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- W3204703501 abstract "We consider the Massera-Schaffer problem for the equation $$ - yprime (x) + q(x)y(x) = f(x),,,,,,x in mathbb{R},$$ where $$f in L_p^{{rm{loc}}}(mathbb{R})$$ , p ∈ [1, ∞) and $$0 leqslant q in L_p^{{rm{loc}}}(mathbb{R})$$ . By a solution of the problem we mean any function y, absolutely continuous and satisfying the above equation almost everywhere in $$mathbb{R}$$ . Let positive and continuous functions μ(x) and θ(x) for $$x in mathbb{R}$$ be given. Let us introduce the spaces $$matrix{{{L_p}(mathbb{R},mu ) = left{ {f in L_p^{{rm{loc}}}(mathbb{R}):left| f right|_{{L_p}(mathbb{R},mu )}^p = int_{ - infty }^infty {{{left| {mu (x)f(x)} right|}^p}{rm{d}}x < infty } } right},} hfill cr {{L_p}(mathbb{R},theta ) = left{ {f in L_p^{{rm{loc}}}(mathbb{R}):left| f right|_{{L_p}(mathbb{R},theta )}^p = int_{ - infty }^infty {{{left| {theta (x)f(x)} right|}^p}{rm{d}}x < infty } } right}.} hfill cr } $$ We obtain requirements to the functions μ θ and q under which (1) for every function $$f in {L_p}(mathbb{R},theta )$$ there exists a unique solution $$y in {L_p}(mathbb{R},mu )$$ of the above equation; (2) there is an absolute constant c(p) ∈ (0, ∞) such that regardless of the choice of a function $$f in {L_p}(mathbb{R},theta )$$ the solution of the above equation satisfies the inequality $${left| y right|_{{L_p}(mathbb{R},mu )}} leqslant c(p){left| f right|_{{L_p}(mathbb{R},theta ).}}$$ ." @default.
- W3204703501 created "2021-10-11" @default.
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- W3204703501 date "2021-10-01" @default.
- W3204703501 modified "2023-10-11" @default.
- W3204703501 title "The Massera-Schäffer problem for a first order linear differential equation" @default.
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- W3204703501 doi "https://doi.org/10.21136/cmj.2021.0548-20" @default.
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