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 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 New Characterizations of Weights in Hardy and Opial Type Inequalities via Solvability of Dynamic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
S. H. Saker ◽  
A. G. Sayed ◽  
A. Sikorska-Nowak ◽  
I. Abohela
Keyword(s):  
Time Scales ◽  
Second Order ◽  
Dynamic Equations ◽  
Special Cases ◽  
Discrete Inequalities

In this paper, we prove that the solvability of dynamic equations of second order is sufficient for the validity of some Hardy and Opial type inequalities with two different weights on time scales. In particular, the results give new characterizations of two different weights in inequalities containing Hardy and Opial operators. The main contribution in this paper is the characterizations of weights in discrete inequalities that will be formulated from our results as special cases.

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

2004 ◽  
Vol 2004 (7) ◽  
pp. 551-565 ◽  
Author(s):  
Pavel Řehák
Keyword(s):  
Time Scales ◽  
Comparison Theorem ◽  
Dynamic Equation ◽  
Second Order ◽  
Comparison Theorems ◽  
Dynamic Equations ◽  
Discrete Case ◽  
Linear Dynamic ◽  
Suitable Function

We obtain comparison theorems for the second-order half-linear dynamic equation[r(t)Φ(yΔ)]Δ+p(t)Φ(yσ)=0, whereΦ(x)=|x|α−1sgn xwithα>1. In particular, it is shown that the nonoscillation of the previous dynamic equation is preserved if we multiply the coefficientp(t)by a suitable functionq(t)and lower the exponentαin the nonlinearityΦ, under certain assumptions. Moreover, we give a generalization of Hille-Wintner comparison theorem. In addition to the aspect of unification and extension, our theorems provide some new results even in the continuous and the discrete case.

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.

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