scholarly journals Mean-field anticipated BSDEs driven by time-changed Lévy noises

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Youxin Liu ◽  
Yang Dai
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Mean Field ◽  
Iterative Sequence ◽  
Uniqueness Of The Solution ◽  
The Fixed Point Theorem

Abstract The objective of this work is to show a new kind of mean-field anticipated backward stochastic differential equation (in short MF-ABSDE) driven by time-changed Lévy noises. We give two methods to prove the existence and uniqueness of the solution of those equations by the fixed point theorem and the Picard iterative sequence. Finally, we obtain a comparison theorem for the solutions.

Unique Solution for Multi-point Fractional Integro-Differential Equations

2020 ◽  
Vol 21 (2) ◽  
pp. 219-226
Author(s):  
Chengbo Zhai ◽  
Lifang Wei
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Unique Solution ◽  
Existence And Uniqueness ◽  
Iterative Sequence ◽  
Point Boundary ◽  
Uniqueness Of Solutions

AbstractWe study a fractional integro-differential equation subject to multi-point boundary conditions: $$\left\{\begin{array}{l} D^\alpha_{0^+} u(t)+f(t,u(t),Tu(t),Su(t))=b,\ t\in(0,1),\\u(0)=u^\prime(0)=\cdots=u^{(n-2)}(0)=0,\\ D^p_{0^+}u(t)|_{t=1}=\sum\limits_{i=1}^m a_iD^q_{0^+}u(t)|_{t=\xi_i},\end{array}\right.$$where $\alpha\in (n-1,n],\ n\in \textbf{N},\ n\geq 3,\ a_i\geq 0,\ 0<\xi_1<\cdots<\xi_m\leq 1,\ p\in [1,n-2],\ q\in[0,p],b>0$. By utilizing a new fixed point theorem of increasing $\psi-(h,r)-$ concave operators defined on special sets in ordered spaces, we demonstrate existence and uniqueness of solutions for this problem. Besides, it is shown that an iterative sequence can be constructed to approximate the unique solution. Finally, the main result is illustrated with the aid of an example.

scholarly journals A Nonlinear Implicit Fractional Equation with Caputo Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ameth Ndiaye
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Implicit Differential Equation ◽  
The Fixed Point Theorem ◽  
Banach Principle ◽  
Fractional Equation

In this paper, we study a nonlinear implicit differential equation with initial conditions. The considered problem involves the fractional Caputo derivatives under some conditions on the order. We prove an existence and uniqueness analytic result by application of Banach principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions for this result are established by means of the fixed point theorem of Schaefer. At the end, we discuss two examples to illustrate the applicability of the main results.

scholarly journals Existence of the Unique Nontrivial Solution for Mixed Fractional Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yujing Liu ◽  
Chenguang Yan ◽  
Weihua Jiang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Monotone Operators ◽  
Uniqueness Of The Solution ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
Mixed Monotone Operators

In this paper, we consider the differential equations with right-sided Caputo and left-sided Riemann-Liouville fractional derivatives. Furthermore, the expression of Green’s function is derived, and its properties are investigated. By the fixed-point theorem for both φ − h , e -concave operators and mixed monotone operators, we get the existence and uniqueness of the solution, respectively. As applications, some examples are provided to illustrate our main results.

scholarly journals A generalized sequential problem of Lane-Emden type via fractional calculus

2020 ◽  
Vol 6 (2) ◽  
pp. 168-183 ◽  
Author(s):  
Yazid Gouari ◽  
Zoubir Dahmani ◽  
Ameth Ndiaye
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Group Property ◽  
Semi Group ◽  
Singular Differential Equation ◽  
Banach Contraction ◽  
The Fixed Point Theorem

AbstractIn this paper, we combine the Riemann-Liouville integral operator and Caputo derivative to investigate a nonlinear time-singular differential equation of Lane Emden type. The considered problem involves n fractional Caputo derivatives under the conditions that neither commutativity nor semi group property is satisfied for these derivatives. We prove an existence and uniqueness analytic result by application of Banach contraction principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions related to this result are established by means of the fixed point theorem of Schaefer. We end the paper by presenting to the reader some illustrative examples.

scholarly journals Generalized BSDE driven by a Lévy process

2006 ◽  
Vol 2006 ◽  
pp. 1-25 ◽  
Author(s):  
Mohamed El Otmani
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Existence Of A Solution ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Independent Brownian Motion ◽  
Teugels Martingales

We study the solution of one-dimensional generalized backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. We prove existence and uniqueness of the solution when the coefficient verifies some conditions of Lipschitz. If the coefficient is left continuous, increasing, and bounded, we prove the existence of a solution.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals Lp (p ≥ 1) solutions of multidimensional BSDEs with monotone generators in general time intervals

2014 ◽  
Vol 15 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Li-Shun Xiao ◽  
Sheng-Jun Fan ◽  
Na Xu
Keyword(s):  
Differential Equation ◽  
Weak Convergence ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Additional Assumption ◽  
Backward Stochastic Differential Equation ◽  
Time Interval ◽  
Time Intervals ◽  
Lipschitz Continuous ◽  
General Time

In this paper, we are interested in solving general time interval multidimensional backward stochastic differential equation in Lp (p ≥ 1). We first study the existence and uniqueness for Lp (p > 1) solutions by the method of convolution and weak convergence when the generator is monotonic in y and Lipschitz continuous in z both non-uniformly with respect to t. Then we obtain the existence and uniqueness for L1 solutions with an additional assumption that the generator has a sublinear growth in z non-uniformly with respect to t.

BSDE with rcll reflecting barrier driven by a Lévy process

2020 ◽  
Vol 28 (1) ◽  
pp. 63-77 ◽  
Author(s):  
Mohamed El Jamali ◽  
Mohamed El Otmani
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Lévy Process ◽  
Backward Stochastic Differential Equation ◽  
American Option ◽  
Levy Process ◽  
Fair Price ◽  
Penalization Method ◽  
Uniqueness Of A Solution

AbstractIn this paper, we study the solution of a backward stochastic differential equation driven by a Lévy process with one rcll reflecting barrier. We show the existence and uniqueness of a solution by means of the penalization method when the coefficient is stochastic Lipschitz. As an application, we give a fair price of an American option.

scholarly journals Uniqueness of Positive Solutions for a Perturbed Fractional Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Chen Yang ◽  
Jieming Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Point Boundary ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem:D0+αu(t)+f(t,u,u',…,u(n-2))+g(t)=0, 0<t<1, n-1<α≤n, n≥2,u(0)=u'(0)=⋯=u(n-2)(0)=u(n-2)(1)=0, whereD0+αis the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem of generalized concave operators. An example is given to illustrate the main result.

Fractional anticipated BSDEs with stochastic Lipschitz coefficients

2018 ◽  
Vol 26 (3) ◽  
pp. 143-161
Author(s):  
Ahmadou Bamba Sow ◽  
Bassirou Kor Diouf
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Comparison Theorem ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Hurst Parameter ◽  
Uniqueness Of A Solution

Abstract In this paper, we deal with an anticipated backward stochastic differential equation driven by a fractional Brownian motion with Hurst parameter {H\in(1/2,1)} . We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and prove a comparison theorem in a specific case.

Sign in / Sign up

Export Citation Format

Share Document