Solutions To Several Classes Of Fractional Differential Equations

Fractional calculus is an extension and expansion of integral calculus. It is the theory of researching the characteristics and applications of differential and inte-gral operators of arbitrary order. Fractional differential equations are increasingly used to model problems in acoustics and thermal systems, rheology materials and mechanical systems, control and robotics, the order of nature and other areas of application. Fractional differential equations have become an important subject, it has been widely concerned and has the extensive theoretical significance and prac-tical research values. This paper makes use of Schauder fixed point theorem, the theory of mixed monotone operator and Banach contraction mapping principle to get the existence and uniqueness of solutions.Chapter1we introduce some background materials from fractional calculus theory and we give some preliminary definitions and lemmas on fractional calculus which are needed in this thesis.In chapter2, we investigates the existence and uniqueness of the positive so-lutions to a class of singular fractional-order differential equations boundary value problemwhere3<α<4, CD0α+is the Caputo’s fractional derivatives,f:(0,1) x (0,+∞) x (0,+∞)â†'[0,+∞) is continuous. The approache used here is the theory of mixed monotone operator. In chapter3,we investigates a class of nonlinear impulsive differential equation of flractional boundary value problemwhere3<σ≤4,f:[0,1]×Râ†'R is continuous,Ik;Jk;Qk:Râ†'R. The approaches used here are Schguder fixed point theory with the Banach contraction mapping principle.In chapter4,we consider a class of impulsive fractional functional differential equationswhere0<σ≤1,0=t0<t1<…<tp-1=b,D={ψ:ψis a map from[—τ,0]to everywhere except for a finite number of points t at which ψ(t) and ψ(t1)exist andψ(t)=ψ(t)},φ∈D,L1([-Ï„,0],R)={u is a map from[-Ï„,0]t0R,and∫-Ï„0,|u(t)|dt＜+∞}J×R×L1([-Ï„,0],R)â†'R and Ik: Râ†'R(k=1,2,…,p)are continuous.The approaches used here are Schauder fixed point theory with the Bangch contraction magpping principle.