A critical oscillation constant as a variable of time scales for half-linear dynamic equations

2010 ◽  
Vol 60 (2) ◽  
Author(s):  
Pavel Řehák
Keyword(s):  
Time Scales ◽  
Type Inequality ◽  
Dynamic Equations ◽  
Hardy Type Inequality ◽  
Linear Dynamic ◽  
Hardy Type ◽  
Special Cases ◽  
Critical Oscillation ◽  
Critical Constant ◽  
Oscillation Constant

AbstractWe present criteria of Hille-Nehari type for the half-linear dynamic equation (r(t)Φ(y Δ))Δ+p(t)Φ(y σ) = 0 on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from q-calculus, a Hardy type inequality with weights, and further possibilities for study are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case.

scholarly journals New Inequalities of Opial's Type on Time Scales and Some of Their Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Samir H. Saker
Keyword(s):  
Time Scales ◽  
Second Order ◽  
Dynamic Equations ◽  
Linear Dynamic ◽  
Special Cases ◽  
Discrete Inequalities ◽  
Zeros Of Solutions ◽  
Yield Conditions ◽  
New Inequalities

We will prove some new dynamic inequalities of Opial's type on time scales. The results not only extend some results in the literature but also improve some of them. Some continuous and discrete inequalities are derived from the main results as special cases. The results will be applied on second-order half-linear dynamic equations on time scales to prove several results related to the spacing between consecutive zeros of solutions and the spacing between zeros of a solution and/or its derivative. The results also yield conditions for disfocality of these equations.

scholarly journals Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales

2018 ◽  
Vol 51 (1) ◽  
pp. 198-210 ◽  
Author(s):  
Douglas R. Anderson ◽  
Masakazu Onitsuka
Keyword(s):  
Time Scales ◽  
Dynamic Equations ◽  
Ulam Stability ◽  
Step Size ◽  
First Order ◽  
Linear Dynamic ◽  
Discrete Step ◽  
Special Cases ◽  
Parameter Values ◽  
Homogeneous Linear

Abstract We establish theHyers-Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, which include the continuous (step size zero) and the discrete (step size constant and nonzero) dynamic equations as important special cases. In particular, for certain parameter values in relation to the graininess of the time scale, we find the minimum HUS constants. A few nontrivial examples are provided. Moreover, an application to a perturbed linear dynamic equation is also included.

scholarly journals Characterizations of weighted dynamic Hardy-type inequalities with higher-order derivatives

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
S. H. Saker ◽  
R. R. Mahmoud ◽  
K. R. Abdo
Keyword(s):  
Time Scales ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Hardy Type Inequality ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Weighted Functions ◽  
Necessary And Sufficient

AbstractIn this paper, we establish some necessary and sufficient conditions for the validity of a generalized dynamic Hardy-type inequality with higher-order derivatives with two different weighted functions on time scales. The corresponding continuous and discrete cases are captured when $\mathbb{T=R}$ T = R and $\mathbb{T=N}$ T = N , respectively. Finally, some applications to our main result are added to conclude some continuous results known in the literature and some other discrete results which are essentially new.

scholarly journals Some New Inequalities of Opial's Type on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Samir H. Saker
Keyword(s):  
Time Scales ◽  
Dynamic Equations ◽  
Linear Dynamic ◽  
Special Cases ◽  
Discrete Inequalities ◽  
New Inequalities

We will prove some new dynamic inequalities of Opial's type on time scales. The results not only extend some results in the literature but also improve some of them. Some continuous and discrete inequalities are derived from the main results as special cases. The results can be applied on the study of distribution of generalized zeros of half-linear dynamic equations on time scales.

scholarly journals Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

scholarly journals Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

2020 ◽  
Vol 6 (2) ◽  
pp. 198-209
Author(s):  
Mohamed Laghzal ◽  
Abdelouahed El Khalil ◽  
My Driss Morchid Alaoui ◽  
Abdelfattah Touzani
Keyword(s):  
Dirichlet Problem ◽  
Variational Approach ◽  
Type Inequality ◽  
Biharmonic Operator ◽  
Nondecreasing Sequence ◽  
Hardy Type Inequality ◽  
Principal Curve ◽  
Hardy Type ◽  
Homogeneous Dirichlet Problem

AbstractThis paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

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