Existence Of Solutions For Some Classes Of Boundary Value Problems For Fractional Differential Equations

Fractional differential equation is a generalization of classical integer differential equation.It has the characteristics of simple modeling,clear physical meaning of parameters and accurate description.It is widely used in engineering,physics,fluid dynamics,finance and other fields.In this paper,the existence of solutions for some kinds of fractional differential equations is studied by means of fixed point theory and topological degree theory.Chapter 1 mainly introduces the research background,research status,and some necessary preparatory knowledge of several kinds of differential equations.Chapter 2 considers the following integral boundary value problems of fractional differential equations with generalized Riemann-Liouville-type derivatives where 2<α<3,1<ν<2,α-ν-1>0,f∈C([0,1]∈R+,R+),g,h∈C([0,1],R+),R+=[0,+∞),g’>0.Applying the fixed point theorem on cone,the existence of multiple solutions for considered system is obtained.The results generalize and improve existing conclusions.Meanwhile,the Ulam stability for above system is also considered.Finally,some examples are worked out to illustrate the main results.Chapter 3 investigates multiple solutions for the following boundary value problem where tCD0+R is the Caputo fractional derivative with t,1<R<2,α>β>0,η>δ>0,εk∈C(R+,R+),ε,ρ1,ρ2≥ 0 a.e.on Q=[0,1],Q’=Q\｛t1,…,tm},ε,ρ1,ρ2∈L1(0,1),f:Q × R+ → R+,R+=[0,+∞),0<t1<t2<…<tm<1.△u|t=tk,△u’|t=tk denote the jump of u(t)and u’(t)at t=tk,respectively.First,the distinctive tool used here is multivalued analysis in the study of discontinuous problems.At the same time,a suitable cone is established by researching properties of Green’s function deeply.Finally,the positive solutions can be obtained by means of Krasnoselskii’s fixed point theorem for discontinuous operators on cones.Chapter 4 investigates a class of boundary value problems of fractional neutral evolution equations with impulses and state-dependent delay where,CDtq denote the Caputo fractional derivative of order q ∈(1,2).A is the infinitesimal generator of the strongly continuous β-order fractional cosine functions of bounded linear operators {Cβ(t)}t≥0 the Banach space(V,‖·‖ν).The history xt:(-∞,0]→ V is given by xt(θ)=x(t+θ),θ ∈(-∞,0]and xt is the element of an phase space D.Let I=[0,T],L0=(-∞,0].The mappings b:I×D→V.f:I×D→V,li:D→V,mi:(ti,si]×D→V,i=1,2,…,N and k1,k2:D→V are appropriate functions and ρ:I × D→(-∞,T]is continuous,where 0=s0=t0<t01<t02<…<t0r1<t0r1+1=t1<s1<t1r1+1<t1r1+2<…<t1r2<t1r2+1=t2<s2<…<sN<tNrN+1<…<tNrN+1<tNrN+1+1=tN+1=T.LetJj:=(tj,sj],j=1,2,…,N,and Iik=(tik-1,tik],i=0,1,…,N,ri≤k≤ri+1+1,k∈N.We investigate the existence of mild solutions for above system by using Hausdorff noncompactness measure and Monch fixed point theorem.