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\begin{document}
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\title{Existence and Uniqueness of Solutions of a Class of Evolution Quantum Stochastic Differential Equations}
\author{ S. A. Bishop^1 . E. O.  Ayoola^2\\1Department of Mathematics,Covenant University,\\ Ota, Ogun Sate, Nigeria\\ sheila.bishop@covenantuniversity.edu.ng\\ 2 Department of Mathematics, University of Ibadan,\\ Ibadan, Oyo Sate, Nigeria}
\date{}
\maketitle  
\date{}
\maketitle
\begin{abstract}
We study the properties of the existence and uniqueness of solutions of a class of evolution quantum stochastic differential equations (QSDEs) defined on a locally convex space whose topology is generated by a family of seminorms defined via the norm of the range space of the operator processes. These solutions are called strong solutions in comparison with the solutions of similar equations defined on the space of operator processes where the topology is generated by the family of seminorms defined via the inner product of the range space. The evolution operator generates a bounded semigroup. We show that under some more general conditions, the unique solution is stable. These results extend some existing results in the literature concerning strong solutions of quantum stochastic differential equations.
\end{abstract}
\textbf{Keywords:} Strong solutions, Stability, Bounded semigroup, General Lipschitz condition.\\
\textbf{AMS(2010) Subject Classification:} 58J65, 81S25, 60H10\\
\section{Introduction}
Several results on weak forms of solutions of the following quantum stochastic differential equation have been studied. See [1, 3-7] and the references therein. The motivations for studying this class of equations have been discussed in the references. We consider equations of the form:
\begin{eqnarray*}
dz(t) &=& U(t,z(t))d\wedge_{\pi}(t) + V(t,z(t))dA_g(t) \\
&&+W(t,z(t))dA_{f^+}(t)+H(t,z(t))dt,\\
 z(t_0)&=& z_0, \; t\in I=[t_0,T] \hspace{3.0in}(1.1)
\end{eqnarray*}
In Eq. (1.1), the coefficients U, V, W, and H lie in a certain class of stochastic processes defined in [1], while the
gauge, creation, annihilation processes $\Lambda_{\Pi}, A_{f^+}, A_g$ and the Lebesgue measure $t$ are well defined in [2]and the references therein. $z \in \tilde {\cal B}$ which is a locally convex space.\\
Quantum stochastic differential equation (QSDE) (1.1) is understood in the framework of the Hudson and Parthasarathy [9] quantum stochastic calculus. It has found applications in many physical systems, especially those that have to do with quantum optics, quantum measure theory, quantum open systems and quantum dynamical systems. Also see the above references.\\ In [6], some properties of solutions of Eq. (1.1) were studied. Results on the existence and uniqueness of solutions of these class of equations were established in the space of the operator processes endowed with weak topologies. In [7], quantum stochastic differential inclusions of hypermaximal monotone type were studied under some general conditions and existence of an evolution operator connected with these inclusions were established. Also see [4] for some results on evolution inclusions where the multivalued map $P_1$ is of hypermaximal monotone type.
Further studies were carried out by [5] on properties of the solution sets of quantum
stochastic differential inclusions of Eq. (1.1) under the weak topologies. However, Ayoola in [1] investigated some existence properties on the space when endowed with strong topology and under a more general Lipschitz conditions on the coefficients ($U, V, W, H$). Some results were obtained including some stability results. The results in [1, 2] generalized some similar results in the classical setting.
This paper is concerned with the study of the properties of solutions of an evolution equation defined on the space with strong topology. In [3, 12] existence of mild solutions of evolution QSDEs were studied under the weak topologies. Evolution problems have found practical applications in virtually all fields of sciences. See the references [11, 13-15] for some applications of evolution problems. The results in the present work extend some existing results on strong solutions of Eq. (1.1) and extend the solution space for which QSDE will be applicable. We will consider some applications in our subsequent work.
\section{Preliminaries}
The following evolution equation is considered in what follows:
\begin{eqnarray*}
dz(t) &=& A(t)z(t)+U(t, z(t))d\wedge_{\pi}(t) + V(t, z(t))dA_g(t) \\
&&+W(t,z(t))dA_{f^+}(t)+H(t,z(t))dt,\\
z(t_0)&=& z_0 ,\; t\in I \hspace{3.0in}(2.1)
\end{eqnarray*}
where $A$ generates a bounded semigroup $\{S(t): t \geq 0\}$. For details on semigroup and their applications , see the references [8,10].
We adopt in most cases the definitions and notations of the spaces used in this paper from the references [1-3]. $\tilde{\cal B}$ is the completion of the topological space $(\tilde{\cal B}, \tau)$, and $\tau$ is the topology generated by the family of seminorms $||\phi||_{\xi} = ||\phi\xi||, \xi\in \D \underline {\otimes}\E$, where $||.||$ is the norm of the space ${\cal R} \otimes \Gamma(L^{2}_{\gamma}(\R_+))$. The space $\cal B$, is the linear space of all linear operators on ${\cal R} \otimes\Gamma(L^{2}_{\gamma}(\R_+))$. $\D, \E,$ and $\cal R$ are well defined in [1].
The notations and structures of the following spaces are from the references [1, 2]: ${\cal R} \otimes \Gamma(L^{2}_{\gamma}(\R_+))$, $Ad(\tilde{\cal B})_{ac}$,
$L^p_{loc}(\tilde{\cal B})$, $L^{2}_{\gamma}(\R_+),$ $L(\tilde{\cal B})$, $\D \underline {\otimes}\E$,  Fin($\D \underline {\otimes}\E$). $\D, \E,$ and $\cal R$ is well defined in [1].\\
\textbf{Definition 2.1.}\\
(i) $\phi : I \rightarrow \tilde{\cal B}$ is a stochastic process indexed by $I =[0,T]\subseteq \R_+$.\\
(ii) If $\phi(t) \in {\tilde{\cal B}}_t$, $t \in I$, then $\phi$ is said to be adapted and we denote set of all such stochastic processes by $Ad(\tilde{\cal B})$ .\\
(iii) $\phi(t) \in Ad(\tilde{\cal B})_{ac}$ is said to be adapted, absolutely continuous.\\
(iv) $\phi(t) \in L^p_{loc}(\tilde{\cal B})$ is said to be locally, absolutely p-integrable, where $p \in (0, \infty)$.\\
(v) Since the evolution operator $A$ generates a bounded semigroup $\{S(t)\}_{t\geq0}$, then for each $t \geq 0$, there exists a constant $M>0$ such that $||S(t)||_{\xi} \leq M$.\\
(vi) Let $\theta \in Fin(\D \underline {\otimes}\E)$ and $z \in \tilde {\cal B}$ then, $||z||_{\theta} = max_{\xi \in \theta}||z||_{\xi}$,  where the set $\{||.||_{\theta} : \theta \in Fin(\D \underline {\otimes}\E)\}$ is a family of seminorms on $\tilde{\cal B}$ and $Fin(\D \underline {\otimes}\E)$ denote the set of all finite subsets of $\D \underline {\otimes} \E$. Also see definitions 2.5 and 2.6 in [2].\\ 
 \textbf{Definition 2.2.}\\
A stochastic process $\phi \in L^2_{loc}(\tilde{\cal B})$ is called a strong solution of the problem (2.1) on $I$ if it is absolutely continuous and satisfies
\begin{eqnarray*}
\phi(t)&=& S(t){\phi}_0+\int^t_{t_0}S(t-s)[U(s, \phi(s))d\wedge_{\pi}(s) + V(s, \phi(s))dA_g(s) \\
&&+W(s, \phi(s))dA_{f^+}(s)+H(s,\phi(s))ds],\\
\phi(t_0)&=& {\phi}_0, \; t\in I \hspace{3.0in}(2.2)
\end{eqnarray*}
\textbf{Definition 2.3.}\\
$\Phi :  I \times \tilde{\cal B} \rightarrow \tilde{\cal B}$ is Lipschitzian if
$$\|\Phi(t,y)-\Phi(t,z)\|_{\xi}\leq K^{\Phi}_{\xi}(t)\|y-z\|_{\theta_{\Phi(\xi)}}$$ is satisfied for each $\xi \in \D\underline{\otimes}\E$,
where $y,z \in \tilde{\cal B}$, $\theta \in  (\D\underline{\otimes}\E, Fin( \D\underline{\otimes}\E))$ and $K^{\Phi}_{\xi}: I \rightarrow (0,\infty)$  is a Lipschitz function lying in $L^1_{loc}(I)$. $I=[0,T] \subseteq \R_+$.\\
\textbf{Remark 1.} Theorem 2.2 and Remark (a) - (c) in [1] hold in this case.\\
For the remaining part of this paper, $\xi \in \D\underline{\otimes}\E$ is arbitrary, except otherwise stated.
The following result established in [1] will be used to establish the major results.\\
\textbf{Theorem 2.3.} (a) Let $p,q,u,v \in L^2_{loc}(\tilde{\cal B})$ and let \textbf{M} be their stochastic integral. If $\et\, \xi \in \D\underline{\otimes}\E$ where $\eta =c\otimes e(\alpha), \xi =d \otimes e(\beta), \alpha, \beta \in L^{\infty}_{\gamma, loc}(\R_+)$ and $t\geq0$, then
\begin{eqnarray*}
<\eta,\textbf{M}(t)\xi> &=& \int^t_0<\eta, \{\alpha(s), \pi(s) \beta(s)>_{\gamma}p(s)+<f(s),\beta(s)>_{\gamma}q(s)\\
&&+<\alpha(s),g(s)>_{\gamma}u(s)+v(s)\}\xi>ds. \hspace{1.3in}(2.3)
\end{eqnarray*}
(b) \; Let
$$ K(T)=\sup_{0\leq s\leq T}\max\{|\left\langle \beta(s),\pi(s)\beta(s)\right\rangle|, |\left\langle f(s),\beta(s)\right\rangle|, |\left\langle \beta(s),g(s)\right\rangle|, ||\pi(s)\beta(s)||^2, ||g(s)||^2\}.$$
Then for $T>0$ and $0\leq t\leq T$,
\begin{eqnarray*}
||\textbf{M}(t)\xi||^2 &\leq& 6K(T)^2 \int^t_0 e^{t-s}\{||p(s)\xi||^2+||q(s)\xi||^2+||u(s)\xi||^2\\
&+&||v(s)\xi||^2\}ds. \hspace{3.3in}(2.4)
\end{eqnarray*}
(c) \; Let  $0 \leq s\leq t\leq T.$ Then
\begin{eqnarray*}
||(\textbf{M}(t)-\textbf{M}(s))\xi||^2 &\leq& 6K(T)^2 \int^t_0 e^{t-\tau}\{||p(\tau)\xi||^2+||q(\tau)\xi||^2+||u(\tau)\xi||^2\\
&+&||v(\tau)\xi||^2\}d\tau. \hspace{2.5in}(2.5)
\end{eqnarray*}
\textbf{Note:} $\textbf{M}$ is absolutely continuous hence, $\textbf{M} \in L^2_{loc}(\tilde{\cal B})$.
\section{Main Results}
This section is dedicated to the main results on existence, uniqueness and stability of strong solutions of (2.1).
Subsequently, except otherwise stated, $t \in I= [t_0, T] \subseteq \R_+$ and  $\xi \in \D \underline{\otimes}\E$ is arbitrary.\\
\textbf{Theorem 3.1.}\\
Suppose that the coefficients $ U, V, W, H \in L^2_{loc}(I \times \tilde{\cal B})$ are Lipschitzian. Then for $(t_0,z_0) \in I \times\tilde{\cal A}$  there exists a unique strong solution $\varphi$ of equation (2.1) satisfying $\varphi(t_0)=z_0.$\\
\textbf{Proof.} To prove the theorem, we make the following assumptions:\\
$H_1.$ \;\ Let $\{\varphi_n(t)\}_{n\geq0}$ be a sequence of successive approximations of $\varphi \in \tilde{\cal B};$\\  
$H_2.$ \;\ let $\varphi_{n}(t), \; n\geq1$ define an absolutely continuous process in $L^2_{loc}(\tilde{\cal A}).$\\
	Let  $T>t_0, \; t \in I$ be fixed. Then, we prove $H_1- H_2$  by induction as follows:
	For $n\geq0$, we have
\begin{eqnarray*}
\varphi_{n+1}(t) &=& S(t)z_0 + \int^t_{t_0}S(t-s)[U(s, \varphi_n(s))d\wedge_{\pi}(s) + V(s, \varphi(s))dA^+_g(s) \\
&&+W(s, \varphi_n(s))dA_f(s)+H(s, \varphi_n(s))ds].
\end{eqnarray*}
By hypothesis, $U(s, z_0),  V(s,z_0),  W(s,z_0), H(s,z_0) \in \tilde{\cal B}_s$ for $s \in [t_0,T]$ while\\
 $U(., z_0),  V(.,z_0),  W(.,z_0), H(.,z_0) \in L^2_{loc}(\tilde{\cal B}).$\\
Therefore the quantum stochastic integral which defines $\varphi_1(t)$ exists for\\ $t \in [t_0,T].$\\
By Theorem 2.3, $\varphi_1(t) \in L^2_{loc}(\tilde{\cal B}).$ Hence it implies that each\\
$U(s, \varphi_n(s)), V(s, \varphi(s)), W(s, \varphi_n(s))$ and $H(s, \varphi_n(s) \in L^2_{loc}(\tilde{\cal B}).$\\ This proves assumptions $H_1- H_2-$.
Next we show that the sequence of successive approximations converges as follows:
\begin{eqnarray*}
\parallel \varphi_{n+1}(t) - \varphi_n(t)\parallel_{\xi}&=& ||\int^t_{t_0}S(t-s)[(U(s, \varphi_n(s))-U(s, \varphi_{n-1}(s)))d\wedge_{\pi}(s)\\
 &&+ (V(s, \varphi(s))-V(s, \varphi_{n-1}(s)))dA^+_g(s)\\
&&+ (W(s, \varphi_n(s))-W(s, \varphi_{n-1}(s)))dA_f(s)\\
&&+(H(s, \varphi_n(s))-H(s, \varphi_{n-1}(s)))ds]||_{\xi}. \hspace{0.5in}(3.1)
\end{eqnarray*}
By Theorem 2.3 and (v) of Definition 2.1, we get
\begin{eqnarray*}
\parallel \varphi_{n+1}(t) - \varphi_n(t)\parallel^2_{\xi}&\leq& 6M^2K(T)^2\int^t_{t_0}e^{t-s}\{||U(s, \varphi_n(s))-U(s, \varphi_{n-1}(s))||^2_{\xi}\\
&&+||V(s, \varphi(s))-V(s, \varphi_{n-1}(s))||^2_{\xi}\\
&&+||W(s, \varphi_n(s))-W(s, \varphi_{n-1}(s))||^2_{\xi}\\
&&+||H(s, \varphi_n(s))-H(s, \varphi_{n-1}(s))||^2_{\xi}\}ds. \hspace{0.3in}(3.2)
\end{eqnarray*}
By definition 2.3, we have
$$\parallel \textbf{M}(s, \varphi_{n}(s)) - \textbf{M}(s,\varphi_{n-1}(s))\parallel_{\xi}\\
\leq  K^{\textbf{M}}_{\xi}(s)\parallel \varphi_{n}(s) - \varphi_{n-1}(s)\parallel_{\theta_{\textbf{M}\xi}},$$
for each $\textbf{M} \in \{U,V.W,H\}$.
Thus, there exists $\xi^1_{\textbf{M}} \in \theta_{\textbf{M}}(\xi) $ satisfying
$$\parallel \varphi_{n}(s) - \varphi_{n-1}(s)\parallel_{\theta_{\textbf{M}\xi}}=\parallel \varphi_{n}(s) - \varphi_{n-1}(s)\parallel_{\xi^1_{\textbf{M}}}. \eqno(3.3)$$
Using (3.2), we obtain
\begin{eqnarray*}
\parallel \varphi_{n+1}(t) - \varphi_n(t)\parallel^2_{\xi}&\leq& NC(T)L_{\xi}\int^t_{t_0}e^{t-s}\parallel \varphi_{n}(s) - \varphi_{n-1}(s)\parallel^2_{\xi_1}ds\\
&& = NC(T)L_{\xi}e^t\int^t_{t_0}e^{-s}\parallel \varphi_{n}(s) - \varphi_{n-1}(s)\parallel^2_{\xi_1}ds.\hspace{0.3in}(3.4)
\end{eqnarray*}
where $$\parallel \varphi_{n}(s) - \varphi_{n-1}(s)\parallel_{\xi_1} =\max_{\textbf{M} \in \{U,V.W,H\}} \parallel \varphi_{n}(s) - \varphi_{n-1}(s)\parallel_{\xi^1_{\textbf{M}}}. \eqno(3.5)$$  and
$$N=M^2, \; C(T)=6K(T)^2, \; L_{\xi}= \mbox{ess} \sup_{s\in [0,T]} \left[K_{\xi}(s)=\sum_{\textbf{M} \in \{U,V.W,H\}}K^{\textbf{M}}_{\xi}(s)^2\right]. \eqno(3.6)$$
 Continuing the iteration and replacing $\xi_2$ with $\xi_1$ in (3.5), yields
\begin{eqnarray*}
\parallel \varphi_{n+1}(t) - \varphi_n(t)\parallel^2_{\xi} &\leq& N^2C(T)^2L_{\xi}L_{\xi_1}e^{t}\\
&&\times\int^t_{t_0}\int^s_{t_0}e^{-s^{\prime}}\parallel \varphi_{n-1}(e^{-s^{\prime}}) - \varphi_{n-2}(e^{-s^{\prime}})\parallel^2_{\xi_2}ds^{\prime}ds\\
&&\leq N^nC(T)^n\textbf{M}(\xi)^ne^t\int^t_{t_0}ds_1\int^{s_1}ds_2...\int^{s_{n-2}}_{t_0}ds_{n-1}\\
&& \times \int^{s_{n-1}}_{t_0}e^{-s_n}\parallel \varphi_{1}(s_n) - \varphi_{0}(s_n)\parallel^2_{\xi}ds_n, \hspace{0.3in}(3.7)
\end{eqnarray*}
where $\textbf{M}_n(\xi)=\max\{L_{\xi,j}, j=0,1,...,n-1\}$, $\textbf{M}(\xi)=\sup_{n\in\N}\{\textbf{M}_n(\xi)\},$ and $L_{\xi,j}, j=0,1,...,n-1$ are positive real numbers.\\
Since the map $s\longrightarrow \|\varphi_1(s)-z_0\|_{\xi}$ is continuous on $I,$ we obtain,\\ $R_{{\xi}_n} = \sup_{s \in I} \|\varphi_1(s)-z_0\|_{{\xi}_n} < \infty$ and put $R_{\xi} =\sup_{n\in \N}\{R_{{\xi}_n}\}$ in (3.7) to get
$$\parallel \varphi_{n+1}(t) - \varphi_n(t)\parallel^2_{\xi} \leq [NC(T)\textbf{M}(\xi)]^ne^{T}\frac{T^n}{n!}R^2_{\xi}, n=0, 1, 2, ....$$
For $n>k$ we get,
$$\parallel \varphi_{n+1}(t) - \varphi_{k+1}(t)\parallel_{\xi} = \|\Sigma^n_{m=k+1}(\varphi_{m+1}(t)-\varphi_m(t))\|_{\xi}$$
$$\leq \Sigma^n_{m=k+1}\|\varphi_{m+1}(t)-\varphi_m(t)\|_{\xi}$$ $$\leq e^{\frac{T}{2}}R_{\xi}\sum^n_{m=k+1}\left(\frac{[NC(T)\textbf{M}(\xi)]^mT^m}{m!}\right)^{\frac{1}{2}} < \infty.$$
Showing that ${\varphi_n(t)}$ is a Cauchy sequence in $\tilde{\cal B}$ and converges uniformly to some $\varphi(t).$\\
 Now since $\varphi_n(t)$ is adapted and absolutely continuous, the same is true for $\varphi(t).$\\
 Next we show that $\varphi(t)$ satisfies Eq. (2.1).
 Let $\varphi(t_0)=z_0$ and by (3.4), there exists $\eta \in \D \underline{\otimes}\E$ such that
 \begin{eqnarray*}
 ||\int^t_{t_0}S(t-s)[U(s, \varphi_n(s))d\wedge_{\pi}(s) &+& V(s, \varphi_n(s))dA^+_g(s)\\
 &&+W(s, \varphi_n(s))dA_f(s)+H(s, \varphi_n(s))ds]||^2_{\xi}\\
&&-||\int^t_{t_0}S(t-s)[U(s, \varphi(s))d\wedge_{\pi}(s)\\
&& + V(s, \varphi(s))dA^+_g(s)+W(s, \varphi(s))dA_f(s)\\
&&+H(s, \varphi(s))ds]||^2_{\xi}\\
&& = ||\int^t_{t_0}S(t-s)(P(s, \varphi_n(s)) - P(s, \varphi(s)))ds||^2_{\xi}\\
&&\leq NC(T)L_{\xi}e^{t}\\
&& \times\int^t_{t_0}e^{-s}\parallel \varphi_{n}(s) - \varphi(s)\parallel^2_{\xi}ds\longrightarrow 0\\
&&\mbox{as} \;\;\ n\longrightarrow\infty.
\end{eqnarray*}
Since $\varphi_n(s) \longrightarrow \varphi(s) \;\;\ \mbox{in} \;\;\ \tilde{\cal B}$ uniformly on $[t_0,T],$
we have
\begin{eqnarray*}
\varphi(t) &=& \lim_{n\rightarrow \infty}\varphi_{n+1}(t)\\
&&=S(t)z_0+\lim_{n\rightarrow \infty}(\int^t_{t_0}S(t-s)(U(s, \varphi_n(s))d\wedge_{\pi}(s) + V(s, \varphi_n(s))dA^+_g(s)\\
&& +W(s, \varphi_n(s))dA_f(s)+H(s, \varphi_n(s))ds)\\
&&=S(t)z_0+\int^t_{t_0}S(t-s)(U(s, \varphi(s))d\wedge_{\pi}(s) + V(s, \varphi(s))dA^+_g(s)\\
&&+W(s, \varphi(s))dA_f(s)+H(s, \varphi(s))ds), , t \in I.
\end{eqnarray*}
This shows that $\varphi(t)$ is a solution of Eq. (2.1).\\
\textbf{Uniqueness}\\
Suppose that $y(t), t \in[t_0,T]$ is another adapted absolutely continuous solution with $y(t_0)=z_0,$ then just as we established the above result, we obtain
$$\parallel \varphi(t) - y(t)\parallel^2_{\xi} \leq [NC(T)\textbf{M}(\xi)]^ne^{T}\frac{T}{n!} \; \sup_{t \in I}\parallel \varphi(t) - y(t)\parallel^2_{\xi} <\infty. \eqno(3.8)$$
 By the right hand side of Eq. (3.8), we conclude that for $n\in \N$, $\parallel \varphi(t) - y(t)\parallel_{\xi}=0$ and $\varphi(t) = y(t)$ on $\D \underline{\otimes} \E$, $t \in I.$ Hence the solution is unique.
\section{Stability}
In this section, we show that under the condition (v) of Definition 2.1, the solutions of Eq.(2.1) is stable. Next, we state the following.\\
(a) let the coefficients U, V, W, H satisfy the conditions of theorem 3.1 and let $z(t) , y(t)$ , $t \in [t_0, T]$ be solutions to Eq. (2.1) such that $z(t_0)=z_0$ and $y(t_0)= y_0,$  $z_0, y_0 \in \tilde{\cal B}.$  The solution z(t) is stable under the changes in the initial condition over a finite time interval as follows:\\
(b) Let $L_{\xi}$, $N$ and $C(T)$ be constants such that
$$L_{\xi}=\mbox{ess} \sup_{s\in I}K_{\xi}(s), \; C(T)=12K(T)^2 \; \mbox{and}\; N=M^2\eqno(4.1)$$
where $K(T)$ is as defined in Theorem 2.3 and $||S(t)||_{\xi}$ by (v) of Definition 2.1.\\
(c) Define the function $K_{\xi}(s)$ as
$$K_{\xi}(s)=\sum_{\textbf{M} \in \{U,V.W,H\}}(K^{\textbf{M}}_{\xi}(s))^2 \eqno(4.2)$$
\textbf{Theorem 4.1.} Let the conditions of Definition 2.1 hold and let $\epsilon > 0,$ be given. Then there exists $ \delta >0$ such that if $\left\|z_0-y_0\right\|_{\xi} < \delta$, then $\left\|z(t)- y(t)\right\|_{\xi} < \epsilon$, $ \forall \; t\in [0,T]$.\\
\textbf{Proof:}\\
Let $z_n(t), y_n(t), \; n=0,1,...$ be the iterates corresponding to $z_0$, $y_0$ respectively. Let $z_0(t)=z_0$  and $y_0(t)=y_0$, $0\leq t \leq T.$  Then we get 
\begin{eqnarray*}
 \parallel z_{n+1}(t) - y_{n+1}(t)\parallel_{\xi}&\leq& ||S(t-s)(z_0-y_0)||_{\xi}\\
&&+||\int^t_{t_0}S(t-s)[(U(s, z_n(s))-U(s, y_n(s)))d\wedge_{\pi}(s)\\
 &&+(V(s, z(s))-V(s, y_n(s)))dA^+_g(s)\\
&&+ (W(s, z_n(s))-W(s, y_n(s)))dA_f(s)\\
&&+ (H(s, z_n(s))-H(s, y_n(s)))ds]||_{\xi}
\end{eqnarray*}
So that by applying Theorem 2.3 and condition (v) of Definition 2.1, we obtain
\begin{eqnarray*}
 \parallel z_{n+1}(t) - y_{n+1}(t)\parallel^2_{\xi}&\leq& 2M^2||z_0-y_0||^2_{\xi}\\
&&+2M^2|| \int^t_{t_0}S(t-s)[(U(s, z_n(s))-U(s, y_n(s)))d\wedge_{\pi}(s)\\
&&+(V(s, z(s))-V(s, y_n(s)))dA^+_g(s)\\
&&+ (W(s, z_n(s))-W(s, y_n(s)))dA_f(s)\\
&&+ (H(s, z_n(s))-H(s, y_n(s)))ds]||^2_{\xi}
\end{eqnarray*}
\begin{eqnarray*}
\parallel z_{n+1}(t) - y_{n+1}(t)\parallel^2_{\xi} &\leq&  2N||z_0-y_0||^2_{\xi}\\
&&+NC(T)\int^t_{t_0}e^{s-t}\{||U(s, z_n(s))-U(s, y_n(s))||^2_{\xi}\\
&&+||V(s, z(s))-V(s, y_n(s))||^2_{\xi}\\
&&+ ||(W(s, z_n(s))-W(s, y_n(s))||^2_{\xi}\\
&&+ (H(s, z_n(s))-H(s, y_n(s)))||^2_{\xi}\}ds \hspace{1.1in}(4.3)
\end{eqnarray*}
Since Definition 2.3 also holds for the coefficients $U, V, W, H$, we find elements $\xi_{\textbf{M}, 1} \in \theta_{\textbf{M}}(\xi) \in Fin(\D \underline{\otimes}\E)$, $ \textbf{M} \in \{U, V, W, H\}$ such that
\begin{eqnarray*}
\parallel z_{n+1}(t) - y_{n+1}(t)\parallel^2_{\xi} &\leq& 2N||z_0-y_0||^2_{\xi}+ NC(T)\\ 
&& \times \int^t_{t_0}e^{t-{s_1}}[\sum_{\textbf{M} \in \{U,V,W,H\}} K^{\textbf{M}}_{\xi}(s_1)^2 ||z_n(s_1))- y_n(s_1))||^2_{\xi_{\textbf{M},1}}]ds_1\\
&&\leq 2N||z_0-y_0||^2_{\xi}\\
&&+  NC(T)L_{\xi}e^t\int^t_{t_0}e^{-s_1}||z_n(s_1)- y_n(s_1)||^2_{\xi}ds_1. \hspace{0.5in}(4.4)
\end{eqnarray*}
where $\xi_1 \in {\xi_{\textbf{M},1} : \textbf{M}} \in \{U,V.W,H\}\}$ satisfies
\begin{eqnarray*}
\parallel \varphi_{n}(s) - \varphi_{n-1}(s)\parallel^2_{\xi_1}&=&\max_{\textbf{M} \in \{U,V.W,H\}} \parallel \varphi_{n}(s) - \varphi_{n-1}(s)\parallel^2_{\xi_{\textbf{M},1}}, \; s \in I. \hspace{0.6in}(4.5)
\end{eqnarray*}
Also, if we have $\xi_2 \in \D \underline{\otimes} \E$ then,
\begin{eqnarray*}
 \parallel z_{n}(s_1) - y_{n}(s_1)\parallel^2_{\xi} & \leq & 2N\left\|(z_0-y_0)\right\|^2_{\xi_1} \\
&&+NC(T)L_{\xi_1}\int^t_{s_1}e^{s_1-s_2}||z_{n-1}(s_2) - y_{n-1}(s_2)||^2_{\xi_2}ds_2.
\end{eqnarray*}
By (4.4), we obtain for $t \in [0,T]$,
\begin{eqnarray*}
 \parallel z_{n+1}(t) - y_{n+1}(t)\parallel^2_{\xi} & \leq & 2N\left\|(z_0-y_0)\right\|^2_{\xi} \\
&&+2NC(T)\left\|z_0-y_0\right\|^2_{\xi}L_{\xi}e^t\int^t_0e^{-s_1}ds_1\\
&&+N^2C(T)^2L_{\xi}L_{\xi_1}e^t\\
&& \times \int^t_0 \int^{s_1}_0e^{-s_2}||z_{n-1}(s_2) - y_{n-1}(s_2)||^2_{\xi_2}ds_2ds_1.
\end{eqnarray*}
Continuous iterations yields,
\begin{eqnarray*}
 \parallel z_{n+1}(t) - y_{n+1}(t)\parallel^2_{\xi} & \leq & 2N\left\|z_0-y_0\right\|^2_{\xi}+2NC(T)\left\|z_0-y_0\right\|^2_{\xi_1}L_{\xi}e^Tt\\
&&+2N^2C(T)^2\left\|z_0-y_0\right\|^2_{\xi_2}L_{\xi}L_{\xi_1}e^T\int^t_0\int^{s_1}_0 ds_2ds_1\\
&&+2N^3C(T)^3\left\|z_0-y_0\right\|^2_{\xi_2}L_{\xi}L_{\xi_1}L_{\xi_2}e^T\int^t_0\int^{s_1}_0\\
&&\times \int^{s_2}_0 \int^{s_1}_0ds_3ds_2ds_1\\
&&+...+N^{n+1}C(T)^{(n+1)}e^TL_{\xi}L_{\xi_1}L_{\xi_2}...L_{\xi_n}\int^t_0\int^{s_1}_0...\\
&&\times \; \int^{s_n}_0||z_{0}(s_{n+1}) - y_{0}(s_{n+1})||^2_{\xi_{n+1}}ds_1ds_2ds_3...ds_{n+1}.
\end{eqnarray*}
Now, by letting $\textbf{K}(\xi) = \sup_{n\in \N}\{L_{\xi}, L_{\xi_1}, L_{\xi_2}, ... , L_{\xi_n}\}, \; \eta_n \in \{\xi, \xi_1, \xi_2,...,\xi_n,\xi_{n+1}\}$ so that if
$$\left\|z_0-y_0\right\|_{\eta_n}=\max\{\left\|z_0-y_0\right\|_{\xi_j}, j=0,1,...,n+1\}, \; \mbox{we obtain}$$
\begin{eqnarray*}
\parallel z_{n+1}(t) - y_{n+1}(t)\parallel^2_{\xi} &\leq& 2e^T\left\|z_0-y_0\right\|^2_{\eta_n}\sum^{n+1}_{m=0}[NC(T)\textbf{K}(\xi)]^m\frac{T^m}{m!}\\
&&\leq 2\left\|z_0-y_0\right\|^2_{\eta_n}e^{(NC(T)\textbf{K}(\xi)+T)}.   \hspace{1.2in}(4.6)
\end{eqnarray*}
Finally, by taking the square root of both sides of (4.6), letting $n \rightarrow \infty,$ we obtain\\
 $\left\|z(t)-y(t)\right\|_{\xi}\leq \epsilon$, take $\delta =\epsilon[2e^{(NC(T)\textbf{K}(\xi)T+T)}]^{-\frac{1}{2}},$
for all $ t \in [0,T],$ and the desired result is obtained.\\
\textbf{Remark 3.} If $N < 1$, then we obtain the results in [1].\\
\textbf{\textbf{Conflict of Interest}:}
The authors declare that there is no conflict of\\
interest.\\
\textbf{References}
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\end{document}
