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Solvability in L p of the Neumann problem for a singular non‐homogeneous Sturm‐Liouville equation
Author(s) -
Chernyavskaya N.,
Shuster L.
Publication year - 1999
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300007919
Subject(s) - mathematics , homogeneous , mathematical analysis , inversion (geology) , constant (computer programming) , sturm–liouville theory , neumann boundary condition , function (biology) , pure mathematics , combinatorics , boundary value problem , paleontology , structural basin , evolutionary biology , computer science , biology , programming language
Consider the equation 1 −( r ( x ) y ′ ( x ) ) ′ + q ( x ) y ( x ) = f ( x ) , x ∈ Rwhere f ( x ) ∈ L s ( R ) , s ∈ [ 1 , ∞ ] , r ( x ) > 0 , q ( x ) ⩾ 0 for x ∈ R , 1 / r ( x ) ∈ L 1 loc( R ) , q ( x ) ∈ L 1 loc( R ) . The inversion problem for (1) is called regular in L p if, uniformly in p ∈[1, ∞] for any f ( x )∈ L p ( R ), equation (1) has a unique solution y ( x )∈ L p ( R ) of the form y ( x ) =∫ − ∞ ∞ G ( x , t ) f ( t ) d t, x ∈ Rwith‖ y ‖ p ⩽ c‖ f ‖ p . Here G ( x , t ) is the Green function corresponding to (1) and c is an absolute constant. For a given s ∈[l, ∞], necessary and sufficient conditions are obtained for assertions (2) and (3) to hold simultaneously: (2) the inversion problem for (1) is regular in L p ; (3)lim| x | → ∞ r ( x ) y ′ ( x ) = 0 for any f ( x )∈ L S ( R ).