On a periodic problem for super-linear second-order ODEs

Abstract The present paper concerns the periodic problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable columnalign="center" rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:msup> <m:mi>u</m:mi> <m:mo>″</m:mo> </m:msup> <m:mo>=</m:mo> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>;</m:mo> <m:mspace width="1em"/> <m:mi>u</m:mi> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>,</m:mo> <m:mspace width="thinmathspace"/> <m:msup> <m:mi>u</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:msup> <m:mi>u</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>,</m:mo> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $$\begin{array}{} \displaystyle u''=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\, u'(0)=u'(\omega), \end{array}$$ where p , f : [0, ω] → ℝ are Lebesgue integrable functions and q : [0, ω] × ℝ → ℝ is a Carathéodory function. We assume that the anti-maximum principle holds for the corresponding linear problem and provide sufficient conditions guaranteeing the existence and uniqueness of a positive solution to the given non-linear problem. The general results obtained are applied to the non-autonomous Duffing type equation with a super-linear power non-linearity.