Remarks on nonlocal boundary value problems at resonance. (English) Zbl 1198.34026

Appl. Math. Comput. 216, No. 2, 497-500 (2010); addendum ibid. 218, No. 10, 6176 (2012).
The author considers the nonlocal boundary value problems
\[ -u''(t)=f(t, u(t), u'(t)),\quad u(0)=0,\;u(1)=\int^1_0 tdA(t), \] and
\[ -(p(t)u')'(t)=f(t, u(t),\;u'(t)),\quad u'(0)=0,\;u(1)=\int^1_0 tdB(t). \]
He shows that it is important to allow the nonlinear term \(f\) to change sign when discussing the existence of positive solutions by providing simple necessary and sufficient conditions for the existence of positive solutions when \(f\) has a fixed sign.
Reviewer: Ruyun Ma (Lanzhou)


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


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