Positive solutions of singular positone dirichlet boundary value problems

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Abstract

The singular Dirichlet boundary value problem x″ + μq(t)f(t,x,x′) = 0, x(0) = x(T) = 0 is considered. Here f(t,x,y) ≥ 0 may be singular at x = 0 and x = A > 0 of the phase variable x and at y = 0 of the phase variable y. Effective sufficient conditions imposed upon μ, q, and f are given for the solvability of the above problem in the set {x : xC1(J) ∩ C2((0,T) β t : t ∈ [0,T], x′(t) = 0}), 0 < x(t) < A, for t ∈ (0,T)}.

Keywords

Dirichlet problem
Singular boundary value problem
Positive solution

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Supported by Grant No. 201/98/0318 of the Grant Agency of Czech Republic and by the Council of Czech Government J14/98:153100011.