Existence Of Solutions For Fractional Differential Equations(Systems)Involving Two Fractional Derivatives With Boundary Conditions

The fractional derivatives are a kind of pseudo differential operator.The fractional differential equations appear naturally in a number of fields such as physics,chemistry,aerodynamics,polymer rheology,etc.Recently,the fractional dofferential equations in-volving left Caputo and right Riemann-Liouville fractional derivatives have gained consid-erable attention.In this thesis,we study the existence of solutions for two kinds of fractional differential equations(systems)involving two fractional derivatives with boundary conditions.Firstly,we study a nonlinear fractional differential equation involving two fractional derivatives with nonlocal boundary conditions as follows:?where ?,?,?+??(0,1),?>0,?>1,?>0,?+?>1 and ?i,??(0,1](i=1,2,…,m).cD1?-is the right-sided Caputo fractional derivative,LD0?+ is the left-sided Riemann-Liouville fractional derivative,I0+1-? is the left-sided Riemann-Liouville fractional integral,?I0+? is a Katugampola fractional integral.By using some new techniques,we introduce a formula of solutions for above problem,which can be regarded as a novelty item.Moreover,under the weak assumptions and using Leray-Schauder degree theory,we obtain the existence result of solutions for above problem.Secondly,we study a nonlinear fractional differential system involving two fractional derivatives with boundary conditions as follows:?where ?>0,??(0,1),a1,a2,a3,a4 ?R\{0},?,p?(0,1),?,q?(0,1),cD1?,cD1p?are the right-sided Caputo derivatives,LD0+?,LD0+q are the Left-sided Riemann-Liouville derivatives,and 1<?+?,?+q,p+q,p+?<2.We present the solutions for above problem and obtain the existence result of solutions under the weak assumptions.