Existence Of Solutions For Several Classes Of Fractional Differential Equations With R-S Integral Boundary Conditions

Early, the research on the theory of fractional calculus focused on the field of pure mathematics, it seemed that the theory was only useful for mathematicians. However, in recent decades, a growing number of fractional differential equations are used to describe the problems of optics and thermal systems, electromagnetics, control engineering and robotic and other fields, attracting the attention of a great many experts at home and abroad. Therefore,both theoretical analysis and applied research for fractional differential equations are particularly urgent. By using nonlinear analysis tools, some scholars got the solutions of nonlinear fractional differential equations, explained and summarized for the theory and application in this area, which had played an important role for promoting the research and development of fractional differential equations.We mainly study the existence of positive solutions for several categories nonlinear fractional differential equation with R-S integral boundary conditions.The paper is organized as follows:In Chapter 1, by means of the fixed point index theory in cones, we consider the existence and multiplicity of positive solutions to nonlinear fractional differential equation with Riemann-Stieltjes integral boundary conditions and parameter: where?> 0 is a parameter, n-1< ?< n,0< ?< 1,0??/???< 1, A(s) denotes bounded variation function, g:[0,1]? [0,+?) and g(s) ? L1[0,1], ?:(0,1)? [0,+?) continuous and ?(t) may be singular at t= 0 and t= 1.f:[0,1] x (0,+?)? [0,+?) and may have singularity at x= 0. D0?+ is the standard Ricmann-Liouville derivative.In Chapter 2, we consider the existence and uniqueness of positive solution to singu-lar problems of nonlinear fractional differential equation with Riemann-Stieltjes integral boundary conditions by using a fixed point theorem for mixed monotone operators: where n-1<??n,0<??1,0??/???<1,A(s)denotes bounded variation function, 9:[0,1]?[0,+?)and g(s)?L1[0,1],f(t,x,y):[0,1]×(0,+?)×(0,+?)?[0,+?) may have singularity at y=0.D0+? is the standard Riemann-Liouville derivative.In Chapter 3,we use a fixed point theorem for monotone operators for considering the existence and uniqueness of positive solution to nonlinear fractional differential equation with Riemann-Stieltj es integral boundary conditions and dependence on the derivatives: where n-1<??n,0<??1,0??/???<1,A(s)denotes bounded variation function, 9:[0,1]?[0,+?)and g(s)?L1[0,1],f(t,x,y):[0,1]×[0,+?)×[0,+?)?[0,+?)is continuous.D0+?is the standard Riemann-Liouville derivative.