Existence Of Solutions To Boundary Problems For Differential Equations With Caputo Fractional-Order Derivatives

The application of fractional caleulus is very broad,the differential equations involv-ing Caputo operators of fractional orders can more accurately describe many scientific disciplines as the mathematical modeling of many phenomena in various fields of science and engineering than the differential equations involving integer order,therefore the study for fractional differential equations is very meaningful and important.In this thesis,we study the existence of solutions for impulsive boundary value prob-lem for nonlinear multiple base points differential equations of fractional order ??(1,2),and the existence of solutions of the nonlinear Langevin fractional differential equations involving the anti-periodic boundary value problem.Mittag-Leffler functions play an important role in the study of fractional differen-tial equations with constant coeffcient.Firstly,we study the properties of Mittag-Leffler fuctions E?(z)and E?,?(z)(?,??(1,2)),which can b e applied to solve he problems infollowing sections.Secondly,we consider the following impulsive problem for fractional differential equa-tions with boundary value,cD*?x(t)=q(t)f(t,x(t),cD*?x(t)),cD*?x(t)),a.e.t?Jl:=(0,1]\{t1,t2,…,tm},?x(tk)=Ik,?xl(tk)=Ik,k=1,2…,m,x(0+)=0,cxl(0+)+dcDtm?+x(1)=0,where ??(1,2),?,?,?-??(0,1),?-??(1,2),cD*? is the Caputo fractional deriva-tive,?x(tk)represents the jump of function x at tk.By using sme well-known fixed pointtheorem,we obtain an existence result on solution under the weak assumption.Thirdly,we study the following anti-periodic boundary value problem of nonlinearfractional Langevin equations:cDo?+(cDo+?+?)u(t)=f(t,u(t)),t?J:=(0,1],(?)where ?,??(0,1),?,?+??(1,2),??0,0??+?-???1.cDo+*is the standardCaputo fractional derivative,HDO+?,?is the Hilfer fractional derivative,Under the weak as-sumptions for nonlinear terms,we obtain the existence result of solutions.