Generalized Prüfer angle and oscillation of half-linear differential equations

Math Methods in App Sciences

2019

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.

“…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%

“…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%

“…A basic result is mentioned, eg, in Došlý and Řehák, (p.43) where it is proved that the equation

{(t^{β} Φ (x^{'}))}^{'} + \frac{γ}{t^{p - β}} Φ (x) = 0, γ \in R,

is oscillatory if β ≠ p − 1 and

γ > {(\frac{| p - 1 - β |}{p})}^{p},

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%

Section: Theorem 13 Let Us Consider An -Periodic Timescale T and Thmentioning

confidence: 99%

See 3 more Smart Citations

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

Math Methods in App Sciences

2019

Oscillation and nonoscillation of perturbed nonlinear equations with p‐Laplacian

Mathematische Nachrichten

2023

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.

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

Math Methods in App Sciences

2018

The purpose of this paper is to describe the oscillatory properties of second-order Euler-type half-linear differential equations with perturbations in both terms.All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non-oscillation of the considered equations, including the so-called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.