In this paper, we consider the unique representation and the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ell _p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-maximal regularity of the solution for the fractional difference equation with finite delay: <Equation ID="Equ35"> <EquationSource Format="TEX">$\begin{aligned} \Delta ^{\alpha }u(n)=Au(n+1)+B u(n+1-\lambda )+f(n),\quad n\in {{\mathbb {N}}}_0 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mi>α</mi> </msup> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>-</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with initial conditions <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ u(i)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$i= -\lambda +1, -\lambda +2, \ldots , 0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mo>-</mo> <mi>λ</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>-</mo> <mi>λ</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0<\alpha <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\lambda \in {{\mathbb {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> are given, A and B are two bounded linear operators defined on a Banach space X and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$f:{{\mathbb {N}}}_0\rightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> is an X-valued sequence. We introduce the notion of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-resolvent sequence of bounded linear operators based on the operator theoretical methods, which gives the unique representation of solution under suitable assumption on the parameters <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$A, B, \lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\alpha .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We also characterize the <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\ell _p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-maximal regularity of solution using Blunck’s operator-valued Fourier multipliers theorems on <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\ell _p ({{{\mathbb {Z}}}}; X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo>;</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$1 <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and X is a UMD space. Moreover, we are able to relax some conditions in the case of Hilbert spaces.

The $l_p$-Maximal Regularity for Fractional Difference Equations with Finite Delay on UMD Spaces

期刊论文
发表日期：

2025年11月24日

查看全文

Jichao Zhang,
Shangquan Bu

摘要

In this paper, we consider the unique representation and the $\ell _p$ $ℓ_{p}$ -maximal regularity of the solution for the fractional difference equation with finite delay: $\begin{aligned} \Delta ^{\alpha }u(n)=Au(n+1)+B u(n+1-\lambda )+f(n),\quad n\in {{\mathbb {N}}}_0 \end{aligned}$ $\begin{matrix} Δ^{α} u (n) = A u (n + 1) + B u (n + 1 - λ) + f (n), n \in N_{0} \end{matrix}$ with initial conditions $ u(i)=0$ $u (i) = 0$ when $i= -\lambda +1, -\lambda +2, \ldots , 0,$ $i = - λ + 1, - λ + 2, \dots, 0,$ where $0<\alpha <1$ $0 < α < 1$ and $\lambda \in {{\mathbb {N}}}$ $λ \in N$ are given, A and B are two bounded linear operators defined on a Banach space X and $f:{{\mathbb {N}}}_0\rightarrow X$ $f : N_{0} \to X$ is an X-valued sequence. We introduce the notion of $\alpha $ $α$ -resolvent sequence of bounded linear operators based on the operator theoretical methods, which gives the unique representation of solution under suitable assumption on the parameters $A, B, \lambda $ $A, B, λ$ and $\alpha .$ $α .$ We also characterize the $\ell _p$ $ℓ_{p}$ -maximal regularity of solution using Blunck’s operator-valued Fourier multipliers theorems on $\ell _p ({{{\mathbb {Z}}}}; X)$ $ℓ_{p} (Z; X)$ when $1< p < \infty $ $1 < p < \infty$ and X is a UMD space. Moreover, we are able to relax some conditions in the case of Hilbert spaces.

The \(l_p\)-Maximal Regularity for Fractional Difference Equations with Finite Delay on UMD Spaces

摘要