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Eighty fifth anniversary of birthday and scientific legacy of Professor Miloš Ráb

Ondřej Došlý, Eva Jansová, Josef Kalas (2014)

Archivum Mathematicum

The paper is concentrated on Professor Miloš Ráb and his contribution to the theory of oscillatory properties of solutions of second and third order linear differential equations, the theory of differential equations with complex coefficients and dynamical systems, and the theory of nonlinear second order differential equations. At the beginning, we take a brief look at the most important moments in his life. Afterwards, we describe his scientific activities on mentioned theories.

Eventual disconjugacy of y ( n ) + μ p ( x ) y = 0 for every μ

Uri Elias (2004)

Archivum Mathematicum

The work characterizes when is the equation y ( n ) + μ p ( x ) y = 0 eventually disconjugate for every value of μ and gives an explicit necessary and sufficient integral criterion for it. For suitable integers q , the eventually disconjugate (and disfocal) equation has 2-dimensional subspaces of solutions y such that y ( i ) > 0 , i = 0 , ... , q - 1 , ( - 1 ) i - q y ( i ) > 0 , i = q , ... , n . We characterize the “smallest” of such solutions and conjecture the shape of the “largest” one. Examples demonstrate that the estimates are sharp.

Existence and non-existence of sign-changing solutions for a class of two-point boundary value problems involving one-dimensional p -Laplacian

Yūki Naito (2011)

Mathematica Bohemica

We consider the boundary value problem involving the one dimensional p -Laplacian, and establish the precise intervals of the parameter for the existence and non-existence of solutions with prescribed numbers of zeros. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.

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