciences

Advanced
May  2018, 17(3): 1023-1052. doi: 10.3934/cpaa.2018050

Spaces admissible for the Sturm-Liouville equation

N. A. Chernyavskaya 1, and  L. A. Shuster 2,

1. 

Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105, Israel
2. 

Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel

Received  August 2016 Revised  November 2017 Published  January 2018

We consider the equation
$-{y}''(x)+q(x)y(x)=f(x),\ \ \ \ x\in \mathbb{R}\text{ }\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$
where
$f∈ L_p^{\text{loc}}(\mathbb R),$
$p∈[1,∞)$
 and
$0≤ q∈ L_1^{\text{loc}}(\mathbb R).$
 By a solution of (1) we mean any function
$y,$
 absolutely continuous together with its derivative and satisfying (1) almost everywhere in
$\mathbb R.$
 Let positive and continuous functions
$μ(x)$
 and
$θ(x)$
 for
$x∈\mathbb R$
 be given. Let us introduce the spaces
$\begin{align} & {{L}_{p}}(\mathbb{R},\mu )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\mu )}^{p}=\int_{-\infty }^{\infty }{|}\mu (x)f(x){{|}^{p}}dx < \infty \right\}, \\ & {{L}_{p}}(\mathbb{R},\theta )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\theta )}^{p}=\int_{-\infty }^{\infty }{|}\theta (x)f(x){{|}^{p}}dx <\infty \right\}. \\ \end{align}$
In the present paper, we obtain requirements to the functions
$μ,θ$
 and
$q$
 under which
1) for every function
$f∈ L_p(\mathbb R,θ)$
 there exists a unique solution (1)
$y∈ L_p(\mathbb R,μ)$
 of (1);
2) there is an absolute constant
$c(p)∈(0,∞)$
 such that regardless of the choice of a function
$f∈ L_p(\mathbb R,θ)$
 the solution of (1) satisfies the inequality
$\|y\|_{L_p(\mathbb R,μ)}≤ c(p)\|f\|_{L_p(\mathbb R,θ)}.$
Keywords: Admissible spaces, Sturm-Liouville equation.
Mathematics Subject Classification: 34B05, 34B24, 34K06.
Citation: N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050
References:
[1]

N. Chernyavskaya and L. Shuster, On the WKB-method, Diff. Uravnenija, 25 (1989), 1826-1829.   Google Scholar

[2]

N. Chernyavskaya and L. Shuster, Estimates for the Green function of a general Sturm-Liouville operator and their applications, Proc. Amer. Math. Soc., 127 (1999), 1413-1426.   Google Scholar

[3]

N. Chernyavskaya and L. Shuster, A criterion for correct solvability of the Sturm-Liouville equation in the space Lp(R), Proc. Amer. Math. Soc., 130 (2002), 1043-1054.   Google Scholar

[4]

N. Chernyavskaya and L. Shuster, Classification of initial data for the Riccati equation, Boll. Unione Mat. Ital., 8 (2002), 511-525.   Google Scholar

[5]

N. Chernyavskaya and L. Shuster, Davies-Harrell representations, Otelbaev's inequalities and properties of solutions of Riccati equations, J. Math. Anal. Appl., 334 (2007), 998-1021.   Google Scholar

[6]

N. Chernyavskaya and L. Shuster, A criteria for correct solvability in Lp(R) of a general Sturm-Liouville equation, J. London Math. Soc. (2), 80 (2009), 99-120.   Google Scholar

[7]

R. Courant, Differential and Integral Calculus, Vol. Ⅱ, Blackie and Son, Glasgow and London, 1936.  Google Scholar

[8]

E. B. Davies and E. M. Harrell, Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66 (1987), 165-188.   Google Scholar

[9]

E. Goursat, A Course in Mathematical Analysis, Vol. 1, Ch. IV, $\S $75, New York, Dover Publications, 1959.  Google Scholar

[10]

L. W. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977.  Google Scholar

[11]

A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., 2003.  Google Scholar

[12]

J. L. Masssera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York -London, 1966.  Google Scholar

[13]

K. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988.  Google Scholar

[14]

M. Otelbaev, A criterion for the resolvent of a Sturm-Liouville operator to be a kernel, Math. Notes, 25 (1979), 296-297.   Google Scholar

[15]

C. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.  Google Scholar

show all references

References:
[1]

N. Chernyavskaya and L. Shuster, On the WKB-method, Diff. Uravnenija, 25 (1989), 1826-1829.   Google Scholar

[2]

N. Chernyavskaya and L. Shuster, Estimates for the Green function of a general Sturm-Liouville operator and their applications, Proc. Amer. Math. Soc., 127 (1999), 1413-1426.   Google Scholar

[3]

N. Chernyavskaya and L. Shuster, A criterion for correct solvability of the Sturm-Liouville equation in the space Lp(R), Proc. Amer. Math. Soc., 130 (2002), 1043-1054.   Google Scholar

[4]

N. Chernyavskaya and L. Shuster, Classification of initial data for the Riccati equation, Boll. Unione Mat. Ital., 8 (2002), 511-525.   Google Scholar

[5]

N. Chernyavskaya and L. Shuster, Davies-Harrell representations, Otelbaev's inequalities and properties of solutions of Riccati equations, J. Math. Anal. Appl., 334 (2007), 998-1021.   Google Scholar

[6]

N. Chernyavskaya and L. Shuster, A criteria for correct solvability in Lp(R) of a general Sturm-Liouville equation, J. London Math. Soc. (2), 80 (2009), 99-120.   Google Scholar

[7]

R. Courant, Differential and Integral Calculus, Vol. Ⅱ, Blackie and Son, Glasgow and London, 1936.  Google Scholar

[8]

E. B. Davies and E. M. Harrell, Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66 (1987), 165-188.   Google Scholar

[9]

E. Goursat, A Course in Mathematical Analysis, Vol. 1, Ch. IV, $\S $75, New York, Dover Publications, 1959.  Google Scholar

[10]

L. W. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977.  Google Scholar

[11]

A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., 2003.  Google Scholar

[12]

J. L. Masssera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York -London, 1966.  Google Scholar

[13]

K. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988.  Google Scholar

[14]

M. Otelbaev, A criterion for the resolvent of a Sturm-Liouville operator to be a kernel, Math. Notes, 25 (1979), 296-297.   Google Scholar

[15]

C. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.  Google Scholar

[1]

Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405
[2]

Peter Howard, Alim Sukhtayev. The Maslov and Morse indices for Sturm-Liouville systems on the half-line. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 983-1012. doi: 10.3934/dcds.2020068
[3]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
[4]

Guglielmo Feltrin. Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1083-1102. doi: 10.3934/cpaa.2017052
[5]

Rashad M. Asharabi, Jürgen Prestin. Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185
[6]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko, Xiao-Chuan Xu. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems & Imaging, 2020, 14 (1) : 153-169. doi: 10.3934/ipi.2019068
[7]

Elimhan N. Mahmudov. Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints. Journal of Industrial & Management Optimization, 2020, 16 (1) : 169-187. doi: 10.3934/jimo.2018145
[8]

Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066
[9]

Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171
[10]

Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407
[11]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255
[12]

SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026
[13]

Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993
[14]

Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043
[15]

Phuong Le. Liouville theorems for an integral equation of Choquard type. Communications on Pure & Applied Analysis, 2020, 19 (2) : 771-783. doi: 10.3934/cpaa.2020036
[16]

Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549
[17]

Sun-Yung Alice Chang, Yu Yuan. A Liouville problem for the Sigma-2 equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 659-664. doi: 10.3934/dcds.2010.28.659
[18]

Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947
[19]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262
[20]

Emile Franc Doungmo Goufo, Abdon Atangana. Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 645-662. doi: 10.3934/dcdss.2020035

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (100)
  • HTML views (318)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences