Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values

Hasil, Petr; Veselý, Michal

doi:10.1515/math-2018-0047

Cited by 17 publications

(19 citation statements)

References 23 publications

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“…Thus (consider Corollary ), Equation is oscillatory if 4 ab > (1 − β ) 2 . As in Example , the oscillation of the treated equation follows from known results for

a > false(1 - β false) false/ \sqrt{2}

, a = b (see, eg, other studies for other relevant results). In addition, for

a > false(1 - β false) false/ \sqrt{2}

, a = b , considering

\underset{t \to \infty}{lim inf} r (t) = a, \underset{t \to \infty}{lim inf} s (t) = \frac{b}{2} = \frac{a}{2},

the oscillation of the equation is guaranteed directly by the well‐known Sturm comparison theorem and Došlý and Řehák, (theorem1.4.4) ie, by the classic Kneser type criterion which says that Equation is oscillatory if

(\underset{t \to \infty}{lim inf} r (t)) \cdot (\underset{t \to \infty}{lim inf} s (t)) > {(\frac{1 - β}{2})}^{2},

and which is known in the half‐linear case as well.…”

Section: Applicationsmentioning

confidence: 74%

“…One can compute that

M (a + \sin t) = M (a + \cos t) = a, M (b + b \sin t) = M (b + b \cos t) = b .

Therefore, the inequality

{((), p M false(r false))}^{p} M false(s false) > {false(p - β - 1 false)}^{p} M false(r false)

from Corollary takes the form

{((), p a)}^{p} b > {false(2 p - 2 false)}^{p} a

for each equation above. Thus, the equations are oscillatory if

b a^{p - 1} > {(\frac{2 p - 2}{p})}^{p} .

For

double-struckT = double-struckR

(see Hasil and Veselý, all equations above are non‐oscillatory if the opposite inequality holds. We illustrate the domains of oscillation in figures below.…”

Section: Applicationsmentioning

confidence: 97%

“…Later, the equation

{((), r false(t false) normalΦ ((), x^{'}))}^{'} + \frac{s false(t false)}{t^{p}} normalΦ false(x false) = 0

was studied for continuous functions r , s such that

0 < \underset{t \to \infty}{lim inf} r (t) \leq \underset{t \to \infty}{lim sup} r (t) < \infty

and that

M ((), r^{1 false/ false(1 - p false)})

, M ( s ) exist, where M ( f ) stands for the mean value of function f (see also Definition in Section ). In Hasil et al, the oscillation of Equation was proved if

M (s) > {(\frac{p - 1}{p})}^{p} {(M (r^{\frac{1}{1 - p}}))}^{1 - p} .

For other relevant results, we refer to other studies …”

Section: Introductionmentioning

confidence: 94%

“…Of course, it is very likely that such a generalization will have some consequences in limitations elsewhere. This point is inspired, among others, by paper . In that paper, differential equations (ie,

double-struckT = double-struckR

) are studied and the coefficient that corresponds to the coefficient s in Equation can change its sign and can be unbounded. (b) Sign of β .…”

Section: Open Problemsmentioning

confidence: 99%

“…Based on similar problems for differential equations (ie,

double-struckT = double-struckR

), we conjecture that the treated equations are non‐oscillatory if inequality holds contrariwise with negative number ε . Concerning this conjecture, we refer, eg, to other studies …”

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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“…Thus (consider Corollary ), Equation is oscillatory if 4 ab > (1 − β ) 2 . As in Example , the oscillation of the treated equation follows from known results for

a > false(1 - β false) false/ \sqrt{2}

, a = b (see, eg, other studies for other relevant results). In addition, for

a > false(1 - β false) false/ \sqrt{2}

, a = b , considering

\underset{t \to \infty}{lim inf} r (t) = a, \underset{t \to \infty}{lim inf} s (t) = \frac{b}{2} = \frac{a}{2},

(\underset{t \to \infty}{lim inf} r (t)) \cdot (\underset{t \to \infty}{lim inf} s (t)) > {(\frac{1 - β}{2})}^{2},

and which is known in the half‐linear case as well.…”

Section: Applicationsmentioning

confidence: 74%

“…One can compute that

M (a + \sin t) = M (a + \cos t) = a, M (b + b \sin t) = M (b + b \cos t) = b .

Therefore, the inequality

{((), p M false(r false))}^{p} M false(s false) > {false(p - β - 1 false)}^{p} M false(r false)

from Corollary takes the form

{((), p a)}^{p} b > {false(2 p - 2 false)}^{p} a

for each equation above. Thus, the equations are oscillatory if

b a^{p - 1} > {(\frac{2 p - 2}{p})}^{p} .

For

double-struckT = double-struckR

(see Hasil and Veselý, all equations above are non‐oscillatory if the opposite inequality holds. We illustrate the domains of oscillation in figures below.…”

Section: Applicationsmentioning

confidence: 97%

“…Later, the equation

{((), r false(t false) normalΦ ((), x^{'}))}^{'} + \frac{s false(t false)}{t^{p}} normalΦ false(x false) = 0

was studied for continuous functions r , s such that

0 < \underset{t \to \infty}{lim inf} r (t) \leq \underset{t \to \infty}{lim sup} r (t) < \infty

and that

M ((), r^{1 false/ false(1 - p false)})

, M ( s ) exist, where M ( f ) stands for the mean value of function f (see also Definition in Section ). In Hasil et al, the oscillation of Equation was proved if

M (s) > {(\frac{p - 1}{p})}^{p} {(M (r^{\frac{1}{1 - p}}))}^{1 - p} .

For other relevant results, we refer to other studies …”

Section: Introductionmentioning

confidence: 94%

double-struckT = double-struckR

) are studied and the coefficient that corresponds to the coefficient s in Equation can change its sign and can be unbounded. (b) Sign of β .…”

Section: Open Problemsmentioning

confidence: 99%

“…Based on similar problems for differential equations (ie,

double-struckT = double-struckR

), we conjecture that the treated equations are non‐oscillatory if inequality holds contrariwise with negative number ε . Concerning this conjecture, we refer, eg, to other studies …”

Section: Open Problemsmentioning

confidence: 99%

See 3 more Smart Citations

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

Hasil

Veselý

2019

Math Methods in App Sciences

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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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Non‐oscillation of linear and half‐linear differential equations with unbounded coefficients

Šišoláková

2020

Math Methods in App Sciences

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We deal with Euler-type half-linear second-order differential equations, and our intention is to derive conditions in order their non-trivial solutions are non-oscillatory. This paper connects to the article P. Hasil, J. Šišoláková, M. Veselý: Averaging technique and oscillation criterion for linear and half-linear equations, Appl. Math. Lett. 92 (2019), 62-69, where the corresponding oscillatory counterpart is studied and an oscillation criterion is established. The used effective technique for this investigation is the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This paper is completed by two corollaries and an example concerning the linear case when we obtain new results as well.

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Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values

Cited by 17 publications

References 23 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

Oscillation and nonoscillation of perturbed nonlinear equations with p‐Laplacian

Non‐oscillation of linear and half‐linear differential equations with unbounded coefficients

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