Well posedness of second-order impulsive fractional neutral stochastic differential equations

1.   Introduction

In recent years, fractional differential equations (FDEs) are an effective mathematical tool to model and analyze many real life problems; it has been used by researchers and scientists to get better results than the integer order differential equations. Classical theory and applications of FDEs are presented in the monographs ^[8,11,13]. Stochastic differential equations (SDEs) are the proper apparatus to model systems with external noise and suffered by uncertain or random facts, for more details on SDEs readers can can refer to ^{[1,2,5,6,10,14]}. Very recently, many researchers were devoted to study impulsive fractional integro differential evolution equations Xie ^[18] investigated as follows,

where $\mathfrak{B}\mathrm{y}(\mathrm{t}) = \int^{\mathrm{t}}_{0}\mathfrak{k}(\mathrm{t}, \mathrm{s})y(\mathrm{s})d\mathrm{s}$, $\mathrm{k}\in \mathfrak{C}(\mathfrak{D}, \mathfrak{R}^{+})$, $\mathfrak{D} = \left\{(\mathrm{t}, \mathrm{s}):0\leq \mathrm{s} \leq \mathrm{t}\leq b\right\}$.

On the other hand, Poisson jumps processes are used in modeling for several real life situations. Moreover, many practical applications are used in the field of market crashes, earthquakes, epidemics, etc, . In dynamical systems, a jump term is included to make the model a realistic one. Many literature have been study SDEs driven by Poisson jumps has ^{[3,4,7,9,12,15,16,17]}. However, there is no literature in IFNSDEs, we use of the successive method and Bihari's inequality. This paper is concerned with IFNSDEs driven by Poisson jump,

Here, $\mathfrak{D}^{\alpha}_{\mathrm{t}}$ denotes the Caputo fractional derivative of order $0 < \alpha < 1$; $\mathfrak{A}:\mathscr{D}(\mathfrak{A})\subset \mathfrak{X}\rightarrow \mathfrak{X}$ denotes sectorial operator. The nonlinear maps $\mathfrak{f, g} :[0, a]\times \mathcal{B}\rightarrow \mathfrak{X}$, $\sigma:[0, a]\times \mathcal{B}\rightarrow \mathcal{L}(\mathfrak{Y}, \mathfrak{X})$ and $\mathfrak{h}:[0, a]\times \mathcal{B}\times\mathcal{U}\rightarrow \mathfrak{X}$ are appropriate mappings. Let $\mathcal{B}$ is an abstract phase space. Let $\mathrm{y}_{\mathrm{t}}:(-\infty, 0]\rightarrow \mathfrak{X}$, $\mathrm{y}_{\mathrm{t}}(\mathrm{s}) = \mathrm{y}(\mathrm{t}+\mathrm{s})$, $\mathrm{s}\leq 0 \in \mathcal{B}$. In $\widetilde{\mathfrak{N}}(d\mathrm{s}, d\mathrm{u}) = \mathfrak{N}(d\mathrm{s}, d\mathrm{u})-\mathrm{v}(d\mathrm{u})d\mathrm{s}.$ the Poisson measure $\tilde{\mathcal{N}}(d\mathrm{t}, d\mathrm{u})$ denotes the Poisson counting measure associated with a characteristic measure $\lambda$. Moreover, $\Delta \mathrm{y}(\mathrm{t}_{\mathrm{k}}) = \mathrm{y}(\mathrm{\mathrm{t}}^{+}_{\mathrm{k}})-\mathrm{y}(\mathrm{t}^{-}_{\mathrm{k}})$ for $0\leq \mathrm{t}_{0} < \mathrm{t}_{1} < \cdots < \mathrm{t}_{n} < \mathrm{t}_{n+1} = a$, and $\mathrm{y}(\mathrm{t}^{+}_{\mathrm{k}})$ and $\mathrm{y}(\mathrm{t}^{-}_{\mathrm{k}})$ denote the right and the left limits of $\mathrm{y}(\mathrm{t})$ at $\mathrm{t} = \mathrm{t}_{\mathrm{k}}$, respectively.

2.   Preliminaries

In this section, we dealt with basic definitions for FC and some of the lemmas that are useful for further derivation, (see ^[13,19]).

We introduce the space $\mathscr{PC}$ formed by all $\mathfrak{X}$-valued stochastic processes $\big\{\mathrm{y}(\mathrm{t}):\mathrm{t}\in [0, a]\big\}$ such that $\mathrm{y}$ is continuous at $\mathrm{t}\neq \mathrm{t}_{\mathrm{k}}$, $\mathrm{y}(\mathrm{t}^{-}_{\mathrm{k}}) = \mathrm{y}(\mathrm{t}_{\mathrm{k}})$ and $\mathrm{y}(\mathrm{t}^{+}_{\mathrm{k}})$ exist for all $\mathrm{k} = 1, 2, ..., m$. When $\mathscr{PC}$ is endowed with the norm $\left\|\mathrm{y}\right\|_{\mathscr{PC}} = \left(\sup_{\mathrm{s}\in [0, a]}\mathbb{E}\|\mathrm{y}(\mathrm{s})\|^{2}\right)^{1/2}$, $(\mathscr{PC}, \left\|\cdot\right\|_{\mathscr{PC}})$ is a Banach space. Next, we present an axiomatic definition of the phase space $\mathcal{B}$ are established for $\Im_{0}$-measurable functions from $(-\infty, 0]\rightarrow \mathfrak{X}$, with a semi norm $\left\|\cdot\right\|_{\mathcal{B}}$ which satisfies:

$({\bf{H1}})$ If $\mathrm{y}:(-\infty, a]\rightarrow \mathfrak{X}$, $a > 0$ is s.t $\mathrm{y}_{0}\in \mathcal{B}$ and $\mathrm{y}|_{[0, a]}\in \mathscr{PC}$, then, for every $\mathrm{t}\in [0, a]$, if the following conditions hold:

$(1) \; \mathrm{y}_{\mathrm{t}}\in \mathcal{B}$

$(2) \; \left|\mathrm{y}(\mathrm{t})\right| < \mathrm{K}\left\|\mathrm{y}_{\mathrm{t}}\right\|_{\mathcal{B}}$,

$(3) \; \left\|\mathrm{y}_{\mathrm{t}}\right\|_{\mathcal{B}}\leq \mathrm{M}(\mathrm{t})\sup_{0\leq \mathrm{s} \leq \mathrm{t}}\left|\mathrm{y}(\mathrm{s})\right|+\mathrm{N}(\mathrm{t})\left\|\mathrm{y}_{0}\right\|_{\mathcal{B}}$

where $\mathrm{K} > 0$, $\mathrm{M, N}:[0, +\infty)\rightarrow [1, +\infty)$ are mappings. $\mathrm{M}$ is continuous and $\mathrm{N}$ is locally bounded.

$({\bf{H2}})$ The space $\mathcal{B}$ is complete.

Lemma 2.1. Let $\mathrm{y}:(-\infty, a]\rightarrow \mathfrak{X}$ be an $\Im_{\mathrm{t}}$-adapted measurable process s.t $\mathrm{y}_{0} = \varphi\in \mathcal{L}_{2}(\Omega, \mathcal{B})$,

where $\mathrm{N}_{a} = \sup_{\mathrm{t}\in [0, a]}\left\{\mathrm{N}(\mathrm{t})\right\}$ and $\mathrm{M}_{a} = \sup_{\mathrm{t}\in [0, a]}\left\{\mathrm{M}(\mathrm{t})\right\}$.

Denoted by $\mathcal{M}^{2}((-\infty, a], \mathfrak{X})$, the space of $\mathfrak{X}$-valued cadlag processes $\mathrm{y} = \left\{\mathrm{y}(\mathrm{t})\right\}_{-\infty < \mathrm{t} < a}$ s.t

(ⅰ) $\mathrm{y}_{0} = \varphi\in \mathcal{B}$, $\mathrm{y}(\mathrm{t})$ is $\Im_{\mathrm{t}}$-adapted on $[0, a]$

(ⅱ) If $\mathcal{M}^{2}((-\infty, a], \mathfrak{X})$ with the norm

Definition 2.2. An $\mathfrak{X}$-valued stochastic process $\mathrm{y}(\mathrm{t}), (\mathrm{t} \in \mathrm{J})$ is called a mild solution of (1.1), if

(i) $\mathrm{y}(\mathrm{t})$ is measurable and $\Im_{\mathrm{t}}$-adapted for $\mathrm{t}\in [0, a]$,

(ii) $\mathrm{y}_{0} = \varphi\in \mathcal{B}$.

(iii) For $\mathrm{t}\in [0, a]$, a.s

where $\mathrm{S}_{\alpha}(\mathrm{t}), \mathrm{T}_{\alpha}(\mathrm{t}):\mathfrak{R}_{+}\rightarrow \mathcal{L}(\mathfrak{X}, \mathfrak{X}) \; (\zeta = 1+\alpha)$ are given by

and $\mathbb{B}_{r}$ denotes the Bromwich path ^[18].

3.   Main results

In order to prove our main results, we enforce the following hypotheses,

$({\bf{H3}}) \; \mathfrak{A}$ is the infinitesimal generator of an $\alpha$-order cosine families $\mathrm{S}_{\alpha}(\mathrm{t}) \ \text{and} \ \mathrm{T}_{\alpha}(\mathrm{t})$ on $\mathfrak{X}$ and $\exists \; \mathrm{L} > 0, \ \mathrm{L}_{a} \geq 1$

$({\bf{H4}}) \; \mathfrak{g, f}:[0, a]\times\mathcal{B}\rightarrow \mathfrak{X}$, $\sigma:[0, a]\times\mathcal{B}\rightarrow \mathcal{L}^{0}_{2}$ and $\mathfrak{h}:[0, a]\times\mathcal{B}\rightarrow \mathfrak{X}$ satisfy

Here $\mathrm{k}(\cdot)$ is a concave, continuous and nondecreasing function from $\mathfrak{R}^{+}$ to $\mathfrak{R}^{+}$ s.t $\mathrm{k}(0) = 0$, $\mathrm{k}(u) > 0$ for $u > 0$ and $\int_{0^{+}}\frac{ds}{\mathrm{k}(\mathrm{s})} = \infty$.

$({\bf{H5}}) \; I_{\mathrm{k}}, J_{\mathrm{k}}:\mathcal{B}\rightarrow \mathfrak{X}$ are continuous and there are positive constants $p_{\mathrm{k}}, q_{\mathrm{k}} > 0$ such that for each $\varphi, \phi \in \mathcal{B}$,

$({\bf{H6}}) \; \left\|\mathfrak{g}(\mathrm{t}, 0)\right\|^{2}\vee \left\|\mathfrak{f}(\mathrm{t}, 0)\right\|^{2}\vee \left\|\sigma(\mathrm{t}, 0)\right\|^{2}\vee\int_{\mathcal{U}}\left\|\mathfrak{h}(\mathrm{t}, o, \mathrm{u})\right\|^{2}\leq \mathrm{k}_{0}$, where $\mathrm{k}_{0}$ is a positive constant, $I_{\mathrm{k}}(0) = 0$, $J_{\mathrm{k}}(0) = 0$, $(\mathrm{k} = 1, 2, ..., \mathrm{m})$.

Now, The successive approximations are considered as follows,

Lemma 3.1. Suppose that $({\bf{H3}})-({\bf{H6}})$ hold, and

then $\mathrm{y}^{n}(\mathrm{t})\in \mathcal{M}^{2}((-\infty, a];\mathfrak{X}), \; \forall \; \mathrm{t}\in (-\infty, a]$, $n\geq 0$,

here $\widetilde{\mathrm{M}} > 0$.

Proof. Let $\mathrm{y}^{0}(\mathrm{t})\in \mathcal{M}^{2}((-\infty, a];\mathfrak{X})$ and

It's easy to get the estimations

Next,

and

By $({\bf{H3}})-({\bf{H6}})$ and $\left\|\mathrm{S}_{\alpha}(\mathrm{t})\right\|_{\mathcal{L}(\mathfrak{X}, \mathfrak{X})}\leq \mathrm{L}$, we have

and

By $({\bf{H3}})-({\bf{H6}})$, we have

Next, by $({\bf{H3}})-({\bf{H6}})$, Holder inequality and B-D-G inequality, we obtain

Finally,

Let

From the above estimations, together yields

By using Lemma 2.1 and $\mathrm{k}(\cdot)$, we have to show that a pair of $+$ve constants $\beta$ and $\lambda$ s.t $\mathrm{k}(u)\leq \beta+\lambda u$, $\forall \ u\geq 0$. Then

and

where $\tilde{\mathrm{k}} > 0$. Let

Then

Using Gronwall inequality,

Due to the arbitrary of $\tilde{\mathrm{k}}$, we have

Consequently,

let $\widetilde{M} = \mathbb{E}\left\|\varphi\right\|^{2}_{\mathcal{B}}+\mathrm{M}^{'}$. Hence the proof.

Theorem 3.2. If $({\bf{H3}})-({\bf{H6}})$ and

hold, then the system (1.1) has a unique mild solution of $(-\infty, a]$.

Proof. Let

By the fact $({\bf{H3}})-({\bf{H6}})$, we have

Let $\bar{p} = \max_{1\leq \mathrm{k} \leq m}p_{\mathrm{k}}$. Since

and $\tilde{\mathrm{k}}\circ c(\cdot) = \mathrm{k}(c(\cdot))$ is a concave function, we get

Hence,

Using Lemma 3.1, we get

Define

Choose $a_{1}\in [0, a)$ such that $Q_{4}\tilde{k}(Q_{5}\mathrm{t})\leq Q_{5}$, $\forall \; 0\leq \mathrm{t} \leq a_{1}$.

For any $\mathrm{t}\in [0, a_{1})$, $\left\{\varphi_{n}(\mathrm{t})\right\}$ is a decreasing sequence. In fact

By induction, we get

Therefore, the statement is true and we can define the function $\phi(t)$ as

By the Bihari's inequality, we have $\phi(\mathrm{t}) = 0 \; \forall \ 0\leq \mathrm{t} \leq a_{1}$. It means that for all $0\leq \mathrm{t} \leq a_{1}$,

By using the condition $8m\mathrm{L}^{2}\sum_{\mathrm{t}_{\mathrm{k}} < \mathrm{t}}p_{\mathrm{k}}+8m\mathrm{L}^{2}a\sum_{\mathrm{t}_{\mathrm{k}} < \mathrm{t}}q_{\mathrm{k}} < 1$ and (3.8), we obtain

this implies that $\left\{\mathrm{y}^{n}(\mathrm{t})\right\}$ is a Cauchy sequence in $\mathcal{L}_{2}(\Omega, \mathfrak{X})$. Let $\lim_{n\rightarrow \infty}\mathrm{y}^{n}(\mathrm{t}) = \mathrm{y}(\mathrm{t})$, obviously,

Taking limits on both side of equation (3.2), $\forall t\in [0, a_{1}]$, we have

so we have presented the existence of the mild solution of (1.1) on $[0, a_{1}]$. By iteration we can get the existence of the mild solution of (1.1) on $[0, a]$.

Suppose that $\mathrm{y}_{1}, \mathrm{y}_{2}$ are two solutions of (1.1). Using the similar discussion as (3.8), we get

the Bihari's inequality implies $\mathbb{E}\left\|\mathrm{y}_{1}- \mathrm{y}_{2}\right\|^{2} = 0$, and we have shown the existence and the uniqueness of the mild solution of (1.1).

4.   Stability results

In this section, we have given the continuous dependence of solutions on the initial values by means of the Bihari's inequality. We first propose the following assumption on $\mathfrak{g}$ instead of $({\bf{H4}})$,

$({\bf{H7}}) \; \mathfrak{g}:[0, a]\times \mathscr{B}\rightarrow \mathfrak{X}$ satisfy

Definition 4.1. A mild solution $\mathrm{y}^{\varphi, \mathrm{y}_{1}}(\mathrm{t})$ of Cauchy problem (1.1) with initial value $(\varphi, \mathrm{y}_{1})$ is known as stable in square if $\forall \ \epsilon > 0, \ \ni \; \delta > 0$ s.t

where $\mathrm{z}^{\varphi, \mathrm{z}_{1}}(\mathrm{t})$ is further solution of (1.1) with initial condition $(\varphi, \mathrm{z}_{1})$.

Theorem 4.2. Assume $3(8m\mathrm{L}^{2}\sum_{\mathrm{t}_{\mathrm{k}} < \mathrm{t}}p_{\mathrm{k}}+8m\mathrm{L}^{2}a\sum_{\mathrm{t}_{\mathrm{k}} < \mathrm{t}}q_{\mathrm{s}}) < 1$, by Theorem 3.2 and $({\bf{H7}})$, are satisfied, then the mild solution of (1.1) is stable.

Proof. Using similar argument of Theorem 3.2, we get $\forall 0\leq t \leq a$,

Then we get

where $v = \max\left\{6\mathrm{L}^{2}, 6\mathrm{L}^{2}a^{2}k_{1}+3\mathrm{L}^{2}\right\}$, $\tilde{v} = 3\big(16\mathrm{L}^{2}a+16m^{2}\mathrm{L}^{2}+16m\mathrm{L}^{2}+8m\mathrm{L}\frac{a^{2\alpha-1}}{2\alpha-1}+8m\mathrm{L}^{2}a^{2\alpha-2}+32m\mathrm{L}^{2}a^{2\alpha-2}\big)$, and $\mathcal{G} = 1-3(8m\mathrm{L}^{2}\sum_{\mathrm{t}_{\mathrm{k}} < \mathrm{t}}p_{\mathrm{k}}+8m\mathrm{L}^{2}a\sum_{\mathrm{t}_{\mathrm{k}} < \mathrm{t}}q_{\mathrm{k}})$. The function $\tilde{\mathrm{k}}(u)$ is defined in (3.6) which has the Lemma 2.4 in ^[19]. For $\epsilon > 0$, letting $\epsilon_{1} = \frac{1}{2}\epsilon$, we have $\lim_{s\rightarrow 0}\int^{\epsilon_{1}}_{s}\frac{1}{\tilde{\mathrm{k}}(u)}du = \infty$. $\exists \; \delta$ and $\delta < \epsilon_{1}$ such that $\int^{\epsilon_{1}}_{\mathrm{s}}\frac{1}{\tilde{\mathrm{k}}(u)}du\geq \mathfrak{T}$. Let $u_{0} = \frac{v}{\Lambda}(\left\|\mathrm{y}_{1}-\mathrm{y}_{1}\right\|^{2}+\left\|\varphi-\phi\right\|_{\mathcal{B}})$, $u(\mathrm{t}) = \mathbb{E}\sup_{0\leq \mathrm{s} \leq \mathrm{t}}\left\|\mathrm{y}^{\varphi, \mathrm{y}_{1}}(\mathrm{s})-\mathrm{z}^{\varphi, \mathrm{y}_{1}}(\mathrm{s})\right\|$, $v(\mathrm{t}) = 1$. If $u_{0}\leq \delta \leq \epsilon_{1}$, the Lemma 2.5 in ^[19], shows that $\int^{\epsilon_{1}}_{u_{0}}\frac{1}{\tilde{\mathrm{k}}(u)}du\geq \int^{\epsilon_{1}}_{\delta}\frac{1}{\tilde{\mathrm{k}}(u)}du\geq \mathfrak{T} = \int^{a}_{0}v(\mathrm{s})d\mathrm{s}$. So for any $\mathrm{t}\in [0, a]$, the estimate $u(\mathrm{t})\leq \epsilon_{1}\leq \epsilon$ holds. Hence the proof.

5.   Conclusions

In this manuscript, we investigate a class of second-order impulsive fractional neutral stochastic differential equations (IFNSDEs) driven by Poisson jumps in Banach space. Firstly, sufficient conditions of the existence and the uniqueness of the mild solution for this type of equations are drived by means of the successive approximation and the Bihari's inequality. Next we get the stability in mean square of the mild solution via continuous dependence on initial value are established.

Acknowledgments

The authors thank the referees for useful comments and suggestion which led to an improvement in the quality of this article.

Conflict of interest

All authors declare no conflicts of interest in this paper.