TY - JOUR
T1 - Conditions for correct solvability of a simplest singular boundary value problem
AU - Chernyavskaya, Nina
PY - 2002/9/24
Y1 - 2002/9/24
N2 - We consider a boundary value problem (0.1) -y′ (x) + q(x)y(x) = f(x), x ∈ R, lim|x|→∞ y(x) = 0, where f(x) ∈ Lp(R), p ∈ [1, ∞] (L∞(R) := C(R)) and 0 ≤ q(x) ∈ L1loc(R). Boundary value problem (0.1) is called correctly solvable in the given space Lp(R) if for any f(x) ∈ Lp(R) there is a unique solution y(x) ∈ Lp(R) and the following inequality ∥y∥p ≤ c(P) ∥f∥p, for all f(x) ∈ Lp(R), holds with absolute constant c(p) ∈ (0, ∞). We find criteria for correct solvability of the problem (0.1) in Lp(R).
AB - We consider a boundary value problem (0.1) -y′ (x) + q(x)y(x) = f(x), x ∈ R, lim|x|→∞ y(x) = 0, where f(x) ∈ Lp(R), p ∈ [1, ∞] (L∞(R) := C(R)) and 0 ≤ q(x) ∈ L1loc(R). Boundary value problem (0.1) is called correctly solvable in the given space Lp(R) if for any f(x) ∈ Lp(R) there is a unique solution y(x) ∈ Lp(R) and the following inequality ∥y∥p ≤ c(P) ∥f∥p, for all f(x) ∈ Lp(R), holds with absolute constant c(p) ∈ (0, ∞). We find criteria for correct solvability of the problem (0.1) in Lp(R).
KW - Correct solvability
KW - First order linear differential equation
UR - http://www.scopus.com/inward/record.url?scp=0036041036&partnerID=8YFLogxK
U2 - 10.1002/1522-2616(200209)243:1<5::AID-MANA5>3.0.CO;2-B
DO - 10.1002/1522-2616(200209)243:1<5::AID-MANA5>3.0.CO;2-B
M3 - Article
AN - SCOPUS:0036041036
SN - 0025-584X
VL - 243
SP - 5
EP - 18
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
ER -