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• W3136290419 abstract "Abstract We consider discrete Sturm–Liouville-type equations of the form $$begin{aligned} varDelta (r_nvarDelta x_n)=a_nf(x_{sigma (n)})+b_n. end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> We present a theory of asymptotic properties of solutions which allows us to control the degree of approximation. Namely, we establish conditions under which for a given sequence y which solves the equation $$varDelta (r_nvarDelta y_n)=b_n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , the above equation possesses a solution x with the property $$x_n=y_n+mathrm {o}(u_n)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where u is a given positive, nonincreasing sequence. The obtained results are applied to the study of asymptotically periodic solutions. Moreover, these results also allow us to obtain some nonoscillation criteria for the classical Sturm–Liouville equation." @default.
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• W3136290419 date "2021-03-19" @default.
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• W3136290419 title "Properties of Solutions of Generalized Sturm–Liouville Discrete Equations" @default.
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