On Regularity of Stochastic Convolutions of Functional Linear Differential Equations with Memory

TL;DR

This paper investigates the regularity of stochastic convolutions in linear stochastic retarded functional differential equations with unbounded operators, developing new estimates and resolvent constructions for Volterra-type equations.

Contribution

It introduces novel estimates for fundamental solutions and constructs resolvent operators for Volterra-type integrodifferential equations, advancing understanding of stochastic convolution regularity.

Findings

Established estimates on fundamental solutions for delay equations

Constructed resolvent operators for Volterra-type equations

Proved regularity properties of stochastic convolutions

Abstract

In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on $C_0$-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates to stochastic convolutions of our equations to obtain the desired regularity property.

Equations

$$\begin{array}{rl}x\cdot h_i(x)& \ge {\gamma }_{i,1}|x{|}^p+{\alpha }_{i,1},\forall x\in \mathbb{R},\\ |h_i(x)|& \le {\gamma }_{i,2}|x{|}^{p-1}+{\alpha }_{i,2},\forall x\in \mathbb{R},\end{array}$$

$$q(t,x)=-h_0(\partial y(t,x)/\partial x)-h_1(y(t-r,x)),t\ge 0,x\in (0,\pi ),$$

$$d_{t}p(t,x)+\frac{\partial q(t,x)}{\partial x}dt=0,t\ge 0,x\in (0,\pi ),$$

$$d_{t}p(t,x)+\frac{\partial q(t,x)}{\partial x}dt=b(x)dw(t,x),t\ge 0,x\in (0,\pi ).$$

$$\{\begin{array}{l}dy(t,x)=\frac{\partial h_0(\partial y(t,x)/\partial x)}{\partial x}dt+\frac{\partial h_1(y(t-r,x))}{\partial x}dt+b(x)dw(t,x),t\ge 0,\\ y(0,\cdot )={\varphi }_0(\cdot )\in L^2(0,\pi ),y(\theta ,\cdot )={\varphi }_1(\theta ,\cdot )\in W_0^{1,p}(0,\pi ),\theta \in [-r,0],\\ y(t,0)=y(t,\pi )=0,t\in (0,\infty ).\end{array}$$

$$\{\begin{array}{l}dy(t)=g_0(y(t))dt+g_1(y(t-r))dt+BdW(t),t\ge 0,\\ y(0)={\varphi }_0,y_0={\varphi }_1,\end{array}$$

$$\begin{array}{rl}⟨u,g_i(u){⟩}_{U,U^{\ast }}& \ge {\gamma }_{i,1}\Vert u{\Vert }_U^p+{\alpha }_{i,1},\forall u\in U,\\ \Vert g_i(u){\Vert }_{U^{\ast }}& \le {\gamma }_{i,2}\Vert u{\Vert }_U^{p-1}+{\alpha }_{i,2},\forall u\in U,\end{array}$$

$$\{\begin{array}{l}dy(t)=\Delta y(t)dt+\gamma (-\Delta {)}^{1/2}y(t-r)dt+BdW(t),t\ge 0,\\ y(0)={\varphi }_0,y_0={\varphi }_1.\end{array}$$

$$\{\begin{array}{l}dy(t)=Ay(t)dt+A_{1}y(t-r)dt+{\int }_{-r}^{0}a(\theta )A_{2}y(t+\theta )d\theta +f(t),t\ge 0,\\ y(0)={\varphi }_0,y(\theta )={\varphi }_1(\theta ),\theta \in [-r,0],\varphi =({\varphi }_0,{\varphi }_1),\end{array}$$

$$\Vert e^{tA}\Vert \le M{e}^{-\mu t},\Vert A{e}^{tA}\Vert \le M/tfor\; allt>0,$$

$$\Vert (-A{)}^{\gamma }{e}^{tA}\Vert \le M_{\gamma }/t^{\gamma }for\; allt>0,$$

$$\begin{array}{rl}\frac{d}{dt}G(t)& =AG(t)+A_{1}G(t-r)+{\int }_{-r}^{0}a(\theta )A_{2}G(t+\theta )d\theta ,\\ & G(0)=I,G(t)=\mathrm{O},t\in (-\infty ,0),\end{array}$$

$$G(t)=\{\begin{array}{ll}{e}^{tA}+{\int }_0^t{e}^{(t-s)A}{A}_{1}G(s-r)ds+{\int }_0^t{\int }_{-r}^{0}a(\theta )e^{(t-s)A}{A}_{2}G(s+\theta )d\theta ds,& t\ge 0,\\ \mathrm{O},& t<0.\end{array}$$

$$y(t)=G(t){\varphi }_0+{\int }_{-r}^0{U}_t(\theta ){\varphi }_1(\theta )d\theta +{\int }_0^{t}G(t-s)f(s)ds,t\ge 0,$$

$$U_t(\theta )=G(t-\theta -r)A_1+{\int }_{-r}^{\theta }G(t-s+\tau )a(\tau )A_{2}d\tau ,\theta \in [-r,0],$$

$$\mathcal{D}((-A{)}^{\mu })\subset \mathcal{D}(A_1),\mathcal{D}((-A{)}^{\nu })\subset \mathcal{D}(A_2)$$

$$\Vert (-A{)}^{\gamma }G(t)\Vert \le \frac{C_{n,\gamma }}{(t-nr{)}^{\gamma }}for\; allt\in (nr,(n+1)r],$$

$$\displaystyle\Big{\|}\displaystyle\int^{t}_{s}(-A)^{\gamma}G(u)du\Big{\|}\leq C_{n,\gamma}\hskip 15.0pt\hbox{for all}\hskip 15.0ptnr\leq s<t\leq(n+1)r,$$

$$\Vert (-A{)}^{\gamma }(G(t)-G(s))\Vert \le C_{n,\beta ,\gamma }\frac{(t-s{)}^{\beta }}{(s-nr{)}^{\beta +\gamma }}for\; allnr<s<t<(n+1)r,$$

$$\mathbb{E}⟨W_Q(t),x{⟩}_K⟨W_Q(s),y{⟩}_K=(t\wedge s)⟨Qx,y{⟩}_{K}for\; allx,y\in K,$$

$$⟨u,v{⟩}_{K_Q}=⟨Q^{-1/2}u,Q^{-1/2}v{⟩}_{K}for\; anyu,v\in K_Q.$$

$$\Vert \Psi {\Vert }_{{\mathcal{L}}_2(K_Q,H)}^2=Tr[\Psi Q^{1/2}(\Psi Q^{1/2}{)}^{\ast }]for\; any\Psi \in {\mathcal{L}}_2(K_Q,H).$$

$$|J|_{t}:=\biggl{\{}\mathbb{E}\int^{t}_{0}Tr\Big{[}J(s,\omega)Q^{1/2}(J(s,\omega)Q^{1/2})^{*}\Big{]}ds\biggr{\}}^{\frac{1}{2}}.$$

$$W(t)=\sum _{j=1}^{\infty }\sqrt{{\lambda }_j}{w}_j(t)e_j,t\ge 0,$$

$${\int }_0^{t}J(s)dW(s)=L^2-\underset{n\to \infty }{\lim }\sum _{j=1}^n{\int }_0^t\sqrt{{\lambda }_j}J(s)e_{j}d{w}_j(s),t\in [0,T].$$

$$\{\begin{array}{l}dy(t)=Ay(t)dt+A_{1}y(t-r)dt+{\int }_{-r}^{0}a(\theta )A_{2}y(t+\theta )d\theta +BdW(t),t\ge 0,\\ y(0)={\varphi }_0,y(\theta )={\varphi }_1(\theta ),\theta \in [-r,0],\varphi =({\varphi }_0,{\varphi }_1),\end{array}$$

$$y(t)=G(t){\varphi }_0+{\int }_{-r}^0{U}_t(\theta ){\varphi }_1(\theta )d\theta +{\int }_0^{t}G(t-s)BdW(s),t\ge 0,$$

$$U_t(\theta )=G(t-\theta -r)A_1+{\int }_{-r}^{\theta }G(t-s+\tau )a(\tau )A_{2}d\tau ,\theta \in [-r,0],$$

$$y(t):=W_G(t)={\int }_0^{t}G(t-s)BdW(s),t\ge 0.$$

$$\|G(t)-G(s)\|\leq C_{n,\kappa}\Big{(}\displaystyle\frac{t-s}{s-nr}\Big{)}^{\kappa}\hskip 15.0pt\hbox{for all}\hskip 15.0ptnr<s<t<(n+1)r,$$

$$\beta <\{\begin{array}{l}1/2if\rho >1/2,\\ \rho if\rho \le 1/2.\end{array}$$

Full text

On Regularity of Stochastic Convolutions of

Functional Linear Differential

Equations with Memory

Kai Liu${}^{a,\,b}$

aCollege of Mathematical Sciences,

Tianjin Normal University,

300387, Tianjin, P. R. China.

bDepartment of Mathematical Sciences,

School of Physical Sciences,

The University of Liverpool,

Peach Street, Liverpool, L69 7ZL, U.K.

E-mail: k.liu@liverpool.ac.uk

Abstract: In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on $C_{0}$-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates to stochastic convolutions of our equations to obtain the desired regularity property.

Keywords: Regularity property; Fundamental solution; Stochastic convolution.

2000 Mathematics Subject Classification(s): 60H15, 60G15, 60H05.

1 Introduction

We begin with an example of stochastic delay heat equations without exterior energy source to motivate our work. Let $h_{i}:\,{\mathbb{R}}\to{\mathbb{R}}$, $i=0,\,1$, be two monotonically differentiable functions which satisfy the following conditions (see, e.g., Coleman and Gurtin [1] and Nunziato [6])

[TABLE]

where $p\geq 2$, $\gamma_{i,j}>0$ and $\alpha_{i,j}\in{\mathbb{R}}$ for $i=0,\,1$ and $j=1,\,2$. The non-Fourier heat conduction model with delay in the conductor $(0,\pi)\subset{\mathbb{R}}$ starts from the following constitutive equation

[TABLE]

and the energy conservative equation without exterior energy sources

[TABLE]

where $y$ denotes the temperature, $q$ is the heat flux and $p$ is the internal energy which can be taken, in most situations, as the form: $p(t,x)=\kappa y(t,x)$, $\kappa>0$. In practice, the assumption of zero exterior energy source is artificial, and a more realistic model is that the null exterior energy source is perturbed by a noise process, for example, a Gaussian white noise $b(x)\dot{w}(t,x)$, $b\in L^{2}(0,\pi)$. In other words, we replace (1.2) by the equation

[TABLE]

Then, by substituting (1.3) into (1.1), we obtain, for simplicity, letting $\kappa=1$, the following equation

[TABLE]

Let $\Delta=d^{2}/dx^{2}$, $H=L^{2}(0,\pi)$, $V=H^{1}_{0}(0,\pi)$, $U=W^{1,p}_{0}(0,\pi)$ and $Bu=bu$, $u\in H$, $W(t)=w(t,\cdot)$, $g_{0}(u)=\displaystyle\frac{dh_{0}(du(x)/dx)}{dx}$, $g_{1}(u)=\displaystyle\frac{dh_{1}(u(x))}{dx}$ for any $u\in W^{1,p}_{0}(0,\pi)$. We have thus a stochastic differential equation with delay in $H$,

[TABLE]

where $U\subset H\subset U^{*}$ and $g_{i}$, $i=0,\,1$, is a continuous monotone operator from $U$ to $U^{*}$ such that

[TABLE]

where $p\geq 2$, $\gamma_{i,j}>0$ and $\alpha_{i,j}\in{\mathbb{R}}$ for $i=0,\,1$ and $j=,\,2$. In particular, if $h_{0}(x)=x$, $h_{1}(x)=\gamma x$, $\gamma>0$, $p=2$, then $g_{0}(u)=\Delta u$, $g_{1}(u)=\gamma(-\Delta)^{1/2}u$, $V=U$ and the equation (1.5) reduces to

[TABLE]

The aim of this work is to investigate the regularity property of such stochastic systems as (1.6).

The organization of this work is as follows. In Section 2, we first introduce the deterministic linear retarded functional differential equation associated in our formulation of stochastic systems. We review the useful variation of constants formula for the equation under consideration by means of its fundamental solution. Also, we state some estimates about fundamental solutions which will play an important role in the subsequent investigation. By employing the main results, we establish in Section 3 the desired regularity property of stochastic convolutions. In Sections 4 and 5, we present the detailed proofs of the main theorem, i.e., Theorem 2.1, by following J. Prüss’s method of constructing the resolvent operators for the integrodifferential equations of Volterra type.

2 Fundamental Solution

We are concerned with the following linear retarded functional differential equation in a Banach space $X$,

[TABLE]

where $r>0$ is some constant incurring the system delay, $a(\cdot)\in L^{2}([-r,0],{\mathbb{R}})$ and $\phi=(\phi_{0},\phi_{1})$ is an appropriate initial datum. Here $A:{\mathscr{D}}(A)\subset X\to X$ is the infinitesimal generator of an analytic semigroup $e^{tA}$, $t\geq 0$, and $A_{1}$, $A_{2}$ are two closed linear operators with domains ${\mathscr{D}}(A_{i})\supset{\mathscr{D}}(A)$, $i=1,\,2$, and $f$ is a continuous function with values in $X$. For simplicity, we assume in this work that the $C_{0}$-semigroup $e^{tA}$ is negative type, i.e., there exist constants $M\geq 1$ and $\mu>0$ such that

[TABLE]

and for $\gamma\in(0,1)$, there exists a constant $M_{\gamma}>0$ such that

[TABLE]

where $(-A)^{\gamma}$ is the standard fractional power of operator $A$.

Equations of the type (2.1) were investigated by Di Blasio, Kunisch and Sinestrari [3], Sinestrari [9, 10] and the fundamental solution to (2.1) was introduced by Jeong, Nakagiri and Tanabe [5]. In particular, it is known that the fundamental solution $G(\cdot):{\mathbb{R}}\to{\mathscr{L}}(X)$ to (2.1) is an operator-valued function which is strongly continuous in $X$ and satisfies

[TABLE]

where ${\rm O}$ is the null operator in $X$. According to the well-known Duhamel’s principle, the problem (2.4) is transformed to the integral equation

[TABLE]

The fundamental solution $G$ enables us to solve the initial value problem for the equation (2.1). In fact, it may be shown that under some reasonable conditions on $f$ and initial datum $\phi=(\phi_{0},\phi_{1})$, the unique mild solution $y$ of (2.1) is represented as

[TABLE]

where

[TABLE]

with the initial condition $y(0)=\phi_{0}$ and $y(\theta)=\phi_{1}(\theta)$, $\theta\in[-r,0)$. This is a time delay version of the usual variation of constants formula without memory.

In order to apply (2.6) to such equations as (1.6) to consider their regularity property of solutions, we need establish some inequalities in association with $G(\cdot)$, which are the main results of this work. To this end, we shall formulate the following condition:

1. (H)

$a(\cdot)\in L^{\infty}([-r,0],{\mathbb{R}})$ and

[TABLE]

for some $0<\mu,\,\nu<1$.

Theorem 2.1**.**

Assume that condition (H) holds. Then

1. (a)

for any $\gamma\in[\nu,1)$, it is true that

[TABLE]

where $C_{n,\gamma}>0$, $n\in{\mathbb{N}}:=\{0,\,1,\,\cdots\}$, are constants depending on $n$ and $\gamma$. 2. (b)

for any $\gamma\in[\nu,1)$ and $0<\beta<1-\gamma$, it is true that

[TABLE]

where $C_{n,\beta,\gamma}>0$, $n\in{\mathbb{N}}$, are constants depending on $n,$ $\beta$ and $\gamma$.

3 Stochastic Convolution

Let $\{\Omega,{\mathscr{F}},{\mathbb{P}}\}$ be a probability space equipped with some filtration $\{\mathscr{F}_{t}\}_{t\geq 0}$. Let $K$ be a separable Hilbert space and $\{W_{Q}(t),\,t\geq 0\}$ denote a $Q$-Wiener process with respect to $\{\mathscr{F}_{t}\}_{t\geq 0}$ in $K$, defined on $\{\Omega,{\mathscr{F}},{\mathbb{P}}\}$, with covariance operator $Q$, i.e.,

[TABLE]

where $Q$ is a positive, self-adjoint and trace class operator on $K$. We frequently call $W_{Q}(t)$, $t\geq 0$, a $K$-valued $Q$-Wiener process with respect to $\{\mathscr{F}_{t}\}_{t\geq 0}$ if the trace $Tr(Q)<\infty$. We introduce a subspace $K_{Q}={\mathscr{R}}(Q^{1/2})\subset K$, the range of $Q^{1/2}$, which is a Hilbert space endowed with the inner product

[TABLE]

Let $H$ be a separable Hilbert space and ${\mathscr{L}}_{2}(K_{Q},H)$ denote the space of all Hilbert-Schmidt operators from $K_{Q}$ into $H$. Then ${\mathscr{L}}_{2}(K_{Q},H)$ turns out to be a separable Hilbert space, equipped with the norm

[TABLE]

For arbitrarily given $T\geq 0$, let $J(t,\omega)$, $t\in[0,T]$, be an ${\mathscr{L}}_{2}(K_{Q},H)$-valued process, and we define the following norm for arbitrary $t\in[0,T]$,

[TABLE]

In particular, we denote by {\cal U}^{2}\big{(}[0,T];\,{\mathscr{L}}_{2}(K_{Q},H)\big{)} the space of all ${\mathscr{L}}_{2}(K_{Q},H)$-valued measurable processes $J$, adapted to the filtration $\{{\mathscr{F}}_{t}\}_{t\leq T}$, satisfying $|J|_{T}<\infty$.

Suppose that $W(\cdot)$ is a $Q$-Wiener process in $K$ such that $Qe_{j}=\lambda_{j}e_{j}$, $\lambda_{j}>0$, $j\geq 1$, where $\{e_{j}\}$ is a complete orthonormal basis in $K$, then it is immediate that

[TABLE]

where $\{w_{j}(\cdot)\}$ is a group of independent real Wiener processes. The stochastic integral $\displaystyle\int^{t}_{0}J(s)dW(s)\in H$, $t\geq 0$, may be defined for all $J\in{\cal U}^{2}([0,T]\times\Omega;\,{\mathscr{L}}_{2}(K_{Q},H))$ by

[TABLE]

The reader is referred to Da Prato and Zabczyk [2] for more details on this topic.

We are concerned about the following linear stochastic retarded functional differential equation on $H$,

[TABLE]

where $A$, $A_{1}$, $A_{2}$ are given as in Section 2, $B\in{\mathscr{L}}_{2}(K_{Q},H)$ and $\phi=(\phi_{0},\phi_{1})$ is an appropriate initial datum. It is well known that the unique mild solution $y$ of (3.3) is represented as

[TABLE]

where

[TABLE]

with the initial condition $y(0)=\phi_{0}$ and $y(\theta)=\phi_{1}(\theta)$, $\theta\in[-r,0)$. In particular, if $\phi=(0,0)$, then the unique mild solution (3.4) is the so-called stochastic convolution process

[TABLE]

For any $T>0$, let $C^{\beta}([0,T];H)$ denote the usual Banach space of all Hölder continuous functions on $H$ with order $\beta\in(0,1)$. The following lemma is referred to Tanabe [11].

Lemma 3.1**.**

Suppose that $a(\cdot)$ in (3.3) is Hölder continuous with order $\rho\in(0,1)$, then operator $G(t)$ is strongly continuous in $H$ on each $[nr,(n+1)r]$, $n=0,\,1,\,\cdots$. Moreover, the following estimates hold:

[TABLE]

where $\kappa\in(0,\rho)$ and $C_{n,\kappa}>0$ are some constants depending on $n,$ $\kappa$.

Theorem 3.1**.**

Suppose that $a(\cdot)$ in (3.3) is Hölder continuous with order $\rho\in(0,1)$. Let $T>0$. Assume that $Tr(Q)<\infty$ and $B\in{\mathscr{L}}(K,H)$, the space of all bounded, linear operators from $K$ into $H$, then the trajectories of $W_{G}$ are in $C^{\beta}([0,T];H)$ where

[TABLE]

Proof. It suffices to show this theorem for any $0\leq s<t\leq T$ with $t-s<r.$ To this end, it is easy to have that

[TABLE]

Since $G(\cdot)$ is strongly continuous on ${\mathbb{R}}$, it is easy to see, by the well-known Principle of Uniform Boundedness, that $G(\cdot)$ is norm bounded on $[0,T]$, and there exists a real number $C(T)>0$ such that

[TABLE]

On the other hand, suppose that $s\in(Nr,(N+1)r)$ for some $N\in{\mathbb{N}}$ and $t-s<r$. Then, for the item $I_{2}$ we have

[TABLE]

By using Lemma 3.1 for those values

[TABLE]

one can further obtain

[TABLE]

where $C_{k,\beta}>0$, $C^{\prime}_{k,\beta}>0$ are two numbers depending on $k$ and $\beta$. Thus, by substituting (3.8) and (3.9) into (3.7), we immediately obtain

[TABLE]

where

[TABLE]

From (3.10), it follows further that for any integer $m\geq 1$,

[TABLE]

So, by the well-known Kolmogorov test, $W_{G}$ is $\alpha_{m}$-Hölder continuous with

[TABLE]

Since $m$ is arbitrary, the trajectories of $W_{G}$ are in $C^{\beta}([0,T];H)$. The proof is complete.

Theorem 3.2**.**

Suppose that condition (H) holds. Let $T>0$ and $Tr(Q)<\infty$. For $\nu\leq\gamma<1$ and $0<\beta<\displaystyle\frac{1}{2}-\gamma$, the trajectories of $W_{G}$ are in $C^{\beta}([0,T],{\mathscr{D}}((-A)^{\gamma}))$.

Proof. Once again, we intend to use the Kolmogorov test. Let $0\leq s<t\leq T$ with $s,\,t\in(Nr,(N+1)r)$ for some $N>0$. Then, by definition, it follows easily that

[TABLE]

Now we estimate $I_{1}$ and $I_{2}$ separately. First, by using Theorem 2.1 we have for $s,\,t\in(Nr,(N+1)r)$ with $s<t$ that

[TABLE]

Since $0<\beta<\displaystyle\frac{1}{2}-\gamma<1-\gamma$, it follows that $1-2\beta-2\gamma>0$, and we thus employ Theorem 2.1 to obtain

[TABLE]

where $[T]$ denotes the biggest integer less than or equal to $T$. Hence, by substituting (3.12), (3.13) into (3.11) and using the Kolmogorov test, we then obtain the desired result.

4 Proof of Theorem 2.1 (a)

We begin with establishing some useful lemmas.

Lemma 4.1**.**

Let $\gamma\in(0,1)$. For any $0<s<t<\infty$, there exists a constant $M_{\gamma}>0$ such that

[TABLE]

[TABLE]

Proof. It is well-known that for any $\gamma\in(0,1)$ and $k=1,\,2$, there exists a constant $M_{\gamma,k}>0$ such that

[TABLE]

Therefore, by using this estimate and the following equalities

[TABLE]

one can easily obtain the desired results. The proof is complete now.

Corollary 4.1**.**

Let $\gamma\in(0,1)$ and $\beta>0$. For any $0<s<t<\infty$, there exists a constant $C_{\beta,\gamma}>0$ such that

[TABLE]

Proof. First note that for any real numbers $a\geq b\geq 0$ and $0<\delta\leq 1$, we have the following inequality

[TABLE]

For $0<\beta\leq\gamma$ and any $0<s<t<\infty$, we have by using (4.4) that

[TABLE]

On the other hand, for $\beta>\gamma$ and $0<s<t<\infty$, we have

[TABLE]

Now we have by virtue of Lemma 4.1 the desired inequality.

To proceed further, let us denote by $|\cdot|_{\infty}$ the essential least upper bound norm in $L^{\infty}([-r,0],{\mathbb{R}})$, i.e.,

[TABLE]

For $\gamma\in(0,1)$, we define

[TABLE]

Proposition 4.1**.**

The mapping $\Gamma(\cdot):[0,r]\to{\mathscr{L}}(X)$, the space of all bounded linear operators on $X$, is uniformly bounded and for any $0<\beta<1-\gamma$, there exists a number $C_{\beta,\gamma}>0$ such that

[TABLE]

i.e., $\Gamma$ is Hölder continuous on $[0,r]$ with order $\beta\in(0,1-\gamma)$.

Proof. First, it is easy to see by virtue of (2.3) that

[TABLE]

which immediately implies

[TABLE]

Thus, $\Gamma$ is uniformly bounded in $[0,r]$. To show the relation (4.7), we have for $0\leq s<t\leq r$ that

[TABLE]

By using the relation (2.3), we easily obtain the inequality

[TABLE]

On the other hand, we can apply Corollary 4.1 to (4.8) to obtain

[TABLE]

Hence, substituting (4.9) and (4.10) into (4.8), we immediately obtain the desired result.

To show Theorem 2.1 (a), we intend to develop an induction scheme. We first consider the case that $n=0$ and set

[TABLE]

Then, the integral equation to be satisfied by $V(t)$ is

[TABLE]

where

[TABLE]

Since ${\mathscr{D}}(A)\subset{\mathscr{D}}(A_{i})$, we have $A_{i}A^{-1}\in{\mathscr{L}}(X)$, $i=1,\,2$. Then for any $0<\beta<1-\gamma$, it follows by virtue of Proposition 4.1 and (2.2) that

[TABLE]

Hence, $V_{0}(\cdot)$ is uniformly bounded on $[0,r]$. Since ${\mathscr{D}}((-A)^{\nu})\subset{\mathscr{D}}(A_{2})$, we have

[TABLE]

Hence, by virtue of the well-known Gronwall lemma and (4.11) that

[TABLE]

Note that

[TABLE]

Hence, for $t\in(0,r]$ we have

[TABLE]

which is (2.8) with $n=0$. In a similar manner, for any $0<s<t\leq r$, we have from (2.3) and (4.13) that

[TABLE]

which is (2.9) with $n=0$.

Now suppose the fundamental solution $G(\cdot)$ satisfies all the estimates in Theorem 2.1 on the intervals $[0,r]$, $\cdots$, $[(n-1)r,nr]$. Then in the interval $[nr,(n+1)r]$, the integral equation to be satisfied by

[TABLE]

is

[TABLE]

where

[TABLE]

Now we estimate each term on the right hand side of (4.14). First, note that

[TABLE]

and

[TABLE]

By virtue of condition (H), for any $\mu\vee\gamma\leq\tau<1$, ${\mathscr{D}}((-A)^{\tau})\subset{\mathscr{D}}(A_{i})$ and $A_{i}(-A)^{-\tau}\in{\mathscr{L}}(X)$, $i=1,\,2$. For $t\in(nr,(n+1)r]$, $j=0,\,1,\,\cdots,\,n-2$, and $\gamma\vee\mu+\delta<1$ with $\delta>0$ sufficiently small, it is easy to see, by the induction assumption and Corollary 4.1, that

[TABLE]

On the other hand, for $j=n-1$ and $t\in(nr,(n+1)r]$, we have by using Corollary 4.1 and induction assumption that for $0<\beta<1-\gamma$ and $\delta>0$ sufficiently small,

[TABLE]

where $C_{\beta,\gamma,\delta},\,M_{\delta,\gamma},\,C^{\prime}_{n-1}>0$ and $B(\cdot,\cdot)$ is the standard Beta function. Combining (4.15), (4.16) and (4.17), we have for $t\in(nr,(n+1)r]$ that

[TABLE]

To estimate $I_{3}(t)$, we first note that

[TABLE]

for any $0<s<t<\infty$ and $\alpha\in(0,1]$. Indeed, by virtue of (2.2), we have for any $0<s<t$ that

[TABLE]

It is elementary to see that

[TABLE]

Hence, (4.19) follows immediately from (4.20). Now we re-write $I_{3}(t)$ as

[TABLE]

For $j=0,\,1,\,\cdots,\,n-2$, we have by the induction assumption and (4.19) that for sufficiently small $\delta>0$,

[TABLE]

In a similar manner, we have for $t\in(nr,(n+1)r]$ and sufficiently small $\delta>0$ that

[TABLE]

By virtue of (4.22), (4.23) and (4.24), we thus obtain for $t\in(nr,(n+1)r]$ that

[TABLE]

Now we intend to estimate $I_{4}(t)$. To this end, we have by virtue of (H), (4.7) and induction assumption that for sufficiently small $\delta>0$,

[TABLE]

where $0<\beta<1-\gamma$ and $C_{\beta,\gamma,\delta},\,\,C^{\prime}_{n-1,\delta}>0$.

Last, we are in a position to estimate $I_{5}(t)$. By assumption, we can obtain in terms of Corollary 4.1 and induction assumption with $\delta>0$ sufficiently small that for $nr<u\leq(n+1)r$,

[TABLE]

where $0<\tilde{\beta}<1-\nu$. Hence, we have for $t\in(nr,(n+1)r]$ that

[TABLE]

By combining (4.18), (4.25) with (4.27) and noticing (4.12), we obtain that $V_{0}(t)$ and $V(t)$ both are uniformly bounded in $[nr,(n+1)r]$.

Finally, for $nr<t\leq(n+1)r$, we have by (H) and induction assumption that for sufficiently small $\delta>0$,

[TABLE]

where $C^{\prime}_{\beta,\gamma,n},\,C^{\prime\prime}_{\beta,\gamma,n},C^{\prime}_{n,\delta},\,M_{\gamma,\delta}>0$, $0<\beta<1-\gamma$ and $B(\cdot,\cdot)$ is the Beta function.

Now the inequality (2.8) for $t\in(nr,(n+1)r]$ follows from (4.28) and the equality

[TABLE]

On the other hand, by using (4.28), we have that

[TABLE]

is uniformly bounded in $[nr,(n+1)r]$, a fact which implies (2.9). The proof is complete now.

5 Proof of Theorem 2.1 (b)

We still want to develop an induction scheme here. First, consider $V(t)$ and $V_{0}(t)$, $t\in[0,r]$, defined in (4.11).

Lemma 5.1**.**

For any $0<s<t<r$ and $0<\beta<1-\gamma$, there exists a constant $M_{\beta,\gamma}>0$ such that

[TABLE]

Proof. We notice, by definition, for $0<s<t<r$ that

[TABLE]

For $0<s<t<r$, it is easy to see from (4.21) that

[TABLE]

where $M>0$ and by Proposition 4.1, we have for $0<s<t<r$ that

[TABLE]

Hence, by combining (5.2), (5.3) and (5.4), we get for $0<s<t<r$ that

[TABLE]

from which the desired result follows by (4.12) and the well-known Gronwall inequality.

By virtue of Corollary 4.1 and (5.1), we obtain the estimates in Theorem 2.1 for $0<s<t<r$:

[TABLE]

Now suppose that $G(t)$ satisfies those estimates in Theorem 2.1 on the intervals $[0,r]$, $\cdots$, $[(n-1)r,nr]$. Then in the interval $[nr,(n+1)r]$, the integral equation to be satisfied by

[TABLE]

is

[TABLE]

where $V_{0}(t)$ is given as in (4.14). We first show the Hölder continuity of $V_{0}(t)$ and $V(t)$. Let $nr<s<t<(n+1)r$. By virtue of (4.14), we have

[TABLE]

First, by virtue of Corollary 4.1 we have

[TABLE]

Now let us consider the item $I_{2}(t,s)$. For sufficiently small $\delta>0$, there exists, by virtue of (4.17) and (4.19), a value $c_{n,\beta,\gamma,\delta}>0$ such that

[TABLE]

As for the third term on the right side of (5.6), we have

[TABLE]

By virtue of (2.3) and induction assumption, we have for sufficiently small $\delta>0$ that

[TABLE]

On the other hand, by virtue of (4.19) we have for $(n-1)r<u<s-r$ and sufficiently small $\delta>0$ that

[TABLE]

In a similar way, we have by virtue of (4.19) that

[TABLE]

where $c^{\prime\prime}_{\beta,\gamma,\delta}>0$. Combining these two inequalities, it follows that

[TABLE]

for $0<\beta<1-\gamma$ and sufficiently small $\delta>0$. With the aid of (5.13), we have for sufficiently small $\delta>0$ that

[TABLE]

In a similar manner, one can have by virtue of (4.20) and (4.21) that

[TABLE]

and

[TABLE]

Combining (5.9)-(5.16), we conclude that for some $M^{\prime\prime}_{\alpha,\gamma}>0$,

[TABLE]

For the item $I_{4}(t,s)$, let $(n-1)r<s-r<t-r<u<nr$ and by Proposition 4.1, (4.21), (5.10) and induction assumption, we may obtain for sufficiently small $\delta>0$ that

[TABLE]

In a similar way, we can have

[TABLE]

Combining (5.6)-(5.18), we conclude that

[TABLE]

and further we have

[TABLE]

for $nr<s<t<(n+1)r$.

On the other hand, we have by assumption that

[TABLE]

for any $nr<s<t<(n+1)r$ and sufficiently small $\delta>0$. Hence, we have by the induction hypothesis and Theorem 2.1 that

[TABLE]

In a similar manner, we have

[TABLE]

With the aid of (5.20)-(5.22), we thus obtain

[TABLE]

for some $C_{n,\beta,\gamma}>0$. Now combining (5.5), (5.19) and (5.23), we finally obtain (2.10). The proof is thus complete.