The Existence Of Positive Solutions For Several Types Of Fractional Nonlocal Boundary Value Problems With Parameters

Fractional order differential equations have been widely used in mathematics,physics and engineering etc.Owing to the importance in both theory and application,fractional order differential equations nonlocal boundary value problems have received highly at-tention of the domestic and foreign mathematics and natural science field.By using the nonlinear functional analysis theory and methods,many researchers established the exis-tence,nonexistence and multiplicity of(positive)solutions to fractional order differential equations nonlocal boundary value problems with parameters.In this paper,we mainly study the existence,nonexistence and multiplicity of positive solutions for several nonlocal boundary value problems of fractional differential equations with parameters.This paper is divided into three chapters:In Chapter 1,we study the singular fractional differential equation with infinite-point boundary conditions and a parameter:where D0+?,D0+?,D0+? denote the Riemann-Liouville fractional derivative,n-1<? ?n,n ?3,1???n-2,0?r??,?>0 is a parameter.? ? 0,0<?1<?2<…<?i-1<?i<…<1(i =1,2…),(?)g may be singular at t = 0 and/or t = 1,f(t,x)may also have singularity at x=0.We obtain the existence of positive solutions by means of the fixed point index theory under some conditions on f concerning the first eigenvalue correponding to the relevant linear operator.In Chapter 2,we consider the fractional thermostat model with a parameter:where 1<? ? 2,?>0,0???1,??(?)-(1-?])?-1>0,cD0?-is Caputo fractional derivative,?>0 is a parameter,g:(0,1)?[0,+?)is continuous and 0<?01g(s)ds<+?,?:[0,+?)?[0,+?)is continuous.The intervals of ? that correspond to at least two and one positive solutions are obtained by means of the first eigenvalue corresponding to the relevant linear operator,fixed point index theory and iterative methods.The uniqueness of solutions and the dependence of solutions on the parameter are also studied.In Chapter 3,we investigate the following fractional integral boundary value problems with a parameter:where D0+?-2 denotes the Riemann-Liouville fractional derivative,n-1<??n,??4,?,?,????0,A(t),B(t)are nodecreasing on[0,1],?01u(s)dA(s)and ?01u(s)dB(s)denote the Riemann-Stieltjes integrals of u with respect to A and B,respectively.f:[0,1]×[0,+?)?[0,+?)is continuous,?>0 is a parameter.Existence and nonexistence results for positive solutions are derived in terms of the parameter lies in some intervals.We base our analysis on the Guo-Krasnoselskii fixed point theorem on cones.