一类分数阶发展方程非局部问题的精确可控性 Exact Controllability for a Class of Fractional Evolution Equations with Nonlocal Conditions

doi:10.12677/PM.2022.125096

Pure Mathematics
Vol. 12 No. 05 ( 2022 ), Article ID: 51878 , 14 pages
10.12677/PM.2022.125096

一类分数阶发展方程非局部问题的精确可控性

苏怡，杨和

●How to Cite this Article

西北师范大学数学与统计学院，甘肃兰州

收稿日期：2022年4月21日；录用日期：2022年5月23日；发布日期：2022年5月30日

摘要

本文讨论了一类分数阶发展方程非局部问题 ${\begin{cases} D^{q} x (t) = A x (t) + f (t, x (t)) + B u (t), t \in [0, b], \\ x (0) = \sum_{k = 1}^{m} c_{k} x (t_{k}) \end{cases}$ 的精确可控性。文中通过引入一个新的非紧性测度，在C₀-半群等度连续的情形下，运用Mönch不动点定理，证明了该问题的精确可控性，并通过一个具体的例子来验证本文的抽象结论。

关键词

Caputo分数阶导数，精确可控性，Mönch不动点定理，非紧性测度，等度连续模

Exact Controllability for a Class of Fractional Evolution Equations with Nonlocal Conditions

Yi Su, He Yang

College of Mathematics and Statistics, Northwest Normal University, Lanzhou Gansu

Received: Apr. 21^st, 2022; accepted: May 23^rd, 2022; published: May 30^th, 2022

ABSTRACT

This paper discusses the exact controllability of nonlocal conditions for a class of fractional evolution equations ${\begin{cases} D^{q} x (t) = A x (t) + f (t, x (t)) + B u (t), t \in [0, b], \\ x (0) = \sum_{k = 1}^{m} c_{k} x (t_{k}) \end{cases}$ . In this paper, by introducing a new measure of non-compactness, the exact controllability of the problem is proved by using Mönch fixed point theorem under the condition that C₀-semigroup is equicontinuous, and an example is given to verify the abstract conclusion of this paper.

Keywords:Caputo Fractional Derivative, Exact Controllability, The Mönch Fixed Point Theorem, The Measure of Non-Compactness, Modulus of Equicontinuity

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

分数阶微分方程由于在数学、物理、工程等方面具有广泛应用而被许多学者关注，但是在实际应用中非局部初始条件相比于经典的初始条件能够更好地描述自然现象(参见文献 [1] - [8] )，因此研究分数阶发展方程非局部问题的可解性和可控性具有重要的理论意义和实际应用价值。Byszewski在文献 [2] 中首次研究了一类具有非局部初始条件的半线性一阶发展方程解的存在唯一性。从此以后，人们开始关注具有非局部初始条件的微分方程。

2015年，梁进等 [9] 在非紧半群的情况下，利用Mönch不动点定理证明了分数阶积微分发展方程非局部问题

${\begin{cases} D^{q} x (t) = A x (t) + f (t, x (t), G x (t)) + B u (t), t \in [0, b], \\ x (0) = \sum_{k = 1}^{m} c_{k} x (t_{k}) \end{cases}$ (1.1)

的精确可控性。Deng在文献 [4] 中指出在(1.1)中使用的非局部函数可以很好地描述少量气体在透明管中的扩散现象。

2020年，陈鹏玉等 [10] 通过引入一个新的格林函数，应用Schauder不动点定理及预解算子理论研究了Banach空间中分数阶发展方程非局部问题

${\begin{cases} D_{t}^{q} x (t) = A x (t) + B u (t) + f (t, x (t)), t \in [0, b], \\ x (0) = \sum_{k = 1}^{m} c_{k} x ( t k ) \end{cases}$

mild解的存在性和近似可控性。

受上述文献的启发，本文研究Banach空间X中半线性分数阶发展方程非局部问题

${\begin{cases} D^{q} x (t) = A x (t) + f (t, x (t)) + B u (t), t \in [0, b], \\ x (0) = \sum_{k = 1}^{m} c_{k} x (t_{k}) \end{cases}$ (1.2)

的精确可控性，其中 $D^{q}$ 为 $q \in (0, 1)$ 阶Caputo分数阶导数， $A : D (A) \subset X \to X$ 是 $C_{0}$ -半群 ${T (t)}_{t \geq 0}$ 的无穷小生成元，控制函数 $u \in L^{2} (J, U)$ ， $J = [0, b]$ ， $b > 0$ 为常数，U为Banach空间， $B : U \to X$ 为有界线性算子，函数f为给定函数。 $0 < t_{1} < t_{2} < \dots < t_{m} < b$ ， $b \in N$ ， $c_{k}$ 是实数且 $c_{k} \neq 0$ ， $k = 1, 2, \dots, m$ 。

本文对非局部函数的紧性和连续性都不做要求。在 $C_{0}$ -半群等度连续的情况下，通过定义一种新的非紧性测度(见文献 [11] )并应用Mönch不动点定理证明了系统(1.2)的精确可控性。在第2节中介绍了讨论问题(1.2)的一些准备工作，主要的结果和证明将在第3节中给出，第4节通过一个具体的例子来验证本文获得的抽象结论。

2. 预备知识

设 $(X, ‖ \cdot ‖)$ 是可分Banach空间，U是另一个Banach空间。设 $C (J, X)$ 表示J上所有X−值连续函数全体按范数 ${‖ u ‖}_{C} = \max_{t \in J} ‖ u (t) ‖$ 构成的Banach空间。 $L^{p} (J, X) (p \geq 1)$ 是J上的X−值p方Bochner可积函数全体按范数 ${‖ x ‖}_{L^{p}}$ 构成的Banach空间。关于Caputo分数阶导数，我们介绍如下定义。

定义1 [12] 函数 $f \in C^{n} [0, \infty)$ 下限为0的Caputo分数阶导数为：

$D^{q} f (t) = \frac{1}{Γ (n - q)} \int_{0}^{t} {(t - s)}^{n - q - 1} f^{(n)} (s) d s, t > 0, n - 1 < q < n, n \in N,$

其中 $Γ (\cdot)$ 为Gamma函数。

对 $\forall h \in C (J, X)$ ，考虑线性发展方程的非局部问题

${\begin{cases} D^{q} x (t) = A x (t) + h (t), t \in [0, b], \\ x (0) = \sum_{k = 1}^{m} c_{k} x (t_{k}) \end{cases}$ (2.1)

引理1 [9] 设条件(H0)成立，其中

(H0) $\sum_{k = 1}^{m} | c_{k} | \leq \frac{1}{M}, M \geq 1,$

则线性系统(2.1)有唯一mild解 $x \in C (J, X)$ 满足

$x (t) = \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) h (s) d s + \int_{0}^{t} {(t - s)}^{q - 1} T_{q} (t - s) h (s) d s, t \in J,$ (2.2)

其中 $P = {(I - \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t_{k}))}^{- 1}$ 满足 $‖ P ‖ \leq \frac{1}{1 - M \sum_{k = 1}^{m} | c_{k} |}$ ，

$\hat{T_{q}} (t) = \int_{0}^{\infty} ξ_{q} (θ) T (t^{q} θ) d θ,$

$T_{q} (t) = q \int_{0}^{\infty} θ ξ_{q} (θ) T (t^{q} θ) d θ,$

其中

$ξ_{q} (θ) = \frac{1}{q} θ^{- 1 - \frac{1}{q}} W_{q} (θ^{- \frac{1}{q}}) \geq 0, θ \in (0, \infty),$

$W_{q} (θ) = \frac{1}{π} \sum_{k = 1}^{m} {(- 1)}^{n - 1} θ^{- q n - 1} \frac{Γ (1 + n q)}{n!} \sin (n π q), θ \in (0, \infty) .$

这里， $ξ_{q} (θ)$ 表示定义在 $(0, \infty)$ 上的概率密度函数，满足

$ξ_{q} (θ) \geq 0, \forall θ \in (0, \infty), \int_{0}^{\infty} ξ_{q} (θ) d θ = 1.$

引理2 [9] [12] 线性算子族 ${\hat{T_{q}} (t)}_{t \geq 0}$ 和 ${T_{q} (t)}_{t \geq 0}$ 具有下列性质：

(1) 对任意给定的 $t \geq 0$ 和 $\forall x \in X$ ，有

$‖ \hat{T_{q}} (t) x ‖ \leq M ‖ x ‖, ‖ T_{q} (t) x ‖ \leq \frac{M q}{Γ (1 + q)} ‖ x ‖;$

(2) 对 $\forall t \geq 0$ ， $\hat{T_{q}} (t)$ 和 $T_{q} (t)$ 都是强连续的；

(3) 如果 ${T (t)}_{t \geq 0}$ 是等度连续半群，则对 $\forall x > 0$ ， $\hat{T_{q}} (t)$ 和 $T_{q} (t)$ 都是等度连续的。

根据引理1，本文采用如下mild解的定义。

定义2对 $\forall u \in L^{2} (J, U)$ ，如果函数 $x \in C (J, X)$ 满足积分方程

$\begin{matrix} x (t) = \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u (s) + f (s, x (s))] d s \\ + \int_{0}^{t} {(t - s)}^{q - 1} T_{q} (t - s) [B u (s) + f (s, x (s))] d s, \end{matrix}$ (2.3)

则称函数x为半线性分数阶发展系统(1.2)的mild解。

定义3 [13] 若对任一给定的 $x_{b} \in X$ ，存在控制函数 $u \in L^{2} (J, U)$ ，使得系统(1.2)的mild解x满足

$x (b) = x_{b},$

则称系统(1.2)在J上是精确可控的。

本文主要采用非紧性测度方法和不动点定理研究半线性分数阶发展系统(1.2) mild解的存在性和精确可控性，关于非紧性测度，我们有如下结论。

引理3 [14] [15] 设 $Ω, Ω_{1}, Ω_{2}$ 是X中的非空有界集， $a \in R$ ，则Hausdroff非紧性测度 $β (\cdot)$ 具有下列性质：

(1) $Ω_{1} \subset Ω_{2} \Rightarrow β (Ω_{1}) \leq β (Ω_{2})$ ；

(2) $β (Ω_{1} + Ω_{2}) \leq β (Ω_{1}) + β (Ω_{2})$ ，其中 $Ω_{1} + Ω_{2} = {x + y : x \in Ω_{1}, y \in Ω_{2}}$ ；

(3) $β (Ω_{1} \cup Ω_{2}) \leq \max {β (Ω_{1}), β (Ω_{2})}$ ；

(4) $β (λ Ω) \leq | λ | β (Ω)$ ，对 $\forall λ \in R$ ；

(5) $β ({0} \cup Ω) = β (Ω)$ ；

(6) $β (Ω) = 0 \Leftrightarrow Ω$ 在X中相对紧。

引理4 [14] 设X为Banach空间， $D \subset C (J, X)$ 有界且等度连续，则 $β (D (t))$ 在J上连续，且 $β (D) = \max_{t \in J} β (D (t))$ 。

引理5 [16] 设X是可分的Banach空间， $D_{0} = {x_{m}} \subset C (J, X)$ 是可数集。若 $\exists ϕ \in L^{1} (J)$ ，使得

$‖ x_{m} (t) ‖ \leq ϕ (t), t \in J, m = 1, 2, \dots,$

则 $β (D (t))$ 在J上Lebesgue可积，且

$β ({\int_{J} x_{m} (t) d t : m = 1, 2, \dots}) \leq 2 \int_{J} β (D_{0} (t)) d t .$

特别地，当 $D_{0}$ 有界时上式成立。

定义4 [11] 如果序列 ${f_{n}}_{n = 1}^{\infty}$ 在X中对 $a . e . t \in J$ 是相对紧的，且对 $a . e . t \in J$ ，存在函数 $μ \in L^{1} (J, X)$ 满足

$\sup_{n \geq 1} ‖ f_{n} (t) ‖ \leq μ (t),$

则称序列 ${f_{n}}_{n = 1}^{\infty} \subset L^{1} (J, X)$ 是半紧的。

引理6 [11] 设 $G : L^{1} (J, X) \to C (J, X)$ 定义为

$(G f) (t) = \int_{0}^{t} T_{q} (t - s) f (s) d s .$

如果序列 ${f_{n}}_{n = 1}^{\infty} \subset L^{1} (J, X)$ 是半紧的，则序列 ${G f_{n}}_{n = 1}^{\infty}$ 在 $C (J, X)$ 上相对紧。此外，如果 $f_{n} \to f$ ，则 $(G f_{n}) (t) \to (G f) (t), n \to \infty$ 。

引理7 [17] 设D为Banach空间X的闭凸子集且 $0 \in D$ 。如果 $F : D \to X$ 是连续映射，且满足如下条件：若 $Ω \subseteq D$ 可数，且 $Ω \subseteq \bar{c o} ({0} \cup F (Ω))$ ，则 $\bar{Ω}$ 是紧的，那么F在D中有不动点。

3. 主要结果及证明

为了证明本文的主要定理，我们先给出如下假设条件：

(H1) 线性算子A生成的 $C_{0}$ -半群 $T (t) (t \geq 0)$ 是等度连续半群，且存在 $M \geq 1$ ，使得 $‖ T (t) ‖ \leq M$ 。

(H2) 函数 $f : J \times X \to X$ 满足：

(1) 对 $a . e . t \in J$ ，函数 $f (t, \cdot) : X \to X$ 连续；对每个 $x \in X$ ，函数 $f (\cdot, x) : J \to X$ 强可测；

(2) 对 $\forall r > 0$ ，存在常数 $q_{1} \in (0, q)$ 和函数 $m \in L^{\frac{1}{q_{1}}} (J, R^{+})$ ，使得对 $\forall t \in J, x \in X$ ，有

$‖ f (t, x) ‖ \leq m (t), t \in J,$

其中 $m (t)$ 满足 $\lim_{k \to + \infty} \inf \frac{{‖ m ‖}_{L^{\frac{1}{q_{1}}}}}{r} = σ < \infty$ ；

(3) 存在常数 $L > 0$ ，使得对X中任意非空有界集D，有

$β (f (t, D)) \leq L β (D) .$

(H3) $B : U \to X$ 是线性有界算子，即存在 $M_{B} > 0$ ，使得

$‖ B ‖ \leq M_{B} .$

(H4) 线性算子 $W : L^{2} (J, U) \to X$ 定义为：

$W u = \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) B u (s) d s + \int_{0}^{b} {(b - s)}^{q - 1} T_{q} (b - s) B u (s) d s,$

满足下列条件：

(1) W存在取值于 $L^{2} (J, U) \ K e r W$ 中的线性逆算子 $W^{- 1}$ ，其中

$K e r W = {x \in L^{2} (J, U) : W x = 0};$

(2) 存在常数 $L_{W}$ ，使得 $‖ W^{- 1} ‖ \leq L_{W}$ ；

(3) 存在常数 $L^{'} > 0$ ，使得对X中任意非空有界集D，有

$β (W^{- 1} (D)) \leq L^{'} β (D) .$

(H5) 存在常数 $R > 0$ ，使得

$p = [2 q M L l_{1} l_{2} (2 q M L^{'} M_{B} l_{1} l_{2} + 1)] \sup_{t \in [0, b]} \int_{0}^{t} e^{- R (t - s)} d s < 1,$

其中，

$l_{1} = \frac{1}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)}, l_{2} = \frac{b^{q - q_{1}}}{{(1 + a)}^{1 - q_{1}}} .$

定理3.1设条件(H1)~(H5)成立。如果

$l_{1} l_{2} q M σ (q l_{1} M M_{B} L_{W} + 1) < 1,$ (3.1)

则半线性分数阶发展系统(1.2)在J上精确可控。

证明：根据条件(H3) (1)，对每个 $x \in X, t \in J$ ，定义控制 $u (t) : = u_{x} (t)$ 如下：

$\begin{matrix} u_{x} (t) = W^{- 1} [x_{1} - \hat{T_{q}} (b) \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) f (s, x (s)) d s \\ - \int_{0}^{b} {(b - s)}^{q - 1} T_{q} (b - s) f (s, x (s)) d s] (t) . \end{matrix}$

按此控制函数，分数阶发展系统(1.2)在J上的精确可控等价于算子 $G : C (J, X) \to C (J, X)$ 存在不动点，其中算子G定义如下：

$\begin{matrix} (G x) (t) = \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u_{x} (s) + f (s, x (s))] d s \\ + \int_{0}^{t} {(t - s)}^{q - 1} T_{q} (t - s) [B u_{x} (s) + f (s, x (s))] d s . \end{matrix}$ (3.2)

下面我们将用引理7证明算子G在 $C (J, X)$ 上存在不动点，证明过程分为4步。

第1步，证明存在 $r > 0$ ，使得 $G (B_{r}) \subseteq B_{r}$ ，其中 $B_{r} = {x \in C (J, X) : {‖ x ‖}_{C} \leq r}$ 。

反设不成立，则对任意 $r > 0$ ，存在函数 $x^{r} (\cdot) \in B_{r}$ ，但 $G (x^{r}) \notin B_{r}$ ，即 $‖ G (x^{r}) (t) ‖ > r, t \in J$ 。易知 ${(t - s)}^{q - 1} \in L^{\frac{1}{1 - q_{1}}} ([0, b], R)$ ，其中 $q_{1} \in [0, q), a = \frac{q - 1}{1 - q_{1}}$ 。由条件(H2) (2)及Hölder不等式，有

$\begin{matrix} ‖ u_{x^{r}} (t) ‖ = ‖ W^{- 1} [x_{1} - \hat{T_{q}} (b) \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) f (s, x^{r} (s)) d s \\ - \int_{0}^{b} {(b - s)}^{q - 1} T_{q} (b - s) f (s, x^{r} (s)) d s] ‖ \\ \leq L_{W} [‖ x_{1} ‖ + \frac{M \sum_{k = 1}^{m} | c_{k} |}{1 - M \sum_{k = 1}^{m} | c k |} \frac{M q}{Γ (1 + q)} ‖ \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} f (s, x^{r} (s)) d s ‖ \\ + \frac{M q}{Γ (1 + q)} ‖ \int_{0}^{b} {(b - s)}^{q - 1} f (s, x^{r} (s)) d s ‖] \end{matrix}$

$\begin{array}{l} \leq L_{W} ‖ x_{1} ‖ + \frac{q L_{W} M^{2} \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} [{(\int_{0}^{t_{k}} {(t_{k} - s)}^{\frac{q - 1}{1 - q_{1}}} d s)}^{1 - q_{1}} {‖ m ‖}_{L^{\frac{1}{q_{1}}}}] \\ + \frac{L_{W} M q}{Γ (1 + q)} [{(\int_{0}^{b} {(b - s)}^{\frac{q - 1}{1 - q_{1}}} d s)}^{1 - q_{1}} {‖ m ‖}_{L^{\frac{1}{q_{1}}}}] \\ \leq L_{W} ‖ x_{1} ‖ + \frac{q L_{W} M}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \frac{b^{q - q_{1}}}{{(1 + a)}^{1 - q_{1}}} {‖ m ‖}_{L^{\frac{1}{q_{1}}}} . \end{array}$

所以

$‖ u_{x^{r}} ‖ \leq L_{W} ‖ x_{1} ‖ + q l_{1} l_{2} L_{W} M {‖ m ‖}_{L^{\frac{1}{q_{1}}}} .$ (3.3)

则

$‖ B u_{x^{r}} (t) ‖ + ‖ f (s, x^{r} (s)) ‖ \leq M_{B} L_{W} ‖ x_{1} ‖ + q l_{1} l_{2} M L_{W} M_{B} {‖ m ‖}_{L^{\frac{1}{q_{1}}}} + m (t) .$

因此，

$\begin{matrix} r < ‖ (G x^{r}) (t) ‖ \\ \leq ‖ \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u_{x^{r}} (s) + f (s, x^{r} (s))] d s ‖ \\ + ‖ \int_{0}^{t} {(t - s)}^{q - 1} T_{q} (t - s) [B u_{x^{r}} (s) + f (s, x^{r} (s))] d s ‖ \\ \leq \frac{q M^{2} \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} [‖ B u_{x^{r}} (s) ‖ + ‖ f (s, x^{r} (s)) ‖] d s \\ + \frac{M q}{Γ (1 + q)} \int_{0}^{b} {(b - s)}^{q - 1} [‖ B u_{x^{r}} (s) ‖ + ‖ f (s, x^{r} (s)) ‖] d s \end{matrix}$

$\begin{array}{l} \leq \frac{b^{q} M^{2} \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} (M_{B} L_{W} ‖ x_{1} ‖ + l_{1} l_{2} q M L_{W} M_{B} {‖ m ‖}_{L^{\frac{1}{q_{1}}}}) \\ + \frac{q M^{2} \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{K} |)} l_{2} {‖ m ‖}_{L^{\frac{1}{q_{1}}}} + \frac{M q}{Γ (1 + q)} l_{2} {‖ m ‖}_{L^{\frac{1}{q_{1}}}} \\ + \frac{M b^{q}}{Γ (1 + q)} (M_{B} L_{W} ‖ x_{1} ‖ + l_{1} l_{2} q M L_{W} M_{B} {‖ m ‖}_{L^{\frac{1}{q_{1}}}}) \\ \leq l_{1} M M_{B} L_{W} b^{q} ‖ x_{1} ‖ + l_{1} l_{2} q M (q l_{1} M M_{B} L_{W} + 1) {‖ m ‖}_{L^{\frac{1}{q_{1}}}} . \end{array}$

两边同时除以r，并令 $r \to \infty$ ，得

$1 \leq l_{1} l_{2} q M σ (q l_{1} M M_{B} L_{W} + 1),$

这与(3.1)式矛盾。因此， $\exists r > 0$ ，使得 $G (B_{r}) \subseteq B_{r}$ 。

第2步，证明 $G : B_{r} \to B_{r}$ 连续。

设 ${x^{n}} \subset B_{r}$ 满足 $x^{n} \to x$ 。由(H2) (1)和(2)及Lebesgue控制收敛定理，有

$\int_{0}^{t} {(t - s)}^{q - 1} ‖ f (s, x^{n} (s)) - f (s, x (s)) ‖ d s \to 0, t \in J, n \to \infty .$

因此，

$\begin{array}{l} ‖ u_{x^{n}} (s) - u_{x} (s) ‖ \\ = ‖ W^{- 1} [x_{1} - \hat{T_{q}} (b) \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) f (s, x^{n} (s)) d s - \int_{0}^{b} {(b - s)}^{q - 1} T_{q} (b - s) f (s, x^{n} (s)) d s] \\ - W^{- 1} [x_{1} - \hat{T_{q}} (b) \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) f (s, x (s)) d s - \int_{0}^{b} {(b - s)}^{q - 1} T_{q} (b - s) f (s, x (s)) d s] ‖ \\ \leq ‖ W^{- 1} \hat{T_{q}} (b) \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [f (s, x^{n} (s)) - f (s, x (s))] d s ‖ \\ + ‖ W^{- 1} \int_{0}^{b} {(b - s)}^{q - 1} T_{q} (b - s) [f (s, x^{n} (s)) - f (s, x (s))] d s ‖ \end{array}$

$\begin{array}{l} \leq \frac{L_{W} q M^{2} \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} ‖ f (s, x^{n} (s)) - f (s, x (s)) ‖ d s \\ + \frac{L_{W} M q}{Γ (1 + q)} \int_{0}^{b} {(b - s)}^{q - 1} ‖ f (s, x^{n} (s)) - f (s, x (s)) ‖ d s \\ \to 0. \end{array}$

因此，由算子G的定义，有

$\begin{array}{l} ‖ (G x^{n}) (t) - (G x) (t) ‖ \\ = ‖ \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u_{x^{n}} (s) + f (s, x^{n} (s))] d s \\ + \int_{0}^{t} {(t - s)}^{q - 1} T_{q} (t - s) [B u_{x^{n}} (s) + f (s, x^{n} (s))] d s \\ - \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u_{x} (s) + f (s, x (s))] d s \\ - \int_{0}^{t} {(t - s)}^{q - 1} T_{q} (t - s) [B u_{x} (s) + f (s, x (s))] d s ‖ \end{array}$

$\begin{array}{l} \leq \frac{q M^{2} \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} ‖ B u_{x^{n}} (s) - B u_{x} (s) ‖ d s \\ + \frac{M q}{Γ (1 + q)} \int_{0}^{b} {(b - s)}^{q - 1} ‖ B u_{x^{n}} (s) - B u_{x} (s) ‖ d s \\ \leq \frac{q M^{2} M_{B} \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} ‖ u_{x^{n}} (s) - u_{x} (s) ‖ d s \\ + \frac{M_{B} M q}{Γ (1 + q)} \int_{0}^{b} {(b - s)}^{q - 1} ‖ u_{x^{n}} (s) - u_{x} (s) ‖ d s \\ \to 0. \end{array}$

所以， $G : B_{r} \to B_{r}$ 连续。

第3步，证明 $G (B_{r})$ 在J上等度连续。

取 $t_{1}, t_{2} \in J, t_{1} < t_{2}$ 。对 $\forall x \in B_{r}$ ，及 $\forall ε \in (0, t_{1})$ ，有

$\begin{array}{l} ‖ (G x) (t_{2}) - (G x) (t_{1}) ‖ \\ = ‖ \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t_{2}) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u (s) + f (s, x (s))] d s \\ + \int_{0}^{t_{2}} {(t_{2} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s \\ - \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u (s) + f (s, x (s))] d s \\ - \int_{0}^{t_{1}} {(t_{1} - s)}^{q - 1} T_{q} (t_{1} - s) [B u (s) + f (s, x (s))] d s ‖ \end{array}$

$\begin{array}{l} \leq ‖ [\hat{T_{q}} (t_{2}) - \hat{T_{q}} (t_{1})] \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{0}^{t_{1}} {(t_{2} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{t_{1}}^{t_{2}} {(t_{2} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{0}^{t_{1}} {(t_{1} - s)}^{q - 1} T_{q} (t_{1} - s) [B u (s) + f (s, x (s))] d s ‖ \\ \leq ‖ [\hat{T_{q}} (t_{2}) - \hat{T_{q}} (t_{1})] \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u (s) + f (s, x (s))] d s ‖ \end{array}$

$\begin{array}{l} + ‖ \int_{t_{1}}^{t_{2}} {(t_{2} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{0}^{t_{1}} {(t_{2} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ - ‖ \int_{0}^{t_{1}} {(t_{1} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{0}^{t_{1}} {(t_{1} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{0}^{t_{1}} {(t_{1} - s)}^{q - 1} T_{q} (t_{1} - s) [B u (s) + f (s, x (s))] d s ‖ \end{array}$

$\begin{array}{l} \leq ‖ [\hat{T_{q}} (t_{2}) - \hat{T_{q}} (t_{1})] \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{t_{1}}^{t_{2}} {(t_{2} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{0}^{t_{1}} [{(t_{2} - s)}^{q - 1} - {(t_{1} - s)}^{q - 1}] T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{0}^{t_{1}} {(t_{1} - s)}^{q - 1} [T_{q} (t_{2} - s) - T_{q} (t_{1} - s)] [B u (s) + f (s, x (s))] d s ‖ \\ : = I_{1} + I_{2} + I_{3} + I_{4} . \end{array}$

由引理2可知，当 $t_{2} \to t_{1}$ 时， $\hat{T_{q}} (t_{2}) - \hat{T_{q}} (t_{1}) \to 0$ ，即 $I_{1} \to 0$ 。

$\begin{matrix} I_{2} \leq \frac{M q}{Γ (1 + q)} \int_{t_{1}}^{t_{2}} {(t_{2} - s)}^{q - 1} [M_{B} L_{W} ‖ x_{1} ‖ + l_{1} q M L_{W} M_{B} \frac{b^{q - q_{1}}}{{(1 + a)}^{1 - q_{1}}} {‖ m ‖}_{L^{\frac{1}{q_{1}}}} + ‖ m (t) ‖] d s \\ \leq \frac{M q}{Γ (1 + q)} [M_{B} L_{W} ‖ x_{1} ‖ + l_{1} q M L_{W} M_{B} l_{2} {‖ m ‖}_{L^{\frac{1}{q_{1}}}}] {(t_{2} - t_{1})}^{q} + \frac{M q}{Γ (1 + q)} \frac{{(t_{2} - t_{1})}^{q}}{{(1 + a)}^{1 - q_{1}}} {‖ m ‖}_{L^{\frac{1}{q_{1}}}} . \end{matrix}$

当 $t_{2} - t_{1} \to 0$ 时， $I_{2} \to 0$ 。

$\begin{matrix} I_{3} \leq \frac{M q}{Γ (1 + q)} [M_{B} L_{W} ‖ x_{1} ‖ + l_{1} l_{2} q M L_{W} M_{B} {‖ m ‖}_{L^{\frac{1}{q_{1}}}}] \int_{0}^{t_{1}} [{(t_{2} - s)}^{q - 1} - {(t_{1} - s)}^{q - 1}] d s \\ + \frac{M q}{Γ (1 + q)} \int_{0}^{t_{1}} [{(t_{2} - s)}^{q - 1} - {(t_{1} - s)}^{q - 1}] m (t) d s \\ \leq \frac{M q}{Γ (1 + q)} [M_{B} L_{W} ‖ x_{1} ‖ + l_{1} l_{2} q M L_{W} M_{B} {‖ m ‖}_{L^{\frac{1}{q_{1}}}}] \int_{0}^{t_{1}} [{(t_{2} - s)}^{q - 1} - {(t_{1} - s)}^{q - 1}] d s \\ + \frac{M q}{Γ (1 + q)} {‖ m ‖}_{L^{\frac{1}{q_{1}}}} {(\int_{0}^{t_{1}} {| {(t_{2} - s)}^{q - 1} - {(t_{1} - s)}^{q - 1} |}^{\frac{1}{1 - q_{1}}} d s)}^{1 - q_{1}} . \end{matrix}$

当 $t_{2} - t_{1} \to 0$ 时， $I_{3} \to 0$ 。

对 $I_{4}$ ，由引理2(3)可知，对 $\forall t > 0$ ， $T_{q} (t)$ 是等度连续算子，当 $t_{2} - t_{1} \to 0$ 且 $ε \to 0$ 时，有

$\begin{matrix} I_{4} \leq ‖ \int_{0}^{t_{1} - ε} {(t_{1} - s)}^{q - 1} [T_{q} (t_{2} - s) - T_{q} (t_{1} - s)] [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{t_{1} - ε}^{t_{1}} {(t_{1} - s)}^{q - 1} [T_{q} (t_{2} - s) - T_{q} (t_{1} - s)] [B u (s) + f (s, x (s))] d s ‖ \\ \leq \sup_{s \in [0, t_{1} - ε]} ‖ T_{q} (t_{2} - s) - T_{q} (t_{1} - s) ‖ \int_{0}^{t_{1} - ε} {(t_{1} - s)}^{q - 1} ‖ B u (s) + f (s, x (s)) ‖ d s \\ + \frac{2 M q}{Γ (1 + q)} \int_{t_{1} - ε}^{t_{1}} {(t_{1} - s)}^{q - 1} ‖ B u (s) + f (s, x (s)) ‖ d s \\ \to 0. \end{matrix}$

所以， $‖ (G x) (t_{2}) - (G x) (t_{1}) ‖ \leq I_{1} + I_{2} + I_{3} + I_{4} \to 0$ ，即 $G (B_{r})$ 在J上等度连续。

第4步，证明由(3.2)式定义的函数 $G : B_{r} \to B_{r}$ 满足M不动点条件。为此，我们设 $W \subseteq B_{r}$ 可数，其中 $W \subseteq \bar{c o} ({0} \cup G (W))$ ，仅证明W相对紧即可。

我们用 $ϕ$ 定义 $B_{r}$ 中的非紧性测度如下：

$ϕ (Ω) = \max_{E \in Δ (Ω)} (α (E), m o d_{c} (E)),$ (3.4)

其中， $Ω$ 是 $B_{r}$ 的所有有界集， $Δ (Ω)$ 是 $Ω$ 的可数子集的集合。 $α$ 是实的非紧性测度，且

$α (E) = \sup_{t \in [0, b]} e^{- L t} β (E (t)),$

其中， $E (t) = {x (t) : x \in E}$ ，L是我们选取的适当的一个常数。 $m o d_{c} (E)$ 是函数集E的等度连续模，且

$m o d_{c} (E) = \lim_{δ \to 0} \sup_{x \in E} \max_{‖ t_{1} - t_{2} ‖ < δ} ‖ x (t_{1}) - x (t_{2}) ‖, t_{1}, t_{2} \in J .$

在文献 [11] 中证明了 $ϕ$ 是有意义的，即存在 $E_{0} \in Δ (Ω)$ ，使得(3.4)式在 $E_{0}$ 处达到最大值，并且 $ϕ$ 是一个单调非奇异正则的非紧性测度。

对于(3.2)式定义的 $G x$ ，由 $ϕ$ 的正则性，只需证明 $ϕ (W) = (0, 0)$ 。因此 $ϕ (G (W))$ 是一个最大值。

$\begin{array}{l} m o d_{c} (W) = \lim_{δ \to 0} \sup_{x \in E} \max_{‖ t_{1} - t_{2} ‖ < δ} ‖ x (t_{1}) - x (t_{2}) ‖ \\ = \lim_{δ \to 0} \sup_{x \in E} \max_{‖ t_{1} - t_{2} ‖ < δ} ‖ \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t_{2}) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u (s) + f (s, x (s))] d s \\ + \int_{0}^{t_{2}} {(t_{2} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s \\ - \sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t_{1}) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u (s) + f (s, x (s))] d s \\ - \int_{0}^{t_{1}} {(t_{1} - s)}^{q - 1} T_{q} (t_{1} - s) [B u (s) + f (s, x (s))] d s ‖ \end{array}$

$\begin{array}{l} \leq \lim_{δ \to 0} \sup_{x \in E} \max_{‖ t_{1} - t_{2} ‖ < δ} (‖ [\hat{T_{q}} (t_{2}) - \hat{T_{q}} (t_{1})] \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{t_{1}}^{t_{2}} {(t_{2} - s)}^{q - 1} T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{0}^{t_{1}} [{(t_{2} - s)}^{q - 1} - {(t_{1} - s)}^{q - 1}] T_{q} (t_{2} - s) [B u (s) + f (s, x (s))] d s ‖ \\ + ‖ \int_{0}^{t_{1}} {(t_{1} - s)}^{q - 1} [T_{q} (t_{2} - s) - T_{q} (t_{1} - s)] [B u (s) + f (s, x (s))] d s ‖) \\ : = \lim_{δ \to 0} \sup_{x \in E} \max_{‖ t_{1} - t_{2} ‖ < δ} (I_{1} + I_{2} + I_{3} + I_{4}) . \end{array}$

由第3步的证明易知当 $‖ t_{1} - t_{2} ‖ < δ$ 且 $ε \to 0$ 时， $I_{1} \to 0, I_{2} \to 0, I_{3} \to 0, I_{4} \to 0$ 。所以 $m o d_{c} (W) = 0$ 。

设 ${z^{n}}_{n = 1}^{\infty} \subseteq G (W)$ 是 $ϕ$ 取得最大值的可数值，那么存在集合 ${x^{n}}_{n = 1}^{\infty} \subseteq W$ ，使得

$z^{n} (t) = (G x^{n}) (t), n \geq 1, t \in J .$ (3.5)

因为

$\begin{array}{l} β ({u_{x^{n}} (s)}_{n = 1}^{\infty}) \\ = β (W^{- 1} [x_{1} - \hat{T_{q}} (b) \sum_{k = 1}^{m} c_{k} P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) {f (s, x^{n} (s))}_{n = 1}^{\infty} d s \\ - \int_{0}^{b} {(b - s)}^{q - 1} T_{q} (b - s) {f (s, x^{n} (s))}_{n = 1}^{\infty} d s]) \\ \leq L^{'} β (\frac{q M \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} {f (s, x^{n} (s))}_{n = 1}^{\infty} d s \\ \begin{matrix} \overset{}{} \\ \underset{}{} \end{matrix} - \frac{M q}{Γ (1 + q)} \int_{0}^{b} {(b - s)}^{q - 1} {f (s, x^{n} (s))}_{n = 1}^{\infty} d s) \end{array}$

$\begin{array}{l} \leq L^{'} [\frac{q M \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \frac{b^{q - q_{1}}}{{(1 + a)}^{1 - q_{1}}} β (\int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} {f (s, x^{n} (s))}_{n = 1}^{\infty} d s) \\ \begin{matrix} \overset{}{} \\ \underset{}{} \end{matrix} + \frac{M q}{Γ (1 + q)} \frac{b^{q - q_{1}}}{{(1 + a)}^{1 - q_{1}}} β (\int_{0}^{b} {(b - s)}^{q - 1} {f (s, x^{n} (s))}_{n = 1}^{\infty} d s)] \\ \leq \frac{q M L^{'}}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \frac{b^{q - q_{1}}}{{(1 + a)}^{1 - q_{1}}} \cdot 2 \int_{0}^{b} β ({f (s, x^{n} (s))}_{n = 1}^{\infty}) d s \\ \leq \frac{2 q M L^{'} L}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \frac{b^{q - q_{1}}}{{(1 + a)}^{1 - q_{1}}} \int_{0}^{b} β ({x^{n} (s)}_{n = 1}^{\infty}) d s . \end{array}$

所以，由算子G的定义，有

$\begin{array}{l} β ({(G x^{n}) (t)}_{n = 1}^{\infty}) \\ = β (\sum_{k = 1}^{m} c_{k} \hat{T_{q}} (t) P \int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} T_{q} (t_{k} - s) [{B u_{x^{n}} (s)}_{n = 1}^{\infty} + {f (s, x^{n} (s))}_{n = 1}^{\infty}] d s \\ + \int_{0}^{b} {(b - s)}^{q - 1} T_{q} (b - s) [{B u_{x^{n}} (s)}_{n = 1}^{\infty} + {f (s, x^{n} (s))}_{n = 1}^{\infty}] d s) \\ \leq \frac{2 q M^{2} \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} [\int_{0}^{t_{k}} {(t_{k} - s)}^{q - 1} β ({B u_{x^{n}} (s)}_{n = 1}^{\infty} + {f (s, x^{n} (s))}_{n = 1}^{\infty}) d s] \\ + \frac{2 M q}{Γ (1 + q)} [\int_{0}^{b} {(b - s)}^{q - 1} β ({B u_{x^{n}} (s)}_{n = 1}^{\infty} + {f (s, x^{n} (s))}_{n = 1}^{\infty}) d s] \end{array}$

$\begin{array}{l} \leq [\frac{2 q M^{2} \sum_{k = 1}^{m} | c_{k} |}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} + \frac{2 M q}{Γ (1 + q)}] [2 q M L^{'} L M_{B} l_{2}^{2} \int_{0}^{b} β ({x^{n} (s)}_{n = 1}^{\infty}) d s + L l_{2} \int_{0}^{b} β ({x^{n} (s)}_{n = 1}^{\infty}) d s] \\ \leq [2 q M L l_{1} l_{2} (2 q M L^{'} M_{B} l_{1} l_{2} + 1)] \int_{0}^{t} β ({x^{n} (s)}_{n = 1}^{\infty}) d s \\ \leq [2 q M L l_{1} l_{2} (2 q M L^{'} M_{B} l_{1} l_{2} + 1)] \int_{0}^{t} e^{R s} (e^{- R s} β ({x^{n} (s)}_{n = 1}^{\infty})) d s \\ = [2 q M L l_{1} l_{2} (2 q M L^{'} M_{B} l_{1} l_{2} + 1)] α ({x^{n}}_{n = 1}^{\infty}) \int_{0}^{t} e^{R s} d s, t \in J . \end{array}$

现在我们给出 $α ({z^{n}}_{n = 1}^{\infty})$ 的一个估计：

$\begin{matrix} α ({z^{n}}_{n = 1}^{\infty}) = \sup_{t \in [0, b]} e^{- R t} β ({(G x^{n}) (t)}_{n = 1}^{\infty}) \\ \leq \sup_{t \in [0, b]} e^{- R t} [2 q M L l_{1} l_{2} (2 q M L^{'} M_{B} l_{1} l_{2} + 1)] α ({x^{n}}_{n = 1}^{\infty}) \int_{0}^{t} e^{R s} d s \\ \leq α ({x^{n}}_{n = 1}^{\infty}) [2 q M L l_{1} l_{2} (2 q M L^{'} M_{B} l_{1} l_{2} + 1)] \sup_{t \in [0, b]} \int_{0}^{t} e^{- R (t - s)} d s \\ = p α ({x^{n}}_{n = 1}^{\infty}) . \end{matrix}$

因此，

$α ({x^{n}}_{n = 1}^{\infty}) \leq α (W) \leq α (\bar{c o} ({0} \cup G (W))) = α ({z^{n}}_{n = 1}^{\infty}) \leq α ({x^{n}}_{n = 1}^{\infty}) p .$

由 $p < 1$ 得

$α ({x^{n}}_{n = 1}^{\infty}) = α (W) = α ({z^{n}}_{n = 1}^{\infty}) = 0.$

根据 $α$ 定义可知，

$β ({x^{n} (t)}_{n = 1}^{\infty}) = 0.$

又因为

$\begin{array}{l} β ({B u_{x^{n}} (t)}_{n = 1}^{\infty} + {f (t, x^{n} (t))}_{n = 1}^{\infty}) \\ \leq \frac{2 q M L L^{'}}{Γ (1 + q) (1 - M \sum_{k = 1}^{m} | c_{k} |)} \frac{b^{q - q_{1}}}{{(1 + a)}^{1 - q_{1}}} \int_{0}^{b} β ({x^{n} (t)}_{n = 1}^{\infty}) d t + L \int_{0}^{b} β ({x^{n} (t)}_{n = 1}^{\infty}) d t \\ = 0. \end{array}$

所以， ${B u_{x^{n}} (t) + f (t, x^{n} (t))}_{n = 1}^{\infty}$ 在X中对几乎所有 $t \in J$ 相对紧。再由(H2) (2)和(3.3)式，易知 ${B u_{x^{n}} (t) + f (t, x^{n} (t))}_{n = 1}^{\infty}$ 对几乎处处 $t \in [0, b]$ 是一致可积的。所以根据定义4， ${B u_{x^{n}} + f (\cdot, x^{n} (\cdot))}_{n = 1}^{\infty}$ 是半紧的。由引理6可知， $G ({x^{n}}_{n = 1}^{\infty})$ 在 $B_{r}$ 中相对紧。通过(3.5)式得， ${z^{n}}_{n = 1}^{\infty}$ 在 $B_{r}$ 中也相对紧。因为 $ϕ$ 是单调非奇异正则的非紧性测度，根据Mönch不动点的条件，所以

$ϕ (W) \leq ϕ (\bar{c o} ({0} \cup G (W))) = ϕ ({z^{n}}_{n = 1}^{\infty}) = 0.$

故W在 $B_{r}$ 中相对紧。

4. 例子

例1设 $X = U : = C ([0, 1])$ 。考虑分数阶发展方程非局部问题

${\begin{cases} \frac{\partial^{\frac{1}{2}}}{\partial t^{\frac{1}{2}}} x (t, z) + \frac{\partial x (t, z)}{\partial t} = \frac{e^{- 2 t}}{1 + e^{t}} x (t, z) + λ μ (t, z), t \in [0, b] = J, z \in (0, 1), \\ x (t, 0) = x (t, 1) = 0, t \in J, \\ x (0, z) = \sum_{k = 1}^{m} \arctan \frac{1}{2 k^{2}} x (k, z), z \in (0, 1), \end{cases}$ (4.1)

其中 $λ > 0, 0 < m < b, μ : J \times (0, 1) \to (0, 1)$ 。

定义算子 $A : D (A) \subset X \to X$ 如下：

$D (A) = {x \in X : x^{'} \in X, x (0) = x (1) = 0} .$

$A x = - x^{'}, x \in D (A),$

则A生成X中的等度连续半群 ${T (t)}_{t \geq 0}$ ，且对任意的 $x \in X$ ， $T (t)$ 满足

$T (t) x (s) = x (t + s),$

那么 $T (t) (t \geq 0)$ 在X中非紧，并且有 $\sup_{t \in J} ‖ T (t) ‖ \leq 1$ 。

定义 $x (t) (z) = x (t, z); f (t, x (t)) = \frac{e^{- 2 t}}{1 + e^{t}} x (t, z); u (t) (z) = μ (t, z); c_{k} = \arctan \frac{1}{2 k^{2}}$ 。对 $\forall x \in B_{r} : = {x \in C (J, X) : {‖ x ‖}_{C (J, X)} \leq r}, t \in J$ ，有

$‖ f (t, x) ‖ \leq ‖ \frac{e^{- 2 t}}{1 + e^{t}} x (t, z) ‖ \leq \frac{r}{e^{2 t} (1 + e^{t})} \leq \frac{1}{2},$

则条件(H2)成立。

又因为 $\sum_{k = 1}^{m} | c_{k} | \leq \sum_{k = 1}^{m} \arctan \frac{1}{2 k^{2}} = \frac{π}{4} < 1$ ，所以条件(H0)成立。

定义算子W如下

$\begin{matrix} (W t) (z) = \hat{T_{q}} (b) \sum_{k = 1}^{m} \arctan \frac{1}{2 k^{2}} {[I - \sum_{k = 1}^{m} \arctan \frac{1}{2 k^{2}} \hat{T_{q}} (t_{k})]}^{- 1} \int_{0}^{k} {(k - s)}^{- \frac{1}{2}} T_{q} (k - s) λ μ (s, z) d s \\ + \int_{0}^{b} {(b - s)}^{- \frac{1}{2}} T_{q} (b - s) λ μ (s, z) d s . \end{matrix}$

${\hat{T_{q}} (t)}_{t \geq 0}$ 与 ${T_{q} (t)}_{t \geq 0}$ 定义如下：

$\hat{T_{q}} (t) x (s) = \int_{0}^{\infty} η_{\frac{1}{2}} (θ) x (t^{\frac{1}{2}} θ + s) d θ,$

$T_{q} (t) x (s) = \frac{1}{2} \int_{0}^{\infty} θ η_{\frac{1}{2}} (θ) x (t^{\frac{1}{2}} θ + s) d s,$

其中 $η_{\frac{1}{2}} (θ) = 2 θ^{- 3} W_{\frac{1}{2}} (θ^{- \frac{1}{2}}), W_{\frac{1}{2}} (θ) = \frac{1}{π} \sum_{n = 1}^{\infty} {(- 1)}^{n - 1} θ^{- \frac{n}{2} - 1} \frac{Γ (1 + \frac{n}{2})}{n!} \sin (\frac{n π}{2}), θ \in (0, \infty)$ 。

我们假设W满足条件(H4)，且不等式条件(H5)和(3.1)式成立，则由定理3.1可知，分数阶发展系统(4.1)在J上精确可控。

基金项目

国家基金委青年科学资助项目(12061062)。

文章引用

苏怡,杨和. 一类分数阶发展方程非局部问题的精确可控性
Exact Controllability for a Class of Fractional Evolution Equations with Nonlocal Conditions[J]. 理论数学, 2022, 12(05): 861-874. https://doi.org/10.12677/PM.2022.125096

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