Admissible pair of spaces for non-correctly solvable linear differential equations

• Autorzy: Chernyavskaya N. A. , Dorel L. S. , Shuster L. A.

Identyfikatory

DOI

10.1515/jaa-2016-0001

• Języki publikacji: EN

Abstrakty

EN

We consider the differential equation −y′(x)+q(x)y(x)=f(x),x∈R, where f∈L_p (R), p∈[1,∞), and 0≤q∈L^loc₁ (R), ∫⁰_−∞q(t)dt=∫^∞₀q(t)dt=∞, q0(a)=inf_x∈R∫^x+a_x−aq(t)dt=0 for any a∈(0,∞). Under these conditions, the above equation is not correctly solvable in L_p (R) for any p∈[1,∞). Let q^∗(x) be the Otelbaev-type average of the function q(t), t∈R, at the point t=x; let θ(x) be a continuous positive function for x∈R, and L_p,θ (R)={f∈L^loc_p (R):∫ ^∞_−∞|θ(x)f(x)|pdx<∞},∥f∥_p,θ:=∥f∥_Lp,θ(R) =(∫ ^∞_−∞|θ(x)f(x)| ^pdx) ^1/p. We show that if there exists a constant c∈[1,∞) such that the inequality c⁻¹q^∗(x)≤θ(x)≤cq^∗(x) holds for all x∈R, then under some additional conditions for q the pair of spaces {L_p,θ (R);L_p (R)} is admissible for the considered equation.

Słowa kluczowe

EN

linear differential equations; admissible pair; non-correctly solvable differential equation

PL

liniowe równanie różniczkowe; para dopuszczalna; równanie bez rozwiązania

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2016
• Tom: Vol. 22, nr 1
• Strony: 1--14
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Chernyavskaya N. A.

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

autor

Dorel L. S.

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

autor

Shuster L. A.

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

Bibliografia

• 1 N. Chernyavskaya, Conditions for correct solvability of a simplest singular boundary value problem, Math. Nachr. 243 (2002), 1, 5–18.
• 2 N. Chernyavskaya, L. Dorel and L. Shuster, Admissible pair of spaces for not correctly solvable linear differential equations, preprint 2014, http://arxiv.org/abs/1409.7823.
• 3 N. Chernyavskaya and L. Shuster, Conditions for correct solvability of a simplest singular boundary value problem of general form, Z. Anal. Anwend. 25 (2006), 2, 205–235.
• 4 N. Chernyavskaya and L. Shuster, Weight estimates for solutions of linear singular differential equations of the first order and the Everitt–Giertz problem, Differential Integral Equations 25 (2012), 5–6, 467–504.
• 5 R. W. Cottle, Mathematical Programming. Essays in Honor of George B. Dantzig. Part II, Springer, Berlin, 1985.
• 6 M. Lukachev and L. Shuster, On uniqueness of the solution of a linear differential equation without boundary conditions, Funct. Differ. Equ. 14 (2007), 2, 337–346.
• 7 J. L. Massera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Pure Appl. Math. 21, Academic Press, New York, 1966.
• 8 K. T. Mynbaev and M. O. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988.
• 9 M. Otelbaev, The smoothness of the solution of differential equations, Izv. Acad. Nauk Kazak. SSR 5 (1977), 45–48.
• 10 E. C. Titchmarsh, The Theory of Functions, Oxford University Press, Oxford, 1939.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.