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Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation

Jaroslav Jaroš, Kusano Takaŝi, Jelena Manojlović (2013)

Open Mathematics

Positive solutions of the nonlinear second-order differential equation ( p ( t ) | x ' | α - 1 x ' ) ' + q ( t ) | x | β - 1 x = 0 , α > β > 0 , are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.

Asymptotic behavior of solutions of a 2 n t h order nonlinear differential equation

C. S. Lin (2002)

Czechoslovak Mathematical Journal

In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for ( - 1 ) n u ( 2 n ) + f ( t , u ) = 0 , in ( α , ) , u ( i ) ( ξ ) = 0 , i = 0 , 1 , , n - 1 , and ξ ( α , ) , must be unbounded, provided f ( t , z ) z 0 , in E × and for every bounded subset I , f ( t , z ) is bounded in E × I . (B) Every bounded solution for ( - 1 ) n u ( 2 n ) + f ( t , u ) = 0 , in , must be constant, provided f ( t , z ) z 0 in × and for every bounded subset I , f ( t , z ) is bounded in × I .

Asymptotic behaviour of oscillatory solutions of n -th order differential equations with quasiderivatives

Miroslav Bartušek (1997)

Czechoslovak Mathematical Journal

Sufficient conditions are given under which the sequence of the absolute values of all local extremes of y [ i ] , i { 0 , 1 , , n - 2 } of solutions of a differential equation with quasiderivatives y [ n ] = f ( t , y [ 0 ] , , y [ n - 1 ] ) is increasing and tends to . The existence of proper, oscillatory and unbounded solutions is proved.

Asymptotic relationship between solutions of two linear differential systems

Jozef Miklo (1998)

Mathematica Bohemica

In this paper new generalized notions are defined: Ψ -boundedness and Ψ -asymptotic equivalence, where Ψ is a complex continuous nonsingular n × n matrix. The Ψ -asymptotic equivalence of linear differential systems y ' = A ( t ) y and x ' = A ( t ) x + B ( t ) x is proved when the fundamental matrix of y ' = A ( t ) y is Ψ -bounded.

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