In this paper, we study the existence of periodic solutions for Rayleigh type p-Laplacian equations <Equation ID="Equ1"> <EquationSource Format="TEX">$(\phi_{p}(x^{\prime}))^{\prime}+f(t,x^{\prime})+g(x)=e(t).$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </math></EquationSource> </Equation> By developing a continuation theorem, we prove that the given equation has at least one T-periodic solution provided that G (a primitive function of g) satisfies <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lim_{x \rightarrow }\inf_{+\infty}{pG(x)\over x^{p}}<({\pi_{p} \over T})^{p}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munder> <mo form="prefix" movablelimits="true">lim</mo> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> </mrow> </munder> <munder> <mo form="prefix" movablelimits="true">inf</mo> <mrow> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mfrac> <mrow> <mi>p</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mrow> <mi>p</mi> </mrow> </msup> </mfrac> </mrow> <mo><</mo> <mo stretchy="false">(</mo> <mrow> <mfrac> <msub> <mi>π</mi> <mrow> <mi>p</mi> </mrow> </msub> <mi>T</mi> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> and f satisfies p-sublinear condition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lim_{\vert x \vert \rightarrow+\infty }{f(t,x)\over\phi_{p}(x)}=0$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munder> <mo form="prefix" movablelimits="true">lim</mo> <mrow> <mo fence="false" stretchy="false">|</mo> <mi>x</mi> <mo fence="false" stretchy="false">|</mo> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>ϕ</mi> <mrow> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </math></EquationSource> </InlineEquation>.

Periodic Solutions for Rayleigh Type p-Laplacian Equations Crossing Resonant Points

期刊论文
发表日期：

2025年5月28日

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Cong-min Yang,
Zai-hong Wang

摘要

In this paper, we study the existence of periodic solutions for Rayleigh type p-Laplacian equations $(\phi_{p}(x^{\prime}))^{\prime}+f(t,x^{\prime})+g(x)=e(t).$ $(ϕ_{p} (x^{'}))^{'} + f (t, x^{'}) + g (x) = e (t) .$ By developing a continuation theorem, we prove that the given equation has at least one T-periodic solution provided that G (a primitive function of g) satisfies $\lim_{x \rightarrow }\inf_{+\infty}{pG(x)\over x^{p}}<({\pi_{p} \over T})^{p}$ $lim_{x \to} inf_{+ \infty} \frac{p G (x)}{x^{p}} < (\frac{π_{p}}{T})^{p}$ and f satisfies p-sublinear condition $\lim_{\vert x \vert \rightarrow+\infty }{f(t,x)\over\phi_{p}(x)}=0$ $lim_{| x | \to + \infty} \frac{f (t, x)}{ϕ_{p} (x)} = 0$ .