<p>In this paper, we mainly investigate new asymptotically bounded solutions to a multi-term semilinear fractional integro-differential equation with infinite delay. We first consider a new class of recurrent functions in Banach spaces, which is called <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S_{v}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>v</mi> </msub> </math></EquationSource> </InlineEquation>-asymptotically <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\omega ,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Bloch periodic functions. This notion generalizes that of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S_{v}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>v</mi> </msub> </math></EquationSource> </InlineEquation>-asymptotically <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-periodic functions in a natural way. We show several properties of such functions as completeness, composition and convolution theorems. We apply the derived results to discuss the existence of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$S_{v}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>v</mi> </msub> </math></EquationSource> </InlineEquation>-asymptotically <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\omega ,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Bloch periodic mild solutions to multi-term semilinear fractional integro-differential equations with infinite delay. The main results are based on the Mönch fixed point theorem and the Kuratowski measure of non-compactness.</p>

New asymptotically bounded solutions to multi-term semilinear fractional integro-differential equations with infinite delay

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发表日期：

2025年11月3日

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Yanyan Wei,
Sanyang Liu,
Yong-Kui Chang

摘要

In this paper, we mainly investigate new asymptotically bounded solutions to a multi-term semilinear fractional integro-differential equation with infinite delay. We first consider a new class of recurrent functions in Banach spaces, which is called $S_{v}$ $S_{v}$ -asymptotically $(\omega ,k)$ $(ω, k)$ -Bloch periodic functions. This notion generalizes that of $S_{v}$ $S_{v}$ -asymptotically $\omega $ $ω$ -periodic functions in a natural way. We show several properties of such functions as completeness, composition and convolution theorems. We apply the derived results to discuss the existence of $S_{v}$ $S_{v}$ -asymptotically $(\omega ,k)$ $(ω, k)$ -Bloch periodic mild solutions to multi-term semilinear fractional integro-differential equations with infinite delay. The main results are based on the Mönch fixed point theorem and the Kuratowski measure of non-compactness.