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A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations

Manabu Naito (2024)

Mathematica Bohemica

The half-linear differential equation ( | u ' | α sgn u ' ) ' = α ( λ α + 1 + b ( t ) ) | u | α sgn u , t t 0 , is considered, where α and λ are positive constants and b ( t ) is a real-valued continuous function on [ t 0 , ) . It is proved that, under a mild integral smallness condition of b ( t ) which is weaker than the absolutely integrable condition of b ( t ) , the above equation has a nonoscillatory solution u 0 ( t ) such that u 0 ( t ) e - λ t and u 0 ' ( t ) - λ e - λ t ( t ), and a nonoscillatory solution u 1 ( t ) such that u 1 ( t ) e λ t and u 1 ' ( t ) λ e λ t ( t ).

A note on the fundamental matrix of variational equations in 3

Ladislav Adamec (2003)

Mathematica Bohemica

The paper is devoted to the question whether some kind of additional information makes it possible to determine the fundamental matrix of variational equations in 3 . An application concerning computation of a derivative of a scalar Poincaré mapping is given.

A note on the oscillation of second order differential equations

Hishyar Kh. Abdullah (2004)

Czechoslovak Mathematical Journal

We give a sufficient condition for the oscillation of linear homogeneous second order differential equation y ' ' + p ( x ) y ' + q ( x ) y = 0 , where p ( x ) , q ( x ) C [ α , ) and α is positive real number.

A note on the oscillation problems for differential equations with p ( t ) -Laplacian

Kōdai Fujimoto (2023)

Archivum Mathematicum

This paper deals with the oscillation problems on the nonlinear differential equation ( a ( t ) | x ' | p ( t ) - 2 x ' ) ' + b ( t ) | x | λ - 2 x = 0 involving p ( t ) -Laplacian. Sufficient conditions are given under which all proper solutions are oscillatory. In addition, we give a-priori estimates for nonoscillatory solutions and propose an open problem.

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