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NONNEGATIVITY OF FUNCTIONALS CORRESPONDING TO THE SECOND
ORDER HALF-LINEAR DIFFERENTIAL EQUATION

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*Robert Marik*

**Address.** Department of Mathematics, Masaryk University, Janackovo
nam. 2a, 662 95 Brno, CZECH REPUBLIC

**E-mail:** marik@math.muni.cz

**Abstract.** In this paper we study extremal properties of functional
associated with the half--linear second order differential equation \Ep.
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.

**AMSclassification.** 34C10, 49K15

**Keywords.** Half--linear differential equation, associated functional,
Picone identity, conjugate points