Existence of almost automorphic mild solutions to non-autonomous neutral stochastic differential equations

Revathi, P.; Sakthivel, R.; Ren, Yong; Anthoni, S. Marshal

doi:10.1016/j.amc.2013.12.075

Cited by 21 publications

(7 citation statements)

References 28 publications

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“…We assume that the filtration be generated by the Poisson point process

\overset{q}{true} (\cdot)

and be augmented; that is,

{frakturF}_{t} = σ ({}, N ((0, s], A); s \leq t, A \in scriptℬ (Z)) \lor scriptN, t \in J

, where

scriptN

is the class of P ‐null sets. Definition A stochastic process

x (t) : double-struckR \to L_{2} (P, H)

is said to be stochastically bounded if there exists

double-struckM > 0

such that

E ∥ x (t) ∥^{2} \leq double-struckM

, for all

t \in double-struckR

.Definition A stochastic process

x (t) : double-struckR \to L_{2} (P, H)

is said to be stochastically continuous if

\lim_{t \to s} E ∥ x (t) - x (s) ∥^{2} = 0

.Definition A continuous stochastic process

x (t) : double-struckR \to L_{2} (P, H)

is said to be square‐mean almost automorphic if for every sequence of real numbers

{ŝ_{n}}_{n \in double-struckN}

, there exists a subsequence

{s_{n}}_{n \in double-struckN}

and a stochastic process

y : double-struckR \to L_{2} (P, H)

such that

\lim_{n \to \infty} E ∥ x (t + s<...>

…”

Section: Preliminariesmentioning

confidence: 99%

“…Revathi et al . investigated the existence of square mean and weighted pseudo almost automorphic mild solutions of non‐autonomous neutral stochastic differential equations, which are nice and interesting results. Liu et al .…”

Section: Introductionmentioning

confidence: 99%

“…Here, we move from deterministic fractional neutral differential equations of order 1 < α < 2 to fractional stochastic neutral differential equations of order 1 < α < 2 for the study of existence of square‐mean almost automorphic solutions by using the successive approximation approach as discussed in . Motivated by few studies , the existence of almost automorphic mild solutions to the following fractional neutral stochastic integro‐differential equations with infinite delay driven by Poisson jumps remains an untreated topic in the literature, and this fact is the main motivation for the present work.

\begin{matrix} D_{t}^{α} [x (t) + f (t, x_{t})] = Ax (t) & + \int_{0}^{t} A_{1} (t - s) x (s) ds + g (t, x_{t}) \\ + \int_{Z} h (t, x (t -), η) \hat{N} (dt, dη), t \in J : = [0, b], \\ x_{0} = φ \in B, x^{'} (0) = 0, \end{matrix}

where α ∈(1,2).…”

Section: Introductionmentioning

confidence: 99%

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Almost automorphic solutions for fractional stochastic differential equations and its optimal control

Rajivganthi

Muthukumar

2015

Optim Control Appl Methods

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Summary Fractional calculus is the field of mathematical analysis that deals with the investigation and applications of integrals, derivatives of arbitrary order. The strength of derivatives of non‐integer order is their ability to describe real situations more adequately than integer order derivatives, especially when the problem has memory or hereditary properties. This paper is mainly concerned with the square‐mean almost automorphic mild solutions to a class of fractional neutral stochastic integro‐differential equations with infinite delay driven by Poisson jumps. The existence of square‐mean almost automorphic mild solutions of the previous fractional dynamical system is proved by using the method of successive approximation. The results are formulated and proved by using the fractional calculus, solution operator, and stochastic analysis techniques. Further, the existence of optimal control of the proposed problem is also presented. An example is provided to illustrate the developed theory. Copyright © 2015 John Wiley & Sons, Ltd.

show abstract

“…We assume that the filtration be generated by the Poisson point process

\overset{q}{true} (\cdot)

and be augmented; that is,

{frakturF}_{t} = σ ({}, N ((0, s], A); s \leq t, A \in scriptℬ (Z)) \lor scriptN, t \in J

, where

scriptN

is the class of P ‐null sets. Definition A stochastic process

x (t) : double-struckR \to L_{2} (P, H)

is said to be stochastically bounded if there exists

double-struckM > 0

such that

E ∥ x (t) ∥^{2} \leq double-struckM

, for all

t \in double-struckR

.Definition A stochastic process

x (t) : double-struckR \to L_{2} (P, H)

is said to be stochastically continuous if

\lim_{t \to s} E ∥ x (t) - x (s) ∥^{2} = 0

.Definition A continuous stochastic process

x (t) : double-struckR \to L_{2} (P, H)

is said to be square‐mean almost automorphic if for every sequence of real numbers

{ŝ_{n}}_{n \in double-struckN}

, there exists a subsequence

{s_{n}}_{n \in double-struckN}

and a stochastic process

y : double-struckR \to L_{2} (P, H)

such that

\lim_{n \to \infty} E ∥ x (t + s<...>

…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

\begin{matrix} D_{t}^{α} [x (t) + f (t, x_{t})] = Ax (t) & + \int_{0}^{t} A_{1} (t - s) x (s) ds + g (t, x_{t}) \\ + \int_{Z} h (t, x (t -), η) \hat{N} (dt, dη), t \in J : = [0, b], \\ x_{0} = φ \in B, x^{'} (0) = 0, \end{matrix}

where α ∈(1,2).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Almost automorphic solutions for fractional stochastic differential equations and its optimal control

Rajivganthi

Muthukumar

2015

Optim Control Appl Methods

View full text Add to dashboard Cite

show abstract

“…By using contraction mapping principle, some sufficient conditions for approximate controllability are established. For more recent works of existence, stability results and approximate controllability, one can see the contributions of Sakthivel et al [35][36][37][38], Yan [39], Ren et al [40], Debbouche and Torres [41], Mahmudov and Zorlu [42] and reference theine. Note that successive approximations method is a powerful approach to find the solution of all kinds of differential equations.…”

Section: Introductionmentioning

confidence: 98%

Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces

Wang

2015

Applied Mathematics and Computation

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“…In particular, Sakthivel et al discussed pseudo almost the automorphic mild solutions [30], existence of solutions [33] and approximate controllability [32,34] of fractional stochastic differential systems. For more recent works about existence, stability results and approximate controllability, we refer the reader to [1,9,27,29,31,35] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and controllability results for stochastic fractional evolution hemivariational inequalities

Lü

Liu

2015

Applied Mathematics and Computation

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Existence of almost automorphic mild solutions to non-autonomous neutral stochastic differential equations

Cited by 21 publications

References 28 publications

Almost automorphic solutions for fractional stochastic differential equations and its optimal control

Almost automorphic solutions for fractional stochastic differential equations and its optimal control

Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces

Existence and controllability results for stochastic fractional evolution hemivariational inequalities

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