Stability for stochastic neutral integro-differential equations with infinite delay and Poisson jumps

ABSTRACT

This paper investigates a kind of stochastic neutral integro-differential equations with infinite delay and Poisson jumps in the concrete-fading memory-phase space $C_{\mu }$. We suppose that the linear part has a resolvent operator and the nonlinear terms are globally Lipschitzian. We introduce sufficient conditions that ensure the existence and uniqueness of mild solutions by using successive approximation. Moreover, we target exponential stability, including moment exponential stability in $q$-th ($q\ge 2$) and almost surely exponential stability of solutions and their maps. An example illustrates the potential of the main result.

KEYWORDS:

• Stochastic systems
• stability
• phase space C_μ
• time-delay systems

1. Introduction

Research on stochastic differential equations with delay has received attention over the last few decades because of their appropriateness to describe physical systems subject to delays, such as the ones found in biology, medicine, epidemiology, chemistry, physics, and economics (see Helge et al., 2010; Intissar., 2020; Kostikov & Romanenkov, 2020; Mao, 2007; Trung, 2020 for a brief overview). The qualitative and quantitative properties of solutions of stochastic differential equations with delay, such as the existence, uniqueness, controllability, and stability, have been considered by several authors (see Bouzahir et al., 2017; Dieye et al., 2017; Diop et al., 2014; Taniguchi et al., 2002; Zouine et al., 2020). One point, in particular, has received a lot of attention: the study of existence and asymptotic behavior of mild solutions of some stochastic differential equations on Hilbert spaces, such as the semigroup approach (Taniguchi et al., 2002), comparison theorem (Govindan, 2003), Razumikhin-type theorem (Kai & Yufeng, 2006), analytic technique (Taniguchi, 1998), and Banach fixed-point principle (Diop et al., 2014).

The literature shows many dynamical systems modeled by neutral stochastic partial differential equations 2(2) $\mathrm{d}\nu (t)=\left(A\nu (t)+{\int }_0^t\mathrm{{\rm Y}}(t-s)\nu (s)ds\right)\mathrm{d}t,\mathrm{\forall }t\ge 0,$(2) (Chen et al., 2014; Cui et al., 2011) [9](9) $I_2=\mathrm{E}\underset{0\le s\le t}{sup}\parallel \gamma (s,u_s^n){\parallel }^q\le K_0\mathrm{E}\underset{0\le s\le t}{sup}\parallel u_s^n){\parallel }_{C_{\mu }}^q.$(9) . For these equations, some contain the derivatives of delayed states, which differ from stochastic partial differential equations with delays that depend on the present and past states only (for more details on this theory and its applications, see Mao et al., 2017; Yue, 2014). Stochastic integro-differential equations have been intensively studied, with special attention paid to qualitative properties, such as stability, regularity, periodicity, control problems, and optimality conditions (see Dieye et al., 2019; Diop et al., 2014). Due to the existence of an integral term in the equations, we use here the theory of the resolvent operator instead of the strongly continuous semigroups operator (see Grimmer, 1982 for further details).

Yet, most of the researchers dealing with exponential stability have limited their research to finite delay (see Dieye et al., 2017; Diop et al., 2014). Regarding the infinite delay, most investigations have been done for the case of continuous dependence of solutions on the initial value, considering exponential and asymptotic estimates (see, for instance, the papers Cui & Yan, 2012; Mao et al., 2017; Ren & Xia, 2009; Yue, 2014 for an account on phase spaces). It is noteworthy that few contributions exist for characterizing the exponential stability of stochastic equations with infinite delays (see Jiang et al., 2016; Wu et al., 2017). Jiang et al. (2016) showed the exponential stability for a class of second-order neutral stochastic partial differential equations with infinite delays and impulses using the integral inequality technique. Wu et al. (2017) showed boundedness in the mean square and convergence for both solutions and their maps in the phase space $C_{\mu }$ by using the Itô formula.

Motivated by the above discussion, consider the following neutral stochastic integro-differential equations with infinite delay and Poisson jumps given on the complete probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$:

$\mathrm{d}[u(t)+\gamma (t,u_t)]=A[u(t)+\gamma (t,u_t)]\mathrm{d}t+f_1(t,u_t)\mathrm{d}t$

$+{\int }_0^t\mathrm{{\rm Y}}(t-s)[u(s)+\gamma (s,u_s)]\mathrm{d}s\mathrm{d}t$

(1) $+g_1(t,u_t)dw(t)+{\int }_{\pounds }{\mathrm{\hslash }}_1(t,u_t,z)\tilde{N}(\mathrm{d}t,dz),\mathrm{\forall }t\ge 0.$(1)

System (1) holds with $u_0(.)=\varphi \in C_{\mu }$, where $\varphi$ is ${\mathcal{F}}_0-$measurable, and the definition of the concrete fading memory-phase space $C_{\mu }$ is detailed in the next section. $A:D(A)\subset \mathbb{H}\mapsto \mathbb{H}$ is a closed linear operator on a separable Hilbert space $\mathbb{H}$ and $\mathrm{{\rm Y}}(t)$ represents a closed linear operator such that $\mathrm{{\rm Y}}(t)$ has $D(\mathrm{{\rm Y}}(t))$ as a domain, where $D(A)\subset D(\mathrm{{\rm Y}}(t))$, for all $t\ge 0$. $\gamma ,f_1:[0,+\mathrm{\infty })\times C_{\mu }\mapsto \mathbb{H}$, $g_1:[0,+\mathrm{\infty })\times C_{\mu }\mapsto {\mathcal{L}}_2^0(\mathbb{K},\mathbb{H})$, ${\mathrm{\hslash }}_1:[0,+\mathrm{\infty })\times C_{\mu }\times \pounds \mapsto \mathbb{H}$ are appropriate functions, the history $u_t:(-\mathrm{\infty },0]\mapsto \mathbb{H},t\ge 0$, such that $u_t(\theta )=u(t+\theta )$ belongs to the phase space $C_{\mu }$. The process $w(t)$ represents a Wiener process on a separable Hilbert space $\mathbb{K}$ and $\tilde{N}$ is a compensated Poisson random measure.

To the best of the authors’ knowledge, this paper is the first to present a study of the existence and exponential stability of neutral stochastic integro-differential equations with infinite delay and Poisson jumps. The main contribution of this paper is to find conditions to ensure existence, uniqueness, exponential stability in $q$-th moment for $q\ge 2$ and almost surely exponential stability of solutions and their maps of (1). We show the result using stochastic techniques and the resolvent operator theory, as defined in Grimmer (1982). It is worth mentioning that Diop et al. (2014) studied the system (1) with finite delay. They focused only on the existence of mild solutions and their exponential stability in mean square (Diop et al., 2014). For this reason, our approach can be seen as an extension of the result of Diop et al. (2014) for the infinite delay case.

The organization of this paper is as follows. Some notations and preliminary results are presented in Section 2. The existence and uniqueness of mild solutions for neutral stochastic integro-differential equations with infinite delay are shown in Section 3. Conditions assuring moment exponential stability in the $q$-th ($q\ge 2$) and almost surely exponential stability of the solution $u(t)$, and the solution maps $u_t$, $t\ge 0$ are shown in Section 4. Finally, an example that illustrates our results is presented in Section 5.

2. Notations and preliminary results

Let $\mathbb{H}$ and $\mathbb{K}$ be two real separable Hilbert spaces (c.f. Taniguchi et al., 2002). We let also $\mathcal{L}(\mathbb{K},\mathbb{H})$ be the space of bounded linear operators from $\mathbb{K}$ into $\mathbb{H}$ associated with $\parallel .\parallel$ to represent the norm operator in $\mathbb{H}$, $\mathbb{K}$, and $\mathcal{L}(\mathbb{K},\mathbb{H})$. We assume that System (1) is equipped with a normal filtration$\{{\mathcal{F}}_t{\}}_{t\ge 0}$.

Denote by $N$, the Poisson random measure induced by the $\sigma -$finite stationary ${\mathcal{F}}_t-$adapted Poisson point process $\tilde{p}(.)$ taking values in a measurable space $(\pounds ,\mathcal{B}(\pounds ))$, and define the compensated Poisson random measure $\tilde{N}$ as $\tilde{N}(\mathrm{d}t,dy)=N(\mathrm{d}t,dy)-\pi (dy)\mathrm{d}t$, where $N((0,t]\times \mathrm{\Delta }):={\sum }_{s\in (0,t]}{1}_{\mathrm{\Delta }}(\tilde{p}(s))$ for $\mathrm{\Delta }\in \pounds$ and $\pi$ is the characteristic measure of $N$.

Let $C((-\mathrm{\infty },0],\mathbb{H})$ represent the space of all continuous functions from $(-\mathrm{\infty },0]$ into $\mathbb{H}$ equipped with the norm defined by $\parallel \varphi \parallel =\underset{\theta \le 0}{sup}\parallel \varphi (\theta )\parallel$. For a given $\mu >0$, consider the fading memory $C_{\mu }$ defined as follows: $C_{\mu }:=\{\varphi \in C((-\mathrm{\infty },0],\mathbb{H}):\underset{\theta \to -\mathrm{\infty }}{lim}{e}^{\mu \theta }\varphi (\theta )\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{n}\mathbb{H}\}$ as the chosen phase space in this paper. It is a Banach space with the norm $\parallel \varphi {\parallel }_{\mu }=\underset{-\mathrm{\infty }<\theta \le 0}{sup}{e}^{\mu \theta }\parallel \varphi (\theta )\parallel$. To know more details, see Hino et al. (1991) and Appendix.

Supposed that $\{\omega (t),t\ge 0\}$ represents a $\mathbb{K}$-valued Wiener process which is independent of the Poisson point process on the probability space $(\mathrm{\Omega },\mathcal{F},\{{\mathcal{F}}_t{\}}_{t\ge 0},\mathbb{P})$ with a positive self-adjoint covariance operator $Q$. In addition, we suppose that there exists a complete orthonormal system $e_i$ in $\mathbb{K}$, a bounded sequence of positive real numbers ${\lambda }_i$ such that $Q{e}_i={\lambda }_i{e}_i,i=1,2,\dots$, and a sequence $\{{\beta }_i(t){\}}_{i>1}$ of independent standard Brownian motions such that $\omega (t)={\sum }_{i=1}^{+\mathrm{\infty }}\sqrt{{\lambda }_i}{\beta }_i(t)e_i$ for $t\ge 0$ and ${\mathcal{F}}_t$ is the $\sigma$-algebra generated by $\{\omega (s):0\le s\le t\}$ (see (Taniguchi et al., 2002)). We consider the subspace ${\mathbb{K}}_0=Q^{1/2}\mathbb{K}$ of $\mathbb{K}$, it is a Hilbert space equipped with the inner product $⟨u,v{⟩}_{{\mathbb{K}}_0}=⟨Q^{-1/2}u,Q^{-1/2}v{⟩}_{\mathbb{K}}$. Let ${\mathcal{L}}_2^0={\mathcal{L}}_2({\mathbb{K}}_0,\mathbb{H})$ be the space of all Hilbert-Schmidt operators from ${\mathbb{K}}_0$ to $\mathbb{H}$. ${\mathcal{L}}_2^0$ is a separable Hilbert space endowed with the norm $\parallel v{\parallel }_{{\mathcal{L}}_2^0}=tr((v{Q}^{1/2})(v{Q}^{1/2}{)}^{\ast })$ for any $v\in {\mathcal{L}}_2^0$.

Hereafter, $A$ and $\mathrm{{\rm Y}}(t)$ are closed linear operators on a Banach space denoted by $\mathbb{X}$, and $\mathbb{Y}$ is the Banach space $D(A)$ endowed with the graph norm

$|y{|}_{\mathbb{Y}}:=|Ay|+|y|$ for $y\in \mathbb{Y}$.

The notations $C([0,+\mathrm{\infty });\mathbb{Y})$, $C^1([0,+\mathrm{\infty });\mathbb{X})$ and $\mathcal{L}(\mathbb{Y},\mathbb{X})$ represent the space of continuous functions from $[0,+\mathrm{\infty })$ into $\mathbb{Y}$, the space of continuously differentiable functions from $[0,+\mathrm{\infty })$ into $\mathbb{X}$ and the set of bounded linear operators from $\mathbb{Y}$ into $\mathbb{X}$, respectively.

2.1. Preliminaries on partial integro-differential equations

We now consider the problem

(2) $\mathrm{d}\nu (t)=\left(A\nu (t)+{\int }_0^t\mathrm{{\rm Y}}(t-s)\nu (s)ds\right)\mathrm{d}t,\mathrm{\forall }t\ge 0,$(2)

with $\nu (0)={\nu }_0\in \mathbb{X}$.

Definition 2.1. (Grimmer, 1982). A bounded linear operator-valued function $\mathrm{\Re }(t)\in \mathcal{L}(\mathbb{X})$, $t\ge 0$ is called a resolvent operator for (2) if the next two conditions are satisfied.

(i) $\mathrm{\Re }(0)=I$ and there exist two constants $\alpha \ge 1$ and $\delta$ such that $|\mathrm{\Re }(t)|\le \alpha exp(\delta t)$ for all $t\ge 0$.

(ii) For each element $x$ in $\mathbb{X}$, the function $t\mapsto \mathrm{\Re }(t)x$ is strongly continuous for each $t\ge 0$ and for $x$ in $\mathbb{Y}$, $\mathrm{\Re }(.)x\in C^1([0,+\mathrm{\infty });\mathbb{X})\cap C([0,+\mathrm{\infty });\mathbb{Y})$ and satisfies

$\mathrm{d}\mathrm{\Re }(t)x=\left(A\mathrm{\Re }(t)x+{\int }_0^t\mathrm{{\rm Y}}(t-s)\mathrm{\Re }(s)xds\right)\mathrm{d}t$

$=\left(\mathrm{\Re }(t)Ax+{\int }_0^t\mathrm{\Re }(t-s)\mathrm{{\rm Y}}(s)xds\right)\mathrm{d}t.$

Remark 1. The resolvent operator is said to be exponentially stable when Definition 2.1(i) holds with $\delta <0$.

The following two conditions, borrowed from Grimmer (1982), are sufficient to assure the existence of solutions for equation (2)(2) $\mathrm{d}\nu (t)=\left(A\nu (t)+{\int }_0^t\mathrm{{\rm Y}}(t-s)\nu (s)ds\right)\mathrm{d}t,\mathrm{\forall }t\ge 0,$(2) .

(A₁) The operator $A$ is an infinitesimal generator of a $C_0$-semigroup on $\mathbb{X}$.

(A₂) For all $t\ge 0$,$\mathrm{{\rm Y}}(t)$ denotes a closed, continuous linear operator from $D(A)$ to $\mathbb{X}$ and $\mathrm{{\rm Y}}(t)$ belongs to $\mathcal{L}(\mathbb{Y},\mathbb{X})$. For any $y\in \mathbb{Y}$, the map $t\mapsto \mathrm{{\rm Y}}(t)y$ is bounded, differentiable, and its derivative $\mathrm{d}\mathrm{{\rm Y}}(t)y/\mathrm{d}t$ is bounded and uniformly continuous on $[0,\mathrm{\infty })$.

Lemma 2.2. (Grimmer, 1982). Under Assumptions $\mathbf{(}{\mathbf{A}}_{\mathbf{1}}\mathbf{)}$ and $\mathbf{(}{\mathbf{A}}_{\mathbf{2}}\mathbf{)}$, the existence of a resolvent operator for (2) is guaranteed, and it is unique.

We now recall conditions that assure existence of solutions for the deterministic, integro-differential equation

(3) $\mathrm{d}\nu (t)=\left(A\nu (t)+{\int }_0^t\mathrm{{\rm Y}}(t-s)\nu (s)ds+m(t)\right)\mathrm{d}t,\mathrm{\forall }t\ge 0,$(3)

with $\nu (0)={\nu }_0\in \mathbb{X}$ and $m:[0,+\mathrm{\infty })\to \mathbb{X}$ is a continuous function.

Lemma 2.3. (Grimmer, 1982). Suppose that Assumptions $\mathbf{(}{\mathbf{A}}_{\mathbf{1}}\mathbf{)}$ and $\mathbf{(}{\mathbf{A}}_{\mathbf{2}}\mathbf{)}$ hold, if $\nu$ is a strict solution of (3) (i.e., $\nu (t)$ satisfies (3), for $t\ge 0$ and it belongs to $C^1([0,+\mathrm{\infty }),\mathbb{X}){\bigcap }_C([0,+\mathrm{\infty }),\mathbb{X})$), then

(4) $\nu (t)=\mathrm{\Re }(t){\nu }_0+{\int }_0^t\mathrm{\Re }(t-s)m(s)ds,\mathrm{\forall }t\ge 0,$(4)

Let us give the concept of solutions for the stochastic system in the next definition.

Definition 2.4. A mild solution of (1) is an $\mathbb{H}$-valued process $\{u(t),0\le t\le T\}$ (with $T>0$) which satisfies the next two conditions.

(i) $u(t)$ is ${\mathcal{F}}_t$-adapted and ${\int }_0^T\parallel u(t){\parallel }^q\mathrm{d}t<+\mathrm{\infty }$ almost surely.

(ii) $u(t)$ is continuous for any $t\in [0,T]$ and equals

$u(t)=\mathrm{\Re }(t)[\varphi (0)+\gamma (0,\varphi )]-\gamma (t,u_t)+{\int }_0^t\mathrm{\Re }(t-s)f_1(s,u_s)ds$

(5) $+{\int }_0^t\mathrm{\Re }(t-s)g_1(s,u_s)dw(s)+{\int }_0^t{\int }_{\pounds }\mathrm{\Re }(t-s){\mathrm{\hslash }}_1(s,u_s,z)\tilde{N}(ds,dz),$(5)

with $u_0(.)=\varphi \in C_{\mu }$, where $\mathrm{\Re }(\cdot )$ in (5) represents the resolvent operator of (2).

To achieve our main goal, we assume the following three assumptions:

(A₃) The resolvent operator $\mathrm{\Re }(\cdot )$ satisfying Lemma 2.3 is exponentially stable, i.e., there exist two constants $\lambda >0$ and $M\ge 1$ such that $\parallel \mathrm{\Re }(t)\parallel \le M{e}^{-\lambda t}$, for all $t\ge 0$.

(A₄) Let $q\ge 2$ be an integer, there exists a real number $K_0>0$, such that

$\parallel \gamma (t,\xi )-\gamma (t,\eta ){\parallel }^q\le K_0\parallel \xi -\eta {\parallel }_{C_{\mu }}^q$

for all $\xi ,\eta \in C_{\mu }$, and all $t\ge 0$,

(A₅) Let $q\ge 2$ be an integer, there exists a real number $K_1>0$, such that

$\parallel f_1(t,\xi )-f_1(t,\eta ){\parallel }^q\vee \parallel g_1(t,\xi )-g_1(t,\eta ){\parallel }_{{\mathcal{L}}_2^0}^q$

$\vee {\int }_{\pounds }\parallel {\mathrm{\hslash }}_1(t,\xi ,z)-{\mathrm{\hslash }}_1(t,\eta ,z){\parallel }^q\pi (dz)\le K_1\parallel \xi -\eta {\parallel }_{C_{\mu }}^q$

for all $\xi ,\eta \in C_{\mu }$ and all $t\ge 0$,

In particular, there holds $\gamma (t,0)\equiv f_1(t,0)\equiv g_1(t,0)\equiv {\mathrm{\hslash }}_1(t,0,\cdot )\equiv 0$.

Remark 2. We consider the assumption $\gamma (t,0)=f_1(t,0)=g_1(t,0)={\mathrm{\hslash }}_1(t,0,\cdot )=0$ for all $t\ge 0$, to guarantee that there exists a zero equilibrium solution to the stochastic equation (1)(1) $+g_1(t,u_t)dw(t)+{\int }_{\pounds }{\mathrm{\hslash }}_1(t,u_t,z)\tilde{N}(\mathrm{d}t,dz),\mathrm{\forall }t\ge 0.$(1) . If this assumption does not hold, the equilibrium solution for equation (1)(1) $+g_1(t,u_t)dw(t)+{\int }_{\pounds }{\mathrm{\hslash }}_1(t,u_t,z)\tilde{N}(\mathrm{d}t,dz),\mathrm{\forall }t\ge 0.$(1) can always be transformed into the zero equilibrium solution of another equation.

3. Existence and uniqueness

In this section, we present sufficient conditions to guarantee the existence and uniqueness of mild solutions of the equation in (1). To do so, we use the method of successive approximations and some stochastic analysis techniques. Still, we have to develop some new techniques to deal with infinite delay. Hereafter, we replace $\mathbb{X}$ by the Hilbert space $\mathbb{H}$ in $\mathbf{(}{\mathbf{A}}_{\mathbf{1}}\mathbf{)}$ and $\mathbf{(}{\mathbf{A}}_{\mathbf{2}}\mathbf{)}$. Now, we present the following main result.

Theorem 3.1. Assumed that Assumptions $\mathbf{(}{\mathbf{A}}_{\mathbf{1}}\mathbf{)}$, $\mathbf{(}{\mathbf{A}}_{\mathbf{2}}\mathbf{)}$, $\mathbf{(}{\mathbf{A}}_{\mathbf{4}}\mathbf{)}$ and $\mathbf{(}{\mathbf{A}}_{\mathbf{5}}\mathbf{)}$ hold, and $K_0<\frac{1}{{10}^{q-1}}$. Then, equation (1)(1) $+g_1(t,u_t)dw(t)+{\int }_{\pounds }{\mathrm{\hslash }}_1(t,u_t,z)\tilde{N}(\mathrm{d}t,dz),\mathrm{\forall }t\ge 0.$(1) has a unique mild solution.

Proof. The proof of this theorem uses the following sequence of successive approximations that is defined for $t\le 0$ by $u^n(t)=\varphi (t)$ for any $n\in \mathbb{N}$ and for $0\le t\le T$ by

$u^n(t)=\mathrm{\Re }(t)[\varphi (0)+\gamma (0,\varphi )]-\gamma (t,u_t^n)+{\int }_0^t\mathrm{\Re }(t-s)f_1(s,u_s^{n-1})ds$

(6) $+{\int }_0^t\mathrm{\Re }(t-s)g_1(s,u_s^{n-1})dw(s)+{\int }_0^t{\int }_{\pounds }\mathrm{\Re }(t-s){\mathrm{\hslash }}_1(s,u_s^{n-1},z)\tilde{N}(ds,dz)$(6)

for any $n\ge 1$ and $u^0(t)=\mathrm{\Re }(t)\varphi (0)$ when $0\le t\le T$. Take $M_T=\underset{0\le t\le T}{sup}\parallel \mathrm{\Re }(t){\parallel }_{\mathcal{L}(\mathbb{H})}$, from the uniform boundedness $M_T<\mathrm{\infty }$. The remaining arguments are divided into three main steps.

Step 1: First, let us show that $u^n(t),n\ge 0$ is a bounded sequence. From (6), for any $t\in (-\mathrm{\infty },0]$, we have $E\parallel u^n(t){\parallel }^q=E\parallel \varphi (t){\parallel }^q\le E\parallel \varphi {\parallel }_{C_{\mu }}^q<\mathrm{\infty }$. For any $t\in [0,T]$, we obtain that $E\underset{0\le s\le t}{sup}\parallel u^0(s){\parallel }^q\le M_T^{q}E\parallel \varphi {\parallel }_{C_{\mu }}^q<\mathrm{\infty }$.

If $n\ge 1$, consider the five terms

$I_1:=\mathrm{E}\underset{0\le s\le t}{sup}\parallel \mathrm{\Re }(s))[\varphi (0)+\gamma (0,\varphi )]{\parallel }^q,I_2:=\mathrm{E}\underset{0\le s\le t}{sup}\parallel \gamma (s,u_s^n){\parallel }^q,$

$I_3:=\mathrm{E}\underset{0\le s\le t}{sup}\parallel {\int }_0^s\mathrm{\Re }(s-r)f_1(r,u_r^{n-1})dr{\parallel }^q,I_4:=\mathrm{E}\underset{0\le s\le t}{sup}\parallel {\int }_0^s\mathrm{\Re }(s-r)g_1(r,u_r^{n-1})dw(r){\parallel }^q,$

$I_5:=\mathrm{E}\underset{0\le s\le t}{sup}\parallel {\int }_0^s{\int }_{\pounds }\mathrm{\Re }(s-r){\mathrm{\hslash }}_1(r,u_r^{n-1},z)\tilde{N}(dr,dz){\parallel }^q.$

Considering the definition of the sequence $u^n(t)$, together with the elementary inequality $({\sum }_{i=1}^5|a_i|{)}^q\le 5^{q-1}{\sum }_{i=1}^5|a_i{|}^q$, we have

(7) $\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^n(s){\parallel }^q\le 5^{q-1}(I_1+I_2+I_3+I_4+I_5),$(7)

From Assumption $\mathbf{(}{\mathbf{A}}_{\mathbf{4}}\mathbf{)}$, we obtain

(8) $I_1=\mathrm{E}\underset{0\le s\le t}{sup}\parallel \mathrm{\Re }(s)[\varphi (0)+\gamma (0,\varphi )]{\parallel }^q\le 2^{q-1}{M}_T^q(1+K_0)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q,$(8)

and

(9) $I_2=\mathrm{E}\underset{0\le s\le t}{sup}\parallel \gamma (s,u_s^n){\parallel }^q\le K_0\mathrm{E}\underset{0\le s\le t}{sup}\parallel u_s^n){\parallel }_{C_{\mu }}^q.$(9)

In addition, combining Assumption $\mathbf{(}{\mathbf{A}}_{\mathbf{5}}\mathbf{)}$ and Holder inequality yields

$I_3\le M_T^q\mathrm{E}\underset{0\le s\le t}{sup}({\int }_0^s\parallel f_1(r,u_r^{n-1})\parallel dr{)}^q$

(10) $\le M_T^q{t}^{q-1}{K}_1{\int }_0^t\mathrm{E}\underset{0\le r\le s}{sup}\parallel u_r^{n-1}{\parallel }_{C_{\mu }}^{q}ds.$(10)

Now, using Lemma 6.2 (see Appendix), and by a similar reasoning on $I_3$, we show that

(11) $I_4\le c_q{M}_T^q{t}^{\frac{q-2}{2}}{K}_1{\int }_0^t\mathrm{E}\underset{0\le r\le s}{sup}\parallel u_r^{n-1}{\parallel }_{C_{\mu }}^{q}ds,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{c}_q=(\frac{q(q-1)}{2}{)}^{\frac{q}{2}}.$(11)

On the other hand, combining Assumption $\mathbf{(}{\mathbf{A}}_{\mathbf{5}}\mathbf{)}$, Lemma 6.3 (see Appendix) and Holder inequality on $I_5$, we find that

$I_5\le D_q\left\{\mathrm{E}{\left({\int }_0^t{M}_T^2{K}_1\parallel u_r^{n-1}{\parallel }_{C_{\mu }}^{2}dr\right)}^{\frac{q}{2}}+\mathrm{E}{\int }_0^t{M}_T^q{K}_1\parallel u_r^{n-1}{\parallel }_{C_{\mu }}^{q}dr\right\}$

$\le D_q\left\{M_T^q{K}_1^{\frac{q}{2}}{t}^{\frac{q-2}{2}}{\int }_0^t\mathrm{E}\parallel u_r^{n-1}{\parallel }_{C_{\mu }}^{q}dr+M_T^q{K}_1{\int }_0^t\mathrm{E}\parallel u_r^{n-1}{\parallel }_{C_{\mu }}^{q}dr\right\}$

(12) $\le D_q{M}_T^q{K}_1((t{K}_1{)}^{\frac{q-2}{2}}+1){\int }_0^t\mathrm{E}\underset{0\le r\le s}{sup}\parallel u_r^{n-1}{\parallel }_{C_{\mu }}^{q}ds.$(12)

Substituting (8)–(12) into (7), we obtain

$\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^n(s){\parallel }^q\le {10}^{q-1}{M}_T^q(1+K_0)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q+5^{q-1}{K}_0\mathrm{E}\underset{0\le s\le t}{sup}\parallel u_s^n{\parallel }_{C_{\mu }}^q$

(13) $+{\tilde{C}}_1(q,T){\int }_0^t\mathrm{E}\underset{0\le r\le s}{sup}\parallel u_r^{n-1}{\parallel }_{C_{\mu }}^{q}ds,$(13)

where ${\tilde{C}}_1(q,T)=5^{q-1}{M}_T^q{K}_1(t^{q-1}+c_q{t}^{\frac{q-2}{2}}+D_q\{(t{K}_1{)}^{\frac{q-2}{2}}+1\}).$ By using the definition of the norm $\parallel .{\parallel }_{C_{\mu }}$ (see Appendix), we can write

$\underset{0\le s\le t}{sup}\parallel u_s^n{\parallel }_{C_{\mu }}^q\le \parallel \varphi {\parallel }_{C_{\mu }}^q+\underset{0\le s\le t}{sup}\parallel u^n(s){\parallel }^q.$

Define

${\tilde{C}}_2(q,T)=\frac{{[10}^{q-1}{M}_T^q(1+K_0)+5^{q-1}{K}_0+{\tilde{C}}_1(q,T)T]}{1-5^{q-1}{K}_0}\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q.$

We can write

(14) $\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^n(s){\parallel }^q\le {\tilde{C}}_2(q,T)+\frac{{\tilde{C}}_1(q,T)}{1-5^{q-1}{K}_0}{\int }_0^t\mathrm{E}\underset{0\le r\le s}{sup}\parallel u^{n-1}(r){\parallel }^{q}ds.$(14)

Therefore, for any $k\ge 1$, one can conclude that

$\underset{1\le n\le k}{max}\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^n(s){\parallel }^q\le {\tilde{C}}_2(q,T)+{\tilde{C}}_3(q,T){\int }_0^t\underset{1\le n\le k}{max}\mathrm{E}\underset{0\le r\le s}{sup}\parallel u^{n-1}(r){\parallel }^{q}ds,$

where ${\tilde{C}}_3(q,T)=(1-5^{q-1}{K}_0{)}^{-1}{\tilde{C}}_1(q,T)$. Besides, we have

$\underset{1\le n\le k}{max}\mathrm{E}\parallel u^{n-1}(r){\parallel }^q\le \mathrm{E}\parallel u^0(r){\parallel }^q+\underset{1\le n\le k}{max}\mathrm{E}\parallel u^n(r){\parallel }^q$

$\le M_T^q\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q+\underset{1\le n\le k}{max}\mathrm{E}\parallel u^n(r){\parallel }^q.$

It follows that

$\underset{1\le n\le k}{max}\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^n(s){\parallel }^q\le {\tilde{C}}_2(q,T)+{\tilde{C}}_3(q,T)T{M}_T^q\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q+{\tilde{C}}_3(q,T){\int }_0^t\underset{1\le n\le k}{max}\mathrm{E}\underset{0\le r\le s}{sup}\parallel u^n(r){\parallel }^{q}ds$

$\le {\tilde{C}}_4(q,T)+{\tilde{C}}_3(q,T){\int }_0^t\underset{1\le n\le k}{max}\mathrm{E}\underset{0\le r\le s}{sup}\parallel u^n(r){\parallel }^{q}ds,$

where ${\tilde{C}}_4(q,T)={\tilde{C}}_2(q,T)+{\tilde{C}}_3(q,T)T{M}_T^q\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q.$The Gronwall inequality gives

$\underset{1\le n\le k}{max}\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^n(s){\parallel }^q\le {\tilde{C}}_4(q,T)e^{{\tilde{C}}_3(q,T)t}\le {\tilde{C}}_4(q,T)e^{{\tilde{C}}_3(q,T)T}<\mathrm{\infty }.$

Since $k$ is arbitrary, we have

$\mathrm{E}\parallel u^n(t){\parallel }^q\le {\tilde{C}}_4(q,T)e^{{\tilde{C}}_3(q,T)T}<\mathrm{\infty },0\le t\le T,$

which assures that the sequence $\{u^n,n\in \mathbb{N}\}$ is bounded.

Step 2: Now we show that $u^n,n\in \mathbb{N}$ is a Cauchy sequence. From the construction of successive approximations, we have $u^n(t)=u^{n-1}(t)$ on $(-\mathrm{\infty },0]$, for $n\ge 1$. For $t\in [0,T]$, we can prove that $\parallel u_t^0{\parallel }_{C_{\mu }}^q\le (1+M_T^q)\parallel \varphi {\parallel }_{C_{\mu }}^q$. Therefore, observe from (6) that

(15) $\mathrm{E}\parallel u^1(t)-u^0(t){\parallel }^q\le 5^{q-1}\sum _{i=1}^5{I}_i,$(15)

where $I_1=\mathrm{E}\parallel \mathrm{\Re }(t)\gamma (0,\varphi ){\parallel }^q$. Using $\mathbf{(}{\mathbf{A}}_{\mathbf{4}}\mathbf{)}$, we obtain

(16) $I_1\le M_T^q{K}_0\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q.$(16)

To show the result for $I_2$, we combine $\mathbf{(}{\mathbf{A}}_{\mathbf{4}}\mathbf{)}$ with the inequality $(a+b{)}^q\le 2^{q-1}(a^q+b^q)$ to obtain

$I_2=\mathrm{E}\parallel \gamma (t,u_t^1){\parallel }^q\le K_0\mathrm{E}\parallel u_t^1{\parallel }_{C_{\mu }}^q\le 2^{q-1}{K}_0(\mathrm{E}\parallel u_t^1-u_t^0{\parallel }_{C_{\mu }}^q+\mathrm{E}\parallel u_t^0{\parallel }_{C_{\mu }}^q)$

(17) $\le 2^{q-1}{K}_0\mathrm{E}\parallel u_t^1-u_t^0{\parallel }_{C_{\mu }}^q+2^{q-1}{K}_0(1+M_T^q)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q.$(17)

Regarding $I_3$, combining $\mathbf{(}{\mathbf{A}}_{\mathbf{5}}\mathbf{)}$ and the Holder inequality produces

$I_3=\mathrm{E}\parallel {\int }_0^t\mathrm{\Re }(t-s)f_1(s,u_s^0)ds{\parallel }^q$

$\le M_T^q\mathrm{E}({\int }_0^t\parallel f_1(s,u_s^0)\parallel ds{)}^q$

$\le M_T^q{T}^{q-1}{K}_1\mathrm{E}{\int }_0^t\parallel u_s^0{\parallel }_{C_{\mu }}^{q}ds$

(18) $\le (M_{T}T{)}^q{K}_1(1+M_T^q)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q.$(18)

Similarly to Step $1$, from Lemma 6.2, we have

$I_4=\mathrm{E}\parallel {\int }_0^t\mathrm{\Re }(t-s)g_1(s,u_s^0)dw(s){\parallel }^q$

$\le \mathrm{E}\underset{t\in [0,T]}{sup}\parallel {\int }_0^t\mathrm{\Re }(t-s)g_1(s,u_s^0)dw(s){\parallel }^q$

$\le c_q{M}_T^q{T}^{\frac{q-2}{2}}{K}_1{\int }_0^T\mathrm{E}\parallel u_s^0{\parallel }_{C_{\mu }}^{q}ds$

(19) $\le c_q{M}_T^q{K}_1{T}^{\frac{q}{2}}(1+M_T^q)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q.$(19)

Finally, by employing $\mathbf{(}{\mathbf{A}}_{\mathbf{5}}\mathbf{)}$ and Lemma 6.3, we can proceed similarly to obtain

$I_5=\mathrm{E}\parallel {\int }_0^t{\int }_{\pounds }\mathrm{\Re }(t-s){\mathrm{\hslash }}_1(s,u_s^0,z)\tilde{N}(ds,dz){\parallel }^q$

$\le \mathrm{E}\underset{t\in [0,T]}{sup}\parallel {\int }_0^t{\int }_{\pounds }\mathrm{\Re }(t-s){\mathrm{\hslash }}_1(s,u_s^0,z)\tilde{N}(ds,dz){\parallel }^q$

$\le D_q{M}_T^q{K}_1((T{K}_1{)}^{\frac{q-2}{2}}+1){\int }_0^T\mathrm{E}\parallel u_s^0{\parallel }_{C_{\mu }}^{q}ds$

(20) $\le D_q{M}_T^q{K}_{1}T(1+M_T^q)[(T{K}_1{)}^{\frac{q-2}{2}}+1]\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q.$(20)

Substituting (16)–(20) into (15) results

$\mathrm{E}\parallel u^1(t)-u^0(t){\parallel }^q\le {10}^{q-1}{K}_0\mathrm{E}\parallel u_t^1-u_t^0{\parallel }_{C_{\mu }}^q+5^{q-1}(M_T^q{K}_0+2^{q-1}{K}_0(1+M_T^q)$

$+(M_{T}T{)}^q{K}_1(1+M_T^q)+c_q{M}_T^q{K}_1{T}^{\frac{q}{2}}(1+M_T^q)$

$+D_q{M}_T^q{K}_{1}T(1+M_T^q)[(T{K}_1{)}^{\frac{q-2}{2}}+1])\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q$

$\le {\tilde{C}}_5(q,T)+{10}^{q-1}{K}_0\mathrm{E}\parallel u_t^1-u_t^0{\parallel }_{C_{\mu }}^q,$

where

${\tilde{C}}_5(q,T)=5^{q-1}(M_T^q{K}_0+(1+M_T^q)\{2^{q-1}{K}_0+(M_{T}T{)}^q{K}_1+c_q{M}_T^q{K}_1{T}^{\frac{q}{2}}$

$+D_q{M}_T^q{K}_{1}T[(T{K}_1{)}^{\frac{q-2}{2}}+1]\})\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q.$

On the other hand, note that

$\parallel u_t^1-u_t^0{\parallel }_{C_{\mu }}^q\le \underset{0\le s\le t}{sup}\parallel u^1(t)-u^0(t){\parallel }^q$

and recalling that $1-{10}^{q-1}{K}_0>0$, we can deduce

$\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^1(t)-u^0(t){\parallel }^q\le \frac{{\tilde{C}}_5(q,T)}{1-{10}^{q-1}{K}_0}=:{\tilde{C}}_6(q,T).$

By similar arguments as above, we get

$\mathrm{E}\parallel u^2(t)-u^1(t){\parallel }^q\le 4^{q-1}\left(\mathrm{E}\parallel \gamma (t,u_t^2)-\gamma (t,u_t^1){\parallel }^q+\mathrm{E}\parallel {\int }_0^t\mathrm{\Re }(t-s)[f_1(s,u_s^1)-f_1(s,u_s^0)]ds{\parallel }^q\right.$

$+\mathrm{E}\parallel {\int }_0^t\mathrm{\Re }(t-s)[g_1(s,u_s^1)-g_1(s,u_s^0)]dw(s){\parallel }^q$

$+\mathrm{E}\parallel {\int }_0^t{\int }_{\pounds }\mathrm{\Re }(t-s)[{\mathrm{\hslash }}_1(s,u_s^1,z)-{\mathrm{\hslash }}_1(s,u_s^0,z)]\tilde{N}(ds,dz){\parallel }^q)$

$\le 4^{q-1}{K}_0\mathrm{E}\parallel u_t^2-u_t^1{\parallel }_{C_{\mu }}^q$

$+4^{q-1}{M}_T^q{K}_1[T^{q-1}+c_q{T}^{\frac{q-2}{2}}+D_q((T{K}_1{)}^{\frac{q-2}{2}}+1)]{\int }_0^t\mathrm{E}\parallel u_s^1-u_s^0{\parallel }_{C_{\mu }}^{q}ds$

$\le 4^{q-1}{K}_0\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^2(s)-u^1(s){\parallel }^q$

$+4^{q-1}{M}_T^q{K}_1[T^{q-1}+c_q{T}^{\frac{q-2}{2}}+D_q((T{K}_1{)}^{\frac{q-2}{2}}+1)]t{\tilde{C}}_6(q,T).$

Therefore

$\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^2(s)-u^1(s){\parallel }^q\le \frac{4^{q-1}{M}_T^q{K}_1[T^{q-1}+c_q{T}^{\frac{q-2}{2}}+D_q((T{K}_1{)}^{\frac{q-2}{2}}+1)]t}{1-4^{q-1}{K}_0}{\tilde{C}}_6(q,T)$

$={\tilde{C}}_7(q,T)t{\tilde{C}}_6(q,T),$

where ${\tilde{C}}_7(q,T):=\frac{4^{q-1}{M}_T^q{K}_1[T^{q-1}+c_q{T}^{\frac{q-2}{2}}+D_q((T{K}_1{)}^{\frac{q-2}{2}}+1)]}{1-4^{q-1}{K}_0}.$

We can also prove that

(21) $\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^3(s)-u^2(s){\parallel }^q\le \frac{{({\tilde{C}}_7(q,T)t)}^2}{2!}{\tilde{C}}_6(q,T).$(21)

Indeed, by repeating the iteration as in, for all $n\ge 0$, we obtain

$\mathrm{E}\underset{0\le s\le t}{sup}\parallel u^{n+1}(s)-u^n(s){\parallel }^q\le \frac{{({\tilde{C}}_7(q,T)t)}^n}{n!}{\tilde{C}}_6(q,T).$

Therefore, for any $m>n\ge 0$, we obtain

$\mathrm{E}\parallel u^m(t)-u^n(t){\parallel }^q\le {\tilde{C}}_6(q,T)\sum _{k=n}^{m-1}\frac{{({\tilde{C}}_7(q,T)t)}^k}{k!}\to 0\mathrm{a}\mathrm{s}n\to +\mathrm{\infty }.$

This argument proves that $u^n(t),n\ge 0$ is a Cauchy sequence in $L^q(\mathrm{\Omega },\mathbb{H}).$

Step 3: Now we prove the existence and uniqueness of the solution of equation (1)(1) $+g_1(t,u_t)dw(t)+{\int }_{\pounds }{\mathrm{\hslash }}_1(t,u_t,z)\tilde{N}(\mathrm{d}t,dz),\mathrm{\forall }t\ge 0.$(1) . One has that $u^n(t)\to u(t)$ as $n\to \mathrm{\infty }$ in $L^q$. The Borel–Cantelli lemma gives us $u^n(t)$ uniformly converge to $u(t)$ as $n\to \mathrm{\infty }$, for $t\in (-\mathrm{\infty },T]$. Using Assumption $\mathbf{(}{\mathbf{A}}_{\mathbf{4}}\mathbf{)}$ and $\mathbf{(}{\mathbf{A}}_{\mathbf{5}}\mathbf{)}$, for all $t\in [0,T]$, we can prove the next inequality holds:

$\mathrm{E}\parallel \gamma (t,u_t^n)-\gamma (t,u_t){\parallel }^q\le K_0\mathrm{E}\parallel u_t^n-u_t{\parallel }_{C_{\mu }}^q$

$\le K_0\mathrm{E}\underset{0\le s\le t}{sup}\parallel u(s{)}^n-u(s){\parallel }^q\to 0,\mathrm{a}\mathrm{s}n\to \mathrm{\infty }.$

Note that

$\mathrm{E}\parallel {\int }_0^t\mathrm{\Re }(t-s)[f_1(s,u_s^n)-f_1(s,u_s)]ds{\parallel }^q$

$\le M_T^q{T}^{q-1}{K}_1{\int }_0^t\mathrm{E}\parallel u_s^n-u_s{\parallel }_{C_{\mu }}^{q}ds\to 0,\mathrm{a}\mathrm{s}n\to \mathrm{\infty },$

Similarly, we have

$\mathrm{E}\parallel {\int }_0^t\mathrm{\Re }(t-s)[g_1(s,u_s^n)-g_1(s,u_s)]dw(s){\parallel }^q$

$\le c_q{M}_T^q{T}^{\frac{q-2}{2}}{K}_1{\int }_0^t\mathrm{E}\parallel u_s^n-u_s{\parallel }_{C_{\mu }}^{q}ds\to 0,\mathrm{a}\mathrm{s}n\to \mathrm{\infty },$

and

$\mathrm{E}\parallel {\int }_0^t{\int }_{\pounds }\mathrm{\Re }(t-s)[{\mathrm{\hslash }}_1(s,u_s^n,z)-{\mathrm{\hslash }}_1(s,u_s,z)]\tilde{N}(dr,dz){\parallel }^q$

$\le D_q{M}_T^q\{(T{K}_1{)}^{\frac{q-2}{2}}+1\}{K}_1{\int }_0^t\mathrm{E}\parallel u_s^n-u_s{\parallel }_{C_{\mu }}^{q}ds\to 0,\mathrm{a}\mathrm{s}n\to \mathrm{\infty }.$

Therefore, we take the limits on both sides of (6) with respect to $n$ to obtain

$u(t)=\mathrm{\Re }(t)[\varphi (0)+\gamma (0,\varphi )]-\gamma (t,u_t)+{\int }_0^t\mathrm{\Re }(t-s)f_1(s,u_s)ds$

$+{\int }_0^t\mathrm{\Re }(t-s)g_1(s,u_s)dw(s)+{\int }_0^t{\int }_{\pounds }\mathrm{\Re }(t-s){\mathrm{\hslash }}_1(s,u_s,z)\tilde{N}(ds,dz).$

We can check the uniqueness of the solution by employing the Gronwall lemma, together with a similar argument as that used in the proof of Step $2$. This argument completes the proof.

Remark 3. We point out that the local solution exists and it is unique on $(-\mathrm{\infty },T]$ for each real number $T>0$, then, existence of the solution to equation (1)(1) $+g_1(t,u_t)dw(t)+{\int }_{\pounds }{\mathrm{\hslash }}_1(t,u_t,z)\tilde{N}(\mathrm{d}t,dz),\mathrm{\forall }t\ge 0.$(1) is global, that is, $u(t)$ is defined in $(-\mathrm{\infty },+\mathrm{\infty })$.

4. Exponential stability

Here, we use the Gronwall lemma and the properties of the concrete-phase space $C_{\mu }$ to obtain the exponential stability for the solutions of the stochastic equation (1)(1) $+g_1(t,u_t)dw(t)+{\int }_{\pounds }{\mathrm{\hslash }}_1(t,u_t,z)\tilde{N}(\mathrm{d}t,dz),\mathrm{\forall }t\ge 0.$(1) and their maps. Other researchers have studied the stability as well (Dieye et al., 2017, 2019), but their results are based on the stochastic convolution, an approach completely detached from ours.

Definition 4.1. The mild solution of (1) is said to be $q$-th moment exponentially stable when $(q\ge 2)$ if, for any initial value $\varphi \in C_{\mu }$, ${\mathcal{F}}_0-$measurable, there exist two positive real numbers ${\alpha }_1>0$ and ${\alpha }_2>0$ such that $\mathrm{E}\parallel u(t){\parallel }^q\le {\alpha }_1\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^{q}exp(-{\alpha }_{2}t)$, for all $t\ge 0.$

For the sake of notational simplicity, we define the function

(22) ${\tilde{C}}_8(q)=5^{q-1}{M}^q{K}_1\left[{\lambda }^{-q+1}+c_q{\left(\frac{q-2}{2(q-1)\lambda }\right)}^{\frac{q-2}{2}}+D_q\left\{{\left(\frac{(q-2)K_1}{2(q-1)\lambda }\right)}^{\frac{q-2}{2}}+1\right\}\right].$(22)

Now, we can introduce the main result of this paper in the following theorem.

Theorem 4.1 Suppose that $\mathbf{(}{\mathbf{A}}_{\mathbf{3}}\mathbf{)}$ and all conditions of Theorem 3.1 hold. Suppose in addition that the next two inequalities hold: $q\mu >\lambda$ and

(23) $\frac{{\tilde{C}}_8(q)}{1-5^{q-1}{K}_0}-\lambda <0.$(23)

Then, the mild solution $u(t)$ and the solution maps $u_t$ to equation (1)(1) $+g_1(t,u_t)dw(t)+{\int }_{\pounds }{\mathrm{\hslash }}_1(t,u_t,z)\tilde{N}(\mathrm{d}t,dz),\mathrm{\forall }t\ge 0.$(1) are $q$-th moments exponentially stable.

Proof. $q$-th moment exponential stability of $u(t)$: Combining (5), $\mathbf{(}{\mathbf{A}}_{\mathbf{3}}\mathbf{)}$, $\mathbf{(}{\mathbf{A}}_{\mathbf{4}}\mathbf{)}$ and $\mathbf{(}{\mathbf{A}}_{\mathbf{5}}\mathbf{)}$, we can write

$\mathrm{E}\parallel u(t){\parallel }^q\le 5^{q-1}\mathrm{E}\parallel \mathrm{\Re }(t)(\varphi (0)+\gamma (0,\varphi )){\parallel }^q+5^{q-1}\mathrm{E}\parallel \gamma (t,u_t){\parallel }^q$

$+5^{q-1}\mathrm{E}\parallel {\int }_0^t\mathrm{\Re }(t-s)f_1(s,u_s)ds{\parallel }^q+5^{q-1}\mathrm{E}\parallel {\int }_0^t\mathrm{\Re }(t-s)g_1(s,u_s)d\omega (s){\parallel }^q$

(24) $+5^{q-1}\mathrm{E}\parallel {\int }_0^t{\int }_{\pounds }\mathrm{\Re }(t-s){\mathrm{\hslash }}_1(s,u_s,z)\tilde{N}(ds,dz){\parallel }^q.$(24)

Note that

$I_1:=\mathrm{E}\parallel \mathrm{\Re }(t)(\varphi (0)+\gamma (0,\varphi )){\parallel }^q\le 2^{q-1}(\mathrm{E}\parallel \mathrm{\Re }(t)\varphi (0){\parallel }^q+\mathrm{E}\parallel \mathrm{\Re }(t)\gamma (0,\varphi ){\parallel }^q)$

(25) $\le 2^{q-1}{M}^q{e}^{-q\lambda t}(1+K_0)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q,$(25)

and that

(26) $I_2:=\mathrm{E}\parallel \gamma (t,u_t){\parallel }^q\le K_0\mathrm{E}\parallel u_t{\parallel }_{C_{\mu }}^q.$(26)

By Holder inequality, it yields that

$I_3:=\mathrm{E}\parallel {\int }_0^t\mathrm{\Re }(t-s)f_1(s,u_s)ds{\parallel }^q\le \mathrm{E}({\int }_0^{t}M{e}^{-\lambda (t-s)}\parallel f_1(s,u_s)\parallel ds{)}^q$

$\le M^q\mathrm{E}{\left({\int }_0^t{e}^{\frac{-\lambda (q-1)(t-s)}{q}}.e^{\frac{-\lambda (t-s)}{q}}\parallel f_1(s,u_s)\parallel ds\right)}^q$

$\le M^q{\left({\int }_0^t{e}^{-\lambda (t-s)}ds\right)}^{q-1}\mathrm{E}{\int }_0^t{e}^{-\lambda (t-s)}\parallel f_1(s,u_s){\parallel }^{q}ds.$

We note that ${\int }_0^t{e}^{-\lambda (t-s)}ds<{\lambda }^{-1}$; therefore, the last inequality becomes

(27) $I_3\le M^q(\frac{1}{\lambda }{)}^{q-1}{K}_1{\int }_0^t{e}^{-\lambda (t-s)}\mathrm{E}\parallel u_s{\parallel }_{C_{\mu }}^{q}ds.$(27)

Recalling Lemma 6.2 and Assumption $\mathbf{(}{\mathbf{A}}_{\mathbf{3}}\mathbf{)}$, as before, we use Holder inequality to get that

(28) $I_4:=\mathrm{E}{∥{\int }_0^t\mathrm{\Re }(t-s)g_1(s,u_s)d\omega (s)∥}^q\le c_q{M}^q{K}_1{\left(\frac{q-2}{2(q-1)\lambda }\right)}^{\frac{q-2}{2}}{\int }_0^t{e}^{-\lambda (t-s)}\mathrm{E}\parallel u_s{\parallel }_{C_{\mu }}^{q}ds.$(28)

Finally, from Lemma 6.3, Assumptions $\mathbf{(}{\mathbf{A}}_{\mathbf{3}}\mathbf{)}$ and $\mathbf{(}{\mathbf{A}}_{\mathbf{5}}\mathbf{)}$ and by Holder inequality, we can write

$I_5:=\mathrm{E}{∥{\int }_0^t{\int }_{\pounds }\mathrm{\Re }(t-s){\mathrm{\hslash }}_1(s,u_s,z)\tilde{N}(ds,dz)∥}^q$

$\le D_q\left\{\mathrm{E}{\left(M^2{\int }_0^t{K}_1{e}^{-2\lambda (t-s)}\parallel u_s{\parallel }_{C_{\mu }}^{2}ds\right)}^{\frac{q}{2}}+\mathrm{E}{\int }_0^t{M}^q{K}_1{e}^{-q\lambda (t-s)}\parallel u_s{\parallel }_{C_{\mu }}^{q}ds\right\}$

$\le D_q{M}^q{K}_1\left\{K_1^{\frac{q-2}{2}}\mathrm{E}{\left({\int }_0^t{e}^{-2\lambda (t-s)}\parallel u_s{\parallel }_{C_{\mu }}^{2}ds\right)}^{\frac{q}{2}}+\mathrm{E}{\int }_0^t{e}^{-q\lambda (t-s)}\parallel u_s{\parallel }_{C_{\mu }}^{q}ds\right\},$

which implies that

$I_5\le D_q{M}^q{K}_1\left\{K_1^{\frac{q-2}{2}}{\left({\int }_0^t{e}^{\frac{-2\lambda (q-1)(t-s)}{q-2}}ds\right)}^{\frac{q-2}{2}}.\mathrm{E}{\int }_0^t{e}^{-\lambda (t-s)}\parallel u_s{\parallel }_{C_{\mu }}^{q}ds+\mathrm{E}{\int }_0^t{e}^{-\lambda (t-s)}\parallel u_s{\parallel }_{C_{\mu }}^{q}ds\right\}$

(29) $\le D_q{M}^q{K}_1\left\{{\left(\frac{(q-2)K_1}{2(q-1)\lambda }\right)}^{\frac{q-2}{2}}+1\right\}{\int }_0^t{e}^{-\lambda (t-s)}\mathrm{E}\parallel u_s{\parallel }_{C_{\mu }}^{q}ds.$(29)

If $q=2$, the last two inequalities hold true with convention $0^0:=1$. Substituting (25)–(29) into (24), we obtain

$\mathrm{E}\parallel u(t){\parallel }^q\le {10}^{q-1}{M}^q{e}^{-q\lambda t}(1+K_0)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q+5^{q-1}{K}_0\mathrm{E}\parallel u_t{\parallel }_{C_{\mu }}^q$

(30) $+{\tilde{C}}_8(q)e^{-\lambda t}{\int }_0^t{e}^{\lambda s}\mathrm{E}\parallel u_s{\parallel }_{C_{\mu }}^{q}ds,$(30)

where ${\tilde{C}}_8(q)$ satisfies (22). Multiplying both sides of (30) by $e^{\lambda t}$ yields

$e^{\lambda t}\mathrm{E}\parallel u(t){\parallel }^q\le {10}^{q-1}{M}^q(1+K_0)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q+5^{q-1}{K}_0{e}^{\lambda t}\mathrm{E}\parallel u_t{\parallel }_{C_{\mu }}^q$

(31) $+{\tilde{C}}_8(q){\int }_0^t{e}^{\lambda s}\mathrm{E}\parallel u_s{\parallel }_{C_{\mu }}^{q}ds.$(31)

From properties of the norm $\parallel .{\parallel }_{C_{\mu }}$ (see Appendix), we can write for any $t\ge 0$

$\parallel u_t{\parallel }_{C_{\mu }}^q\le e^{-q\mu t}\parallel \varphi {\parallel }_{C_{\mu }}^q+\underset{0\le s\le t}{sup}\parallel u(s){\parallel }^q,$

which implies that

$\mathrm{E}{e}^{\lambda t}\parallel u(t){\parallel }^q\le {10}^{q-1}{M}^q(1+K_0)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q+5^{q-1}{K}_0\left(e^{(\lambda -q\mu )t}\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q+\mathrm{E}\underset{0\le s\le t}{sup}{e}^{\lambda s}\parallel u(s){\parallel }^q\right)$

$+{\tilde{C}}_8(q){\int }_0^t\left(e^{(\lambda -q\mu )s}\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q+\mathrm{E}\underset{0\le r\le s}{sup}{e}^{\lambda r}\parallel u(r){\parallel }^q\right)ds.$

Recall that $K_0<\frac{1}{{10}^{q-1}}$ and $\lambda -q\mu <0$, hence

$\mathrm{E}\underset{0\le s\le t}{sup}{e}^{\lambda s}\parallel u(s){\parallel }^q\le \frac{1}{1-5^{q-1}{K}_0}\left[{10}^{q-1}{M}^q(1+K_0)+5^{q-1}{K}_0\right]\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q$

$+\frac{{\tilde{C}}_8(q)}{(1-5^{q-1}{K}_0)(q\mu -\lambda )}\left[1-e^{(\lambda -q\mu )t}\right]\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q$

$+\frac{{\tilde{C}}_8(q)}{1-5^{q-1}{K}_0}{\int }_0^t\mathrm{E}\underset{0\le r\le s}{sup}{e}^{\lambda r}\parallel u(r){\parallel }^{q}ds$

$\le \frac{1}{1-5^{q-1}{K}_0}\left[{10}^{q-1}{M}^q(1+K_0)+5^{q-1}{K}_0+\frac{{\tilde{C}}_8(q)}{q\mu -\lambda }\right]\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q$

$+\frac{{\tilde{C}}_8(q)}{1-5^{q-1}{K}_0}{\int }_0^t\mathrm{E}\underset{0\le r\le s}{sup}{e}^{\lambda r}\parallel u(r){\parallel }^{q}ds$

$={\tilde{C}}_9(q)+{\tilde{C}}_{10}(q){\int }_0^t\mathrm{E}\underset{0\le r\le s}{sup}{e}^{\lambda r}\parallel u(r){\parallel }^{q}ds,$

where ${\tilde{C}}_9(q):=\frac{{10}^{q-1}{M}^q(1+K_0)+5^{q-1}{K}_0+\frac{{\tilde{C}}_8(q)}{q\mu -\lambda }}{1-5^{q-1}{K}_0}\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q$ and ${\tilde{C}}_{10}(q):=\frac{{\tilde{C}}_8(q)}{1-5^{q-1}{K}_0}$. Using the Gronwall lemma, we get

$\mathrm{E}\underset{0\le s\le t}{sup}{e}^{\lambda s}\parallel u(s){\parallel }^q\le {\tilde{C}}_9(q)e^{{\tilde{C}}_{10}(q)t},$

which implies that

(32) $\mathrm{E}\parallel u(t){\parallel }^q\le {\tilde{C}}_9(q)e^{({\tilde{C}}_{10}(q)-\lambda )t}.$(32)

Therefore, from the condition in (23), the result of the $q$-th moment exponential stability of solution $u(t)$ is satisfied.

$q$-th moment exponential stability of $u_t$: For any $t\ge 0$, we have

$\parallel u_t{\parallel }_{C_{\mu }}^q\le e^{-q\mu t}\parallel \varphi {\parallel }_{C_{\mu }}^q+\underset{0\le s\le t}{sup}\parallel u(s){\parallel }^q,$

multiplying both sides of the last inequality by $e^{\lambda t}$, we obtain

$e^{\lambda t}\parallel u_t{\parallel }_{C_{\mu }}^q\le \parallel \varphi {\parallel }_{C_{\mu }}^q+\underset{0\le s\le t}{sup}{e}^{\lambda s}\parallel u(s){\parallel }^q.$

From (31), we have

$\mathrm{E}\underset{0\le s\le t}{sup}{e}^{\lambda s}\parallel u_s{\parallel }_{C_{\mu }}^q\le \frac{1}{1-5^{q-1}{K}_0}\left[\left(1+{10}^{q-1}{M}^q(1+K_0)\right)\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q\right]$

$+{\tilde{C}}_{10}(q){\int }_0^t\mathrm{E}\underset{0\le r\le s}{sup}{e}^{\lambda r}\parallel u_r{\parallel }_{C_{\mu }}^{q}ds.$

Using the Gronwall lemma, we obtain

$\mathrm{E}\underset{0\le s\le t}{sup}{e}^{\lambda s}\parallel u_s{\parallel }_{C_{\mu }}^q\le \frac{1}{1-5^{q-1}{K}_0}\left([1+{10}^{q-1}{M}^q(1+K_0)]\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q\right)e^{{\tilde{C}}_{10}(q)t},$

which results in

(33) $\mathrm{E}\parallel u_t{\parallel }_{C_{\mu }}^q\le \frac{1}{1-5^{q-1}{K}_0}\left([1+{10}^{q-1}{M}^q(1+K_0)]\mathrm{E}\parallel \varphi {\parallel }_{C_{\mu }}^q\right)e^{({\tilde{C}}_{10}(q)-\lambda )t}.$(33)

The inequality in (33) assures the exponential stability in $q$-th moment of the solution maps $u_t$.□

4.1 Almost surely exponential stability

Definition 4.2. The mild solution of (1) is said to be almost surely exponentially stable if the following inequality is guaranteed almost surely

$\underset{t\to \mathrm{\infty }}{limsup}\frac{1}{t}log\parallel u(t)\parallel <0,$

for any ${\mathcal{F}}_0-$measurable initial value $\varphi \in C_{\mu }$.

We present the main result of this section.

Theorem 4.2. Assume that all the conditions of Theorem 4.1 hold true. Then

$(i)\underset{t\to \mathrm{\infty }}{limsup}\frac{1}{t}log\parallel u(t)\parallel \le \frac{({\tilde{C}}_{10}(q)-\lambda )\epsilon }{q}\mathrm{a}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{y},$

$(ii)\underset{t\to \mathrm{\infty }}{limsup}\frac{1}{t}log\parallel u_t{\parallel }_{C_{\mu }}\le \frac{({\tilde{C}}_{10}(q)-\lambda )\epsilon }{q}\mathrm{a}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{y},$

for any $\epsilon \in (0,1)$, which implies that $u(t)$ and $u_t$ are almost surely exponentially stable.

Proof. Now we show $(i)$ for all $n\ge 0$. It follows from Theorem 4.1 that

$\mathrm{E}\underset{n\le t\le n+1}{sup}\parallel u(t){\parallel }^q\le {\tilde{C}}_9(q)e^{({\tilde{C}}_{10}(q)-\lambda )(n+1)}={\tilde{C}}_9(q)e^{{\tilde{C}}_{10}(q)-\lambda }.e^{({\tilde{C}}_{10}(q)-\lambda )n}.$

Let $I_n$ be the interval $[n,n+1]$, for any $\epsilon \in (0,1)$. Define

${\tilde{C}}_{10}(q)=\frac{{\tilde{C}}_8(q)}{1-5^{q-1}{K}_0}.$

Since, from assumption, $1-\epsilon >0$ and ${\tilde{C}}_{10}(q)-\lambda <0$, we can use the Markov inequality to write

(34) $\mathbb{P}\left(\underset{t\in I_n}{sup}\parallel u(t){\parallel }^q>e^{({\tilde{C}}_{10}(q)-\lambda )n\epsilon }\right)\le \frac{\mathrm{E}\parallel u(t){\parallel }^q}{e^{({\tilde{C}}_{10}(q)-\lambda )n\epsilon }}$(34)

(35) $\le {\tilde{C}}_9(q)e^{{\tilde{C}}_{10}(q)-\lambda }{e}^{({\tilde{C}}_{10}(q)-\lambda )n(1-\epsilon )},$(35)

The rightmost term of (34) is bounded from above by ${\sum }_{n=0}^{\mathrm{\infty }}{e}^{(1-\epsilon )({\tilde{C}}_{10}(q)-\lambda )n}<\mathrm{\infty }$. Therefore, the Borel-Cantelli lemma assures that there exists an integer $n_0$ such that, for all $n\ge n_0$,

$\underset{t\in I_n}{sup}\parallel u(t){\parallel }^q\le e^{({\tilde{C}}_{10}(q)-\lambda )n\epsilon }\mathrm{a}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{y}.$

Thus, if $t\in I_n$ and $n\ge n_0$, we get

$\frac{1}{t}log\parallel u(t){\parallel }^q\le \frac{1}{n}({\tilde{C}}_{10}(q)-\lambda )n\epsilon =({\tilde{C}}_{10}(q)-\lambda )\epsilon \mathrm{a}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{y}.$

It follows that

$\underset{t\to \mathrm{\infty }}{limsup}\frac{1}{t}log\parallel u(t)\parallel \le \frac{({\tilde{C}}_{10}(q)-\lambda )\epsilon }{q}\mathrm{a}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{y},$

which shows that $(i)$ is satisfied. The argument to prove $(ii)$ follows analogous reasoning. Namely, one can use the fact that the solution maps $u_t$ are $q$-th moment exponentially stable and can conclude the result by repeating that previous reasoning. The details are omitted.□

Remark 4. The author of (Grimmer, 1982) presents sufficient conditions for the exponential stability of the resolvent operator $(R(t){)}_{t\ge 0}$. The paper (Grimmer, 1982) shows $\lambda$ and $M$ from the contraction of the $C_0$-semigroup $(S(t){)}_{t\ge 0}$ and the properties of the function $b$ by using the infinitesimal generator of the translation semigroup.

5. Example

Set $\mu >0$ and $\theta \in C({\mathbb{R}}^{+},(-\mathrm{\infty },0])$. Consider the following neutral stochastic integro-differential equation with infinite delay and Poisson jumps of the form:

(36) $\left(\begin{array}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}[x(t,\xi )+e^{\mu \theta (t)}\mathrm{\Gamma }(t,x(t+\theta (t),\xi ))]\\ =\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\xi }^2}[x(t,\xi )+e^{\mu \theta (t)}\mathrm{\Gamma }(t,x(t+\theta (t),\xi ))]dt\\ +{\int }_0^{t}b(t-s)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\xi }^2}\left[x(t,\xi )+e^{\mu \theta (t)}\mathrm{\Gamma }(t,x(t+\theta (t),\xi ))\right]dsdt\\ +{\int }_{-\mathrm{\infty }}^0{e}^{2\mu s}f(t,s,x(t+s,\xi ))ds\mathrm{d}t+e^{\mu \theta (t)}g(t,x(t+\theta (t),\xi ))d\omega (t)\\ +{\int }_{\pounds }{e}^{\mu \theta (t)}\mathrm{\hslash }(t,x(t+\theta (t),\xi ),z)\tilde{N}(\mathrm{d}t,dz),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\ge 0\mathrm{a}\mathrm{n}\mathrm{d}\xi \in [0,\pi ],\\ x(t,0)=x(t,\pi )=0,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\ge 0\\ x(\theta ,\xi )=x_0(\theta ,\xi ),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\theta \in (-\mathrm{\infty },0]\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{l}\mathrm{l}\xi \in [0,\pi ],\end{array}\right.$(36)

Here, $\omega (t)$ denotes the standard $\mathbb{R}$-valued Wiener process; $\mathrm{\Gamma },g:{\mathbb{R}}^{+}\times \mathbb{R}\to \mathbb{R},f:{\mathbb{R}}^{+}\times (-\mathrm{\infty },0]\times \mathbb{R}\to \mathbb{R}$ and $\mathrm{\hslash }:{\mathbb{R}}^{+}\times \mathbb{R}\times \pounds \to \mathbb{R}$ are continuous functions, $b\in C^1({\mathbb{R}}^{+},\mathbb{R})$ and $x_0\in C_{\mu }$.

Let $\mathbb{H}=L^2(0,\pi )$ with the norm $\parallel .\parallel$, we define $A:D(A)\subset \mathbb{H}\mapsto \mathbb{H}$ by $A=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}$ with domain $D(A)=H^2(0,\pi )\cap H_0^1(0,\pi )$. It is well known that $A$ is the infinitesimal generator of a strongly continuous contraction semigroup $(S(t){)}_{t\ge 0}$ on $\mathbb{H}$. Thus, $\mathbf{(}{\mathbf{A}}_{\mathbf{1}}\mathbf{)}$ holds. Let $\mathrm{{\rm Y}}:D(A)\subset \mathbb{H}\mapsto \mathbb{H}$ be the operator defined by $\mathrm{{\rm Y}}(t)(y)=b(t)Ay$ for $t\ge 0$ and $y\in D(A)$. The resolvent operator $(R(t){)}_{t\ge 0}$ decays exponentially, i.e. $\parallel R(t)\parallel \le M{e}^{-\lambda t}$, see Remark for more details. We now suppose that the next three conditions are valid.

(i) For $t\ge 0,s\le 0$ and $z\in \pounds ,\mathrm{\Gamma }(t,0)=f(t,s,0)=g(t,0)=\mathrm{\hslash }(t,0,z)=0$.

(ii) Let $q\ge 2$, there exist real numbers ${\alpha }_0\in (0,\frac{1}{{10}^{q-1}}),{\alpha }_1>0$ such that

$|\mathrm{\Gamma }(t,x_1)-\mathrm{\Gamma }(t,x_2){|}^q\le {\alpha }_0|x_1-x_2{|}^q$

,

$|g(t,x_1)-g(t,x_2){|}^q\vee {\int }_{\pounds }|\mathrm{\hslash }(t,x_1,z)-\mathrm{\hslash }(t,x_2,z){|}^q\pi (dz)\le {\alpha }_1|x_1-x_2{|}^q$

,

for $t\ge 0$ and $x_1,x_2\in \mathbb{R}$, where $|.|$ denotes the norm of $\mathbb{R}$.

(iii) Let $q\ge 2$, there exists an integrable function $r:(-\mathrm{\infty },0]\to [0,+\mathrm{\infty })$ such that

$|f(t,s,x_1)-f(t,s,x_2){|}^q\le r(s)|x_1-x_2{|}^q$ for $t\ge 0,s\le 0$ and $x_1,x_2\in \mathbb{R}$.

In addition, we assume that ${\left(\frac{q-1}{\mu q}\right)}^{q-1}{\int }_{-\mathrm{\infty }}^{0}r(s)ds\le {\alpha }_1$.

For $t\ge 0,\xi \in [0,\pi ]$ and ${\varphi }_1\in C_{\mu }$ define the operators $\gamma ,f_1:{\mathbb{R}}^{+}\times C_{\mu }\to \mathbb{H},g_1:{\mathbb{R}}^{+}\times C_{\mu }\to {\mathcal{L}}_2^0(\mathbb{R},\mathbb{H})$ and ${\mathrm{\hslash }}_1:{\mathbb{R}}^{+}\times C_{\mu }\times \pounds \to \mathbb{H}$ as follows:

$\gamma (t,{\varphi }_1)(\xi ):=e^{\mu \theta (t)}\mathrm{\Gamma }(t,{\varphi }_1(\theta (t))(\xi ))$

,

$g_1(t,{\varphi }_1)(\xi ):=e^{\mu \theta (t)}g(t,{\varphi }_1(\theta (t))(\xi ))$

,

${\mathrm{\hslash }}_1(t,{\varphi }_1,z)(\xi ):=e^{\mu \theta (t)}\mathrm{\hslash }(t,{\varphi }_1(\theta (t))(\xi ),z)$

,

which contain a variable delay, and

$f_1(t,{\varphi }_1)(\xi ):={\int }_{-\mathrm{\infty }}^0{e}^{2\mu s}f(t,s,{\varphi }_1(s)(\xi ))ds$

contains a distributed delay. Set $u(t)\xi =x(t,\xi )$ for all $t\ge 0$ and all $\xi \in [0,\pi ]$, and $\varphi (\theta )(\xi )=x_0(\theta ,\xi )$ for all $\theta \in (-\mathrm{\infty },0]$ and all $\xi \in [0,\pi ]$. Then, (36) takes the form of the system (1).

Applying $(ii)$ together with the definition of the norm $\parallel .{\parallel }_{C_{\mu }}$, we have that

$\parallel \gamma (t,{\varphi }_1)-\gamma (t,{\varphi }_2){\parallel }^q\le {\alpha }_0\parallel {\varphi }_1-{\varphi }_2{\parallel }_{C_{\mu }}^q$

,

$\parallel g_1(t,{\varphi }_1)-g_1(t,{\varphi }_2){\parallel }_{{\mathcal{L}}_2^0}^q\vee {\int }_{\pounds }\parallel {\mathrm{\hslash }}_1(t,{\varphi }_1,z)-{\mathrm{\hslash }}_1(t,{\varphi }_2,z){\parallel }^q\pi (dz)\le {\alpha }_1\parallel {\varphi }_1-{\varphi }_2{\parallel }_{C_{\mu }}^q$

,

and by using $(iii)$ together with the Holder inequality, we obtain

$\parallel f_1(t,{\varphi }_1)-f_1(t,{\varphi }_2){\parallel }^q\le {\alpha }_1\parallel {\varphi }_1-{\varphi }_2{\parallel }_{C_{\mu }}^q$

,

for any $t\ge 0$ and ${\varphi }_1,{\varphi }_2\in C_{\mu }$. It then follows that all the assumptions of Theorem 3.1 are satisfied with $K_0={\alpha }_0$ and $K_1={\alpha }_1$. As a result, (36) has a mild solution.

For the case $q=2$, by the convention $0^0=1$, we have the following constants

$c_q={\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}=1,D_q=\frac{1}{2},$

${\tilde{C}}_{10}(q)=\frac{5M^2{\alpha }_1\left(\frac{1}{\lambda }+2\right)}{1-5{\alpha }_0}$

.

Thus, by Theorems 4.1 and 4.2, these solutions and their maps are mean square exponentially stable and almost surely exponentially stable provided that

$5M^2{\alpha }_1\left(\frac{1}{{\lambda }^2}+\frac{2}{\lambda }\right)<1-5{\alpha }_0\mathrm{a}\mathrm{n}\mathrm{d}\lambda <2\mu$

.

For a numerical illustration, take $q=4$ and $c_q=36,D_q=D_4$. It follows that

${\tilde{C}}_{10}(q)=\frac{125M^4{\alpha }_1\left(\frac{1}{{\lambda }^3}+\left(12+\frac{D_4{\alpha }_1}{3}\right)\frac{1}{\lambda }+D_4\right)}{1-125{\alpha }_0}.$

The mild solutions and their maps are fourth moment exponentially stable and almost surely exponentially stable provided that

$125M^4{\alpha }_1\left(\frac{1}{{\lambda }^4}+\left(12+\frac{D_4{\alpha }_1}{3}\right)\frac{1}{{\lambda }^2}+\frac{D_4}{\lambda }\right)<1-125{\alpha }_0\mathrm{a}\mathrm{n}\mathrm{d}\lambda <4\mu$

.

6. Conclusion

In this paper, we have studied neutral stochastic integro-differential equations with infinite delay and Poisson jumps under global Lipschitz conditions. In this study, we have used successive approximations to show the existence of mild solutions. We also prove the exponential stability of solutions and their maps. It is uncertain whether our approach copes with weaker conditions, such as local Lipschitz and non-Lipschitz conditions. The results in this paper can be seen as an extension of the ones in (Diop et al., 2014) because we consider the infinite-delay case here; in contrast, the authors of (Diop et al., 2014) have considered the finite delay case.

PUBLIC INTEREST STATEMENT

Stochastic processes are much used to represent mathematical models for phenomena and systems that vary randomly. These phenomena and systems may include the growth of bacterial population, price changes in the stock market, extinction and persistence of diseases, movement of a gas molecule and the number of phone calls. Representing such random phenomena motivates the study of stochastic differential equations. Characterizing the stability of stochastic systems has become a central topic in systems sciences. In this paper, we focus on the stability of stochastic integro-differential equations with noise.

AppendixThe next result is immediate from the definition of the phase space $C_{\mu }$; see (Hino et al., 1991) for more details

Proposition 6.1. The phase space $C_{\mu }$ satisfies the next proprieties.

$(a)$ If $u:(-\mathrm{\infty },T)\to \mathbb{H},T>0$ is such that $u_0\in C_{\mu }$ and $u$ is continuous on $[0,T)$, then for every $t\in [0,T)$, the conditions below hold:

$(i)$ $u_t\in C_{\mu }$$(ii)$ $\parallel u(t)\parallel \le \parallel u_t{\parallel }_{C_{\mu }}$,$(iii)$ $\parallel u_t{\parallel }_{C_{\mu }}\le e^{-\mu t}\parallel u_0{\parallel }_{C_{\mu }}+e^{-\mu t}\underset{0\le s\le t}{sup}{e}^{\mu s}\parallel u(t)\parallel$. $(b)$The corresponding history $t\to u_t$ of the function $u$ in $(a)$ is a $C_{\mu }$-valued continuous function in $[0,T)$. $(c)$ The space $C_{\mu }$ is complete. $(d)$ Suppose that $\{{\phi }^n\}$ is a Cauchy sequence in $C_{\mu }$, and if $\{{\phi }^n(\theta )\}$ converges to $\phi (\theta )$ for $\theta$ on any compact subset of the interval $(-\mathrm{\infty },0]$, then $\phi \in C_{\mu }$ and $\parallel {\phi }^n-\phi {\parallel }_{C_{\mu }}\to 0$ as $n\to 0$.

Lemma 6.2. ([20, Thm. 4.36, p. 114]). For a ${\mathcal{L}}_2^0$-valued predictable process $\psi$ and for any $q\ge 1$, we have the following inequality

$\underset{0\le s\le t}{sup}\mathrm{E}{∥{\int }_0^s\psi (l)dw(l)∥}^{2q}\le (q(2q-1){)}^q{\left({\int }_0^t{(\mathrm{E}\parallel \psi (s){\parallel }_{{\mathcal{L}}_2^0}^{2q})}^{\frac{1}{q}}ds\right)}^q.$

Lemma 6.3. ([18]). Let $\psi :[0,\mathrm{\infty })\times \pounds \mapsto \mathbb{H}$ and suppose that ${\int }_0^t{\int }_{\pounds }\parallel \psi (s,z){\parallel }^q\pi (\mathrm{d}z)\mathrm{d}s<\mathrm{\infty }$, for any $q\ge 2$. Then, there exists a positive real number $D_q>0$ such that

$\mathrm{E}\left(\underset{0\le t\le T}{sup}{∥{\int }_0^t{\int }_{\pounds }\psi (s,z)\tilde{N}(\mathrm{d}s,\mathrm{d}z)∥}^q\right)\le D_q\left\{\mathrm{E}{\left({\int }_0^T{\int }_{\pounds }\parallel \psi (s,z){\parallel }^2\pi (\mathrm{d}z)\mathrm{d}s\right)}^{\frac{q}{2}}\right.$

$\left.+\mathrm{E}{\int }_0^T{\int }_{\pounds }\parallel \psi (s,z){\parallel }^q\pi (\mathrm{d}z)\mathrm{d}s\right\}.$

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Aziz Zouine

Aziz Zouine is a Ph.D. student at the Ibn Zohr University, Morocco. His main research area is stochastic (hybrid) nonlinear differential equations with delay. He holds a Master’s degree in Mathematics and Applications from the Cadi Ayyad University and a Bachelor's degree in Mathematics from the Ibn Zohr University, Morocco.

Hassane Bouzahir

Dr. Hassane Bouzahir is a Full Professor of Applied Mathematics/Statistics at the National School of Applied Sciences, Ibn Zohr University in Agadir, Morocco. He is the Director of a Research Laboratory on Systems Engineering and Information Technology. His research interests include linear operators theory, functional differential equations, partial functional differential equations, stochastic systems and control, modeling, and engineering simulation.

Alessandro N. Vargas

Alessandro N. Vargas received the BS degree in computer engineering from the Universidade Federal do Esprito Santo, Vitória, in 2002, and Master and Ph.D. degrees in electrical engineering from the School of Electrical and Computer Engineering, University of Campinas, Campinas, Brazil, in 2004 and 2009, respectively. Since 2007, he has been a Control Systems Professor with the Universidade Tecnológica Federal do Paraná, Paraná, Brazil. His research interests include stochastic systems and control with applications in electronics, mechatronics, and electrical engineering.