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 α,β,α+β∈(0,1),λ>0,γ>1,ρ>0,α+ρ>1 and ξi,η∈(0,1](i=1,2,…,m).cD1β-is the right-sided Caputo fractional derivative,LD0α+ is the left-sided Riemann-Liouville fractional derivative,I0+1-α is the left-sided Riemann-Liouville fractional integral,ρI0+γ 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 λ>0,η∈(0,1),a1,a2,a3,a4 ∈R\{0},α,p∈(0,1),β,q∈(0,1),cD1α,cD1p－are the right-sided Caputo derivatives,LD0+β,LD0+q are the Left-sided Riemann-Liouville derivatives,and 1<α+β,α+q,p+q,p+β<2.We present the solutions for above problem and obtain the existence result of solutions under the weak assumptions.