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A note on the solvability of singular boundary value problems. (English) Zbl 0853.34028

The author considers the singular boundary value problem

1 / q ( t )p (t) u ' ' =ft , u , p (t) u ' ,t(0,1),
u(0)=u(1)=0,when 0 1 1 / p ( t )dt<,


u(1)=0,u(t)C[ 0 , 1 ],when 0 1 1 / p ( t )dt=,

where p(t), q(t)>0 on (0,1], qC(0,1], pC[0,1] or pC(0,1] and fC([0,1]× 2 ) or fC((0,1]× 2 ). Under the assumption that f is bounded or f fulfills the classical sign conditions in the second variable and the Nagumo-type conditions in the third variable, he proves the existence. The typical conditions imposed on p and q are

0 1 q(t)dt<, 0 1 1 / p ( t ) 0 1 q(s)dsdt<·

The proof is based on the upper and lower solutions method and the Schauder fixed point theorem. The results complete the earlier ones by the author in [Nonlinear Anal., Theory Methods Appl. 21, 153-159 (1993; Zbl 0790.34027)].

34B15Nonlinear boundary value problems for ODE