scholarly journals Nonoscillatory Solutions for Higher-Order Neutral Dynamic Equations on Time Scales

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
Taixiang Sun ◽  
Hongjian Xi ◽  
Xiaofeng Peng ◽  
Weiyong Yu
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Dynamic Equations ◽  
Bounded Solutions ◽  
Nonoscillatory Solutions ◽  
Neutral Dynamic Equation ◽  
Necessary And Sufficient ◽  
Neutral Dynamic Equations

We study the higher-order neutral dynamic equation{a(t)[(x(t)−p(t)x(τ(t)))Δm]α}Δ+f(t,x(δ(t)))=0fort∈[t0,∞)Tand obtain some necessary and sufficient conditions for the existence of nonoscillatory bounded solutions for this equation.

scholarly journals Oscillation and nonoscillation theorems of neutral dynamic equations on time scales

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yong Zhou ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Dynamic Equations ◽  
Nonoscillatory Solutions ◽  
Oscillation Criteria ◽  
Neutral Dynamic Equation ◽  
The Given ◽  
Neutral Dynamic Equations ◽  
Oscillation And Nonoscillation

Abstract We present the oscillation criteria for the following neutral dynamic equation on time scales: $$ \bigl(y(t)-C(t)y(t-\zeta )\bigr)^{\Delta }+P(t)y(t-\eta )-Q(t)y(t-\delta )=0, \quad t\in {\mathbb{T}}, $$ ( y ( t ) − C ( t ) y ( t − ζ ) ) Δ + P ( t ) y ( t − η ) − Q ( t ) y ( t − δ ) = 0 , t ∈ T , where $C, P, Q\in C_{\mathit{rd}}([t_{0},\infty ),{\mathbb{R}}^{+})$ C , P , Q ∈ C rd ( [ t 0 , ∞ ) , R + ) , ${\mathbb{R}} ^{+}=[0,\infty )$ R + = [ 0 , ∞ ) , $\gamma , \eta , \delta \in {\mathbb{T}}$ γ , η , δ ∈ T and $\gamma >0$ γ > 0 , $\eta >\delta \geq 0$ η > δ ≥ 0 . New conditions for the existence of nonoscillatory solutions of the given equation are also obtained.

scholarly journals Existence of nonoscillatory solutions tending to zero of fourth-order nonlinear neutral dynamic equations on time scales

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yang-Cong Qiu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Time Scales ◽  
Point Theorem ◽  
Fourth Order ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Nonoscillatory Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Neutral Dynamic Equations

AbstractIn this paper, a class of fourth-order nonlinear neutral dynamic equations on time scales is investigated. We obtain some sufficient conditions for the existence of nonoscillatory solutions tending to zero with some characteristics of the equations by Krasnoselskii’s fixed point theorem. Finally, two interesting examples are presented to show the significance of the results.

scholarly journals Oscillation for Higher Order Dynamic Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Taixiang Sun ◽  
Qiuli He ◽  
Hongjian Xi ◽  
Weiyong Yu
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Higher Order ◽  
Dynamic Equations ◽  
Positive Integers

We investigate the oscillation of the following higher order dynamic equation:{an(t)[(an-1(t)(⋯(a1(t)xΔ(t))Δ⋯)Δ)Δ]α}Δ+p(t)xβ(t)=0, on some time scaleT, wheren≥2,ak(t)  (1≤k≤n)andp(t)are positive rd-continuous functions onTandα,βare the quotient of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.

scholarly journals Nonoscillatory Solutions for System of Neutral Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Zhanhe Chen ◽  
Taixiang Sun ◽  
Qi Wang ◽  
Hongjian Xi
Keyword(s):  
Time Scales ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Functional System ◽  
Dynamic Equations ◽  
Nonoscillatory Solutions ◽  
Neutral Dynamic Equations

We will discuss nonoscillatory solutions to then-dimensional functional system of neutral type dynamic equations on time scales. We will establish some sufficient conditions for nonoscillatory solutions with the propertylimt→∞⁡xit=0, i=1, 2, …,n.

scholarly journals Nonoscillatory solutions to fourth-order neutral dynamic equations on time scales

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yang-Cong Qiu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Time Scales ◽  
Fourth Order ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Nonoscillatory Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Neutral Dynamic Equations

AbstractIn this paper, we present some sufficient conditions and necessary conditions for the existence of nonoscillatory solutions to a class of fourth-order nonlinear neutral dynamic equations on time scales by employing Banach spaces and Krasnoselskii’s fixed point theorem. Two examples are given to illustrate the applications of the results.

scholarly journals Oscillation of nonlinear neutral dynamic equations on time scales

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
G. N. Chhatria ◽  
Said R. Grace ◽  
John R. Graef
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Time Scales ◽  
Sufficient Conditions ◽  
Nonlinear Function ◽  
Dynamic Equations ◽  
Krasnosel’Skii’S Fixed Point Theorem ◽  
Necessary And Sufficient ◽  
Strongly Sublinear ◽  
Neutral Dynamic Equations

AbstractThe authors present necessary and sufficient conditions for the oscillation of a class of second order non-linear neutral dynamic equations with non-positive neutral coefficients by using Krasnosel’skii’s fixed point theorem on time scales. The nonlinear function may be strongly sublinear or strongly superlinear.

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