S-Contractive Mappings on Vector-Valued White Noise Functional Space and Their Applications

Abstract

In this paper, we propose a new notion, which we name S-contractive mapping, in a framework of vector-valued white noise functionals ${\mathcal{W}}_{\omega }\otimes \mathcal{N}\subset \Gamma \left(H\right)\otimes K\subset {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$. And we give concrete definitions of S-contractive mappings for vector-valued white noise functionals. We establish the fixed-point theorems of S-contractive mappings. As applications, by applying the fixed-point theorems of generalized S-contractive mappings, we prove the existence and uniqueness of a generalized form of differential equations of vector-valued white noise functionals with weak conditions and investigate Wick-type differential equations of vector-valued white noise functionals with generalized conditions.

1. Introduction

Since the famous Banach fixed-point theorem was established in 1922 [1] and Brouwer fixed-point theorem was formulated in 1990 [2], fixed-point theory and its applications have experienced rapid growth over the past century. This field has played a pivotal role in various domains including nonlinear analysis, economics, game theory, integral–differential equations, optimization theory, dynamical systems theory, signal and image processing, and numerous other areas of applied mathematics.

Particularly, the fixed-point theorems offer a rigorous theoretical foundation for the study of differential equations [3,4,5,6,7]. It enables the transformation of many intricate nonlinear systems of differential equations or boundary value problems into equivalent integral equation forms. Subsequently, these transformed problems can be further examined and solved using the methodologies provided by fixed-point theory. This approach not only facilitates the proof of existence and uniqueness of solutions but also aids in the development of numerical methods and stability analysis of the systems involved. Many people have studied white noise differential equations [8,9,10,11] and quantum differential equations [12,13,14,15,16,17,18,19,20,21]. Hassan Ranjbar [22] focused on the numerical integration of generalized n-dimensional second-order differential equations with initial value conditions, influenced by additive Gaussian white noise. Huang [23] investigated new fixed-point theorems for generalized Meir–Keeler-type nonlinear mappings satisfying a certain condition. Ji [24] obtained the unique solution of the quantum continuity equation by solving the quantum evolution system of the white noise operator. In recent years, Ji and Ma [25] have investigated an abstract differential equation

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d\Phi \left(t\right)}{dt}}& =F(t,\Phi (t\left)\right),\hfill \\ \hfill {\Phi |}_{t=0}& ={\Phi }_0,\hfill \end{array}\right.$$

and Wick-type differential equations

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{\Phi }_t}{dt}}& =\Psi {\diamond }_{\beta }{\Phi }_t,\hfill \\ \hfill {\Phi |}_{t=0}& ={\Phi }_0,t\in [0,T],\hfill \end{array}\right.$$

in the realm of white noise analysis based on the theoretical framework of vector-valued white noise functionals with certain conditions, where ${\diamond }_{\beta }$ is the $\beta$-Wick product of vector-valued white noise functionals; for more details, see Section 3 or refer to reference [20]. Fixed-point theorems are invaluable tools for solving equations. Usually, it provides a generalized theory with more weak conditions for solving equations. However, few studies about fixed-point theorems have been conducted in the field of white noise differential equations. In this paper, the main purpose is to propose a new notion, which we name S-contractive mapping, in a framework of vector-valued white noise functionals ${\mathcal{W}}_{\omega }\otimes \mathcal{N}\subset \Gamma \left(H\right)\otimes K\subset {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$ and to establish fixed-point theorems of S-contractive mappings. As applications, by applying the fixed-point theorems of S-contractive mappings, we prove the existence and uniqueness of a generalized form of differential equations of vector-valued white noise functionals with weak conditions and investigate Wick-type differential equations of vector-valued white noise functionals with generalized conditions.

The framework of this article is outlined below: In Section 2, we review the foundational concepts related to the (scalar-valued) Cochran–Kuo–Sengupta (CKS) space, a widely recognized construction within white noise theory, particularly suited for the solution space of white noise differential equations. Furthermore, we revisit key aspects of vector-valued Gaussian white noise functionals, encompassing the analytic characteristic theorem of the S-transform and the convergence criteria for generalized white noise functionals as expressed through the S-transform. In Section 3, we propose a new notion, which we name S-contractive mapping, in a framework of vector-valued white noise functionals ${\mathcal{W}}_{\omega }\otimes \mathcal{N}\subset \Gamma \left(H\right)\otimes K\subset {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$, and we establish the fixed-point theorems of S-contractive mappings. In Section 4, by applying the fixed-point theorems of generalized S-contractive mappings, we investigate the existence and uniqueness of a generalized form of differential equations and the $\beta$-Wick-type differential equation of vector-valued white noise functionals with weak conditions.

2. Preliminaries

Let H be a separate Hilbert space with norm ${\left|·\right|}_0$, and A be an operator on H satisfying the following conditions:

(A1) 

A is a densely defined operator on H;

(A2) 

A is a self-adjoint operator on H.

The domain of $A^p,p>0$, denoted by $Dom\left(A^p\right)$, is a dense subspace of H, denoted by $S_p$. Then, $S_p$ is a Hilbert space with norm ${\left|·\right|}_p$, where the norm ${\left|·\right|}_p$ is defined by the following:

$${\left|\xi \right|}_p:={\left|A^p\xi \right|}_0<\infty ,\xi \in Dom\left(A^p\right).$$

Let $S_{-p}$ be the completion of H with respect to the norm ${\left|\xi \right|}_p={\left|A^{-p}\xi \right|}_0,\xi \in H$. In fact, $A^{-1}$ is a bounded operator, since $\rho :={∥A^{-q}∥}_{op}={\{infSpec\left(A\right)\}}^{-1}<\infty$. It is clear that, for any $p\in \mathbb{R}$, $q\in {\mathbb{R}}^{+},$

$$\begin{array}{c}\hfill {\left|\xi \right|}_p\le \rho {\left|\xi \right|}_{p+q}.\end{array}$$

It follows that

$$\underset{p\to \infty }{\mathrm{proj}\lim }{S}_p=:S\subset \cdots \subset S_p\subset S_0=H\subset S_{-p}\subset \cdots \subset S^{*}:=\underset{p\to \infty }{\mathrm{ind}\lim }{S}_{-p}.$$

$\left(S,\left\{{\left|·\right|}_p;p=0,1,\cdots \right\}\right)$ becomes a countable Hilbert space by selecting a countable set of defining norms. And $S\subset S_0=H\subset S^{*}$ is a Gel’fand triple if $A^{-r}$ is of Hilbert–Schmidt type for some $r>0$; i.e., the natural injection $S_{p+r}\to S_p$ is of Hilbert–Schmidt type.

Consider the sequence ${\left\{\omega \left(n\right)\right\}}_{n=0}^{\infty }$ to be an increasing sequence of positive real numbers. The exponential generating function for this sequence is defined as

$$G_{\omega }\left(p\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{\omega \left(n\right)}{n!}}{p}^n.$$

We assert that ${\left\{\omega \left(n\right)\right\}}_{n=0}^{\infty }$ meets the following conditions (see [16,26]):

(W1) 

$1=\omega \left(0\right)\le \omega \left(1\right)\le \omega \left(2\right)\le \cdots$;

(W2) 

The generating function $G_{\omega }\left(p\right)$ possesses an infinite radius of convergence;

(W3) 

The power series

$${\tilde{G}}_{\omega }\left(p\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{n^{2n}}{n!\omega \left(n\right)}}\left(\underset{q>0}{inf}{\displaystyle \frac{G_{\omega }\left(q\right)}{q^n}}\right)p^n$$

is convergent with a positive radius;

(W4) 

There exists a constant $C_{1\omega }>0$ such that for all $i,j\in \mathbb{N}$,

$$\omega \left(i\right)\omega \left(j\right)\le C_{1\omega }^{i+j}\omega (i+j);$$

(W5) 

Given any $i,j\in \mathbb{N}$, there exists a constant $C_{2\omega }>0$ satisfying

$$\omega (i+j)\le C_{2\omega }^{i+j}\omega \left(i\right)\omega \left(j\right).$$

The weighted (Boson) Fock space over the Hilbert space $S_p$ is defined by

$${\Gamma }_{\omega }\left(S_p\right)=\left\{\varphi ={\left(f_n\right)}_{n=0}^{\infty }:f_n\in S_p^{\widehat{\otimes }n},{∥\varphi ∥}_{\omega ,p}^2:=\sum _{n=0}^{\infty }n!\omega \left(n\right){\left|f_n\right|}_p^2<\infty \right\},$$

for each $p\ge 0$, where $S_p^{\widehat{\otimes }n}$ denotes the n-fold symmetric tensor power of the Hilbert space $S_p$. When $\omega \equiv 1$, the space ${\Gamma }_{\omega }\left(S_p\right)$ is denoted by $\Gamma \left(S_p\right)$, i.e.,

$$\Gamma \left(S_p\right):={\Gamma }_{\omega }\left(S_p\right),$$

especially, $\Gamma \left(H\right)=\Gamma \left(S_0\right)$. We denote the second quantization operator of A by $\Gamma \left(A\right)$, which is defined by the following:

$$\Gamma \left(A\right)\varphi ={\left(A^{\otimes n}{f}_n\right)}_{n=0}^{\infty },\varphi ={\left(f_n\right)}_{n=0}^{\infty }\in \Gamma \left(H\right),$$

satisfying

$$\sum _{n=0}^{\infty }n!{\left|A^{\otimes n}{f}_n\right|}_0^2<\infty .$$

It follows from the Riesz representation theorem that $\Gamma \left(H\right)$ is identified with its dual. Then, we obtain the following continuous inclusion maps:

$$\cdots \subset {\Gamma }_{\omega }\left(S_q\right)\subset {\Gamma }_{\omega }\left(S_p\right)\subset \cdots \subset \Gamma \left(H\right)\subset \cdots \subset {\Gamma }_{\omega }\left(S_{-p}\right)\subset {\Gamma }_{\omega }\left(S_{-q}\right)\subset \cdots ,p<q.$$

We define the limit spaces as follows:

$${\mathcal{W}}_{\omega }:=\underset{p\to \infty }{\mathrm{proj}\lim }{\Gamma }_{\omega }\left(S_p\right)=\bigcap _{p\ge 0}{\Gamma }_{\omega }\left(S_p\right),$$

$${\mathcal{W}}_{\omega }^{*}:=\underset{p\to \infty }{\mathrm{ind}\lim }{\Gamma }_{{\omega }^{-1}}\left(S_{-p}\right)=\bigcup _{p\ge 0}{\Gamma }_{{\omega }^{-1}}\left(S_{-p}\right).$$

Since ${\mathcal{W}}_{\omega }$ is a countable Hilbert nuclear space [16],

$${\mathcal{W}}_{\omega }\subset \Gamma \left(H\right)\subset {\mathcal{W}}_{\omega }^{*}$$

is a Gel’fand triple.

According to the Bochner–Minlos theorem, there exists a unique probability measure $\mu$ on $S_{\mathbb{R}}^{*}$, ensuring that

$${\int }_{S_{\mathbb{R}}^{*}}{e}^{i〈x,\xi 〉}d\mu \left(x\right)=e^{-{\displaystyle \frac{1}{2}}{\left|\xi \right|}_0^2},\xi \in S_{\mathbb{R}}.$$

The measure $\mu$ in the above expression is defined as the standard Gaussian measure on $S_{\mathbb{R}}^{*}$, and the probability space $\left(S_{\mathbb{R}}^{*},\mu \right)$ is a standard Gaussian space.

Let $L^2(S_{\mathbb{R}}^{*},\mu )$ represent the complex Hilbert space of $\mu$-square integrable functions on $S^{*}$. According to the Wiener–Itô decomposition theorem, every $\varphi \in L^2(S_{\mathbb{R}}^{*},\mu )$ can be uniquely expressed as follows:

$$\varphi =\sum _{n=0}^{\infty }\langle :{·}^{\otimes n}:,f_n\rangle ,$$

where $f_n\in S_p^{\widehat{\otimes }n}$, and $:{·}^{\otimes n}:$ is the Wick tensor. By the Wiener–Itô–Segal isomorphism, $L^2(S_{\mathbb{R}}^{*},\mu )$ is unitarily isomorphic to the (Boson) Fock space $\Gamma \left(H\right)$, i.e.,

$$\Gamma \left(H\right)\cong L^2(S_{\mathbb{R}}^{*},\mu ).$$

For any vector $x\in S^{*}$, the exponential function ${\varphi }_{\xi }\left(x\right)\in L^2(S_{\mathbb{R}}^{*},\mu )$ can be written as $e^{〈x,\xi 〉-{\displaystyle \frac{1}{2}}〈x,x〉}$. This exponential function corresponds to the exponential vector ${\varphi }_{\xi }=\left(1,\xi ,{\displaystyle \frac{{\xi }^{\otimes 2}}{2!}},\cdots ,{\displaystyle \frac{{\xi }^{\otimes n}}{n!}},\cdots \right)\in \Gamma \left(H\right)$. By this correspondence, $\varphi \in L^2(S_{\mathbb{R}}^{*},\mu )$ is uniquely determined by $\varphi ={\left(f_n\right)}_{n=0}^{\infty }\in \Gamma \left(H\right)$. We identify $\varphi \in L^2(S_{\mathbb{R}}^{*},\mu )$ with $\varphi ={\left(f_n\right)}_{n=0}^{\infty }\in \Gamma \left(H\right)$ without confusions. Thus, we can obtain the Gel’fand triple as shown below:

$${\mathcal{W}}_{\omega }\subset \Gamma \left(H\right)\cong L^2(S_{\mathbb{R}}^{*},\mu )\subset {\mathcal{W}}_{\omega }^{*}.$$

This space is known as the Cochran–Kuo–Sengupta space, abbreviated as CKS space. Moreover, the Hida–Kubo–Takenaka space [27] is an instance of CKS spaces with $\omega \equiv 1$ for $0\le \beta <1$, while the Kontratiev–Streit space [28] is an example of CKS spaces with $\omega \left(n\right)={(n!)}^{\beta }$ for $0\le \beta <1$.

Within this framework, the canonical $\mathbb{C}$-bilinear ${\mathcal{W}}^{*}\times \mathcal{W}$ is rigorously defined by the following expression:

$$〈〈\Phi ,\varphi 〉〉=\sum _{n=0}^{\infty }n!〈F_n,f_n〉,$$

where $\Phi =\left(F_n\right)$ belongs to the space ${\mathcal{W}}^{*}$ and $\varphi =\left(f_n\right)$ belongs to the space $\mathcal{W}$.

For any $\Phi =\left(F_n\right)\in {\mathcal{W}}^{*}$, the S-transform of $\Phi =\left(F_n\right)$ is given by

$$S\Phi \left(\xi \right)=\sum _{n=0}^{\infty }〈F_n,{\xi }^{\otimes n}〉=〈〈\Phi ,{\varphi }_{\xi }〉〉\xi \in S.$$

By S-transform, the Wick product of $\Phi ,\Psi \in {\mathcal{W}}^{*}$, denoted by $\Phi \diamond \Psi$, is defined by

$$S(\Phi \diamond \Psi )\left(\xi \right)=S\Phi \left(\xi \right)S\Psi \left(\xi \right)\xi \in S.$$

Here are several lemmas utilized in our research endeavors.

Lemma 1

([26]). We define $\omega =\left\{\omega \right(n\left)\right\}$ as a positive sequence that fulfills the criteria outlined in conditions (W1) and (W2). Let $G_{\omega }\left(t\right)$ denote the generating function. Consequently,

(i)  

$G_{\omega }\left(0\right)=1$ and $G_{\omega }\left(p\right)\le G_{\omega }\left(q\right)$ for $0\le p\le q$;

(ii) 

$e^p{G}_{\omega }\left(q\right)\le G_{\omega }(p+q)$ and $e^q\le G_{\omega }\left(q\right)$ for $p,q\ge 0$;

(iii)

$c[G_{\omega }\left(q\right)-1]\le G_{\omega }\left(cq\right)-1$ for any $c\ge 1$ and $q\ge 0$.

Lemma 2

([26]). Let $\omega =\left\{\omega \right(n\left)\right\}$ denote a sequence consisting of positive numbers, with $G_{\omega }\left(t\right)$ representing its generating function. If ω meets the criteria specified in conditions (W1), (W2), and (W4), it can be stated that

$$G_{\omega }\left(p\right)G_{\omega }\left(q\right)\le G_{\omega }\left(C_{1\omega }(p+q)\right),p,q\ge 0.$$

If conditions (W1), (W2), and (W5) are fulfilled, then

$$G_{\omega }\left(C_{1\omega }(p+q)\right)\le G_{\omega }\left(C_{2\omega }p\right)G_{\omega }\left(C_{2\omega }q\right),p,q\ge 0.$$

Next, we proceed to construct the space of vector-valued white Gaussian noise functionals. Let us introduce a complex Hilbert space, designated as the initial Hilbert space, and denoted by K, with norm ${|·|}_0$. Let B be a densely defined self-adjoint operator on K with condition $infSpec\left(B\right)>1$. It is essential that the operator B does not have to be unbounded. The triple $\mathcal{N}\subset K\subset {\mathcal{N}}^{*}$ is derived from $(K,B)$ by the same way of constructing the Gel’fand triple $S\subset H\subset S^{*}$.

By referencing Proposition 1.3.8 in [17], the following conclusions can be drawn:

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\omega }\otimes \mathcal{N}& \cong \underset{p\to \infty ,q\to \infty }{\mathrm{proj}\lim }{\Gamma }_{\omega }\left(S_p\right)\otimes {\mathcal{N}}_q\cong \underset{p\to \infty }{\mathrm{proj}\lim }{\Gamma }_{\omega }\left(S_p\right)\otimes {\mathcal{N}}_p,\hfill \end{array}$$

(1)

and then by the kernel theorem, we perceive that

$${\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}\cong {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}\cong \mathcal{L}\left({\mathcal{W}}_{\omega },{\mathcal{N}}^{*}\right);$$

(2)

for more details, see Theorem 1.3.10 in [29]. Consequently, it is clear that

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}& \cong \underset{p\to \infty ,q\to \infty }{\mathrm{ind}\lim }{\Gamma }_{{\omega }^{-1}}\left(S_{-p}\right)\otimes {\mathcal{N}}_{-q}\cong \underset{p\to \infty }{\mathrm{ind}\lim }{\Gamma }_{{\omega }^{-1}}\left(E_{-p}\right)\otimes {\mathcal{N}}_{-p}.\hfill \end{array}$$

Next, we construct a specialized structured system for the vector-valued Gaussian white noise functional, which is embedded within a Gel’fand triple that specifically deals with the vector-valued Gaussian white noise functional:

$${\mathcal{W}}_{\omega }\otimes \mathcal{N}\subset \Gamma \left(H\right)\otimes K\subset {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}\cong {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}.$$

(3)

Remark 1.

In [30], according to Proposition 1.3.7, it follows that if $B^{-1}$ is of Hilbert–Schmidt type, then the space ${\mathcal{W}}_{\omega }\otimes \mathcal{N}$ is nuclear. Therefore, it follows that the triple defined in (3) constitutes a Gel’fand triple. As a trivial case, if the operator B is the identity operator, then $\mathcal{N}=K={\mathcal{N}}^{*}$. The elements within the space ${\mathcal{W}}_{\omega }\otimes K$ constitute the Hilbert space K-valued test white noise functionals.

From (1), the notations previously employed for scalar-valued functionals are naturally extended to encompass those for vector-valued functionals. The canonical bilinear form associated with ${({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}\times {\mathcal{W}}_{\omega }\otimes \mathcal{N}$ is symbolized by $〈〈·,·〉〉$. The norms ${∥·∥}_{\omega ,p,q}$ on ${\mathcal{W}}_{\omega }\otimes \mathcal{N}$ are expressed as

$${∥\varphi ∥}_{\omega ,p,q}^2:=\sum _{n=0}^{\infty }n!\omega \left(n\right){\left|{\left(A^{\otimes n}\right)}^p\otimes B^q)f_n\right|}_0^2={∥\Gamma {\left(A\right)}^p\otimes B^q)\varphi ∥}_{\omega ,0,0}^2$$

for $\varphi ={\left(f_n\right)}_{n=0}^{\infty }\in {\mathcal{W}}_{\omega }\otimes \mathcal{N}$ with $f_n\in S^{\widehat{\otimes }n}\otimes \mathcal{N}$ ($n=0,1,2,\cdots$). Here, ${∥·∥}_{\omega ,0,0}$ denotes the Hilbertian norm within ${\Gamma }_{\omega }\left(H\right)\otimes K$. If $p=q$, then the norm ${∥·∥}_{\omega ,p,q}$ is simplified to ${∥·∥}_{\omega ,p}$.

Given any $\Phi \in {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}\cong {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ and every $\xi \in S$, the S-transform of $\Phi$ is an ${\mathcal{N}}^{-}$valued function defined on S as follows:

$$〈S\Phi \left(\xi \right),\zeta 〉=〈〈\Phi ,{\varphi }_{\xi }\otimes \zeta 〉〉,\zeta \in \mathcal{N},\xi \in S,$$

where $〈·,·〉$ denotes the canonical bilinear form on ${\mathcal{N}}^{*}\times \mathcal{N}$.

Theorem 1.

Suppose that $\gamma :S\to {\mathcal{N}}^{*}$ is a ${\mathcal{N}}^{*}$-valued function. Then, the function γ on S is the S-transform of some $\Phi \in {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$, i.e., $\gamma =S\Phi \left(\xi \right),\xi \in S$, if and only if the conditions (S1) and (S2) are satisfied (see [30]):

(S1) 

(Analytic condition) for any fixed $\eta ,\xi \in S$ and $\zeta \in \mathcal{N}$, the function

$$\mathbb{C}\ni z\mapsto 〈\gamma (\xi +z\eta ),\zeta 〉\in \mathbb{C}$$

is entire holomorphic;

(S2) 

(Growth condition) there exist constants $p,q\ge 0$ and $K\ge 0$ such that

$$|〈\gamma \left(\xi \right),\zeta 〉|\le K{\left|\zeta \right|}_q{G}_{\omega }\left({\left|\xi \right|}_p^2\right),\xi \in S,\zeta \in \mathcal{N}.$$

Theorem 2

([12,31,32]). Suppose that $\gamma :S\to {\mathcal{N}}^{*}$ is an ${\mathcal{N}}^{*}$-valued function. Then, the ${\mathcal{N}}^{*}$-valued function γ is the S-transform of some $\Phi \in {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$ if and only if γ satisfies conditions (S1) and (S2) in Theorem 1. For any $\tau >0$, if the condition ${∥A^{-\tau }∥}_{\mathrm{HS}}<R_{\omega }$ holds, then

$${∥\Phi ∥}_{-(s+\tau )}\le C{\left|\zeta \right|}_q\tilde{G}\left({∥A^{-\tau }∥}_{\mathrm{HS}}^2\right),s\ge 0,$$

(4)

with positive radius $R_{\omega }$ satisfying the convergence of the series

$${\displaystyle {\tilde{G}}_{\omega }\left(t\right)=\sum _{n=0}^{\infty }\frac{n^{2n}}{n!\omega \left(n\right)}\left(\underset{\tau >0}{inf}\frac{G_{\omega }\left(\tau \right)}{{\tau }^n}\right)t^n.}$$

Theorem 3

([25]). Let $\left\{{\Phi }_n\right\}$ be a sequence in ${({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$ and ${\gamma }_n\left(\xi \right)=S{\Phi }_n\left(\xi \right),$ and $\xi \in S$ is the S-transform of ${\Phi }_n,n=1,2,\cdots$. Then, the sequence ${\Phi }_n$ strongly converges to some $\Phi \in {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$ if and only if the conditions (D1) and (D2) are satisfied:

(D1) 

The limit of $〈{\gamma }_n\left(\xi \right),\zeta 〉$ exists as $n\to \infty$ in $\mathbb{C}$, for each $\xi \in S,\zeta \in \mathcal{N}$;

(D2) 

For any $\xi \in S,\zeta \in \mathcal{N}$, there exist constants $p,q\ge 0$ and $K\ge 0$, such that

$$|{\gamma }_n\left(\xi \right)| \le K{\left|\zeta \right|}_q{G}_{\omega }\left({\left|\xi \right|}_p^2\right),n\in \mathbb{N}.$$

Theorem 4.

Suppose that the time interval set T is a locally compact space. Then, a function ${\Phi }_t:T\to {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$ is continuous if and only if for any $\tau \in T$, there exist an open neighborhood $U_0$ of τ and constants $q,p\ge 0$, $K\ge 0$, such that

$$|〈S{\Phi }_t\left(\xi \right),\zeta 〉| \le C{\left|\zeta \right|}_q{G}_{\omega }\left({\left|\xi \right|}_p^2\right),\xi \in S,\zeta \in \mathcal{N},t\in U_0$$

and

$$\underset{t\to \tau }{lim}S{\Phi }_t\left(\xi \right)=S{\Phi }_{\tau }\left(\xi \right),\xi \in S.$$

The space of all locally integrable $\mathbb{C}$-valued functions on $\mathbb{R}$ is denoted by $L_{\mathrm{loc}}^1\left(\mathbb{R}\right)$.

Theorem 5

([25]). Suppose that ${\left\{{\Phi }_t\right\}}_{t\in \mathbb{R}}$ is a stochastic process in the space ${\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$. Then, for any function $f\in L_{\mathrm{loc}}^1\left(\mathbb{R}\right)$ and real numbers $t_0,t\in \mathbb{R}$, there exists a unique white noise functional ${\Psi }_{t_0,t}\in {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$ holding the relation

$$〈S{\Psi }_{t_0,t}\left(\xi \right),\zeta 〉={\int }_{t_0}^{t}f\left(s\right)〈S{\Phi }_s\left(\xi \right),\zeta 〉ds,\zeta \in \mathcal{N},\xi \in S.$$

(5)

Moreover, the function ${\Psi }_{t_0,t}:T\to {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$ is continuous.

We can write

$${\Psi }_{t_0,t}={\int }_{t_0}^{t}f\left(s\right){\Phi }_{s}ds,$$

where the white noise functional ${\Psi }_{t_0,t}$ is given by (5).

Theorem 6

([25]). Let $\left\{{\Phi }_t\right\}\subset {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$ and $\left\{{\Psi }_t\right\}\subset {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$ be two stochastic processes satisfying the relation

$${\Psi }_t={\int }_{t_0}^t{\Phi }_{s}ds,t\in \mathbb{R}.$$

Then, the function ${\Psi }_t:T\to {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$ is differentiable, and the relation

$${\displaystyle \frac{d}{dt}}{\Psi }_t={\Phi }_t$$

holds in the space ${\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$.

3. Fixed-Point Theorems of $\mathit{S}$-Contractive Mappings

Since ${\varphi }_{\xi }\otimes \zeta ,\xi \in S,\zeta \in \mathcal{N}$ spans a dense subspace of ${\mathcal{W}}_{\omega }\otimes \mathcal{N}$, the S-transform determines a white noise distribution uniquely [32]. For any $\Phi$ in the space ${\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$, which is a locally convex space, by applying $〈S\Phi \left(\xi \right),\zeta 〉$, $\Phi$ is transformed to a scalar value in complex space. Technically, by using the S-transform and the modulus of complex numbers, we propose S-contractive mappings as follows:

Definition 1.

Let A be an operator from ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ to ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, i.e., $A:{\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}\to {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$. For any vectors $\Phi ,\Psi \in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, if there exists a constant k, $0<k<1$, such that the following inequality

$$|〈S(A\Phi )\left(\xi \right),\zeta 〉-〈S(A\Psi )\left(\xi \right),\zeta 〉|\le k|〈S\Phi \left(\xi \right),\zeta 〉-〈S\Psi \left(\xi \right),\zeta 〉|,\xi \in S,\zeta \in \mathcal{N}$$

holds, then A is called an S-contractive mapping on ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ for vectors.

Theorem 7.

Suppose the mapping A is an S-contractive on ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ for vectors. Then, A has a unique fixed point in ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$.

Proof. 

For any vector ${\Phi }_0\in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, let

$${\Phi }_1=A{\Phi }_0,{\Phi }_2=A{\Phi }_1,\cdots ,{\Phi }_n=A{\Phi }_{n-1},\cdots ,n=1,2,\cdots .$$

Since A is an S-contractive mapping on ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ for vectors, then

$$\begin{array}{cc}\hfill |〈S{\Phi }_n\left(\xi \right),\zeta 〉-〈S{\Phi }_{n-1}\left(\xi \right),\zeta 〉|& = |〈S\left(A{\Phi }_{n-1}\right)\left(\xi \right),\zeta 〉-〈S\left(A{\Phi }_{n-2}\right)\left(\xi \right),\zeta 〉|\hfill \\ \hfill & \le k|〈S{\Phi }_{n-1}\left(\xi \right),\zeta 〉-〈S{\Phi }_{n-2}\left(\xi \right),\zeta 〉|\hfill \\ \hfill & = |〈S\left(A{\Phi }_{n-2}\right)\left(\xi \right),\zeta 〉-〈S\left(A{\Phi }_{n-3}\right)\left(\xi \right),\zeta 〉|\hfill \\ \hfill & \le k^2|〈S{\Phi }_{n-2}\left(\xi \right),\zeta 〉-〈S{\Phi }_{n-3}\left(\xi \right),\zeta 〉|\hfill \\ \hfill & =\cdots \hfill \\ \hfill & \le k^{n-1}|〈S{\Phi }_1\left(\xi \right),\zeta 〉-〈S{\Phi }_0\left(\xi \right),\zeta 〉|.\hfill \end{array}$$

When $n\to \infty$ and $k^{n-1}\to 0$, then

$$|〈S{\Phi }_n\left(\xi \right),\zeta 〉-〈S{\Phi }_{n-1}\left(\xi \right),\zeta 〉|=|〈S({\Phi }_n-{\Phi }_{n-1})\left(\xi \right),\zeta 〉|\to 0.$$

This implies $\Phi =A\Phi$; i.e., the mapping A has a fixed point in ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$. Next, we prove the fixed point of $\left\{{\Phi }_n\right\}$ is unique. If there exist two fixed points $\Phi \in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ and ${\Phi }^{\prime }\in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, then $\Phi \ne {\Phi }^{\prime }$. We have

$$\begin{array}{cc}\hfill |〈S\Phi \left(\xi \right),\zeta 〉-〈S{\Phi }^{\prime }\left(\xi \right),\zeta 〉|& =|〈SA\Phi \left(\xi \right),\zeta 〉-〈SA{\Phi }^{\prime }\left(\xi \right),\zeta 〉|\hfill \\ \hfill & \le k|〈S\Phi \left(\xi \right),\zeta 〉-〈S{\Phi }^{\prime }\left(\xi \right),\zeta 〉|.\hfill \end{array}$$

Since $k\in \left(0,1\right)$, it leads to a contradiction, so $\Phi ={\Phi }^{\prime }$; i.e., the mapping A has a unique fixed point $\Phi$. □

Let two weight sequences $\omega =\left\{\omega \right(n\left)\right\}$ and $\nu =\left\{\nu \right(n\left)\right\}$ satisfy conditions (W1)–(W5). And their relation is fulfilled with the following equation

$$exp\left(ϵ\{G_{\nu }\left(t\right)-1\}\right)=G_{\omega }\left(t\right),$$

(6)

where $ϵ>0$ is a constant. It follows that the following continuous inclusions

$${\mathcal{W}}_{\omega }\otimes \mathcal{N}\subset {\mathcal{W}}_{\nu }\otimes \mathcal{N}\subset \Gamma \left(H\right)\otimes K\subset {({\mathcal{W}}_{\nu }\otimes \mathcal{N})}^{*}\subset {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$$

hold. There is an example as follows for weight sequences $\omega =\left\{\omega \right(n\left)\right\}$ and $\nu =\left\{\nu \right(n\left)\right\}$. It is well known that the m-th order Bell numbers $\left\{B_m\left(n\right)\right\}$ are denoted by

$$\begin{array}{cc}\hfill G_{\mathrm{Bell}\left(m\right)}\left(s\right)& =\stackrel{m-\mathrm{times}}{\overbrace{{\displaystyle \frac{exp(exp(\cdots (exps)\cdots \left)\right)}{exp(exp(\cdots (exp0)\cdots \left)\right)}}}}\hfill \\ & =\sum _{n=0}^{\infty }{\displaystyle \frac{B_m\left(n\right)}{n!}}{s}^n.\hfill \end{array}$$

which are of the form of the exponential generating function for weight sequences; i.e., we can consider Bell numbers $\left\{B_m\left(n\right)\right\}$ as a weight sequence. In fact, we have a concise recursive relation which is of the form of relation (6),

$$G_{\mathrm{Bell}\left(m\right)}\left(s\right)=\left\{\begin{array}{c}exp{\gamma }_m\{G_{\mathrm{Bell}(m-1)}\left(s\right)-1\},m>1\hfill \\ e^s,m=1\hfill \end{array}\right.$$

(7)

where ${\gamma }_{m+1}=exp{\gamma }_m$ for $m\ge 1$ and ${\gamma }_1=1$.

Naturally, by the idea of Definition 7, we have the following definition:

Definition 2.

Let A be an operator from ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ to ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, i.e., $A:{\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}\to {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$. For any processes ${\Phi }_t,{\Psi }_t\in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, if there exists a constant $K\in \left(0,1\right)$, such that the following inequality

$$\left|〈S\left(A{\Phi }_t\right)\left(\xi \right),\zeta 〉-〈S\left(A{\Psi }_t\right)\left(\xi \right),\zeta 〉\right|\le K\left|〈S{\Phi }_t\left(\xi \right),\zeta 〉-〈S{\Psi }_t\left(\xi \right),\zeta 〉\right|$$

(8)

holds, for any $\xi \in S,\zeta \in \mathcal{N}$, and $t\in [0,T]$, A is called an S-contractive mapping on ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ for processes.

Similarly, we have a fixed-point theorem of the S-contractive mapping, which is defined in Definition 2.

Theorem 8.

Suppose the mapping A is an S-contractive mapping on ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ for processes. If inequality (8) holds, then A has a unique fixed point in ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$.

Proof. 

For any process ${\Phi }_t\in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, let

$${\Phi }_t^{\left(1\right)}=A{\Phi }_t^{\left(0\right)},{\Phi }_t^{\left(2\right)}=A{\Phi }_t^{\left(1\right)},\cdots ,{\Phi }_t^{\left(n\right)}=A{\Phi }_t^{(n-1)},\cdots ,n=1,2,\cdots .$$

For any given $t\in [0,T]$,

$$\begin{array}{cc}\hfill \left|〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-1)}\left(\xi \right),\zeta 〉\right|& =\left|〈S\left(A{\Phi }_t^{(n-1)}\right)\left(\xi \right),\zeta 〉-〈S\left(A{\Phi }_t^{(n-2)}\right)\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & \le K\left|〈S{\Phi }_t^{(n-1)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-2)}\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & =\left|〈S\left(A{\Phi }_t^{(n-2)}\right)\left(\xi \right),\zeta 〉-〈S\left(A{\Phi }_t^{(n-3)}\right)\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & \le K^2\left|〈S{\Phi }_t^{(n-2)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-3)}\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & =\cdots \hfill \\ \hfill & \le K^{(n-1)}\left|〈S{\Phi }_t^{\left(1\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{\left(0\right)}\left(\xi \right),\zeta 〉\right|.\hfill \end{array}$$

When $n\to \infty$ and $K^{n-1}\to 0$,

$$\left|〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-1)}\left(\xi \right),\zeta 〉\right|=\left|〈S({\Phi }_t^{\left(n\right)}-{\Phi }_t^{(n-1)})\left(\xi \right),\zeta 〉\right|\to 0.$$

This implies $\Phi =A\Phi$; i.e., the mapping A has a fixed point in ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$. Next, we prove the fixed point of $\left\{{\Phi }_n\right\}$ is unique. If there exist two fixed points $\Phi \in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ and ${\Phi }^{\prime }\in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, then $\Phi \ne {\Phi }^{\prime }$. We have

$$\begin{array}{cc}\hfill \left|〈S\Phi \left(\xi \right),\zeta 〉-〈S{\Phi }^{\prime }\left(\xi \right),\zeta 〉\right|& =\left|〈SA\Phi \left(\xi \right),\zeta 〉-〈SA{\Phi }^{\prime }\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & \le K\left|〈S\Phi \left(\xi \right),\zeta 〉-〈S{\Phi }^{\prime }\left(\xi \right),\zeta 〉\right|.\hfill \end{array}$$

Since $K\in \left(0,1\right)$, it leads to a contradiction, so $\Phi ={\Phi }^{\prime }$; i.e., the mapping A has a unique fixed point $\Phi$. □

Notice that, in Definition 8, for any $t\in [0,T]$, there is a $K>0$ such that the inequality in (8) holds. It is a very strong condition. Moreover, in white noise analysis, $G_{\omega }\left({\left|\xi \right|}_p^2\right),p\ge 0$ is an important coefficient. For example, for any ${\Phi }_1,{\Phi }_2,$ and $\Psi \in {\mathcal{W}}_{\omega }^{*}$, there exists some positive number p, such that

$$\begin{array}{cc}\hfill S(\Psi \diamond ({\Phi }_1-{\Phi }_2))\left(\xi \right)& =S\Psi \left(\xi \right)S({\Phi }_1-{\Phi }_2))\left(\xi \right)\hfill \\ \hfill & \le G_{\omega }\left({\left|\xi \right|}_p^2\right)S({\Phi }_1-{\Phi }_2))\left(\xi \right),\xi \in S\hfill \end{array}$$

holds, where ⋄ is the Wick product. Based on the above analysis, to achieve broader applications in white noise theory, we propose a generalized definition of S-contractive mapping for processes, and prove a fixed-point theorem for S-contractive mapping for processes. Moreover, we give two examples on differential equations of white noise functionals as applications.

Definition 3.

Let two weight sequences $\omega =\left\{\omega \right(n\left)\right\}$ and $\nu =\left\{\nu \right(n\left)\right\}$ satisfy conditions (W1)–(W5). And their generating functions satisfy the relation in (6). Let A be an operator from ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ to ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, i.e., $A:{\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}\to {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$. For any processes ${\Phi }_t,{\Psi }_t\in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, if there exists a constant $p\ge 0$ and a non-negative function $k\in L^1[0,T]$, such that the following inequality

$$\left|〈S\left(A{\Phi }_t\right)\left(\xi \right),\zeta 〉-〈S\left(A{\Psi }_t\right)\left(\xi \right),\zeta 〉\right|\le G_{\nu }\left({\left|\xi \right|}_p^2\right){\int }_0^{t}k\left(s\right)\left|〈S{\Phi }_s\left(\xi \right),\zeta 〉-〈S{\Psi }_s\left(\xi \right),\zeta 〉\right|ds$$

(9)

holds, for any $\xi \in S,\zeta \in {\mathcal{N}}^{*}$, and $s,t\in [0,T]$, A is called a generalized S-contractive mapping on ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ for processes.

Theorem 9.

Let two weight sequences $\omega =\left\{\omega \right(n\left)\right\}$ and $\nu =\left\{\nu \right(n\left)\right\}$ satisfy conditions (W1)–(W5). And their generating functions are fulfilled with the relation in (6). Let A be a generalized S-contractive mapping on ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ for processes satisfying the inequality in (9). For any initial value ${\Phi }_0$ in ${\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*}$, we define a sequence $\left\{{\Phi }_t^{\left(n\right)}\right\}$ as follows:

$$\begin{array}{cc}\hfill {\Phi }_t^{\left(0\right)}& ={\Phi }_0\in {\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*},\hfill \\ \hfill {\Phi }_t^{\left(n\right)}& =A{\Phi }_t^{(n-1)},n\ge 1.\hfill \end{array}$$

If for any process ${\Phi }_t\in {\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*}$, there exist constants $q,p\ge 0$, and $K\ge 0$ such that $|〈S\left(A{\Phi }_t\right)\left(\xi \right),\zeta 〉|\le K{\left|\zeta \right|}_q{G}_{\nu }{\left(\right|\xi |}_p^2)$, then the sequence $\left\{{\Phi }_t^{\left(n\right)}\right\}$ converges to the unique fixed point of A in ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$.

Proof. 

Step1 

Firstly, we prove that ${\Phi }_t^{\left(n\right)}\in {\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*}$ for $n=1,2,\cdots$

Since ${\Phi }_t\in {\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*}$, there exist constants $q,p\ge 0$, and $K\ge 0$, and the equation

$$\left|〈S{\Phi }_t^{\left(1\right)}\left(\xi \right),\zeta 〉\right|=|S\left(A{\Phi }_t^{\left(0\right)}\right)\left(\xi \right)|\le K{\left|\zeta \right|}_q{G}_{\nu }{\left(\right|\xi |}_p^2)$$

holds. It implies that ${\Phi }_t^{\left(1\right)}\in {\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*}$. Similarly, we can conclude ${\Phi }_t^{\left(n\right)}\in {\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*}$, $n\ge 2$. Then, ${\Phi }_t^{\left(n\right)}\in {\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*}$, $n=0,1,2,\cdots$.

Step2 

We prove that $U_t(\zeta ,\xi )=〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉,$ and $\xi \in S,\zeta \in \mathcal{N}$ converges uniformly in t as n tends to infinity. By the condition in (8), for any $\xi \in S,\zeta \in \mathcal{N}$, we have

$$\begin{array}{cc}\hfill & \left|〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-1)}\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & =\left|〈S\left(A{\Phi }_t^{(n-1)}\right)\left(\xi \right),\zeta 〉-〈S\left(A{\Phi }_t^{(n-2)}\right)\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & \le G_{\nu }\left({\left|\xi \right|}_p^2\right){\int }_0^{t}k\left(t_1\right)\left|〈S{\Phi }_{t_1}^{(n-1)}\left(\xi \right),\zeta 〉-〈S{\Phi }_{t_1}^{(n-2)}\left(\xi \right),\zeta 〉\right|d{t}_1\hfill \\ \hfill & =G_{\nu }\left({\left|\xi \right|}_p^2\right){\int }_0^{t}k\left(t_1\right)\left|〈S\left(A{\Phi }_{t_1}^{(n-2)}\right)\left(\xi \right),\zeta 〉-〈S\left(A{\Phi }_{t_1}^{(n-3)}\right)\left(\xi \right),\zeta 〉\right|d{t}_1\hfill \\ \hfill & \le {\left(G_{\nu }\left({\left|\xi \right|}_p^2\right)\right)}^2{\int }_0^t{\int }_0^{t_1}k\left(t_1\right)k\left(t_2\right)\hfill \\ \hfill & \times \left|〈S\left({\Phi }_{t_2}^{(n-2)}\right)\left(\xi \right),\zeta 〉-〈S\left({\Phi }_{t_2}^{(n-3)}\right)\left(\xi \right),\zeta 〉\right|d{t}_{1}d{t}_2\hfill \\ \hfill & =\cdots \hfill \\ \hfill & \le {\left(G_{\nu }\left({\left|\xi \right|}_p^2\right)\right)}^{n-1}{\int }_0^t\cdots {\int }_0^{t_{n-2}}k\left(t_1\right)\cdots k\left(t_{n-1}\right)\hfill \\ \hfill & \times \left|〈S{\Phi }_{t_{n-1}}^{\left(1\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_{t_{n-1}}^{\left(0\right)}\left(\xi \right),\zeta 〉\right|d{t}_1\cdots d{t}_{n-1}.\hfill \end{array}$$

Let ${\int }_0^{T}k\left(t\right)dt=\overline{K}$, since $t\in [0,T]$. Then, the iterated integral

$${\int }_0^t\cdots {\int }_0^{t_{n-2}}k\left(t_1\right)\cdots k\left(t_{n-1}\right)d{t}_1\cdots d{t}_{n-1}\le {\displaystyle \frac{1}{(n-1)!}}{\left(\overline{K}\right)}^{n-1}.$$

For all $\xi \in S$ and $\zeta \in \mathcal{N}$, there exist positive constants $p_0,p_1,q_0,q_1,K_0,$ and $K_1$, such that

$$\begin{array}{cc}\hfill \left|〈S{\Phi }_t^{\left(1\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{\left(0\right)}\left(\xi \right),\zeta 〉\right|& \le \left|〈S{\Phi }_t^{\left(1\right)}\left(\xi \right),\zeta 〉\right|+\left|〈S{\Phi }_t^{\left(0\right)}\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & =\left|〈S\left(A{\Phi }_t^{\left(0\right)}\right)\left(\xi \right),\zeta 〉\right|+\left|〈S{\Phi }_t^{\left(0\right)}\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & \le K_1{\left|\zeta \right|}_{q_1}{G}_{\nu }\left({\left|\xi \right|}_{p_1}^2\right)+K_0{\left|\zeta \right|}_{q_0}{G}_{\nu }\left({\left|\xi \right|}_{p_0}^2\right).\hfill \end{array}$$

Take $p^{\prime }=max\{p_0,p_1\}$ and $q^{\prime }=max\{q_0,q_1\}$; then,

$$\left|〈S{\Phi }_t^{\left(1\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{\left(0\right)}\left(\xi \right),\zeta 〉\right|\le \left(K_0+K_1\right){\left|u\right|}_{q^{\prime }}{G}_{\nu }\left({\left|\xi \right|}_{p^{\prime }}^2\right)=:H(\zeta ,\xi ).$$

Then,

$$\begin{array}{c}\hfill \left|〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-1)}\left(\xi \right),\zeta 〉\right|\le {\displaystyle \frac{1}{(n-1)!}}{\left(\overline{K}{G}_{\nu }{\left({\left|\xi \right|}_p\right)}^2\right)}^{n-1}H(\zeta ,\xi ).\end{array}$$

It implies that

$$\begin{array}{c}\hfill \sum _{n=1}^{\infty }\left|〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-1)}\left(\xi \right),\zeta 〉\right|\le exp\left(\overline{K}{G}_{\beta }\left({\left|\xi \right|}_p^2\right)\right)H(\zeta ,\xi ),\end{array}$$

which proves that

$$\begin{array}{cc}\hfill U_t(\zeta ,\xi )& =\underset{n\to \infty }{lim}〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉\hfill \\ \hfill & =\sum _{n=1}^{\infty }\left(〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-1)}\left(\xi \right),\zeta 〉\right)+〈S{\Phi }_t^{\left(0\right)}\left(\xi \right),\zeta 〉\hfill \end{array}$$

(10)

uniformly converges in t when n tends to infinity, for any fixed $\xi \in S,\zeta \in \mathcal{N}$.

Step3 

We prove that there exists a process ${\left\{{\Phi }_t\right\}}_{t\in [0,T]}\subset {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ satisfying $U_t(\zeta ,\xi )=〈S{\Phi }_t\left(\xi \right),\zeta 〉$ for $\xi \in S,\zeta \in \mathcal{N}$. By applying Morera’s theorem and the dominated convergence theorem, it is easy to check that condition (S1) of Theorem 1 is fulfilled.

Next, we check condition (S2) in Theorem 1. Using the elementary inequality $t{e}^{\lambda t}\le e^{(1+\lambda )t}$ for $t\ge 0$ and the property given as in (iii) of Lemma 1, for $\gamma >0$ and $\tilde{p}\ge max\{p^{\prime },p\}$, there exists $K^{\prime }\ge \overline{K}$ with $(1+K^{\prime })/\gamma \ge 1$ such that we obtain

$$\begin{array}{cc}\hfill G_{\nu }\left({\left|\xi \right|}_{p^{\prime }}^2\right)exp\left(\overline{K}{G}_{\nu }\left({\left|\xi \right|}_p^2\right)\right)& \le exp\left((1+\overline{K})G_{\nu }\left({\left|\xi \right|}_{\tilde{p}}^2\right)\right)\hfill \\ \hfill & \le exp\left((1+K^{\prime })G_{\nu }\left({\left|\xi \right|}_{\tilde{p}}^2\right)\right)\hfill \\ \hfill & \le e^{1+K^{\prime }}exp\left(\gamma \left\{G_{\nu }\left({\displaystyle \frac{1+K^{\prime }}{\gamma }}{\left|\xi \right|}_{\tilde{p}}^2\right)-1\right\}\right)\hfill \\ \hfill & \le e^{1+K^{\prime }}exp\left(\gamma \left\{G_{\nu }\left({\left|\xi \right|}_{\tilde{p}+r}^2\right)-1\right\}\right)\hfill \\ \hfill & =e^{1+K^{\prime }}{G}_{\omega }\left({\left|\xi \right|}_{\tilde{p}+r}^2\right),\hfill \end{array}$$

where the positive number r satisfies the relation ${\displaystyle \frac{1+K^{\prime }}{\gamma }}{∥A^{-1}∥}_{\mathrm{OP}}^{2r}\le 1$. It follows from the fact that $G_{\nu }\left(s\right)\le {\gamma }^{-1}{e}^{\gamma }{G}_{\omega }\left(s\right)$, for all $s\ge 0$, that for some $\tilde{p}\ge max(p_0,p,p^{\prime })$,

$$\begin{array}{cc}\hfill \left|〈U_t\left(\xi \right),\zeta 〉\right|& =\left|\sum _{n=1}^{\infty }\left(〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-1)}\left(\xi \right),\zeta 〉\right)+〈S{\Phi }_t^{\left(0\right)}\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & \le \left|〈S{\Phi }_t^{\left(0\right)}\left(\xi \right),\zeta 〉\right|+\sum _{n=1}^{\infty }\left|〈S{\Phi }_t^{\left(n\right)}\left(\xi \right),\zeta 〉-〈S{\Phi }_t^{(n-1)}\left(\xi \right),\zeta 〉\right|\hfill \\ \hfill & \le K_0{\left|\zeta \right|}_q{G}_{\nu }\left({\left|\xi \right|}_{p_0}^2\right)+(K_0+K_1){\left|\zeta \right|}_q{G}_{\nu }\left({\left|\xi \right|}_{p^{\prime }}^2\right)exp\left(\overline{K}{G}_{\nu }\left({\left|\xi \right|}_p^2\right)\right)\hfill \\ \hfill & \le K_0{\left|\zeta \right|}_q{G}_{\nu }\left({\left|\xi \right|}_{p_0}^2\right)+(K_0+K_1){\left|\zeta \right|}_q{G}_{\omega }\left({\left|\xi \right|}_{\tilde{p}+r}^2\right)\hfill \\ \hfill & \le \left[{\gamma }^{-1}{e}^{\gamma }{K}_0+(K_0+K_1)\right]{\left|\zeta \right|}_q{G}_{\omega }\left({\left|\xi \right|}_{\tilde{p}+r}^2\right),\hfill \end{array}$$

where the positive number r satisfies the relation ${\displaystyle \frac{1+K^{\prime }}{\gamma }}{∥A^{-1}∥}_{OP}^2\le 1.$ It follows that the growth condition (S2) in Theorem 1 is fulfilled. Then, for $t\in [0,T]$, by Theorem 1, it follows that there exists a process ${\Phi }_t\in {\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$, such that $〈S{\Phi }_t\left(\xi \right),\zeta 〉=U_t(\zeta ,\xi )$.

Step4 

And then, we prove the uniqueness of the convergence. Suppose there exist two fixed points ${\Phi }_t$ and ${\Psi }_t$ of the operator A satisfying $〈S{\Phi }_t\left(\xi \right),\zeta 〉=U_t(\zeta ,\xi )$ and $〈S{\Psi }_t\left(\xi \right),\zeta 〉=U_t(\zeta ,\xi )$. Then, it holds that

$$\left|〈S{\Phi }_t\left(\xi \right),\zeta 〉-〈S{\Psi }_t\left(\xi \right),\zeta 〉\right|\le G_{\nu }\left({\left|\xi \right|}_p^2\right){\int }_0^{t}k\left(s\right)\left|〈S{\Phi }_s\left(\xi \right),\zeta 〉-〈S{\Psi }_s\left(\xi \right),\zeta 〉\right|ds.$$

Then, by the Grownwall inequality, $〈S{\Phi }_t\left(\xi \right),\zeta 〉=〈S{\Psi }_t\left(\xi \right),\zeta 〉$ holds, which implies ${\Phi }_t={\Psi }_t$.

It follows from the above steps that the sequence $\left\{{\Phi }_t^{\left(n\right)}\right\}$ converges to the unique fixed point of A in ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$.

□

As an application, we consider a general form of white noise differential problems as follows:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d\Phi \left(t\right)}{dt}}& =F(t,\Phi (t\left)\right),\hfill \\ \hfill \Phi \left(0\right)& ={\Phi }_0,\hfill \end{array}\right.$$

(11)

where F is a continuous function from $[0,T]\times {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$ to ${({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$, and ${\Phi }_0$ is an initial value. It follows from Theorem 6 that Equation (11) is equivalent to the following integral equation:

$$\Phi \left(t\right)={\Phi }_0+{\int }_0^{t}F(s,\Phi \left(s\right))ds.$$

(12)

Let us study the existence and uniqueness of the solution of Equation (11).

Theorem 10.

Let $\omega =\left\{\omega \right(n\left)\right\}$ and $\nu =\left\{\nu \right(n\left)\right\}$ be two weight sequences satisfying conditions (W1)–(W5) such that their generating functions satisfy relation (6). Let $F:[0,T]\times {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}\to {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$ be a continuous function. If for some $p\ge 0$, the conditions

$$\left|〈SF(s,\Phi )\left(\xi \right),\zeta 〉-〈SF(s,\Psi )\left(\xi \right),\zeta 〉\right|\le K\left(s\right)G_{\nu }\left({\left|\xi \right|}_p^2\right)\left|〈S\Phi \left(\xi \right),\zeta 〉-〈S\Psi \left(\xi \right),\zeta 〉\right|$$

(13)

and

$$\left|〈SF(s,\Phi )\left(\xi \right),\zeta 〉\right|\le K\left(s\right)\left|〈S\Phi \left(\xi \right),\zeta 〉\right|$$

(14)

hold, for any $\Phi ,\Psi \in {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$, $\xi \in S,\zeta \in \mathcal{N}$, and $s\in [0,T]$, where $K\in L^1[0,T]$ is a non-negative function; then, for any ${\Phi }_0\in {\left({\mathcal{W}}_{\nu }\otimes \mathcal{N}\right)}^{*}$, Equation (11) has a unique solution in ${\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$.

Proof. 

We define an operator A as follows:

$$A{\Phi }_t={\Phi }_0+{\int }_0^{t}F(s,{\Phi }_s)ds.$$

For any $\Phi ,\Psi \in {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$, $\xi \in S,\zeta \in \mathcal{N}$, and $s\in [0,T]$, by condition (13), there exists some $p\ge 0$, such that

$$\begin{array}{cc}\hfill \left|〈S(A\Phi )\left(\xi \right),\zeta 〉-〈S(A\Psi )\left(\xi \right),\zeta 〉\right|& ={\int }_0^t\left|〈SF(s,\Phi )\left(\xi \right),\zeta 〉-〈SF(s,\Psi )\left(\xi \right),\zeta 〉\right|ds\hfill \\ \hfill & \le {\int }_0^{t}K\left(s\right)G_{\nu }\left({\left|\xi \right|}_p^2\right)\left|〈S\Phi \left(\xi \right),\zeta 〉-〈S\Psi \left(\xi \right),\zeta 〉\right|ds\hfill \\ \hfill & =G_{\nu }\left({\left|\xi \right|}_p^2\right){\int }_0^{t}K\left(s\right)\left|〈S\Phi \left(\xi \right),\zeta 〉-〈S\Psi \left(\xi \right),\zeta 〉\right|ds\hfill \end{array}$$

holds. It is clear that the operator A is a generalized S-contractive mapping on ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ for processes.

For some constants $p_0,q_0,p^{\prime },q^{\prime },K_0,K^{\prime }\ge 0$, by condition (14), we have

$$\begin{array}{cc}\hfill \left|〈SA{\Phi }_t\left(\xi \right),\zeta 〉\right|& =\left|〈S{\Phi }_0\left(\xi \right),\zeta 〉+{\int }_0^t〈SF(s,{\Phi }_s)\left(\xi \right),\zeta 〉ds\right|\hfill \\ \hfill & \le \left|〈S{\Phi }_0\left(\xi \right),\zeta 〉\right|+{\int }_0^t\left|〈SF(s,{\Phi }_s)\left(\xi \right),\zeta 〉\right|ds\hfill \\ \hfill & \le \left|〈S{\Phi }_0\left(\xi \right),\zeta 〉\right|+{\int }_0^{t}K\left(s\right)\left|〈S{\Phi }_s\left(\xi \right),\zeta 〉\right|ds\hfill \\ \hfill & \le K_0{\left|\zeta \right|}_{q_0}{G}_{\nu }\left({\left|\xi \right|}_{p_0}^2\right)+{\int }_0^{t}K\left(s\right)K^{\prime }{\left|\zeta \right|}_{q^{\prime }}{G}_{\nu }\left({\left|\xi \right|}_{p^{\prime }}^2\right)ds\hfill \\ \hfill & \le (K_0+\overline{K}{K}^{\prime }){\left|\zeta \right|}_q{G}_{\nu }\left({\left|\xi \right|}_p^2\right),\hfill \end{array}$$

where $p\ge max\{p_0,p^{\prime }\}$, $q\ge max\{q_0,q^{\prime }\}$, and $\overline{K}={\int }_0^{T}K\left(s\right)ds$.

We construct the sequence $\left\{{\Phi }_t^{\left(n\right)}\right\}$ as follows:

$$\begin{array}{cc}\hfill {\Phi }_t^{\left(0\right)}& ={\Phi }_0\in {\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*},\hfill \\ \hfill {\Phi }_t^{\left(n\right)}& =A{\Phi }_t^{(n-1)},n\ge 1.\hfill \end{array}$$

It follows from Theorem 9 that $A\Phi =\Phi$ and $\Phi \in {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$; i.e., Equation (11) has a unique solution in ${\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$. □

Let us recall an operation, which is called the $\beta$-Wick product, in the paper [20]. For all $\zeta ,\theta \in {\mathcal{N}}^{*}$, we define a continuous bilinear map $\beta$ from ${\mathcal{N}}^{*}\times {\mathcal{N}}^{*}$ to ${\mathcal{N}}^{*}$ by $\beta (\zeta ,\theta ):=L(\zeta \otimes \theta )$, where the linear operator $L:{\mathcal{N}}^{*}\otimes {\mathcal{N}}^{*}\to {\mathcal{N}}^{*}$ is continuous. Then, there are constants $q\ge 0$ and $K\ge 0$, for any $p\ge 0$, such that

$${\left|L(\zeta \otimes \theta )\right|}_{-(q+p)}={\left|\beta (\zeta ,\theta )\right|}_{-(q+p)}\le K{\left|\zeta \right|}_{-q}{\left|\theta \right|}_{-q},\zeta ,\theta \in {\mathcal{N}}^{*}.$$

Then, we call $\Theta$ the $\beta$-Wick product of $\Psi$ and $\Phi$ in the space ${\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$, denoted by $\Psi {\diamond }_{\beta }\Phi$, if for any $\zeta \in \mathcal{N}$,

$$〈S\Theta \left(\xi \right),\zeta 〉=〈S(\Psi {\diamond }_{\beta }\Phi )\left(\xi \right),\zeta 〉=〈\beta (S\Psi (\xi ),S\Phi (\xi \left)\right),\zeta 〉.$$

Now, let us consider the $\beta$-Wick-type differential equation as follows:

$${\displaystyle \frac{d{\Phi }_t}{dt}}=\Psi {\diamond }_{\beta }{\Phi }_t{,\Phi |}_{t=0}={\Phi }_0,t\in [0,T].$$

In fact, this problem has been investigated in the paper [20] with a very strong condition: for any $\Phi ,\Psi \in {({\mathcal{W}}_{\nu }\otimes {\mathcal{N}}_q)}^{*}$, there exists a constant $K\ge 0$ such that

$${\left|\beta (S\Psi (\xi ),S\Phi (\xi \left)\right)\right|}_{-q}\le K{\left|S\Psi \left(\xi \right)\right|}_{-q}{\left|S\Phi \left(\xi \right)\right|}_{-q},\xi \in \mathcal{N},0\le s\le T.$$

In this paper, applying the fixed-point theorem of generalized S-contractive mappings, we investigate $\beta$-Wick-type differential equations again.

Theorem 11.

Let two weight sequences $\omega =\left\{\omega \right(n\left)\right\}$ and $\nu =\left\{\nu \right(n\left)\right\}$ satisfy conditions (W1)–(W5). And their relation is fulfilled with Equation (6). Let Ψ be in the space ${({\mathcal{W}}_{\nu }\otimes \mathcal{N})}^{*}$. Then, for any initial value ${\Phi }_0\in {({\mathcal{W}}_{\nu }\otimes \mathcal{N})}^{*}$, the β-Wick-type differential equation

$${\displaystyle \frac{d{\Phi }_t}{dt}}=\Psi {\diamond }_{\beta }{\Phi }_t{,\Phi |}_{t=0}={\Phi }_0,t\in [0,T]$$

has a unique solution in ${({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$

Proof. 

We define an operator A as follows:

$$A{\Phi }_t={\Phi }_0+{\int }_0^t\Psi {\diamond }_{\beta }{\Phi }_{s}ds,$$

where the process ${\Phi }_t\in {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$ and the vector $\Psi \in {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$, $\xi \in S,\zeta \in \mathcal{N}$, and $s\in [0,T]$.

$$\begin{array}{cc}\hfill \left|〈S\left(A{\Phi }_t\right)\left(\xi \right),\zeta 〉-〈S\left(A{\Phi }_t^{\prime }\right)\left(\xi \right),\zeta 〉\right|& ={\int }_0^t\left|〈S\Psi {\diamond }_{\beta }{\Phi }_s\left(\xi \right),\zeta 〉-〈S\Psi {\diamond }_{\beta }{\Phi }_s^{\prime }\left(\xi \right),\zeta 〉\right|ds\hfill \\ \hfill & \le G_{\nu }\left({\left|\xi \right|}_p^2\right){\int }_0^t\left|〈S{\Phi }_s\left(\xi \right),\zeta 〉-〈S{\Phi }_s^{\prime }\left(\xi \right),\zeta 〉\right|ds\hfill \end{array}$$

which implies A is a generalized S-contractive mapping on ${\mathcal{W}}_{\omega }^{*}\otimes {\mathcal{N}}^{*}$ for processes.

For some constants $p_0,q_0,p_1,p^{\prime },q^{\prime },K_0,K^{\prime },C\ge 0$, by condition (14), we have

$$\begin{array}{cc}\hfill \left|〈SA{\Phi }_t\left(\xi \right),\zeta 〉\right|& =\left|〈S{\Phi }_0\left(\xi \right),\zeta 〉+{\int }_0^t〈S(\Psi {\diamond }_{\beta }{\Phi }_s)\left(\xi \right),\zeta 〉ds\right|\hfill \\ \hfill & \le \left|〈S{\Phi }_0\left(\xi \right),\zeta 〉\right|+{\int }_0^t\left|〈S(\Psi {\diamond }_{\beta }{\Phi }_s)\left(\xi \right),\zeta 〉\right|ds\hfill \\ \hfill & \le \left|〈S{\Phi }_0\left(\xi \right),\zeta 〉\right|+G_{\nu }\left({\left|\xi \right|}_{p_1}^2\right){\int }_0^t\left|〈S{\Phi }_s\left(\xi \right),\zeta 〉\right|ds\hfill \\ \hfill & \le K_0{\left|\zeta \right|}_{q_0}{G}_{\nu }\left({\left|\xi \right|}_{p_0}^2\right)+G_{\nu }\left({\left|\xi \right|}_{p_1}^2\right){\int }_0^t{K}^{\prime }{\left|\zeta \right|}_{q^{\prime }}{G}_{\nu }\left({\left|\xi \right|}_{p^{\prime }}^2\right)ds\hfill \\ \hfill & \le (K_0+C\overline{K}{K}^{\prime }){\left|\zeta \right|}_q{G}_{\nu }\left({\left|\xi \right|}_p^2\right),\hfill \end{array}$$

where $p\ge max\{p_0,p_1+p^{\prime }\}$, $q\ge max\{q_0,q^{\prime }\}$, and $\overline{K}={\int }_0^{T}K\left(s\right)ds$. We construct the sequence $\left\{{\Phi }_t^{\left(n\right)}\right\}$ as follows:

$$\begin{array}{cc}\hfill {\Phi }_t^{\left(0\right)}& ={\Phi }_0\in {\mathcal{W}}_{\nu }^{*}\otimes {\mathcal{N}}^{*},\hfill \\ \hfill {\Phi }_t^{\left(n\right)}& =A{\Phi }_t^{(n-1)},n\ge 1.\hfill \end{array}$$

It follows from Theorem 9 that $A\Phi =\Phi$ and $\Phi \in {\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$; i.e., Equation (11) has a unique solution in ${\left({\mathcal{W}}_{\omega }\otimes \mathcal{N}\right)}^{*}$. □

4. Conclusions

In this study, we propose a new notion, which is named S-contractive mappings, and give concrete definitions of S-contractive mappings for vector-valued white noise functionals in the Gel’fand triple ${\mathcal{W}}_{\omega }\otimes \mathcal{N}\subset \Gamma \left(H\right)\otimes K\subset {({\mathcal{W}}_{\omega }\otimes \mathcal{N})}^{*}$. We prove fixed-point theorems of S-contractive mappings. This study presents an innovative approach to the investigation of differential equations of vector-valued white noise functionals. By means of the demonstration and implementation of the fixed-point theorems of S-contractive mappings, it not only establishes the existence and uniqueness of solutions to differential equations under milder conditions but also furnishes novel theoretical instruments for the white noise differential equations. While in our research, we pay more attention to discussing fixed-point theorems of S-contractive mappings, little attention has been paid to studying the properties of S-contractive mappings. Future research should aim to study the properties of S-contractive mappings to develop the theories of S-contractive mappings.