scholarly journals Generalized Variational Principles on Oscillation for Nonlinear Nonhomogeneous Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Jing Shao ◽  
Fanwei Meng
Keyword(s):  
Differential Equation ◽  
Variational Principle ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Variational Principles ◽  
Second Order ◽  
Generalized Variational Principle ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

Using the generalized variational principle and the Riccati technique, new oscillation criteria are established for the forced second-order nonlinear differential equation, which improves and generalizes some of the new results in literature.

scholarly journals Generalized Variational Oscillation Principles for Second-Order Differential Equations with Mixed-Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Jing Shao ◽  
Fanwei Meng ◽  
Xinqin Pang
Keyword(s):  
Differential Equation ◽  
Variational Principle ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Second Order ◽  
Generalized Variational Principle ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Second Order Differential Equations ◽  
Mixed Nonlinearities

Using generalized variational principle and Riccati technique, new oscillation criteria are established for forced second-order differential equation with mixed nonlinearities, which improve and generalize some recent papers in the literature.

scholarly journals A New Oscillation Criterion for Forced Second-Order Quasilinear Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Shao Jing
Keyword(s):  
Differential Equation ◽  
Variational Principle ◽  
Differential Equations ◽  
Second Order ◽  
Oscillation Criterion ◽  
Generalized Variational Principle ◽  
Quasilinear Differential Equation ◽  
Riccati Technique ◽  
Forcing Term

By using the generalized variational principle and Riccati technique, a new oscillation criterion is established for second-order quasilinear differential equation with an oscillatory forcing term, which improves and generalizes some of new results in the literature.

scholarly journals Oscillatory behavior of a second order nonlinear advanced differential equation with mixed neutral terms

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hongwei Shi ◽  
Yuzhen Bai
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

AbstractIn this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form $$ \bigl(r(t) \bigl(z'(t)\bigr)^{\alpha }\bigr)'+q(t)x^{\beta } \bigl(\sigma (t)\bigr)=0,\quad t\geq t_{0}, $$(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0, where $z(t)=x(t)+p_{1}(t)x(\tau (t))+p_{2}(t)x(\lambda (t))$z(t)=x(t)+p1(t)x(τ(t))+p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.

On Oscillation and Nonoscillation for Differential Equations with 𝑝-Laplacian

2007 ◽  
Vol 14 (2) ◽  
pp. 239-252
Author(s):  
Miroslav Bartušek ◽  
Mariella Cecchi ◽  
Zuzana Došlá ◽  
Mauro Marini
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Oscillatory Solution ◽  
Second Order ◽  
Asymptotic Estimates ◽  
Nonoscillatory Solutions ◽  
Order Nonlinear Differential Equation ◽  
Monotonicity Properties ◽  
Oscillation And Nonoscillation

Abstract The existence of at least one oscillatory solution of a second order nonlinear differential equation with 𝑝-Laplacian is considered. The global monotonicity properties and asymptotic estimates for nonoscillatory solutions are investigated as well.

Oscillation of Second-Order Nonlinear Differential Equations

2014 ◽  
Vol 548-549 ◽  
pp. 1007-1010
Author(s):  
Qing Zhu ◽  
Zhi Bin Ma
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Oscillation Criterion ◽  
Order Nonlinear Differential Equation

A new oscillation criterion is established for a certain class of second-order nonlinear differential equation x"(t)-b(t)x'(t)+c(t)g(x)=0, x"(t)+c(t)g(x)=0 that is different from most known ones. Some applications of the result obtained are also presented. Our results are sharper than some previous ones.

scholarly journals Oscillation and nonoscillation theorems for second order nonlinear differential equations

2001 ◽  
Vol 32 (2) ◽  
pp. 95-102
Author(s):  
Jiang Jianchu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Order Nonlinear Differential Equation ◽  
Oscillation And Nonoscillation

New oscillation and nonoscillation theorems are obtained for the second order nonlinear differential equation $$ (|u'(t)|^{\alpha -1} u'(t))' + p(t)|u(t)|^{\alpha -1} u(t) = 0 $$ where $ p(t) \in C [0, \infty) $ and $ p(t) \ge 0 $. Conditions only about the integrals of $ p(t) $ on every interval $ [2^n t_0, 2^{n+1} t_0] $ ($ n = 1, 2, \ldots $) for some fixed $ t_0 >0 $ are used in the results.

Oscillation Criteria for Second Order Nonlinear Differential Equations Involving Integral Averages

1993 ◽  
Vol 45 (5) ◽  
pp. 1094-1103 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Equation ◽  
Nonlinear Differential Equation ◽  
General Equation ◽  
Nonlinear Differential Equations ◽  
Nonlinear Function ◽  
Second Order ◽  
Oscillation Criteria ◽  
Fowler Equation

AbstractConsider the second order nonlinear differential equationy" + a(t)f(y) = 0where a(t) ∈ C[0,∞),f(y) GC1 (-∞, ∞),ƒ'(y) ≥ 0 and yf(y) > 0 for y ≠ 0. Furthermore, f(y) also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function f(y) = |γ|γ sgny with γ > 1 and 0 < γ < 1 known as the Emden-Fowler case. The coefficient a(t) is allowed to be negative for arbitrarily large values of t. Oscillation criteria involving integral averages of a(t) due to Wintner, Hartman, and recently Butler, Erbe and Mingarelli for the linear equation are shown to remain valid for the general equation, subject to certain nonlinear conditions on f(y). In particular, these results are therefore valid for the Emden-Fowler equation.

scholarly journals On the Oscillatory Behavior of a Class of Fourth-Order Nonlinear Differential Equation

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 524 ◽  
Author(s):  
Osama Moaaz ◽  
Poom Kumam ◽  
Omar Bazighifan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Fourth Order ◽  
Oscillatory Behavior ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

In this work, we study the oscillatory behavior of a class of fourth-order differential equations. New oscillation criteria were obtained by employing a refinement of the Riccati transformations. The new theorems complement and improve a number of results reported in the literature. An example is provided to illustrate the main results.

scholarly journals Set of Oscillation Criteria for Second Order Nonlinear Forced Differential Equations with Damping

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Ambarka Abdalla Salhin ◽  
Ummul Khair Salma Din ◽  
Rokiah Rozita Ahmad ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Second Order ◽  
Riccati Technique ◽  
Averaging Technique ◽  
Oscillation Criteria ◽  
Integral Averaging Technique ◽  
Generalized Riccati Technique

By employing a generalized Riccati technique and an integral averaging technique, some new oscillation criteria are established for the second order nonlinear forced differential equation with damping. These results extend, improve, and unify some known oscillation criteria in the existing literature.

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