In this paper, we establish a new continuation theorem for rotating periodic nonlinear differential system with generalized variable exponents operators. This framework subsumes both ordinary and partial differential equations involving operators such as the p-Laplace operator, the so-called <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2239_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation>-Laplace operator, operators with variable exponents, and double-phase operators. And our results do not require the function defining the differential operator or the right-hand side function in the problem to be gradients. We then use this new theorem to obtain several existence results of rotating periodic solutions for several types differential systems. The rotating periodic solutions u(t) satisfying <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2239_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="175" /> </InlineMediaObject> <EquationSource Format="TEX">$u(t + T) = Qu(t)(t \in \mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Q</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where Q is an orthogonal matrix, can be periodic, anti-periodic, subharmonic, or quasi periodic depending on the specific form of the orthogonal matrix Q. Our results extend some existing relevant works.

A New Continuation Theorem for Nonlinear Rotating Periodic Differential Systems with Generalized Variable Exponents Operators and Its Applications

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2025年10月22日

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Tiefeng Ye,
Tengfei Shen,
Wenbin Liu,
Taiyong Chen

摘要

In this paper, we establish a new continuation theorem for rotating periodic nonlinear differential system with generalized variable exponents operators. This framework subsumes both ordinary and partial differential equations involving operators such as the p-Laplace operator, the so-called $\phi $ $ϕ$ -Laplace operator, operators with variable exponents, and double-phase operators. And our results do not require the function defining the differential operator or the right-hand side function in the problem to be gradients. We then use this new theorem to obtain several existence results of rotating periodic solutions for several types differential systems. The rotating periodic solutions u(t) satisfying $u(t + T) = Qu(t)(t \in \mathbb {R})$ $u (t + T) = Q u (t) (t \in R)$ , where Q is an orthogonal matrix, can be periodic, anti-periodic, subharmonic, or quasi periodic depending on the specific form of the orthogonal matrix Q. Our results extend some existing relevant works.