On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations

2015 ◽  
Vol 145 (5) ◽  
pp. 1007-1028 ◽  
Author(s):  
Jaroslav Jaroš ◽  
Kusano Takaŝi
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Nonlinear Differential Equations ◽  
Radial Symmetry ◽  
Second Order ◽  
Accurate Information ◽  
Regularly Varying Functions ◽  
Cyclic System ◽  
Regularly Varying ◽  
Image Position

The n-dimensional cyclic system of second-order nonlinear differential equationsis analysed in the framework of regular variation. Under the assumption that αi and βi are positive constants such that α1 … αn > β1 … βn and pi and qi are regularly varying functions, it is shown that the situation in which the system possesses decreasing regularly varying solutions of negative indices can be completely characterized, and moreover that the asymptotic behaviour of such solutions is governed by a unique formula describing their order of decay precisely. Examples are presented to demonstrate that the main results for the system can be applied effectively to some classes of partial differential equations with radial symmetry to provide new accurate information about the existence and the asymptotic behaviour of their radial positive strongly decreasing solutions.

scholarly journals Regularly varying solutions of perturbed Euler differential equations and related functional differential equations

2010 ◽  
Vol 88 (102) ◽  
pp. 1-20 ◽  
Author(s):  
Kusano Takaŝi ◽  
Vojislav Maric
Keyword(s):  
Differential Equations ◽  
Asymptotic Analysis ◽  
Critical Case ◽  
Functional Differential Equations ◽  
Second Order ◽  
Regularly Varying Functions ◽  
Order Linear ◽  
Regularly Varying ◽  
Functional Differential ◽  
Linear Differential

An asymptotic analysis in the framework of Karamata regularly varying functions is performed for the solutions of second order linear differential and functional differential equations in the critical case i.e., when condition (1.5) as given below, holds.

Oscillation Criteria for Second Order Nonlinear Delay Equations

1973 ◽  
Vol 16 (1) ◽  
pp. 49-56 ◽  
Author(s):  
Lynn Erbe
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Delay Equations ◽  
Retarded Argument ◽  
Oscillation Criteria ◽  
Image Position ◽  
Nonlinear Delay

It is the purpose of this paper to establish oscillation criteria for second order nonlinear differential equations with retarded argument. Specifically, we consider the equation1.1where f ∊ C[0, + ∞) x R2, g ∊ C[0, + ∞), and1.2We shall restrict attention to solutions of (1.1) which exist on some ray [T, + ∞). A solution of (1.1) is called oscillatory if it has no largest zero.

Asymptotic behaviour of solutions of some general nonlinear differential equations and integral inequalities

1984 ◽  
Vol 98 (1-2) ◽  
pp. 49-67 ◽  
Author(s):  
Paul R. Beesack
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Nonlinear Differential Equations ◽  
Upper Bounds ◽  
Integral Inequalities ◽  
Asymptotic Behaviour Of Solutions ◽  
Type Integral ◽  
Image Position ◽  
Complex Valued

SynopsisWe deal with the asymptotic behaviour, as t→∞, of complex-valued solutions of nonlinear differential equationsUpper bounds for ∣x(l)(t)∣, 0≦j≦n, are obtained by obtaining upper bounds for solutions u(t) of Bihari-type integral inequalities of the form

scholarly journals Oscillatory behavior of nonlinear differential equations with deviating arguments

1985 ◽  
Vol 31 (1) ◽  
pp. 127-136 ◽  
Author(s):  
S.R. Grace ◽  
B.S. Lalli
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Deviating Arguments ◽  
Oscillation Criteria ◽  
Image Position

New oscillation criteria for nonlinear differential equations with deviating arguments of the formn even, are established.

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