Fractional Differential Equations

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Stochastic equations in infinite dimensions

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

Existence results for fractional order functional differential equations with infinite delay

In this paper, we present fractional versions of the Filippov theorem and the Filippov–Wazewski theorem, as well as an existence result, compactness of the solution set and Hausdorff continuity of operator solutions for functional differential inclusions with fractional order, D α y ( t ) ∈ F ( t , y t ) , a.e. t ∈ [ 0 , b ] , 0 α 1 , y ( t ) = ϕ ( t ) , t ∈ [ − r , 0 ] , where J = [ 0 , b ] , D α is the standard Riemann–Liouville fractional derivative, and F is a set-valued map.

Fractional functional differential inclusions with finite delay

In this paper, we study fractional differential inclusions with Dirichlet boundary conditions. We prove the existence of a solution under both convexity and nonconvexity conditions on the multi-valued right-hand side. The proofs rely on nonlinear alternative Leray–Schauder type, Bressan–Colombo selection theorem and Covitz and Nadler’s fixed point theorem for multi-valued contractions. The compactness of the set solutions and relaxation results is also established. In the last section we consider the fractional boundary value problem with infinite delay.

Some results for fractional boundary value problem of differential inclusions

In this paper, we first present an impulsive version of the Filippov–Wazewski theorem and a continuous version of the Filippov theorem for fractional differential inclusions of the form 
D∗αy(t)∈F(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c,
where J=[0,b],D∗α denotes the Caputo fractional derivative, and F is a set-valued map. The functions Ik,I¯k characterize the jump of the solutions at impulse points tk(k=1,…,m). Additional existence results are obtained under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on the nonlinear alternative of Leray–Schauder type, a Bressan–Colombo selection theorem, and Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of the solution set is also investigated. Finally, some geometric properties of solution sets, Rδ sets, acyclicity and contractibility, corresponding to Aronszajn–Browder–Gupta type results, are obtained. We also consider the impulsive fractional differential equations 
D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=Ik(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c,
and 
D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(0,1],y(tk+)=Ik(y(tk−)),k=1,…,m,y(0)=a,
where f:J×R→R is a single map. Finally, we extend the existence result for impulsive fractional differential inclusions with periodic conditions, 
D∗αy(t)∈φ(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=y(b),y′(0)=y′(b),
where φ:J×R→P(R) is a multivalued map. The study of the above problems use an approach based on the topological degree combined with a Poincare operator.

Impulsive differential inclusions with fractional order

In this article, a recent nonlinear alternative for contraction maps in Frechet spaces due to Frigon and Granas [1998, Resultats de type Leray-Schauder pour des contractions sur des espaces de Frechet, Ann. Sci. Math. Quebec 22, 161–168] is used to investigate the existence and uniqueness of solutions for fractional order functional differential equations with infinite delay.

Abdelghani Ouahab

Papers

Existence results for fractional order functional differential equations with infinite delay

Fractional functional differential inclusions with finite delay

Some results for fractional boundary value problem of differential inclusions

Impulsive differential inclusions with fractional order

Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces