A role of the coefficient of the differential term in qualitative theory of half-linear equations

Pavel Řehák

Displaying similar documents to “A role of the coefficient of the differential term in qualitative theory of half-linear equations”

How the constants in Hille-Nehari theorems depend on time scales.

Řehák, Pavel (2006)

Advances in Difference Equations [electronic only]

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On certain comparison theorems for half-linear dynamic equations on time scales.

Řehák, Pavel (2004)

Abstract and Applied Analysis

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Hille-Kneser-type criteria for second-order dynamic equations on time scales.

Erbe, Lynn, Peterson, Allan C., Saker, Samir H. (2006)

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Bounded oscillation of nonlinear neutral differential equations of arbitrary order

Yeter Ş. Yilmaz, Ağacik Zafer (2001)

Czechoslovak Mathematical Journal

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The paper is concerned with oscillation properties of $n$ -th order neutral differential equations of the form ${[x (t) + c x (τ (t))]}^{(n)} + q (t) f (x (σ (t))) = 0, t \geq t_{0} > 0,$ where $c$ is a real number with $| c | \neq 1$ , $q \in C ([t_{0}, \infty), ℝ)$ , $f \in C (ℝ, ℝ)$ , $τ, σ \in C ([t_{0}, \infty), ℝ_{+})$ with $τ (t) < t$ and ${lim}_{t \to \infty} τ (t) = {lim}_{t \to \infty} σ (t) = \infty$ . Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations...

Oscillatory properties of second order half-linear difference equations

Pavel Řehák (2001)

Czechoslovak Mathematical Journal

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We study oscillatory properties of the second order half-linear difference equation $Δ (r_{k} | Δ y_{k} |^{α - 2} Δ y_{k}) - p_{k} {| y_{k + 1} |}^{α - 2} y_{k + 1} = 0, α > 1 . (HL)$ It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation $Δ (r_{k} Δ y_{k}) - p_{k} y_{k + 1} = 0 .$ We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.

Forced oscillation of second order nonlinear dynamic equations on time scales.

Huang, M., Feng, W. (2008)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

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Oscillation of second-order forced nonlinear dynamic equations on time scales.

Saker, S.H. (2005)

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On the oscillation of solutions of third order linear difference equations of neutral type

Anna Andruch-Sobiło, Małgorzata Migda (2005)

Mathematica Bohemica

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In this note we consider the third order linear difference equations of neutral type $Δ^{3} [x (n) - p (n) x (σ (n))] + δ q (n) x (τ (n)) = 0, n \in N (n_{0}), (E)$ where $δ = \pm 1$ , $p, q N (n_{0}) \to ℝ_{+};$ $σ, τ N (n_{0}) \to ℕ$ , ${lim}_{n \to \infty} σ (n) = lim_{n \to \infty} τ (n) = \infty .$ We examine the following two cases: $\begin{matrix} {0 < p (n) & \leq 1, σ (n) = n + k, τ (n) = n + l}, {p (n) & > 1, σ (n) = n - k, τ (n) = n - l}, \end{matrix}$ where $k$ , $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.