Oscillation and non-oscillation of Euler type half-linear differential equations

Došlý, Ondřej; Veselý, Michal

doi:10.1016/j.jmaa.2015.04.030

Cited by 27 publications

(29 citation statements)

References 13 publications

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“…We have (see (54)) r = If this inequality is valid, the considered equation is oscillatory (see Figure 2). Based on known non-oscillation results for = 0 in the continuous case (see, eg, other studies [46][47][48][49][50][51], we conjecture that the equation is non-oscillatory if the opposite inequality holds. It is an open problem.…”

Section: Corollary 51 Let Us Consider the Equationmentioning

confidence: 65%

“…We have (see )

r_{α} = \int_{0}^{2} {normale}^{- t} d t + {normale}^{- 2} \cdot (3 - 2) + \frac{1}{10} \cdot (5 - 3) + a \cdot (6 - 5) = \frac{6}{5} + a

and

s_{α} = \int_{0}^{2} \arctan t d t + \arctan 2 \cdot (3 - 2) + 0 \cdot (5 - 3) + e \cdot (6 - 5) = e - \frac{\ln 5}{2} + 3 \arctan 2 .

Consequently, we can write as

r_{α} s_{α} = (\frac{6}{5} + a) (e - \frac{\ln 5}{2} + 3 \arctan 2) > \frac{α^{2} {(1 - β)}^{2}}{4} = 9 {(1 - β)}^{2},

ie,

a > \frac{18 {(1 - β)}^{2}}{2 e - \ln 5 + 6 \arctan 2} - \frac{6}{5} .

If this inequality is valid, the considered equation is oscillatory (see Figure ). Based on known non‐oscillation results for β = 0 in the continuous case (see, eg, other studies…”

Section: Applicationsmentioning

confidence: 99%

“…For the corresponding results about differential equations in the form of Equation 74, we refer, eg, to other studies. 46,47,63,64 ORCID Petr Hasil https://orcid.org/0000-0002-8039-688X Michal Veselý https://orcid.org/0000-0001-5306-7127…”

Section: Open Problemsmentioning

confidence: 99%

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Oscillation result for half‐linear dynamic equations on timescales and its consequences

Hasil

Veselý

2019

Math Methods in App Sciences

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We study oscillatory properties of half‐linear dynamic equations on timescales. Via the combination of the Riccati technique and an averaging method, we find the domain of oscillation for many equations. The presented main result is not the conversion of a known result from the theory of differential or difference equations, ie, we obtain new results for the timescales double-struckT=double-struckR (for differential equations) and double-struckT=double-struckZ (for difference equations). Half‐linear equations generalize linear equations (in fact, they coincide with certain one‐dimensional PDEs with p‐Laplacian), but the main result is new also for linear differential and difference equations. The corresponding corollaries and examples are given as well.

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Section: Corollary 51 Let Us Consider the Equationmentioning

confidence: 65%

“…We have (see )

r_{α} = \int_{0}^{2} {normale}^{- t} d t + {normale}^{- 2} \cdot (3 - 2) + \frac{1}{10} \cdot (5 - 3) + a \cdot (6 - 5) = \frac{6}{5} + a

and

s_{α} = \int_{0}^{2} \arctan t d t + \arctan 2 \cdot (3 - 2) + 0 \cdot (5 - 3) + e \cdot (6 - 5) = e - \frac{\ln 5}{2} + 3 \arctan 2 .

Consequently, we can write as

r_{α} s_{α} = (\frac{6}{5} + a) (e - \frac{\ln 5}{2} + 3 \arctan 2) > \frac{α^{2} {(1 - β)}^{2}}{4} = 9 {(1 - β)}^{2},

ie,

a > \frac{18 {(1 - β)}^{2}}{2 e - \ln 5 + 6 \arctan 2} - \frac{6}{5} .

If this inequality is valid, the considered equation is oscillatory (see Figure ). Based on known non‐oscillation results for β = 0 in the continuous case (see, eg, other studies…”

Section: Applicationsmentioning

confidence: 99%

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Oscillation result for half‐linear dynamic equations on timescales and its consequences

Hasil

Veselý

2019

Math Methods in App Sciences

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“…Hence, we combine the adapted Riccati equation with the Prüfer technique and involve the averaging method. Then we use the equivalence of the non-oscillation of the given equation and the boundedness of the Prüfer angle (for variations of such approach, see, e.g., [17,33,39,45]).…”

Section: Corollary 54 Consider the Equationmentioning

confidence: 99%

“…The main idea is the equivalence of the boundedness of ϕ and the non-oscillation of Eq. (2.4) (see directly[17, Corollary 4.1] or, e.g.,[12,45,53]). Regarding a solution ϕ of Eq.…”

mentioning

confidence: 98%

Euler type linear and half-linear differential equations and their non-oscillation in the critical oscillation case

et al. 2019

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This paper is devoted to the analysis of the oscillatory behavior of Euler type linear and half-linear differential equations. We focus on the so-called conditional oscillation, where there exists a borderline between oscillatory and non-oscillatory equations. The most complicated problem involved in the theory of conditionally oscillatory equations is to decide whether the equations from the given class are oscillatory or non-oscillatory in the threshold case. In this paper, we answer this question via a combination of the Riccati and Prüfer technique. Note that the obtained non-oscillation of the studied equations is important in solving boundary value problems on non-compact intervals and that the obtained results are new even in the linear case.

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Oscillation and nonoscillation of perturbed nonlinear equations with p‐Laplacian

Hasil

Veselý

2023

Mathematische Nachrichten

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In this paper, we analyze oscillatory properties of perturbed half-linear differential equations (i.e., equations with one-dimensional 𝑝-Laplacian). The presented research covers the Euler and Riemann-Weber type equations with very general coefficients. We prove an oscillatory result and a nonoscillatory one, which show that the studied equations are conditionally oscillatory (i.e., there exists a certain threshold value that separates oscillatory and nonoscillatory equations). The obtained criteria are easy to use. Since the number of perturbations is arbitrary, we solve the oscillation behavior of the equations in the critical setting when the coefficients give exactly the threshold value. The results are new for linear equations as well.

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Oscillation and non-oscillation of Euler type half-linear differential equations

Cited by 27 publications

References 13 publications

Oscillation result for half‐linear dynamic equations on timescales and its consequences

Oscillation result for half‐linear dynamic equations on timescales and its consequences

Euler type linear and half-linear differential equations and their non-oscillation in the critical oscillation case

Oscillation and nonoscillation of perturbed nonlinear equations with p‐Laplacian

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