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On the existence of positive solution for a kind of multi-point boundary value problem at resonance. (English) Zbl 1200.34018

The paper deals with the existence of positive solutions for a second-order multi-point boundary value problem at resonance

x '' (t)+f(t,x(t))=0,t(0,1),
x(0)= i=1 m-2 α i x(ξ i ),x(1)= i=1 m-2 β i x(ξ i )·

The key tool is the Leggett-Williams norm-type theorem for coincidence equations due to D. O’Regan and M. Zima [Arch. Math. 87, No. 3, 233–244 (2006; Zbl 1109.47051)].

Reviewer’s remark: There is a gap in the proof of one of the main results of the paper. Namely, the assumptions of Theorem 2 do not imply the condition (C2) of Lemma 2.1.

34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations