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Nonnegativity of functionals corresponding to the second order half-linear differential equation

Robert Mařík — 1999

Archivum Mathematicum

In this paper we study extremal properties of functional associated with the half–linear second order differential equation E p . Necessary and sufficient condition for nonnegativity of this functional is given in two special cases: the first case is when both points are regular and the second is the case, when one end point is singular. The obtained results extend the theory of quadratic functionals.

Discrete singular functionals

Robert Mařík — 2005

Archivum Mathematicum

In the paper the discrete version of the Morse’s singularity condition is established. This condition ensures that the discrete functional over the unbounded interval is positive semidefinite on the class of the admissible functions. Two types of admissibility are considered.

Positive solutions of inequality with p -Laplacian in exterior domains

Robert Mařík — 2002

Mathematica Bohemica

In the paper the differential inequality Δ p u + B ( x , u ) 0 , where Δ p u = div ( u p - 2 u ) , p > 1 , B ( x , u ) C ( n × , ) is studied. Sufficient conditions on the function B ( x , u ) are established, which guarantee nonexistence of an eventually positive solution. The generalized Riccati transformation is the main tool.

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