Abstract
In this chapter we present the maximal discrete regularity approach to first-order linear difference equations in general Banach spaces. In the first section we introduce the general frame for first-order linear difference equations. The entire linear theory of maximal regularity is not only important on its own, but it is also the indispensable basis for the theory of nonlinear difference equations, which we present in the next chapter.
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Notes
- 1.
If (3.1.1) has discrete maximal l p -regularity, then T is analytic. In fact, put \(M:=\sup _{n}\vert \vert T^{n}\vert \vert \), we consider for all \(b \in \mathbb{N}\) and x ∈ X the sequence \(f \in l_{p}(\mathbb{Z}_{+},X)\) defined by f j = T j x for j = 1, …, b and f j = 0 otherwise. We have the following estimates:
$$\displaystyle\begin{array}{rcl} \vert \vert (T - I)T^{b}x\vert \vert _{ X}& \leq &\vert \vert T^{b-n}(T - I)T^{n}x\vert \vert _{ X} \leq \vert \vert T^{b-n}\vert \vert \|(T - I)T^{n}x\vert \vert _{ X} \leq M\|(T - I)T^{n}x\vert \vert _{ X}, {}\\ \vert \vert K_{T}f\vert \vert _{p}& \leq &\|K_{T}\vert \vert _{\mathcal{B}(l_{p}(\mathbb{Z}_{+};X))}\vert \vert f\vert \vert _{p} \leq \vert \vert K_{T}\vert \vert _{\mathcal{B}(l_{p}(\mathbb{Z}_{+};X))}Mb^{1/p}\vert \vert x\vert \vert _{ X}. {}\\ \end{array}$$Using the first one, we get
$$\displaystyle\begin{array}{rcl} \vert \vert K_{T}f\vert \vert _{p}& \geq &\Big(\sum _{n=1}^{b}\vert \vert (K_{ T}f)_{n}\vert \vert _{X}^{p}\Big)^{1/p} =\Big (\sum _{ n=1}^{b}n^{p}\|(T - I)T^{n}x\vert \vert _{ X}^{p}\Big)^{1/p} {}\\ & \geq & M^{-1}\Big(\sum _{ n=1}^{b}n^{p}\Big)^{1/p}\vert \vert (T - I)T^{b}x\vert \vert _{ X} \geq (2M)^{-1}b^{1+1/p}\vert \vert (T - I)T^{b}x\vert \vert _{ X}. {}\\ \end{array}$$Therefore \(\vert \vert b(T - I)T^{b}x\vert \vert _{X} \leq 2M^{2}\|K_{T}\vert \vert _{\mathcal{B}(l_{p}(\mathbb{Z}_{+};X))}\vert \vert x\vert \vert _{X}.\)
- 2.
The subpositivity of a contraction T on Lp is defined by the existence of a dominating positive contraction S, i.e., \(\vert Tf\vert \leq S\vert f\vert \) for all f ∈ Lp.
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Agarwal, R.P., Cuevas, C., Lizama, C. (2014). First-Order Linear Difference Equations. In: Regularity of Difference Equations on Banach Spaces. Springer, Cham. https://doi.org/10.1007/978-3-319-06447-5_3
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