On periodic problems for non-autonomous Duffing equations and the equation arising from an energy harvester model

Abstract We study a periodic problem for the non-autonomous Duffing equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>u</m:mi> <m:mo>′′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mi>p</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>-</m:mo> <m:mi>h</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mo stretchy="false">|</m:mo> <m:mi>λ</m:mi> </m:msup> <m:mi>sgn</m:mi> <m:mi>u</m:mi> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>λ</m:mi> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> see text u^{\prime\prime}=p(t)u-h(t)|u|^{\lambda}\operatorname{sgn}u\quad(\lambda&gt;1) and unforced undamped periodic motions of the mechanical part of the energy harvester described by the equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>y</m:mi> <m:mo>′′</m:mo> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>y</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>⁢</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mi>mag</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> see text my^{\prime\prime}+by^{\prime}+ky+F_{\mathrm{mag}}(t,y)=f(t).