Necessary and sufficient conditions for oscillation to second-order half-linear delay differential equations

2019 ◽  
Vol 21 (3) ◽  
Author(s):  
Shyam S. Santra
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Linear Delay ◽  
Delay Differential ◽  
Necessary And Sufficient

scholarly journals Necessary and Sufficient Conditions for Oscillation of Second-Order Delay Differential Equations

2020 ◽  
Vol 75 (1) ◽  
pp. 135-146
Author(s):  
Shyam Sundar Santra
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
All Solutions ◽  
Linear Delay ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractIn this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form {\left( {r{{\left( {x'} \right)}^\gamma }} \right)^\prime }\left( t \right) + q\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right) = 0Under the assumption ∫∞(r(n))−1/γdη=∞, we consider the two cases when γ > α and γ < α. Further, some illustrative examples showing applicability of the new results are included, and state an open problem.

scholarly journals Second-Order Differential Equation: Oscillation Theorems and Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Shyam S. Santra ◽  
Omar Bazighifan ◽  
Hijaz Ahmad ◽  
Yu-Ming Chu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Delay ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Oscillation Theorems

Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form ς y u ′ y a ′ + p y u c ϑ y = 0 ,  for  y ≥ y 0 , under the assumption ∫ ∞ ς η − 1 / a = ∞ . Two cases are considered for a < c and a > c , where a and c are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.

Oscillation of the Riemann–Weber Version of Euler Differential Equations with Delay

2000 ◽  
Vol 7 (3) ◽  
pp. 577-584
Author(s):  
Jitsuro Sugie ◽  
Mitsuru Iwasaki
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Equations With Delay ◽  
Oscillation Of Solutions

Abstract Our concern is to consider delay differential equations of Euler type. Necessary and sufficient conditions for the oscillation of solutions are given. The results extend some famous facts about Euler differential equations without delay.

scholarly journals The Oscillation of a Class of the Fractional-Order Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qianli Lu ◽  
Feng Cen
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Analysis Method ◽  
Constant Coefficients ◽  
Delay Differential ◽  
Necessary And Sufficient

Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this,α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.

scholarly journals Second-Order Differential Equation with Multiple Delays: Oscillation Theorems and Applications

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Shyam Sundar Santra ◽  
Omar Bazighifan ◽  
Hijaz Ahmad ◽  
Shao-Wen Yao
Keyword(s):  
Differential Equations ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Acoustic Vibrations ◽  
Linear Delay ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Oscillation Theorems

Differential equations of second order appear in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations, and quantum mechanics. In this paper, necessary and sufficient conditions are established of the solutions to second-order half-linear delay differential equations of the form ς y u ′ y a ′ + ∑ j = 1 m p j y u c j ϑ j y = 0  for  y ≥ y 0 , under the assumption ∫ ∞ ς η − 1 / a d η = ∞ . We consider two cases when a < c j and a > c j , where a and c j are the quotient of two positive odd integers. Two examples are given to show effectiveness and applicability of the result.

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