Saïd Abbas

Topics in Fractional Differential Equations

Preface.- 1. Preliminary Background.- 2. Partial Hyperbolic Functional Differential Equations.- 3. Partial Hyperbolic Functional Differential Inclusions.- 4. Impulsive Partial Hyperbolic Functional Differential Equations.- 5. Impulsive Partial Hyperbolic Functional Differential Inclusions.- 6. Implicit Partial Hyperbolic Functional Differential Equations.- 7. Fractional Order Riemann-Liouville Integral Equations.- References.- Index.

ON FRACTIONAL ORDER DERIVATIVES AND DARBOUX PROBLEM FOR IMPLICIT DIFFERENTIAL EQUATIONS

In this paper we prove some relations between the Riemann-Liouville and the Caputo fractional order derivatives, and we investigate the existence and uniqueness of solutions for the initial value problems (IVP for short), for a class of functional hyperbolic differential equations by using some fixed point theorems.

Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses

In this paper, we investigate some uniqueness and Ulams type stability concepts of fixed point equations for a class of partial functional differential equations with not instantaneous impulses in Banach spaces. An example is also provided to illustrate our results.

Existence and stability results for nonlinear fractional order Riemann-Liouville Volterra-Stieltjes quadratic integral equations

Our aim in this paper is to study the existence and the stability of solutions for Riemann-Liouville Volterra-Stieltjes quadratic integral equations of fractional order. Our results are obtained by using some fixed point theorems. Some examples are provided to illustrate the main results.

Fractional order integral equations of two independent variables

In this paper, we present some results concerning the existence, the uniqueness and the attractivity of solutions for some functional integral equations of Riemann-Liouville fractional order, by using some fixed point theorems.

Advanced Functional Evolution Equations and Inclusions

1. Preliminary Background.- 2. Partial Functional Evolution Equations with Finite Delay.- 3. Partial Functional Evolution Equations with Infinite Delay.- 4. Perturbed Partial Functional Evolution Equations.- 5. Partial Functional Evolution Inclusions with Finite Delay.- 6. Partial Functional Evolution Inclusions with Infinite Delay.- 7. Densely Defined Functional Differential Inclusions with Finite Delay.- 8. Non-Densely Defined Functional Differential Inclusions with Finite Delay.- 9. Impulsive Semi-linear Functional Differential Equations.- 10. Impulsive Functional Differential Inclusions with Unbounded Delay.- 11. Functional Differential Inclusions with Multi-valued Jumps.- 12. Global Existence Results for Functional Differential Equations and Inclusions with Delay.- 13. Global Existence Results of Second Order Functional Differential Equations with Delay.- References.- Index.

New Stability Results for Partial Fractional Differential Inclusions with Not Instantaneous Impulses

Abstract In this work, we discuss the existence and Ulams type stability concepts for a class of partial functional differential inclusions with not instantaneous impulses and a nonconvex valued right hand side in Banach spaces. An example is also provided to illustrate our results.

Darboux problem for fractional-order discontinuous hyperbolic partial differential equations in Banach algebras

In this article, we prove an existence result for partial hyperbolic differential equations of fractional order in Banach algebras under Lipschitz and Carathéodory conditions. The existence of extremal solutions is also proved under certain monotonicity conditions.

Global attractivity for fractional order delay partial integro-differential equations

Our aim in this work is to study the existence and the attractivity of solutions for a system of delay partial integro-differential equations of fractional order. We use the Schauder fixed point theorem for the existence of solutions, and we prove that all solutions are locally asymptotically stable.AMS (MOS) Subject Classifications: 26A33.

Darboux problem for fractional order neutral functional partial hyperbolic differential equations

In this paper, we prove an existence result for initial value problems (IVP for short), for neutral partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative. Our works will be considered by using Krasnoselskiis fixed point theorem.