Existence of solutions for fractional neutral functional differential equations driven by fBm with infinite delay

Lakhel, El Hassan; McKibben, Mark A.

doi:10.1080/17442508.2017.1346657

Cited by 26 publications

(7 citation statements)

References 19 publications

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“…Thus, A can be written as

A x = \sum_{n = 1}^{\infty} n^{2} ⟨ x, x_{n} ⟩ x_{n}, x \in D false(A false),

where

x_{n} false(x false) = \sqrt{\frac{2}{π}} \sin n x, n = 1, 2, …

, is an orthogonal basis of H . The bounded linear operator

A^{\frac{2}{3}}

is given by (for more details, see the works of Lakhel and McKibben and Pazy)

A^{\frac{2}{3}} x = \sum_{n = 1}^{\infty} n^{\frac{4}{3}} ⟨ x, x_{n} ⟩ x_{n},

with the domain

D false(A^{\frac{2}{3}} false) = false{x \in H, \sum_{n = 1}^{\infty} n^{\frac{4}{3}} ⟨ x, x_{n} ⟩ x_{n}, x_{n} \in H false}

. Hence, the infinitesimal generator A generates a compact analytical semigroup { S ( t ): t >0} in H and

false‖ S false(t false) {false‖}_{scriptL false(H, H false)} \leq e^{- 1} < 1 = M, t \geq 0

.…”

Section: Applicationsmentioning

confidence: 99%

Optimal control of fractional neutral stochastic differential equations with deviated argument governed by Poisson jumps and infinite delay

Durga

Muthukumar

2019

Optim Control Appl Methods

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Summary In this work, the optimal control for a class of fractional neutral stochastic differential equations with deviated arguments driven by infinite delay and Poisson jumps is studied in Hilbert space involving the Caputo fractional derivative. The sufficient conditions for the existence of mild solution results are formulated and proved by the virtue of fractional calculus, characteristic solution operator, fixed‐point theorem, and stochastic analysis techniques. Furthermore, the existence of optimal control of the proposed problem is presented by using Balder's theorem. Finally, the obtained theoretical results are applied to the fractional stochastic partial differential equations and a stochastic river pollution model.

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“…Thus, A can be written as

A x = \sum_{n = 1}^{\infty} n^{2} ⟨ x, x_{n} ⟩ x_{n}, x \in D false(A false),

where

x_{n} false(x false) = \sqrt{\frac{2}{π}} \sin n x, n = 1, 2, …

, is an orthogonal basis of H . The bounded linear operator

A^{\frac{2}{3}}

is given by (for more details, see the works of Lakhel and McKibben and Pazy)

A^{\frac{2}{3}} x = \sum_{n = 1}^{\infty} n^{\frac{4}{3}} ⟨ x, x_{n} ⟩ x_{n},

with the domain

D false(A^{\frac{2}{3}} false) = false{x \in H, \sum_{n = 1}^{\infty} n^{\frac{4}{3}} ⟨ x, x_{n} ⟩ x_{n}, x_{n} \in H false}

. Hence, the infinitesimal generator A generates a compact analytical semigroup { S ( t ): t >0} in H and

false‖ S false(t false) {false‖}_{scriptL false(H, H false)} \leq e^{- 1} < 1 = M, t \geq 0

.…”

Section: Applicationsmentioning

confidence: 99%

Optimal control of fractional neutral stochastic differential equations with deviated argument governed by Poisson jumps and infinite delay

Durga

Muthukumar

2019

Optim Control Appl Methods

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“…In recent decades, the existence and uniqueness of mild solutions for fractional neutral SDEs have attracted much attention. For instance, Lakhel and Mckibben [19] established the existence of mild solutions for a class of FNSDEs driven by fBm with infinite delay; Dhanalakshmi and Balasubramaniam [20] showed the stability result of higher-order FNSDEs with infinite delay driven by Poisson jumps and the Rosenblatt process; Ramkumar et al [21] obtained the existence and optimal control for a class of Caputo FNSDEs driven by fBm and Poisson jumps; Alnafisah and Ahmed [22] proved the existence and controllability for neutral delay Hilfer fractional integrodifferential equations driven with fBm by means of the fixed point theorem and semigroup theory.…”

Section: Introductionmentioning

confidence: 99%

Mixed Caputo Fractional Neutral Stochastic Differential Equations with Impulses and Variable Delay

et al. 2021

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In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results.

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“…Hence, the classical stochastic analysis techniques are invalid to study it. Recently, the research of stochastic differential equations driven by fractional Brownian motion has been investigated by many authors, see [4,9,21,28] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion

Yang

2021

J Inequal Appl

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This article is devoted to the study of the existence and uniqueness of mild solution to a class of Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion. Our results are obtained by using fractional calculus, stochastic analysis, and the fixed-point technique. Moreover, an example is provided to illustrate the application of the obtained abstract results.

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Existence of solutions for fractional neutral functional differential equations driven by fBm with infinite delay

Cited by 26 publications

References 19 publications

Optimal control of fractional neutral stochastic differential equations with deviated argument governed by Poisson jumps and infinite delay

Optimal control of fractional neutral stochastic differential equations with deviated argument governed by Poisson jumps and infinite delay

Mixed Caputo Fractional Neutral Stochastic Differential Equations with Impulses and Variable Delay

Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion

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