scholarly journals Study of nonlinear Pachpatte’s inequalities on time scales

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Sarah Sarfaraz ◽  
Naveed Ahmad ◽  
Ghaus ur Rahman
Keyword(s):  
Qualitative Analysis ◽  
Time Scales ◽  
Comparison Theorem ◽  
Higher Order ◽  
Dynamic Equations ◽  
Arbitrary Time ◽  
Dynamic Inequality ◽  
Continuous Analogue ◽  
Discrete Analogues

Abstract In this paper, we develop a fundamental dynamic inequality, a generalization of comparison theorem and reproduce the proofs of some nonlinear integral Pachpatte’s inequalities by using their continuous analogue. We also unify and extend these improved integral Pachpatte’s inequalities and their corresponding discrete analogues on arbitrary time scales. The results are used to make qualitative analysis of higher order dynamic equations.

Oscillation criteria for higher order nonlinear delay dynamic equations on time scales

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Xin Wu ◽  
Taixiang Sun
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Higher Order ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
Arbitrary Time ◽  
Delay Dynamic Equation ◽  
Delay Dynamic Equations ◽  
Nonlinear Delay

AbstractIn this paper, we study the oscillation criteria of the following higher order nonlinear delay dynamic equationon an arbitrary time scalewith

scholarly journals Some Oscillation Criteria for Higher-Order Neutral Emden-Fowler Delay Dynamic Equations on Time Scales

2018 ◽  
Vol 228 ◽  
pp. 01003
Author(s):  
Ying Sui ◽  
Yulong Shi ◽  
Yibin Sun ◽  
Shurong Sun
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Higher Order ◽  
Delay Equation ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Delay Dynamic Equations

New oscillation criteria are established for higher-order Emdn-Fowler dynamic equation $ q(v)x^{\beta } (\delta (v)) + (r(v)(z^{{\Delta ^{{n - 1}} }} (v))^{\alpha } )^{\Delta } = 0 $ on time scales, $ z(v): = p(v)x(\tau (v)) + x(v) $ Our results extend and supplement those reported in literatures in the sense that we study a more generalized neutral delay equation and do not require $ r^{\Delta } (v) \ge 0 $ and the commutativity of the jump and delay operators.

Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales

2018 ◽  
Vol 68 (6) ◽  
pp. 1397-1420 ◽  
Author(s):  
Chao Wang ◽  
Ravi P. Agarwal ◽  
Donal O’Regan
Keyword(s):  
Periodic Solutions ◽  
Time Scales ◽  
Time Scale ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Arbitrary Time ◽  
First Time ◽  
Composition Theorem

Abstract In this paper, by using the concept of changing-periodic time scales and composition theorem of time scales introduced in 2015, we establish a local phase space for functional dynamic equations with infinite delay (FDEID) on an arbitrary time scale with a bounded graininess function μ. Through Krasnoseľskiĭ’s fixed point theorem, some sufficient conditions for the existence of local-periodic solutions for FDEID are established for the first time. This research indicates that one can extract a local-periodic solution for dynamic equations on an arbitrary time scale with a bounded graininess function μ through some index function.

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.

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