<p>We provide sufficient conditions over <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(f\in L^1({\mathbb {R}}/T{\mathbb {Z}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">/</mo> <mi>T</mi> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> assuring the existence of a positive periodic solution of the second-order differential equation <Equation ID="Equ51"> <EquationSource Format="TEX">\(\begin{aligned} x''+x=x^{p-1}(x^2+x'^2)^{\frac{2-q}{2}}f(t), \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi>x</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo>′</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mfrac> <mrow> <mn>2</mn> <mo>-</mo> <mi>q</mi> </mrow> <mn>2</mn> </mfrac> </msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>which is associated with the planar <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> dual Minkowski problem in convex geometry.</p>