On the existence of positive solutions of ordinary differential equations

Abstract

We study the existence of positive solutions of the equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Superscript Baseline plus a left-parenthesis t right-parenthesis f left-parenthesis u right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{u^{}} + a(t)f(u) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with linear boundary conditions. We show the existence of at least one positive solution if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either superlinear or sublinear by a simple application of a Fixed Point Theorem in cones.