A Second Order Superlinear Oscillation Criterion

1984 ◽  
Vol 27 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Oscillation Criterion ◽  
Oscillation Result ◽  
Special Case

AbstractA new oscillation criterion is given for general superlinear ordinary differential equations of second order of the form x″(t)+ a(t)f[x(t)]=0, where a ∈ C([t0∞,)), f∈C(R) with yf(y)>0 for y≠0 and and f is continously differentiable on R-{0} with f'(y)≥0 for all y≠O. In the special case of the differential equation (γ > 1) this criterion leads to an oscillation result due to Wong [9].

scholarly journals Oscillation of second order linear ordinary differential equations with alternating coefficients

1983 ◽  
Vol 27 (2) ◽  
pp. 307-313 ◽  
Author(s):  
Ch.G. Philos
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Oscillation Criterion ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Oscillation Result

A new result is obtained for the oscillation of second order linear ordinary differential equations with alternating coefficients. This oscillation result extends a recent oscillation criterion due to Kamenev [.Mat. Zametki 23 (1978), 249–251].

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 Positive decreasing solutions of second order quasilinear ordinary differential equations in the framework of regular variation

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1995-2010 ◽  
Author(s):  
Jelena Milosevic ◽  
Jelena Manojlovic
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Asymptotic Analysis ◽  
Ordinary Differential Equations ◽  
Regular Variation ◽  
Second Order ◽  
Precise Information ◽  
Varying Coefficients

This paper is concerned with asymptotic analysis of positive decreasing solutions of the secondorder quasilinear ordinary differential equation (E) (p(t)?(|x'(t)|))'=q(t)?(x(t)), with the regularly varying coefficients p, q, ?, ?. An application of the theory of regular variation gives the possibility of determining the precise information about asymptotic behavior at infinity of solutions of equation (E) such that lim t?? x(t)=0, lim t?? p(t)?(-x'(t))=?.

ON THE ASYMPTOTIC EXPANSIONS OF NUMERICAL SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS WITH A REGULAR SINGULAR POINT

2001 ◽  
Vol 11 (01) ◽  
pp. 163-177
Author(s):  
RICHARD WEISS ◽  
FRANK R. de HOOG ◽  
ROBERT S. ANDERSSEN
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Approximation ◽  
Numerical Solutions ◽  
Second Order ◽  
Regular Singular Point ◽  
Second Order Differential Equations ◽  
Uniform Asymptotic Expansion

When difference schemes with uniformly spaced gridpoints are applied to second order ordinary differential equations with a regular singular point, it is often the case that the resulting numerical approximation does not have a uniform asymptotic expansion. As a consequence, postprocessing, such as h2-extrapolation is not an option. This paper examines the cause of this phenomenon and finds that the existence of such expansions requires the discretization of the boundary conditions at the singular point to be compatible with the discretization of the differential equation. In addition, it is shown how an understanding of the need for compatible discretization can assist in the construction of schemes for several classes of equations that arise when symmetry is used to reduce partial differential equations to ordinary differential equations with a regular singular point.

A non-standard numerical approach to the solution of some second-order ordinary differential equations

2015 ◽  
Vol 08 (04) ◽  
pp. 1550076 ◽  
Author(s):  
A. Adesoji Obayomi ◽  
Michael Olufemi Oke
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Approach ◽  
Local Approximation ◽  
Second Order ◽  
Discrete Models ◽  
Order Ordinary Differential Equation ◽  
Non Local

In this paper, a set of non-standard discrete models were constructed for the solution of non-homogenous second-order ordinary differential equation. We applied the method of non-local approximation and renormalization of the discretization functions to some problems and the result shows that the schemes behave qualitatively like the original equation.

On One Problem with the Condition at Infinity for Second Order Singular Ordinary Differential Equations

2008 ◽  
Vol 15 (4) ◽  
pp. 753-758
Author(s):  
Nino Partsvania
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Singular Differential Equation

Abstract For the second order nonlinear singular differential equation 𝑢″ + 𝑓(𝑡, 𝑢, 𝑢′) = 0, the unimprovable sufficient conditions for the solvability of the problem with the condition at infinity are established.

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 LINEAR ASYMPTOTIC BEHAVIOUR OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

2007 ◽  
Vol 49 (1) ◽  
pp. 105-120 ◽  
Author(s):  
MATS EHRNSTRÖM
Keyword(s):  
Differential Equation ◽  
Initial Data ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Ordinary Differential Equations ◽  
Growth Conditions ◽  
Second Order ◽  
Half Line ◽  
Semilinear Differential Equation

Abstract.We study the semilinear differential equation u″ + F(t,u,u′)=0 on a half-line. Under different growth conditions on the function F, equations with globally defined solutions asymptotic to lines are characterized. Both fixed initial data and fixed asymptote are considered.

Integrating Factors of Ordinary Differential Equations of the Second Order: A Geometrical Interpretation

1943 ◽  
Vol 27 (276) ◽  
pp. 159-165
Author(s):  
H. Wallis Chapman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Arbitrary Constant ◽  
Second Order ◽  
Geometrical Interpretation ◽  
Integrating Factor ◽  
Integrating Factors

A Well-Known method of integrating the simple differential equation of the second order y″ + k 2 y = 0 .....1.1 consists in multiplying by an integrating factor 2y′ and integrating directly, when we obtain y′2 + k 2 y 2 = k 2 a 2, .....1.2 where a is an arbitrary constant.

Oscillation Criteria For Second Order Superlinear Differential Equations

1989 ◽  
Vol 41 (2) ◽  
pp. 321-340 ◽  
Author(s):  
CH. G. Philos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Continuous Function ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real Line ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Order ◽  
Oscillation Criteria ◽  
Image Position

This paper is concerned with the question of oscillation of the solutions of second order superlinear ordinary differential equations with alternating coefficients.Consider the second order nonlinear ordinary differential equationwhere a is a continuous function on the interval [t0, ∞), t0 > 0, and / is a continuous function on the real line R, which is continuously differentia t e , except possibly at 0, and satisfies.

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