Existence of Mild Solutions for an Impulsive Fractional Integro-differential Equations with Non-local Condition

In this paper, we establish the existence of piece wise (PC)-mild solutions (defined in Section 2) for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained using techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we used the distributed characteristic operators to define a mild solution of the system. We also discussed the controversy related to the solution operator for the fractional order system using weak and strong Caputo derivatives. Examples are given to illustrate the theory.

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Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion

Open Mathematics ◽

10.1515/math-2020-0063 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1097-1112

Author(s):

Louk-Man Issaka ◽

Mamadou Abdoul Diop ◽

Hasna Hmoyed

Keyword(s):

Brownian Motion ◽

Differential Equations ◽

Fractional Brownian Motion ◽

Stochastic Analysis ◽

Mild Solutions ◽

Hurst Parameter ◽

Resolvent Operators ◽

Final Point ◽

Analysis Theory ◽

Non Local

Abstract This paper deals with the existence of mild solutions for a class of non-local stochastic integro-differential equations driven by a fractional Brownian motion with Hurst parameter H\in \left(\tfrac{1}{2},1\right) . Discussions are based on resolvent operators in the sense of Grimmer, stochastic analysis theory and fixed-point criteria. As a final point, an example is given to illustrate the effectiveness of the obtained theory.

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Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces

10.20944/preprints201810.0589.v1 ◽

2018 ◽

Author(s):

Dimplekumar Chalishajar ◽

C. Ravichandran ◽

S. Dhanalakshmi ◽

R. Murugesu

Keyword(s):

Fixed Point ◽

Group Theory ◽

Differential Equations ◽

Banach Spaces ◽

Mild Solution ◽

Fixed Point Theorems ◽

Mild Solutions ◽

Semi Group ◽

Finite Delay ◽

Non Local

In this paper, we established the existence of PC-mild solutions for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained by using the techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we have used the distributed characteristic operators to define the mild solution of the system. Results obtained here improve and extend some known results.

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On fractional differential equations with state-dependent delay via Kuratowski measure of noncompactness

Filomat ◽

10.2298/fil1702451b ◽

2017 ◽

Vol 31 (2) ◽

pp. 451-460 ◽

Cited By ~ 1

Author(s):

Mohammed Belmekki ◽

Kheira Mekhalfi

Keyword(s):

Differential Equations ◽

Measure Of Noncompactness ◽

Mild Solutions ◽

Functional Differential Equations ◽

Kuratowski Measure Of Noncompactness ◽

State Dependent ◽

State Dependent Delay ◽

Liouville Fractional Derivative ◽

Fractional Differential ◽

Functional Differential

This paper is devoted to study the existence of mild solutions for semilinear functional differential equations with state-dependent delay involving the Riemann-Liouville fractional derivative in a Banach space and resolvent operator. The arguments are based upon M?nch?s fixed point theoremand the technique of measure of noncompactness.

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Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-014-0219-8 ◽

2014 ◽

Vol 17 (4) ◽

Cited By ~ 4

Author(s):

Shengli Xie

Keyword(s):

Differential Equations ◽

Differential Evolution ◽

Existence Theorem ◽

Banach Spaces ◽

Existence And Uniqueness ◽

Evolution Equations ◽

Infinite Delay ◽

Mild Solutions ◽

Existence Results ◽

Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

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Existence results for some integro-differential equations with state-dependent nonlocal conditions in Fréchet Spaces

Nonautonomous Dynamical Systems ◽

10.1515/msds-2020-0121 ◽

2020 ◽

Vol 7 (1) ◽

pp. 272-280

Author(s):

Mamadou Abdoul Diop ◽

Kora Hafiz Bete ◽

Reine Kakpo ◽

Carlos Ogouyandjou

Keyword(s):

Differential Equations ◽

Resolvent Operator ◽

Mild Solutions ◽

Linear Part ◽

Nonlocal Conditions ◽

Existence Results ◽

Fréchet Spaces ◽

Local Conditions ◽

State Dependent ◽

Darbo Fixed Point Theorem

AbstractIn this work, we present existence of mild solutions for partial integro-differential equations with state-dependent nonlocal local conditions. We assume that the linear part has a resolvent operator in the sense given by Grimmer. The existence of mild solutions is proved by means of Kuratowski’s measure of non-compactness and a generalized Darbo fixed point theorem in Fréchet space. Finally, an example is given for demonstration.

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Existence and exponential stability in the pth moment for impulsive neutral stochastic integro-differential equations driven by mixed fractional Brownian motion

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2213-5 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

Xia Zhou ◽

Dongpeng Zhou ◽

Shouming Zhong

Keyword(s):

Brownian Motion ◽

Differential Equations ◽

Exponential Stability ◽

Fractional Brownian Motion ◽

Mild Solutions ◽

Semigroup Theory ◽

Sufficient Conditions ◽

Fractional Power ◽

Banach Fixed Point Theorem ◽

Mixed Fractional Brownian Motion

Abstract This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.

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Non-local Solvable Birth–Death Processes

Journal of Theoretical Probability ◽

10.1007/s10959-021-01087-4 ◽

2021 ◽

Author(s):

Giacomo Ascione ◽

Nikolai Leonenko ◽

Enrica Pirozzi

Keyword(s):

Differential Equations ◽

Orthogonal Polynomials ◽

Limit Distribution ◽

Spectral Decomposition ◽

Strong Solutions ◽

Stochastic Representation ◽

Discrete Approximations ◽

Local Difference ◽

Non Local ◽

Birth Death

AbstractIn this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes.

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