1 Introduction

Stochastic Differential Equations (SDEs) is widely used in population dynamics, physics, automation engineering, economy, biomedicine and other related fields (see [1,2,3,4,5]). Moreover, it is also considered to be the best choice to simulate the behavior of key variables in modern financial theory, such as asset return, price, instantaneous short-term interest rate and its volatility, (see [6] and [7])

We note that the future state of stochastic differential system relies not only on the current state, but also on its past historical state, which led to the emergence of Stochastic Functional Differential Equations(SFDEs). For instance, under Lipschitz condition and linear growth condition, Albeverio [8] and Hausenblas [9] established the existence and uniqueness of solution of the following SDEs with random measure:

$$\begin{aligned} dx\left( t \right) = b\left( {t,x\left( t \right) } \right) dt + \sigma \left( {t,x\left( t \right) } \right) d\omega \left( t \right) + \int _Z {c\left( {t,x\left( s \right) ,z} \right) \tilde{N}\left( {dt,dz} \right) }. \end{aligned}$$

Subsequently, Rong [10] studied the weak solution of the equation and its applications, and Applebaum [11] discussed the martingale problem of the equation.

Impulsive differential systems, as a very active research topic in recent years, have attracted more and more attention to study the system with state mutation at some time points, (see [12] and [21]). As is well known, there are some classical methods to investigate the existence and uniqueness of SDEs, such as successive approximation, Bihair inequality, Banach fixed point method and Picard approximation technique, see [14,15,16,17,18,19,20] for details. Based on the research described above, this paper considers to use advanced approximation method to prove the existence and uniqueness of mild solution.

In real systems, it usually occurs that a stochastic system jumps from a “normal state” or “good state” to a “bad state,” and the strength of system is random. In view of this, it is natural and necessary to consider the system with Poisson jump terms(see [21] and [22]). This paper will discuss we establish the existence of mild solution for the doubly perturbed impulsive neutral stochastic functional differential equations (INSFDEs) with Poisson random measure. The functional term is introduced into the system ,which not only makes the studied equation object more specific, but also provides theoretical support for the application of SFDEs in engineering. As we know, there are few researches on the theory of doubly pertured INSFDEs with jumps. The difficulty lies in how to deal with the influence of the existence of interference terms. By referring to relevant literature [23, 24], it is decided to construct a successive approximation sequence on interference terms using a Picard type iteration. Secondly, our model contains an infinitesimal generator A on a strongly continuous semigroup. The related properties of semigroup theory and the boundedness of linear operators are involved in the process of processing. Therefore, there is a little troubles in the process of Picard iteration by the boundedness of linear operators. Due to the existence of random interference term, the treatment methods of the existence, uniqueness and stability of the mild solution of this kind of SFDEs are often limited by the emergence of the project.

2 Preliminaries

Let \(\left\{ {\Omega ,\mathcal {F},P} \right\} \) be the given compact probility space with the corresponding filtration \( \left\{ \mathcal {F}_{t} \right\} _{t\geqslant 0}\) satisfying the usual conditions (i.e. right continuous and \(\mathcal {F}_{0}\) containing all P-null sets). Let K, H be two real separable Hilbert spaces and \(\pounds \left( K,H \right) \) the set of all linear bounded operators from K into H, equipped with the norm \(\left\| \cdot \right\| \). Suppose \(\left\{ p\left( t \right) :t\ge 0 \right\} \) is a \(\sigma -\)finite stationary \({\mathcal {F}_{t}}\)-adapted Poisson point process taking values in a measurable space \(\left( U,\mathcal {B}\left( U\right) \right) \). For any function \(p\left( \cdot \right) \), define \(N_{p}\left( \left( 0,t \right] \times {\Lambda } \right) =\sum _{s\in (0,t]}1_{\Lambda } \left( p\left( s \right) \right) \) for \(\Lambda \in \mathcal {B}\left( U\right) \), where \(N_{p}\) is the Poisson random measure. Moreover, we also define the measure \(\widetilde{N}\) by \(\widetilde{N}(dt,dy)=N_{p}\left( dt,dy \right) -\nu \left( dy \right) dt\), where \(\upsilon \) is the characteristic measure on \(N_{p}\). Let \(\omega =\left( \omega _{t} \right) _{t\geqslant 0}\) be a K-valued Wiener process defined on \(\left\{ \Omega ,\mathcal {F},\left\{ \mathcal {F}_{t}\right\} _{t\ge 0},P \right\} \) with covariance operator \(\mathcal {Q}\), that is

$$\begin{aligned} E\left\langle \omega \left( t \right) ,x\right\rangle _{K}\left\langle \omega \left( s \right) ,y \right\rangle _{K}=\left( t\wedge s \right) \left\langle \mathcal {Q} x,y \right\rangle _{K}, \; \text {for each }\;x,y\in K, \end{aligned}$$

where \(\mathcal {Q}\) is a positive, self-adjoint, trace class operator on K. As usual, we assume \(\omega \) is independent of the process \(\left\{ p\left( t \right) :t\ge 0 \right\} \).

Next, we introduce the concepts of infinitesimal generators and continuous semigroup. A family \(S=\left\{ S\left( t \right) :t\ge 0 \right\} \) is said to be a semigroup of class \(\mathcal {C}_{0} \), if

  1. (a)

    \(S\left( 0 \right) =I\),    (I is the identity operator);

  2. (b)

    \(S\left( t+s \right) =S\left( t \right) S\left( s \right) \), for every \( t,s\ge 0 \)    (the semigroup property);

  3. (c)

    \(t\rightarrow S\left( t \right) x\) is continuous from \(\left[ 0,\infty \right] \) into X, for each \(x\in X\)    (the \(C_{0} \) property);

In addtion, if we have

  1. (d)

    \(\left\| S\left( t \right) \right\| \le 1 \) for any \(t\ge 0\)    (contraction property),

then it is called for the contraction semigroup. Furthermore, define \(D\subset X\), such that for \(x \in D\subset X \), \(S\left( t \right) x\) is differentiable at \(t=0\) from the right, i.e.,

$$\begin{aligned} D=\left\{ x\in X \mid \lim _{h\rightarrow +0 }\frac{S\left( h \right) x-x }{h}\quad \text {exists} \right\} , \end{aligned}$$

and

$$\begin{aligned} -Ax=\lim _{h \rightarrow + 0} \frac{S\left( h \right) x-x }{h}. \end{aligned}$$

It is clear that \(-A\) is a linear operator defined on D, and it is usually called the infinitesimal generator of \(S\left( t \right) \). Then for any \(\delta \in (\frac{1}{2},1)\), define the fractional power \(\left( -A \right) ^{\delta }\), which is a closed linear operator with its subspace \(D\big ( \left( -A \right) ^{\delta }\big )\) and the subspace is dense in H. (see [25, 26] for detailed information).

For any Poisson process on a Borel set \( \mathcal {B}\left( U-\left\{ 0 \right\} \right) \), we consider the following doubly perturbed INSFDEs with Poisson jumps:

$$\begin{aligned} {\left\{ \begin{array}{ll} d\big [ x\left( t \right) +u\left( t,x\left( t \right) ,x_{t}\right) -\big ( \alpha \underset{s\in \left[ 0,t\right] }{\sup }x\left( s \right) +\beta \underset{s\in \left[ 0,t\right] }{\inf }x\left( s \right) \big ) \big ]\\ = \!Ax\left( t \right) dt\!+f\left( t,x\left( t \right) ,x_{t} \right) dt \!+g\left( t,x\left( t \right) ,x_{t} \right) d\omega \left( t \right) \! +\int _{Z}h\left( t,x\left( t \right) ,x_{t} ,y\right) \widetilde{N}(dt,dy),\\ \Delta x\left( t_{k} \right) =x\left( t_{k}^{+}\right) - x\left( t_{k}^{-}\right) =I_{k}\left( x\left( t_{k}^{-} \right) \right) ,\;k=1,2,\cdots , \\ x\left( \vartheta \right) =\phi \in C\left( \left[ -\lambda ,0 \right] ;H \right) ,\;\vartheta \in \left[ -\lambda ,0 \right] ,\qquad \qquad \; \quad a.s. ,\\ x_{0}\left( t \right) =\varphi \in PC\left( \left[ -\lambda ,0 \right] ;H\right) ,\qquad \qquad \qquad \qquad \quad \;\;\; a.s. \end{array}\right. } \end{aligned}$$
(2.1)

Let \(\lambda >0\), \(PC\equiv PC\left( \left[ -\lambda ,0\right] ;H\right) \) be the family of all right continous maps with left-hand limits from \(\left[ -\lambda ,0\right] \) to H, equipped with the norm \(\left\| \varphi \right\| = \sup _{t\in \left[ -\lambda ,0\right] }\left\| \varphi \left( t\right) \right\| \), and \(x_{t}=\left\{ x\left( t+\theta \right) :-\lambda \leqslant \theta \leqslant 0 \right\} \), which can be regarded as a \(PC\left( \left[ -\lambda ,0 \right] ;H \right) \) -valued stochastic process with norm \(\left| \left\| x_{t} \right\| \right| =\underset{t-\lambda \le s\le t}{\sup }\left\| x\left( s \right) \right\| \). \(\alpha ,\beta \) are two fixed constants, such that \(\left| \alpha \right| +\left| \beta \right| < 1 \). Furthermore, \(I_{k}\in C\left( K,K \right) \), is called impulse mapping function, and u, f, g, h are four Borel measurable functions, satisfy:

$$\begin{aligned} \begin{aligned}&u:R_{+}\times H \times C\left( \left[ -\lambda ,0 \right] ;H \right) \rightarrow H,\\&f:R_{+}\times H \times C\left( \left[ -\lambda ,0 \right] ;H \right) \rightarrow H,\\&g:R_{+}\times H \times C\left( \left[ -\lambda ,0 \right] ;H \right) \rightarrow \pounds _{2}^{0}\left( K,H \right) ,\\&h:R_{+}\times H\times C\left( \left[ -\lambda ,0 \right] ;H \right) \times \mathbb {Z} \rightarrow H \end{aligned} \end{aligned}$$

Here, \(t_{k}\) is called impulse jump time, and the pulse jump amplitude for \(\Delta x\left( t_{k}\right) =x\left( t_{k}^{+}\right) -x\left( t_{k}^{-}\right) \), where \(x\left( t_{k}^{+}\right) =\lim _{h\rightarrow 0^{+}}x\left( t_{k}+h\right) \), \(x\left( t_{k}^{-}\right) =\lim _{h\rightarrow 0^{+}}x\left( t_{k}-h\right) \).

The model (2.1) provides an important contribution to the field of stochastic recurrent neural networks with time delays. Naturally, it is an important topic to discuss the properties of solutions of SFDEs, such as existence, uniqueness and stability. In addition, the research on the stability of this system can successfully apply the neural network to the fields of pattern recognition, image processing, associative memory, optimization calculation and secure communication, especially for the correctness of circuit design and super large scale circuit implementation.

Definition 2.1

An H-valued stochastic process \(\left\{ x\left( t \right) ,t\in \left[ 0,T \right] \right\} \), is called the mild solution of \(\left( 1 \right) \), if

  1. (a)

    x(t) is adapted to \(\mathcal {F}_t\), and \(\int _{0}^{T}\left\langle x\left( t \right) ,x\left( t \right) \right\rangle _{H}< \infty \) almost surely;

  2. (b)

    x(t) has \(c\grave{a}dl\grave{a}g\) path on \(t\in \left[ 0,T \right] \) almost surely, and for \(t\in \left[ 0,T \right] \), x(t) satisfies the following integral equation.

    $$\begin{aligned} {\left\{ \begin{array}{ll} x\left( t \right) =S\left( t \right) \left( \phi \left( 0 \right) +u\left( 0,\phi ,\varphi \right) -u\left( t,x\left( t \right) ,x_{t} \right) \right) -\int _{0}^{t}AS\left( t-s \right) u\left( s,x\left( s \right) ,x_{s} \right) ds\\ \quad \quad \quad \quad +\int _{0}^{t}S\left( t-s \right) f\left( s,x\left( s \right) ,x_{s} \right) ds+\int _{0}^{t}S\left( t-s \right) g\left( s,x\left( s \right) ,x_{s} \right) d\omega \left( s \right) \\ \quad \quad \quad \quad +\int _{0}^{t}\int _{Z}S\left( t-s \right) h\left( s,x\left( s \right) ,x_{s},y\right) \widetilde{N}\left( ds,dy \right) +\sum _{0<t_{k}<t}S\left( t-t_{k} \right) I_{k}\left( x\left( t_{k}^{-} \right) \right) \\ \quad \quad \quad \quad + \alpha \underset{s\in \left[ 0,t\right] }{\sup }x\left( s \right) +\beta \underset{s\in \left[ 0,t\right] }{\inf }x\left( s \right) ,\\ x\left( \vartheta \right) =\phi \in PC,\quad \vartheta \in \left[ -\lambda ,0\right] ,\qquad \qquad \quad a.s.,\\ x_{0}\left( \cdot \right) =\varphi \in C_{\mathcal {F}_{0}}^{b}\left( \left[ -\lambda ,0 \right] ;H \right) ,\;\,\qquad \qquad \quad a.s. \end{array}\right. } \end{aligned}$$
    (2.2)

To obtain our main results, we impose the following necessary assumptions and lemmas:

  1. (A1)

    A is the infinitesimal generator of an analytic semigroup of bounded linear operators \(\left\{ S\left( t \right) ,t\ge 0 \right\} \) in X, such that the resolvent set \(\rho \left( -A \right) \) contains zero. Since S(t) is uniformly bounded, then there exists constants \(\gamma >0\) and \(0<\tilde{C}< 1\), satisfy

    $$\begin{aligned} \left\| S\left( t \right) \right\| \le \tilde{C}e^{-\gamma t}. \end{aligned}$$

    In fact, there is an \(\eta >0 \) such that \(\left\| S\left( t \right) \right\| \) is bounded for any \(0\le t\le \eta \). If this is false, then there exists a positive sequence \(\left\{ t_{n} \right\} \), satisfying \(\lim _{n \rightarrow \infty }t_{n}=0 \) and \( \left\| S\left( t_{n} \right) \right\| \ge n\). From the uniformly boundedness theorem, then it follows that for some \(x_{0}\in X\), \( \left\| S\left( t_{n} \right) x_{0} \right\| \) is unbounded. Thus, \(\left\| S\left( t \right) \right\| \le \tilde{C}\) for any \(0\le t\le \eta \). Since \(\left\| S\left( 0 \right) \right\| =1\), \(0<\tilde{C}< 1\). Let \(\gamma =\eta ^{-1}ln\,\tilde{C}^{-1}\), for any \(t\ge 0 \), set \(t=n\eta +\xi \), where \(0\le \xi < \eta \). From the semigroup property, we have

    $$\begin{aligned} \left\| S\left( t \right) \right\| =\left\| S\left( n\eta +\xi \right) \right\| =\left\| S\left( \xi \right) S\left( \eta \right) ^{n} \right\| \le \tilde{C}^{n+1}\le \tilde{C}\,\tilde{C}^{\frac{t}{\eta } }= \tilde{C}e^{-\gamma t}. \end{aligned}$$
  2. (A2)

    The mapping \(\left( -A\right) ^{\delta }u\left( t,\cdot ,\cdot \right) \) satisfies the uniformly Lipschitz condition: there exists a positive constant L, for any \(x,y\in K\),

    $$\begin{aligned} \big \Vert \left( -A\right) ^{\delta }u\left( t,x,x_{t}\right) -\left( -A\right) ^{\delta }u\left( t,y,y_{t}\right) \big \Vert \le L\left( \left\| x-y\right\| +\left| \left\| x_{t}-y_{t}\right\| \right| \right) , \end{aligned}$$

    and \(\iota :=\big \Vert \left( -A\right) ^{-\delta }\big \Vert L<1\). Here, \(\frac{1}{2}<\delta < 1\) and \( u\left( t,\cdot ,\cdot \right) \in D\big ( \left( -A \right) ^{\delta } \big )\).

  3. (A3)

    The mappings \(f\left( t,\cdot ,\cdot \right) , g\left( t,\cdot ,\cdot \right) ,h\left( t,\cdot ,\cdot , \cdot \right) \) satisfy the following Lipschitz and linear growth conditions : \(\left\| f\left( t,x,x_{t}\right) -f\left( t,y,y_{t}\right) \right\| \leqslant L_{1}\left( \left\| x-y\right\| +\left| \left\| x_{t}-y_{t}\right\| \right| \right) , \quad L_{1}>0,\) \(\left\| g\left( t,x,x_{t}\right) -g\left( t,y,y_{t}\right) \right\| \leqslant L_{2}\left( \left\| x-y\right\| +\left| \left\| x_{t}-y_{t}\right\| \right| \right) , \quad L_{2}>0,\) \(\int _{Z}\left\| h\left( t,x,x_{t},z\right) \!-\!h\left( t,y,y_{t},z\right) \right\| ^{2}\upsilon \left( dz\right) \!\leqslant \! L_{3}\left( \left\| x\!-\!y\right\| ^{2}\!+\!\left| \left\| x_{t}\!-\!y_{t}\right\| \right| ^{2}\right) , \quad L_{3}\!>\!0,\) for any \(x,y\in K\).

  4. (A4)

    \(I_{k}\left( 0 \right) =0\), for any \(k\ge 1\), and for each \(x,y\in K\), there exists a positive constant \(n_{k}\), such that

    $$\begin{aligned} \left\| I_{k}\left( x \right) -I_{k}\left( y \right) \right\| \leqslant n_{k}\left\| x-y \right\| ,\quad \sum _{k=1}^{+\infty }n_{k}<+\infty . \end{aligned}$$

Lemma 2.2

(Theorem 6.13 of [26]) Suppose that Assumption (A1) holds, then for any \(\beta \in (0,1]\), we have

  1. (i)

    for each \(x\in \mathcal {D}\big ( \left( -A\right) ^{\beta }\big ),\)

    $$\begin{aligned} S\left( t\right) \left( -A\right) ^{\beta }x=\left( -A\right) ^{\beta }S\left( t\right) x. \end{aligned}$$
  2. (ii)

    there exists a positive constant \( \tilde{C}_{\beta }> 0\) such that

    $$\begin{aligned} \left\| \left( -A\right) ^{\beta }S\left( t\right) \right\| \leqslant \tilde{C}_{\beta }t^{-\beta }e^{-\gamma t} ,\quad t>0. \end{aligned}$$

3 Existence and Uniqueness

In this section, we present the existence and uniqueness of the mild solution of system (2.2) by means of a successive approximation method. Firstly, let’s introduce the following lemmas :

Lemma 3.1

Fixed any seven real numbers \(a_{i} \), \(1\le i\le 7\), and \( p\ge 1\), then for any \(\varrho > 0 \), we have

$$\begin{aligned} \begin{aligned} \big ( \sum _{i=1}^{7} a_{i} \big )^{p}&\le 3^{p-1}\big ( 1+\tfrac{1}{\varrho } \big )^{p-1} a_{1} ^{p}+9^{p-1}\left( 1+\varrho \right) ^{p-1}\sum _{i=2}^{4}a_{i}^{p}\\&\quad +3^{p-1}\big ( 1+\tfrac{1}{\varrho }\big )^{p-1}a_{5}^{p}+3^{p-1}\left( 1+\varrho \right) ^{p-1} a_{6}^{p}+3^{p-1} a_{7}^{p}. \end{aligned} \end{aligned}$$
(3.1)

Proof

Apply the basic inequality: \( \left( a+b \right) ^{p} \le \upsilon ^{1-p} +\left( 1-\upsilon \right) ^{1-p} b^{p} \), where \(p\in \left[ 1,+\infty \right) \) and \(\upsilon \in \left( 0,1 \right) \), then we obtain

$$\begin{aligned} \big ( \sum _{i=1}^{7} a_{i} \big )^{p}&\le 3^{p-1}\big ( \sum _{i=1}^{4} a_{i} \big )^{p}+3^{p-1}\big ( \sum _{i=5}^{6} a_{i} \big )^{p}+ 3^{p-1}a_{7}^{p}\nonumber \\&\le 3^{p-1}\big [ \big ( 1+\tfrac{1}{\varrho }\big )^{p-1} a_{1} ^{p}+\left( 1+\varrho \right) ^{p-1}3^{p-1}\big ( \sum _{i=2}^{4}a_{i}^{p}\big )\big ]\nonumber \\&\quad +3^{p-1}\big ( 1+\tfrac{1}{\varrho }\big )a_{5}^{p}+3^{p-1}\left( 1+\varrho \right) ^{p-1} a_{6}^{p}+3^{p-1} a_{7}^{p}\nonumber \\&\le 3^{p-1}\big ( 1+\tfrac{1}{\varrho } \big )^{p-1} a_{1} ^{p}+9^{p-1}\left( 1+\varrho \right) ^{p-1}\big ( \sum _{i=2}^{4}a_{i}^{p}\big )\nonumber \\&\quad +3^{p-1}\big ( 1+\tfrac{1}{\varrho }\big )^{p-1}a_{5}^{p}+3^{p-1}\left( 1+\varrho \right) ^{p-1} a_{6}^{p}+3^{p-1} a_{7}^{p}. \end{aligned}$$
(3.2)

Lemma 3.2

Let \(0<s<t\) and \(y=t-s\), where \(\delta \in \left( \frac{1}{2},1 \right) \) and \(\gamma >0\), then

$$\begin{aligned} \begin{aligned} \int _{0}^{t}\left( t-s\right) ^{2\left( \delta -1\right) }e^{-\gamma \left( t-s\right) }ds=\gamma ^{1-2\delta }\Gamma \left( {1+2\left( \delta -1 \right) } \right) . \end{aligned} \end{aligned}$$
(3.3)

Proof

It is easy to see the result.

Lemma 3.3

Assuming that Assumption (A2) holds, then we have

$$\begin{aligned}{} & {} \underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| u\left( s,x^{n}\left( s\right) ,x_{s}^{n}\right) \right\| ^{2}\nonumber \\\le & {} \iota ^{2}\underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left\| x^{n}\left( s \right) \right\| ^{2} \!+\! \iota ^{2} \underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left| \left\| x_{s}^{n} \right\| \right| ^{2}+C\Big (\! \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2} \!+\!\underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left\| u\left( s,0,\varphi \right) \right\| ^{2} \Big ),\nonumber \\ \end{aligned}$$
(3.4)

where \(x_{0}\left( s\right) =\varphi (s), \quad s \in \left[ -\lambda ,0\right] .\)

Proof

By Assumption (A2) , we get

$$\begin{aligned}{} & {} \underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| u\left( s,x^{n}\left( s\right) ,x_{s}^{n}\right) \right\| ^{2}\nonumber \\= & {} \underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| \left( -A \right) ^{-\delta } \left( -A \right) ^{\delta } u\left( s,x^{n}\left( s\right) ,x_{s}^{n}\right) \right\| ^{2}\nonumber \\\le & {} \left\| \left( \! -A \right) ^{-\delta }\right\| ^{2}\underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| \left( -A \right) ^{\delta }u\left( s,x^{n}\left( s \right) ,x_{s}^{n}\right) \!-\!\left( -A \right) ^{\delta }u\left( s,0,\varphi \right) \right\| ^{2} \!+\!\underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| u\left( s,0,\varphi \right) \right\| ^{2} \nonumber \\\le & {} \left\| \left( -A \right) ^{-\delta } \right\| ^{2}\underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left( \left\| x^{n}\left( s\right) \right\| +\left| \left\| x_{s}^{n}-\varphi \right\| \right| \right) ^{2}+\underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| u\left( s,0,\varphi \right) \right\| ^{2}\nonumber \\\le & {} L^{2}\left\| \left( -A \right) ^{-\delta }\right\| ^{2}\underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| x^{n}\left( s \right) \right\| ^{2}+L^{2}\left\| \left( -A \right) ^{-\delta }\right\| ^{2}\underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left| \left\| x_{s}^{n} \right\| \right| ^{2}\nonumber \\{} & {} +L^{2}\left\| \left( -A \right) ^{-\delta }\right\| ^{2}\underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}+2\underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| u\left( s,0,\varphi \right) \right\| ^{2}\nonumber \\\le & {} \iota ^{2}\underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left\| x^{n}\left( s \right) \right\| ^{2} \!+ \iota ^{2} \underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left| \left\| x_{s}^{n} \right\| \right| ^{2}\!+C\Big ( \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2} \!+\underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left\| u\left( s,0,\varphi \right) \right\| ^{2} \Big ),\nonumber \\ \end{aligned}$$
(3.5)

where C in the last inequality is a certain constant.

Lemma 3.4

Suppose that Assumption (A2) holds, we have

$$\begin{aligned}{} & {} \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| \int _{0}^{s}AS\left( s-\sigma \right) u\left( \sigma ,x^{n}\left( \sigma \right) ,x_{\sigma }^{n}\right) d\sigma \right\| ^{2}\nonumber \\\le & {} C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta }\Gamma \left( 1+2\left( \delta -1 \right) \right) \big [ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n} \right| \right\| ^{2}\nonumber \\{} & {} + \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2} + \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| \left( -A \right) ^{\delta }u\left( s,0,\varphi \right) \right\| ^{2} \big ]. \end{aligned}$$
(3.6)

Proof

Applying Lemma 2.2, we have \(\left\| \left( \!-A \right) ^{1-\delta }S\left( t\!-\!s \right) \right\| \!\le \! \tilde{C}_{1-\delta }\left( t\!-\!s \right) ^{\delta -1}e^{-\gamma \left( t-s \right) }\), and proceed further, through Assumption (A2), we can draw

$$\begin{aligned}{} & {} \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| \int _{0}^{t}AS\left( s-\sigma \right) u\left( \sigma ,x^{n}\left( \sigma \right) ,x_{\sigma }^{n}\right) d\sigma \right\| ^{2}\nonumber \\\le & {} E\left\| \int _{0}^{t}\left\| \left( -A \right) ^{1-\delta }S\left( t-s \right) \right\| \cdot \left\| \left( -A \right) ^{\delta }u\left( s,x^{n}\left( s \right) ,x_{s}^{n}\right) \right\| ds \right\| ^{2}\nonumber \\\le & {} E\left| \int _{0}^{t} \tilde{C}_{1-\delta }\left( t\!-s\right) ^{\delta -1}e^{-\gamma \left( t-s\right) }\left[ L\left( \left\| x^{n}\left( s\right) \right\| \!+\!\left| \left\| x_{s}^{n}\!-\varphi \right\| \right| \right) \!+\left\| \left( \!-A\right) ^{\delta }u\left( s,0,\varphi \right) \right\| \right] ds\right| ^{2}\nonumber \\\le & {} 2\left( \int _{0}^{t}L \tilde{C}_{1-\delta }\left( t-s\right) ^{\delta -1}e^{-\gamma \left( t-s\right) }E\left( \left\| x^{n}\left( s\right) \right\| +\left| \left\| x_{s}^{n}-\varphi \right\| \right| \right) ds\right) ^{2} \end{aligned}$$
(3.7)
$$\begin{aligned}{} & {} +2\left( \int _{0}^{t} \tilde{C}_{1-\delta }\left( t-s\right) ^{\delta -1}e^{-\gamma \left( t-s\right) }E\left\| \left( -A\right) ^{\delta }u\left( s,0,\varphi \right) \right\| ds\right) ^{2}. \end{aligned}$$
(3.8)

Applying Holder inequality on Eq. 3.7, which yields the following result:

$$\begin{aligned} (3.7)&\le L^{2} \tilde{C}_{1-\delta }^{2} \big [\big (\int _{0}^{t}\left( t\!-s\right) ^{2\left( \delta -1\right) }e^{-\gamma \left( t-s\right) \cdot \frac{1}{2}\cdot 2}ds\big )^{\frac{1}{2}}\big (\int _{0}^{t}e^{-\gamma \left( t\!-s \right) }E\left( \left\| x^{n}\left( s\right) \right\| \!+\left| \left\| x_{s}^{n-1}\!-\varphi \right\| \right| \right) ^{2}ds\big )^{\frac{1}{2}}\big ]^{2} \nonumber \\&=\tilde{C}_{1-\delta }^{2}L^{2}\int _{0}^{t}\left( t-s\right) ^{2\left( \delta -1 \right) }\cdot e^{-\gamma \left( t-s\right) }ds\cdot \int _{0}^{t}e^{-\gamma \left( t-s\right) }E\left( \left\| x^{n}\left( s\right) \right\| +\left| \left\| x_{s}^{n}-\varphi \right\| \right| \right) ^{2}ds. \end{aligned}$$

Furthermore, from Lemma 3.2, we get

$$\begin{aligned} (3.8) \le \tilde{C}_{1-\delta }^{2}\gamma ^{1-2\delta }\Gamma \left( 1+2\left( \delta -1 \right) \right) \int _{0}^{t}e^{-\gamma \left( t-s\right) }E\left\| \left( -A\right) ^{\delta }u\left( s,0,\varphi \right) \right\| ^{2}ds. \end{aligned}$$

Therefore,

$$\begin{aligned}{} & {} \underset{ s\in \left[ -\lambda ,t \right] }{\sup }E\left\| \int _{0}^{s}AS\left( s-\sigma \right) u\left( \sigma ,x^{n}\left( \sigma \right) ,x_{\sigma }^{n}\right) d\sigma \right\| ^{2}\nonumber \\\le & {} C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta }\Gamma \left( 1+2\left( \delta -1 \right) \right) \Big [ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n} \right| \right\| ^{2}\nonumber \\{} & {} + \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s\right) \right\| ^{2} + \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| \left( -A \right) ^{\delta }u\left( s,0,\varphi \right) \right\| ^{2} \Big ]. \end{aligned}$$
(3.9)

Lemma 3.5

Suppose that Assumption (A1) holds, then

$$\begin{aligned}{} & {} \underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| \int _{0}^{s}S\left( s-\sigma \right) f\left( \sigma ,x^{n-1}\left( \sigma \right) ,x_{\sigma }^{n-1}\right) d\sigma \right\| ^{2} \nonumber \\\le & {} C^{2}L_{1}^{2}\gamma ^{-2}\Big [ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n-1}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n-1}\right| \right\| ^{2}+ \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s\right) \right\| ^{2}\Big ]\nonumber \\{} & {} +C^{2}\gamma ^{-2} \underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| f\left( s,0,\varphi \right) \right\| ^{2}. \end{aligned}$$

Proof

By Assumption (A1), we can see that \(S(t-s)\) is uniformly bounded, then \(\left\| S\left( t-s\right) \right\| \le \tilde{C}e^{-\gamma \left( t-s\right) }\), for \(s\le t\), and since \(f\left( t,\cdot ,\cdot \right) \) satisfies Lipschitz and linear growth conditions, so

$$\begin{aligned} \left\| f\left( s,x^{n-1}\left( s\right) ,x_{s}^{n-1}\right) -f\left( s,0,\varphi \right) \right\| \le L_{1}\left( \left\| x^{n-1}\left( s\right) \right\| +\left| \left\| x_{s}^{n-1}-\varphi \right\| \right| \right) , \end{aligned}$$

which yields the folllowing inequality

$$\begin{aligned}{} & {} \underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left\| \int _{0}^{s}S\left( s-\sigma \right) f\left( \sigma ,x^{n-1}\left( \sigma \right) ,x_{\sigma }^{n-1}\right) d\sigma \right\| ^{2}\nonumber \\\le & {} E \left\{ \int _{0}^{t} \tilde{C}e^{-\gamma \left( t-s\right) } \left[ \left\| f\left( s,x^{n-1}\left( s \right) ,x_{s}^{n-1}\right) -f\left( s,0,\varphi \right) \right\| +\left\| f\left( s,0,\varphi \right) \right\| \right] ds\right\} ^{2}\nonumber \\\le & {} E \left\{ \int _{0}^{t} \tilde{C}e^{-\gamma \left( t-s\right) }\left[ L_{1} \left( \left\| x^{n-1}\left( s\right) \right\| +\left| \left\| x_{s}^{n-1}-\varphi \right\| \right| \right) +\left\| f\left( s,0,\varphi \right) \right\| \right] ds\right\} ^{2}\nonumber \\\le & {} 2 \tilde{C}^{2}L_{1}^{2}E\left[ \int _{0}^{t}e^{-\frac{\gamma }{2}\left( t-s \right) } \cdot e^{-\frac{\gamma }{2}\left( t-s \right) } \left( \left\| x^{n-1}\left( s\right) \right\| +\left| \left\| x_{s}^{n-1}-\varphi \right\| \right| \right) ds\right] ^{2}\nonumber \\{} & {} +C^{2}\gamma ^{-1}\cdot \int _{0}^{t}e^{-\gamma \left( t-s \right) }E\left\| f\left( s,0,\varphi \right) \right\| ^{2}ds\nonumber \\\le & {} C^{2}L_{1}^{2}\gamma ^{-2}\Big [ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n-1}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n-1}\right| \right\| ^{2}+ \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}\Big ]\nonumber \\{} & {} +C^{2}\gamma ^{-2} \underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| f\left( s,0,\varphi \right) \right\| ^{2}. \end{aligned}$$
(3.10)

Lemma 3.6

Suppose that Assumption (A1) and (A2) hold, we have

$$\begin{aligned}{} & {} \underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left\| \int _{0}^{s}S\left( s-\sigma \right) g\left( \sigma ,x^{n-1}\left( \sigma \right) ,x_{\sigma }^{n-1}\right) d\omega \left( \sigma \right) \right\| ^{2}\nonumber \\\le & {} C^{2}L_{2}^{2 } \gamma ^{-1} \Big [ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n-1}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n-1} \right| \right\| ^{2}+ \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}\Big ]\nonumber \\{} & {} +C^{2}\gamma ^{-1} \underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| g\left( s,0,\varphi \right) \right\| ^{2}. \end{aligned}$$
(3.11)

The proof process is paralla to the demonstration of Lemma 3.5.

Lemma 3.7

Suppose that Assumption (A1) and (A2) hold, then

$$\begin{aligned}{} & {} \underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| \int _{0}^{t}\int _{\mathbb {Z}}S\left( s-\sigma \right) h\left( \sigma ,x^{n-1}\left( \sigma \right) ,x_{\sigma }^{n-1},\nu \right) \widetilde{N}\left( d\sigma ,d\nu \right) \right\| ^{2}\nonumber \\\le & {} C^{2}L_{3}^{2}\gamma ^{-1} \Big [ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n-1}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n-1} \right| \right\| ^{2}+ \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}\Big ] \nonumber \\{} & {} +C^{2} \gamma ^{-1}\underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| h\left( s,0,\varphi \right) \right\| ^{2}. \end{aligned}$$
(3.12)

Proof

From (A1) and (A2), we can get

$$\begin{aligned}{} & {} \underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| \int _{0}^{t}\int _{\mathbb {Z}}S\left( s-\sigma \right) h\left( \sigma ,x^{n-1}\left( \sigma \right) ,x_{\sigma }^{n-1},\nu \right) \widetilde{N}\left( d\sigma ,d\nu \right) \right\| ^{2}\nonumber \\\le & {} E\left\| \int _{0}^{t}\int _{\mathbb {Z} }S\left( t-s \right) h\left( s,x^{n-1}\left( s \right) ,x_{s}^{n-1} \right) \widetilde{N}\left( ds,dz \right) \right\| ^{2}\nonumber \\\le & {} E \int _{0}^{t}\int _{\mathbb {Z}}\left\| S\left( t-s \right) h\left( s,x^{n-1}\left( s\right) ,x_{s}^{n-1} \right) \right\| ^{2}ds\nu \left( dz \right) \nonumber \\\le & {} \tilde{C}^{2} E \int _{0}^{t}\int _{\mathbb {Z}}e^{-2\gamma \left( t-s \right) } \!\left\| h\left( s,x^{n-1}\left( s\right) ,x_{s}^{n-1} \right) \!-\!h\left( s,0,\varphi \right) \!+\!h\left( s,0,\varphi \right) \right\| ^{2}\!ds\nu \left( dz \right) \nonumber \\\le & {} 2 \tilde{C}^{2}L_{3}^{p} \int _{0}^{t}e^{-2\gamma \left( t-s\right) }E\left( \left\| x^{n-1}\left( s\right) \right\| ^{2}+\left| \left\| x_{s}^{n-1}-\varphi \right\| \right| ^{2}\right) ds\end{aligned}$$
(3.13)
$$\begin{aligned}{} & {} +2 \tilde{C}^{2}\int _{0}^{t}e^{-2\gamma \left( t-s\right) }E\left\| h\left( s,0,\varphi \right) \right\| ^{2}\nu \left( dz \right) ds. \end{aligned}$$
(3.14)

Applying Holder inequality on Eq. 3.13, we get

$$\begin{aligned} (3.13) \le C^{2}L_{3}^{2}\gamma ^{-1} \Big [\! \underset{s\in \left[ -\lambda ,t \right] }{\sup }\!E\left\| x^{n-1}\left( s \right) \right\| ^{2}\!+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n-1}\right| \right\| ^{2}\!+\! \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}\Big ]. \end{aligned}$$

In summary from the formula derived above, we can conclude that

$$\begin{aligned}{} & {} \underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| \int _{0}^{t}\int _{\mathbb {Z}}S\left( t-s \right) h\left( s,x^{n-1}\left( s \right) ,x_{s}^{n-1}\right) \widetilde{N}\left( ds,dz\right) \right\| ^{2}\nonumber \\\le & {} C^{2}L_{3}^{2}\gamma ^{-1} \Big [ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n-1}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n-1}\right| \right\| ^{2}+ \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}\Big ] \nonumber \\{} & {} +C^{2} \gamma ^{-1}\underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| h\left( s,0,\varphi \right) \right\| ^{2}. \end{aligned}$$
(3.15)

Notice: we mention that the proof process of Lemmas 3.3 to 3.7 comes from [22]. However, their method is not completely applicable to this paper, because the functional term and double perturbed term are added to the model in this paper, which lead to the coefficient changes slightly when applying successive approximation method to shrink.

Theorem 3.8

Assume that conditions (A1)-(A4) hold, then the system (2.2) has a unique mild solution.

Proof

Let \(\iota <1 \) , and \(\gamma \) goes to infinity, fix \(C, L,L_{1},L_{2},L_{3}\) to a smaller positive integer, note that \(\frac{1}{2}<\delta < 1\), \(0<\underset{t_{k}<t}{\sum }q_{k}<1\), \(0<\left| \alpha \right| +\left| \beta \right| <1\), then the following inequality

$$\begin{aligned}{} & {} 18\left( 1-\iota \right) ^{-2}\big [C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta }\Gamma \left( 1+2\left( \delta -1 \right) \right) +C^{2}\big ( \underset{t_{k}<t}{\sum }q_{k}\big )^{2}+C^{2}L_{3} ^{2}\gamma ^{-1}\big ] \nonumber \\{} & {} +3\left( 1-\iota \right) ^{-2}\big [C^{2}L_{1} ^{2}\gamma ^{-2}+C^{2}L_{2} ^{2}\gamma ^{-1}+\left( \left| \alpha \right| +\left| \beta \right| \right) ^{2}] <1 \end{aligned}$$
(3.16)

is satisfied.

In order to obtain the existence of the mild solution in Eq. 2.2, we set \(x^{0}\left( t \right) =S\left( t \right) \left( \phi \left( 0 \right) +u\left( 0,\phi ,\varphi \right) \right) ,\) \(t\in \left[ 0,T \right] \) and \(x_{0}^{n}\left( t \right) =\phi \left( t \right) ,t\in \left[ -\lambda ,0 \right] , \) for \(n\ge 1\). Define the following successive approximating procedure,

$$\begin{aligned} x^{n}\left( t\right)= & {} x^{0}\left( t \right) -S\left( t \right) u\left( t,x^{n}\left( t\right) ,x_{t}^{n}\right) -\int _{0}^{t}AS\left( t-s\right) u\left( s,x^{n}\left( s\right) ,x_{s}^{n}\right) ds\nonumber \\{} & {} +\int _{0}^{t}S\left( t\!-\!s\right) f\left( s,x^{n-1}\left( s\right) ,x_{s}^{n-1}\right) ds\!+\!\int _{0}^{t}S\left( t\!-\!s\right) g\left( s,x^{n-1}\left( s\right) ,x_{s}^{n-1}\right) d\omega \left( s\right) \nonumber \\{} & {} +\int _{0}^{t}\int _{Z}S\left( t\!-\!s \right) h\left( s,x^{n-1}\left( s \right) ,x_{s}^{n-1},y\right) \widetilde{N}\left( ds,dy \right) \!+\!\underset{0<t_{k}<t}{\sum }S\left( t\!-\!t_{k} \right) I_{k}x^{n-1}\left( t_{k} \right) \nonumber \\{} & {} +\alpha \underset{s\in \left[ 0,t\right] }{\sup }x^{n}\left( s \right) +\beta \underset{s\in \left[ 0,t\right] }{\inf }x^{n}\left( s \right) . \end{aligned}$$
(3.17)

Step 1. We claim that the sequence \(\left\{ x^{n}\left( t \right) ,n\ge 0\right\} \) is bounded for each \(t\in \left[ 0,T \right] \).

Firstly, by Eq. 3.17,

$$\begin{aligned} \left\| x^{n} \left( s \right) \right\|&\le \left\| S\left( s \right) \left( \phi \left( 0 \right) +u\left( 0,\phi ,\varphi \right) \right) \right\| +\left\| u\left( \sigma ,x^{n}\left( \sigma \right) ,x_{\sigma }^{n} \right) \right\| \nonumber \\&\quad +\big \Vert \int _{0}^{s}AS\left( s-\sigma \right) u\left( \sigma ,x^{n}\left( \sigma \right) ,x_{\sigma }^{n} \right) d\sigma \big \Vert \nonumber \\&\quad +\big \Vert \int _{0}^{s}S\left( s-\sigma \right) f\left( \sigma ,x^{n-1} \left( \sigma \right) ,x_{\sigma }^{n-1} \right) d \sigma \big \Vert \nonumber \\&\quad +\big \Vert \int _{0}^{s} S\left( s-\sigma \right) g\left( \sigma ,x^{n-1} \left( \sigma \right) ,x_{\sigma }^{n-1} \right) d\omega \left( \sigma \right) \big \Vert \nonumber \\&\quad +\big \Vert \int _{0}^{s} \int _{\mathbb {Z} } S\left( s-\sigma \right) h\left( \sigma ,x^{n-1} \left( \sigma \right) ,x_{\sigma }^{n-1} ,y\right) \widetilde{N} \left( d\sigma , dy\right) \big \Vert \nonumber \\&\quad +\big \Vert \underset{0< t_{k}< t }{\sum } S \left( s-s_{k} \right) I_{k} x^{n-1} \left( s_{k} \right) \big \Vert +\big ( \left| \alpha \big | +\left| \beta \right| \right) \underset{\sigma \in \left[ 0,s \right] }{\sup }\ \left\| x^{n} \left( \sigma \right) \right\| . \end{aligned}$$
(3.18)

Furthermore, by Assumption (A4), we obtain

$$\begin{aligned} E\Big \Vert \underset{0<t_{k}<t}{\sum }S\left( t\!-\!t_{k}\right) I_{k}\left( x^{n-1}\left( t_{k}\right) \right) \Big \Vert ^{2}\le & {} E\Big \Vert \underset{0<t_{k}<t}{\sum } \tilde{C}e^{-\gamma \left( t-t_{k}\right) }\left( I_{k}\left( x^{n-1}\left( t_{k}\right) \!-\!I_{k}\left( 0\right) \right) \right) \Big \Vert ^{2} \\\le & {} \tilde{C}^{2}\big ( \underset{t_{k}<t}{\sum }q_{k}\big )^{2}e^{-2\gamma \left( t-t_{k}\right) }E\left\| x^{n-1}\left( t_{k}\right) \right\| ^{2}. \end{aligned}$$

By Assumption (A1) and (A2), we also have

$$\begin{aligned} \left\| S\left( t\right) \left( \phi \left( 0\right) +u\left( 0,\phi ,\varphi \right) \right) \right\|\le & {} \tilde{C}e^{-\gamma t}\left\| \phi \left( 0 \right) +u\left( 0,\phi ,\varphi \right) -u\left( 0,\phi ,\varphi \right) \right\| \\\le & {} \tilde{C}e^{-\gamma t}\left( \phi \left( 0 \right) +L\big \Vert \left( -A \right) ^{-\delta }\big \Vert \left( \left\| \phi \right\| +\left\| \varphi \right\| \right) \right) . \end{aligned}$$

Furthermore, we can get

$$\begin{aligned} E\left\| S\left( t\right) \left( \phi \left( 0\right) +u\left( 0,\phi ,\varphi \right) \right) \right\| ^{2}\le & {} C^{2} e^{-2\gamma t}\big ( 1+\iota \big \Vert \left( -A \right) ^{-\delta }\big \Vert \big )^{2}\underset{s\in \left[ -\lambda ,0 \right] }{\sup }E\left\| \phi \left( s \right) \right\| ^{2}\\{} & {} +C^{2} e^{-2\gamma t}\iota ^{2}\big \Vert \left( -A \right) ^{-\delta }\big \Vert ^{2}\underset{s\in \left[ -\lambda ,0 \right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}. \end{aligned}$$

Consequently, from Lemmas 3.1 to 3.7 , we simplify (3.18) and draw that

$$\begin{aligned}{} & {} \underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{n}\left( s\right) \right\| ^{2}\nonumber \\\le & {} \iota ^{-1}\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| u\left( s ,x^{n}\left( s \right) ,x_{s }^{n} \right) \right\| ^{2}+3\big (1+\varrho \big )\left( 1-\iota \right) ^{-1} \left( \left| \alpha \right| +\left| \beta \right| \right) ^{2}\underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{n}\left( s\right) \right\| ^{2}\nonumber \\{} & {} + 3\left( 1-\iota \right) ^{-1}\big (1+\tfrac{1}{\varrho } \big ) \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| S\left( s \right) \left( \phi \left( 0 \right) +u\left( 0,\phi ,\varphi \right) \right) \right\| ^{2}\nonumber \\{} & {} +9\left( 1-\iota \right) ^{-1}\left( 1+\varrho \right) \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \int _{0}^{s}AS\left( s-\sigma \right) u\left( \sigma ,x^{n}\left( \sigma \right) ,x_{\sigma }^{n} \right) d\sigma \big \Vert ^{2} \nonumber \\{} & {} +3\left( 1-\iota \right) ^{-1}\big ( 1+\tfrac{1}{\varrho }\big )\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \int _{0}^{s}S\left( s-\sigma \right) f\left( \sigma ,x^{n-1} \left( \sigma \right) ,x_{\sigma }^{n-1} \right) \sigma \big \Vert ^{2}\nonumber \\{} & {} +9\left( 1-\iota \right) ^{-1}\left( 1+\varrho \right) \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \int _{0}^{s} \int _{\mathbb {Z} } S\left( s-\sigma \right) h\left( \sigma ,x^{n-1} \left( \sigma \right) ,x_{\sigma }^{n-1} ,y\right) \widetilde{N} \left( d\sigma , dy\right) \big \Vert ^{2} \nonumber \\{} & {} +3\left( 1-\iota \right) ^{-1}\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \int _{0}^{s} S\left( s-\sigma \right) g\left( \sigma ,x^{n-1} \left( \sigma \right) ,x_{\sigma }^{n-1} \right) d\omega \left( \sigma \right) \big \Vert ^{2}\nonumber \\{} & {} +9\left( 1-\iota \right) ^{-1}\left( 1+\varrho \right) \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \underset{0< t_{k}< t }{\sum } S \left( s-s_{k} \right) I_{k} x^{n-1} \left( s_{k} \right) \big \Vert ^{2}\nonumber \\\le & {} \iota \underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left\| x^{n}\left( s \right) \right\| ^{2} + \iota \underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left| \left\| x_{s}^{n} \right\| \right| ^{2}+C\Big ( \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2} +\underset{ s\in \left[ -\lambda ,t\right] }{\sup }E\left\| u\left( s,0,\varphi \right) \right\| ^{2} \Big )\nonumber \\{} & {} +3\big (1+\varrho \big )\left( 1-\iota \right) ^{-1} \left( \left| \alpha \right| +\left| \beta \right| \right) ^{2}\underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{n}\left( s\right) \right\| ^{2}+ 9\left( 1-\iota \right) ^{-1}\left( 1+\varrho \right) C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta }\nonumber \\{} & {} \cdot \Gamma \left( 1+2\left( \delta -1 \right) \right) \Big [ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n} \right\| \right| ^{2}+ \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}\nonumber \\{} & {} + \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| \left( -A \right) ^{\delta }u\left( s,0,\varphi \right) \right\| ^{2} \Big ]+3\left( 1-\iota \right) ^{-1}\left( 1+\varrho \right) C^{2}\gamma ^{-2} \underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| f\left( s,0,\varphi \right) \right\| ^{2}\nonumber \\{} & {} +3\left( 1-\iota \right) ^{-1}C^{2}L_{1}^{2}\gamma ^{-2}\Big [\underset{s\in \left[ -\lambda ,t\right] }{\sup }E\left\| x^{n-1}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda , t \right] }{\sup }E\left| \left\| x_{s}^{n-1}\right| \right\| ^{2}+\underset{s\in \left[ -\lambda ,0 \right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}\Big ]\nonumber \\{} & {} +3\left( 1-\iota \right) ^{-1} C^{2}L_{2}^{2} \gamma ^{-1} \Big [ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n-1}\left( s \right) \right\| ^{2}+\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n-1} \right| \right\| ^{2}+ \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}\Big ]\nonumber \\{} & {} +3\left( 1\!-\iota \right) ^{-1}C^{2}\gamma ^{-1} \underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| g\left( s,0,\varphi \right) \right\| ^{2}\!+9\left( 1\!-\iota \right) ^{-1}\left( 1\!+\varrho \right) C^{2} \gamma ^{-1}\underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| h\left( s,0,\varphi \right) \right\| ^{2}\nonumber \\{} & {} +9\left( 1\!-\iota \right) ^{-1}\left( 1\!+\varrho \right) C^{2}L_{3}^{2}\gamma ^{-1} \Big [\! \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n-1}\left( s \right) \right\| ^{2}\!+\!\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left| \left\| x_{s}^{n-1} \right| \right\| ^{2}\!+\! \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}\Big ] \nonumber \\{} & {} +9\left( 1-\iota \right) ^{-1}\left( 1+\varrho \right) C^{2} \big ( \underset{t_{k}< t}{\sum }q_{k}\big )^{2}e^{-2\gamma \left( t-t_{k}\right) }\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x^{n-1}\left( s\right) \right\| ^{2}. \end{aligned}$$
(3.19)

Note that

$$ \underset{s\in \left[ -\lambda ,t \right] }{\sup }E\left\| x\left( s \right) \right\| ^{2}\le \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \phi \left( s \right) \right\| ^{2}+\underset{s\in \left[ 0 ,t \right] }{\sup }E\left\| x\left( s \right) \right\| ^{2}, $$

and

$$\begin{aligned} \begin{aligned} E\left| \left\| x_{t}\right| \right\| ^{2}&= E\big \{ \underset{s\in \left[ t-\lambda ,t \right] }{\sup }\left\| x\left( s\right) \right\| ^{2}\big \}\\&\le \underset{s\in \left[ 0,t \right] }{\sup }E\left\| x\left( s\right) \right\| ^{2}+\underset{s\in \left[ t-\lambda ,0 \right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2}. \end{aligned} \end{aligned}$$

So the following inequality can be obtained by combining the same terms in Eq. 3.19:

$$\begin{aligned} (3.19)\le & {} \left[ 1-\iota -3\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )\big (\left( \left| \alpha \right| +\left| \beta \right| \right) ^{2}+ 6 C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta } \Gamma \left( 1+2\left( \delta -1 \right) \right) \big ) \right] ^{-1}\nonumber \\{} & {} \times \Big \{6\left( 1-\iota \right) ^{-1}\big [C^{2} L_{1}^{2} \gamma ^{-2}+\big (1+\varrho \big )C^{2}L_{2}^{2} \gamma ^{-1}+3\big (1+\varrho \big )\big ( C^{2}L_{3}^{2}\gamma ^{-1}+C^{2} \big ( \underset{t_{k}<t}{\sum }q_{k}\big )^{2}\nonumber \\{} & {} \cdot e^{-2\gamma \left( t-t_{k}\right) }\!+C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta } \Gamma \left( 1\!+2\left( \delta \!-1 \right) \right) \big ) \big ] \underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{n-1}\left( s\right) \right\| ^{2} \!+\big [3\left( 1-\!\iota \right) ^{-1}C^{2}\big ( \big ( 1\!+\tfrac{1}{\varrho } \big )\nonumber \\{} & {} \cdot e^{-2\gamma t}+L_{1}^{2}\gamma ^{-2}+L_{2}^{2}\gamma ^{-1} \big )+9\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )\big (C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta } \Gamma \left( 1+2\left( \delta -1 \right) \right) \nonumber \\{} & {} +C^{2}L_{3}^{2}\gamma ^{-1} \big )\!+C^{2}\underset{t_{k}<t}{\sum }q_{k}\big )^{2} e^{-2\gamma \left( t-t_{k}\right) }\big ]\underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \phi \left( s\right) \right\| ^{2} \!+\big [3\left( 1\!-\iota \right) ^{-1}C^{2}\big (\big ( 1\!+\tfrac{1}{\varrho } \big ) e^{-2\gamma t}\iota ^{2} \nonumber \\{} & {} + L_{1}^{2}\gamma ^{-2}+L_{2}^{2}\gamma ^{-1}\big ) +9\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )\big ( C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta }\Gamma \left( 1+2\left( \delta -1 \right) \right) + C^{2} \gamma ^{-1}L_{3}^{2}\big )\big ]\nonumber \\{} & {} \cdot \underset{s\in \left[ -\lambda ,0\right] }{\sup }E\left\| \varphi \left( s\right) \right\| ^{2}+9\left( 1-\iota \right) ^{-1}\big (1+\varrho \big ) C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta }\Gamma \left( 1+2\left( \delta -1 \right) \right) \nonumber \\{} & {} \cdot \underset{ s\in \left[ -\lambda ,t \right] }{\sup }E\left\| \left( -A\right) ^{\delta } u\left( s,0,\varphi \right) \right\| ^{2}+3\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )C^{2}\big (\gamma ^{-2} \underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| f\left( s,0,\varphi \right) \right\| ^{2}\nonumber \\{} & {} + \gamma ^{-1} \underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| g\left( s,0,\varphi \right) \right\| ^{2}\big )+9\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )C^{2} \gamma ^{-1}\underset{s\in \left[ -\lambda ,t \right] }{\sup }E \left\| h\left( s,0,\varphi \right) \right\| ^{2} \Big \} .\nonumber \\ \end{aligned}$$
(3.20)

By Eq. 3.16, we set a number \(\varrho >0 \) small enough such that

$$\begin{aligned} \begin{aligned} N_{1}:=&\iota +3\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )\left( \left| \alpha \right| +\left| \beta \right| \right) ^{2}+18\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )C_{1-\delta }^{2}\\&\cdot L^{2}\gamma ^{-2\delta } \Gamma \left( 1+2\left( \delta -1 \right) \right) <1, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} M:= & {} 6\left( 1\!-\iota \right) ^{-1}\big [C^{2} L_{1}^{2} \gamma ^{-2}\!+\big (1+\varrho \big )C^{2}L_{2}^{2} \gamma ^{-1}\!+3\big (1+\varrho \big )\big ( C^{2}L_{3}^{2}\gamma ^{-1}\!+C^{2} \big ( \underset{t_{k}<t}{\sum }q_{k}\big )^{2}\\{} & {} \cdot e^{-2\gamma \left( t-t_{k}\right) }+C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta } \Gamma \left( 1+2\left( \delta -1 \right) \right) \big ) \big ]<1. \end{aligned}$$

Using mathematical induction to Eq. 3.20, we shall prove the the sequence \(\left\{ x^{n}\left( t \right) ,n\ge 0\right\} \) is bounded for every \(t\in \left[ 0,T \right] \).

In fact, due to \(E\left\| \phi \left( s \right) \right\| ^{2}< \infty ,\, E\left\| \varphi \left( s \right) \right\| ^{2}< \infty \), and functions \( E \Vert \left( -A \right) ^{\alpha }u( s,0,\)\(\varphi )\Vert ^{2}\), \(E\left\| h\left( s,0,\varphi \right) \right\| ^{2}\), \(E\left\| f\left( s,0,\varphi \right) \right\| ^{2},E\left\| g\left( s,0,\varphi \right) \right\| ^{2}\) are uniformly bounded. So, when \(n=1\), \(E\left\| x\left( s \right) \right\| ^{2}< \infty \), when \(n>1\), assuming \(E\left\| x^{n-1} \left( s \right) \right\| ^{2}< \infty \). Then, from Eq. 3.20, \(E\left\| x^{n} \left( s \right) \right\| ^{2}< \infty \).

Step 2. We claim that \(\left\{ x^{n}\left( t \right) ,n\ge 0\right\} \) is a Cauchy sequence. From Eq. 3.18, them,

$$\begin{aligned}{} & {} \underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{n+1}\left( s \right) -x^{n}\left( s \right) \right\| ^{2}\nonumber \\\le & {} \left[ C^{-1}-\left( \left| \alpha \right| +\left| \beta \right| ^{2} \right) \right] ^{-1} \Big \{\underset{s\in \left[ 0,t \right] }{\sup }E\left\| u\left( s,x^{n+1}\left( s \right) ,x_{s}^{n+1} \right) -u\left( s,x^{n}\left( s \right) ,x_{s}^{n} \right) \right\| ^{2}\nonumber \\{} & {} +\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \int _{0}^{s}AS\left( s-\sigma \right) \left( u\left( \sigma ,x^{n+1}\left( \sigma \right) ,x_{\sigma }^{n+1} \right) -u\left( \sigma ,x^{n}\left( \sigma \right) ,x_{\sigma }^{n} \right) \right) d\sigma \big \Vert ^{2}\nonumber \\{} & {} +\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \int _{0}^{s}S\left( s-\sigma \right) \left( f\left( \sigma ,x^{n-1} \left( \sigma \right) ,x_{\sigma }^{n} \right) - f\left( \sigma ,x^{n} \left( \sigma \right) ,x_{\sigma }^{n-1} \right) \right) \sigma \big \Vert ^{2} \nonumber \\{} & {} +\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \int _{0}^{s} \int _{\mathbb {Z} } S\left( s-\sigma \right) \left( h\left( \sigma ,x^{n} \left( \sigma \right) ,x_{\sigma }^{n} ,y\right) - h\left( \sigma ,x^{n-1} \left( \sigma \right) ,x_{\sigma }^{n-1} ,y\right) \right) \widetilde{N} \left( d\sigma , dy\right) \big \Vert ^{2} \nonumber \\{} & {} +\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \int _{0}^{s} S\left( s-\sigma \right) \left( g\left( \sigma ,x^{n} \left( \sigma \right) ,x_{\sigma }^{n} \right) -g\left( \sigma ,x^{n-1} \left( \sigma \right) ,x_{\sigma }^{n-1} \right) \right) d\omega \left( \sigma \right) \big \Vert ^{2}\nonumber \\{} & {} +\underset{s\in \left[ -\lambda ,t \right] }{\sup }E\big \Vert \underset{0< t_{k}< t }{\sum } S \left( s-s_{k} \right) I_{k}\left( x^{n}\left( s_{k} \right) - x^{n-1}\left( s_{k} \right) \right) \big \Vert ^{2}\Big \} \nonumber \\\le & {} \big [ 1-\iota -3\left( 1-\iota \right) ^{-1}\big (1+\tfrac{1}{\varrho } \big )\left( \left| \alpha \right| +\left| \beta \right| \right) ^{2}-6\left( 1-\iota \right) ^{-1}\big (1+\tfrac{1}{\varrho } \big )C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta } \Gamma \left( 1+2\left( \delta -1 \right) \right) \big ]^{-1} \nonumber \\{} & {} \times \big \{6\left( 1\!-\iota \right) ^{-1}\big [C^{2} L_{1}^{2} \gamma ^{-2}\!+\big (1+\varrho \big )\tilde{C}^{2}L_{2}^{2} \gamma ^{-1}\!+2\big (1\!+\varrho \big )C^{2}\big ( L_{3}^{2}\gamma ^{-1}\!+ \big ( \underset{t_{k}<t}{\sum }q_{k}\big )^{2}e^{-2\gamma \left( t-t_{k}\right) }\big )\big ] \big \} \nonumber \\{} & {} \cdot \underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{n}\left( s\right) - x^{n-1}\left( s\right) \right\| ^{2}\nonumber \\&\vdots&\nonumber \\\le & {} \frac{\epsilon ^{n} \underset{s\in \left[ 0 ,t \right] }{\sup }E\left\| x^{1}\left( s \right) -x ^{0}\left( s \right) \right\| ^{2} }{\left( 1-N \right) ^{n} } , \end{aligned}$$
(3.21)

where

$$\begin{aligned} \epsilon =6\left( 1-\iota \right) ^{-1}\big [C^{2} L_{1}^{2} \gamma ^{-2}+\big (1+\varrho \big )\tilde{C}^{2}L_{2}^{2} \gamma ^{-1}+2\big (1+\varrho \big )C^{2}\big ( L_{3}^{2}\gamma ^{-1}+ \big ( \underset{t_{k}<t}{\sum }q_{k}\big )^{2}e^{-2\gamma \left( t-t_{k}\right) }\big )\big ], \end{aligned}$$

and

$$\begin{aligned} N=\iota +3\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )\left( \left| \alpha \right| +\left| \beta \right| \right) ^{2}+6\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta } \Gamma \left( 1+2\left( \delta -1 \right) \right) . \end{aligned}$$

Denote

$$\begin{aligned} \kappa&: =&\left( 1-N \right) ^{-1}; \\ \mu&: =&\iota +3\left( 1\!-\iota \right) ^{-1}\big (1\!+\tfrac{1}{\varrho } \big )C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta }\Gamma \left( 1\!+2\left( \delta \! -1 \right) \right) \!+6 \left( 1\!-\iota \right) ^{-1}\big (1\!+\varrho \big )C^{2}L_{3}^{2}\gamma ^{-1}\\{} & {} +6\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )C^{2}\big ( \underset{t_{k}<t }{\sum }q_{k} \big )^{2}+3\left( 1-\iota \right) ^{-1} C^{2}L_{1}^{2}\gamma ^{-2} +3\left( 1-\iota \right) ^{-1} C^{2}L_{2}^{2}\gamma ^{-1} ;\\ \varsigma&: =&3\left( 1\!-\iota \right) ^{-1}\big (1+\tfrac{1}{\varrho } \big )C_{1-\delta }^{2}L^{2}\gamma ^{-2\delta }\Gamma \left( 1\!+2\left( \delta -1 \right) \right) \!+6 \left( 1\!-\iota \right) ^{-1}\big (1\!+\varrho \big )C^{2}L_{3}^{2}\gamma ^{-1}\\{} & {} +6\left( 1-\iota \right) ^{-1}\big (1+\varrho \big )C^{2}\big ( \underset{t_{k}<t }{\sum }q_{k} \big )^{2}+3\left( 1-\iota \right) ^{-1} C^{2}L_{1}^{2}\gamma ^{-2} +3\left( 1-\iota \right) ^{-1}C^{2}L_{2}^{2}\gamma ^{-1} .\\ \end{aligned}$$

Due to \(\underset{s\in \left[ 0 ,t \right] }{\sup }E\left\| x^{0}\left( s \right) \right\| ^{2}= \underset{s\in \left[ 0 ,t \right] }{\sup }E\left\| S\left( s \right) \phi \left( 0 \right) \right\| ^{2} \le \tilde{C}^{2}E\left\| \phi \left( 0 \right) \right\| ^{2},\) and

$$\begin{aligned} \underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{1}\left( s \right) -x^{0}\left( s \right) \right\| ^{2}\le & {} \kappa \big \{ \epsilon \underset{s\in \left[ 0 ,t \right] }{\sup }E\left\| x^{0}\left( s \right) \right\| ^{2} +\mu \underset{s\in \left[ -\lambda ,0 \right] }{\sup }E\left\| \phi \left( s \right) \right\| ^{2}\\{} & {} +\varsigma \underset{s\in \left[ -\lambda ,0 \right] }{\sup }E\left\| \varphi \left( s \right) \right\| ^{2} \big \} . \end{aligned}$$

Hence, we can see that \(\underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{1}\left( s \right) -x^{0}\left( s \right) \right\| ^{2} \) is bounded. Furthermore, for any \(m>n\ge 1, \)

$$\begin{aligned} \begin{aligned} \underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{m}\left( s \right) -x^{n}\left( s \right) \right\| ^{2}&\le \sum _{k=n}^{m-1 } \underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{k+1}\left( s \right) -x^{k} \left( s \right) \right\| ^{2}\\&\le \sum _{k=n}^{m-1 }\tfrac{\epsilon ^{k} }{\left( 1-N \right) ^{k} } E\left\| x^{1}\left( s \right) -x^{0}\left( s \right) \right\| ^{2}. \end{aligned} \end{aligned}$$

On account of \(E\left\| x^{1}\left( s \right) -x^{0}\left( s \right) \right\| ^{2}< \infty \) , and \(\frac{\epsilon }{1-N} < 1\), then we have

$$\begin{aligned} \sum _{k=n}^{\infty }\big (\tfrac{\epsilon }{\left( 1-N\right) } \big )^{k}E\left\| x^{1}\left( s \right) -x^{0}\left( s \right) \right\| ^{2}<\infty , \end{aligned}$$

which means that the sequence \(\big (\tfrac{\epsilon }{\left( 1-N\right) } \big )^{k}E\left\| x^{1}\left( s \right) -x^{0}\left( s \right) \right\| ^{2}\) is converged. Therefore, for any \(\epsilon >0\), exists a \( N^{*}>0 \), when \(n,m> N^{*} \), we have \(\underset{0\le s\le t}{\sup }E\left\| x^{m} \left( s \right) -x^{n}\left( s \right) \right\| ^{2}\)\(<\epsilon .\) This shows that \(\left\{ x^{n}\left( t \right) ,n\ge 0 \right\} \) is a Cauchy sequence.

Step 3. We shall prove the existence and uniqueness of the mild solution of the system (2.2). According to Step 2., it follows that there exists a solution of \(x\left( t \right) \!\in \! H\), such that

$$\begin{aligned} \lim _{n \rightarrow \infty }E\underset{s\in \left[ 0 ,t \right] }{\sup }\left\| x^{n}\left( s \right) -x\left( s \right) \right\| ^{2}=0. \end{aligned}$$

Borel-Cantelli lemma implies that \( x^{n}\left( t \right) \) converges to \( x\left( t \right) \) almost surely uniformly on \( 0\le t\le T\). Hence, taking limits on both sides of Eq. 3.18, we can get that

$$\begin{aligned} \lim _{n \rightarrow \infty } x^{n}\left( t\right)= & {} S\left( t\right) \left( \phi \left( 0\right) +u\left( 0,\phi ,\varphi \right) \right) -\lim _{n \rightarrow \infty } u\left( t,x^{n}\left( t\right) ,x_{t}^{n}\right) \nonumber \\{} & {} -\lim _{n \rightarrow \infty } \int _{0}^{t}AS\left( t-s\right) u\left( s,x^{n}\left( s\right) ,x_{s}^{n}\right) ds \\{} & {} +\lim _{n \rightarrow \infty } \int _{0}^{t}S\left( t-s\right) f\left( s,x^{n-1}\left( s\right) ,x_{s}^{n-1}\right) ds\nonumber \\{} & {} +\lim _{n \rightarrow \infty }\int _{0}^{t}S\left( t-s\right) g\left( s,x^{n-1}\left( s\right) ,x_{s}^{n-1}\right) d\omega \left( s\right) \\{} & {} +\lim _{n \rightarrow \infty } \underset{0<t_{k}<t}{\sum }S\left( t-t_{k} \right) I_{k}x^{n-1}\left( t_{k}^{-} \right) \nonumber \\{} & {} +\lim _{n \rightarrow \infty } \int _{0}^{t}\int _{Z}S\left( t-s \right) h\left( s,x^{n-1}\left( s \right) ,x_{s}^{n-1},y\right) \widetilde{N}\left( ds,dy \right) \nonumber \\{} & {} +\alpha \lim _{n \rightarrow \infty }\underset{s\in \left[ 0 ,t \right] }{\sup }x^{n} \left( s \right) + \beta \lim _{n \rightarrow \infty }\underset{s\in \left[ 0 ,t \right] }{\inf }x^{n} \left( s \right) . \end{aligned}$$

So, \(x\left( t \right) \) is a solution of the system (2.2), which yields the existence of the equation. The uniqueness of the solutions could be obtained by the similar procedure as Step 2. Assuming that \(x\left( t \right) \) and \(v\left( t \right) \) are the solutions of the system (2.2) respectively, then we have

$$\begin{aligned} x^{n}\left( t\right) -v^{n}\left( t \right)= & {} u\left( t,v^{n}\left( t\right) ,v_{t}^{n}\right) -u\left( t,x^{n}\left( t\right) ,v_{t}^{n}\right) \nonumber \\{} & {} + \int _{0}^{t}AS\left( t-s\right) \left[ u\left( s,v^{n}\left( s\right) ,v_{s}^{n}\right) -u\left( s,x^{n}\left( s\right) ,x_{s}^{n}\right) \right] ds\nonumber \\{} & {} +\int _{0}^{t}S\left( t-s\right) \left[ f\left( s,x^{n-1}\left( s\right) ,x_{s}^{n-1}\right) \!-\!f\left( s,v^{n-1}\left( s\right) ,v_{s}^{n-1}\right) \right] ds\nonumber \\{} & {} +\int _{0}^{t}S\left( t-s\right) \left[ g\left( s,x^{n-1}\left( s\right) ,x_{s}^{n-1}\right) \!-\!g\left( s,v^{n-1}\left( s\right) ,v_{s}^{n-1}\right) \right] d\omega \left( s\right) \nonumber \\{} & {} +\int _{0}^{t}\int _{Z}S\left( t\!-s \right) \left[ h\left( s,x^{n-1}\left( s \right) ,x_{s}^{n-1},y\right) \!-\!h\left( s,v^{n-1}\left( s \right) ,v_{s}^{n-1},y\right) \right] \nonumber \\{} & {} \cdot \widetilde{N}\left( ds,dy \right) +\underset{0<t_{k}<t}{\sum }S\left( t-t_{k} \right) I_{k}\left[ x^{n-1}\left( t_{k}^{-} \right) -v^{n-1}\left( t_{k}^{-} \right) \right] \nonumber \\{} & {} +\alpha \underset{s\in \left[ 0 ,t \right] }{\sup }\left( x^{n}\left( s \right) -v^{n}\left( s \right) \right) +\beta \underset{s\in \left[ 0,t \right] }{\inf }\left( x^{n}\left( s \right) -v^{n}\left( s \right) \right) . \end{aligned}$$

According to Step 2. we can obtain that

$$\underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{n}\left( s \right) -v^{n}\left( s \right) \right\| ^{2} \le \frac{\epsilon ^{n} }{\left( 1-N \right) ^{n}}\underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{0}\left( s \right) -v^{0}\left( s \right) \right\| ^{2}. $$

Note that \(\underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{0}\left( s \right) -v^{0}\left( s \right) \right\| ^{2}< \infty \). Let \(n\rightarrow \infty \), then \(\frac{\epsilon ^{n} }{\left( 1-N \right) ^{n}}\rightarrow 0 \). Therefore, we can derive \(\lim _{n \rightarrow \infty }\underset{s\in \left[ 0,t \right] }{\sup }E\left\| x^{n}\left( s \right) -v^{n}\left( s \right) \right\| ^{2}= 0\). Applying Borel-Cantelli lemma, we know \(x^{n} \left( t \right) \) converges to \( x\left( t \right) \), and \( v^{n} \left( t \right) \) converges to \( v\left( t \right) \) almost surely uniformly on \( 0\le t\le T\). Thus, \(x\left( t\right) =v\left( t\right) \), which means that the solution of the system (2.2) is unique.

4 Example

We shall give an example of doubly perturbed stochastic differential equations with delay

$$\begin{aligned} x\left( t \right)= & {} x_{0} +\int _{0}^{t} f\left( s,x\left( s-\tau \right) \right) ds+\int _{0}^{t}g\left( s,x\left( t-\tau \right) \right) d\omega \left( s \right) \\{} & {} +\int _{0}^{t}\int _{\mathbb {Z} } h\left( s,x\left( s-\tau \right) ,y \right) \widetilde{N}\left( ds,dy \right) +\alpha \underset{s\in \left[ 0 ,t \right] }{\max }x\left( s \right) +\beta \underset{s\in \left[ 0 ,t \right] }{\min }x\left( s \right) , \end{aligned}$$

to explain the existence and uniqueness of the model in Eq. 3.16. Here, set the initial data \(x\left( 0 \right) =x_{0}\), and the mappings \(f\left( t,\cdot \right) \), \( g\left( t,\cdot \right) \) and \(h\left( t,\cdot ,\cdot \right) \) satisfy the following Lipschitz and linear growth conditions:

$$\begin{aligned} \begin{aligned}&\left\| f\left( t,x \right) -f\left( t,y \right) \right\| \le L_{1}\left\| x-y \right\| ,\quad L_{1}>0,\\&\left\| g\left( t,x \right) -g\left( t,y \right) \right\| \le L_{2}\left\| x-y \right\| ,\quad L_{2}>0,\\&\int _{\mathbb {Z} } \left\| h\left( t,x,z \right) -h\left( t,y,z \right) \right\| ^{2}\nu \left( dz \right) \le L_{3}\left\| x-y \right\| ^{2},\quad L_{3}>0. \end{aligned} \end{aligned}$$

Notice that this example is a special case of Eq. 3.16, where there is no linear operator term in the system. This means C is equivalent to 0 in Eq. 3.16.

Using the successive approximation method and the Lipschitz property of the function, for each \(n\ge 1\), there exsits a continuous function \( m\left( t \right) \in C\left[ 0,T \right] \), such that \(E\underset{s\in \left[ 0 ,t \right] }{\max }\ \left\| x^{n} \left( s \right) \right\| ^{2}\le m\left( t \right) \), so we can get \(E\underset{s\in \left[ 0 ,t \right] }{\max }\ \left\| x^{n} \left( s \right) \right\| ^{2}\le C< \infty \). Furthermore, by the Holder inequality, the Doob’s inequality and Assumption (A3), we can immediately obtain that

$$\begin{aligned} \lim _{n,m \rightarrow \infty }\sup E \left( \underset{s\in \left[ 0 ,t \right] }{\max }\left\| x^{n} \left( s \right) -x^{m}\left( s \right) \right\| ^{2} \right) =0. \end{aligned}$$

Lastly, Borel-Cantelli lemma implies that \(x^{n} \left( t \right) \rightarrow x\left( t \right) , v^{n} \left( t \right) \rightarrow v\left( t \right) \) almost surely. Thus, \(x\left( t\right) =v\left( t\right) \), which means that the solution of the system (2.2) is unique.

5 Conclusions and Discussions

Impulsive stochastic differential equations have played an important role in modeling many practical processes, so it is essential to study the properties of such equations, many researchers have proved the existence of mild solution of such equations by using the noncompact measurement strategy and the Mönch fixed point theorem (see, for details, [27,28,29]). In this artical, we studied the existence and uniqueness of mild solution of doubly perturbed INSFDEs with Poisson jumps by successive approximation method, and used Holder inequality and Borel-Cantelli lemma to complete the proof. Since the study of the large deviation theory of SFDEs is a relatively new research direction and has some difficulties, the research literature in this field is relatively rare. In the subsequent research process, we will consider the estimation required to establish the well posedness of the SFDEs based on the variational representation of the general PRM nonnegative functional, so as to obtain the large deviation results of the stochastic functional differential equation.