Existence Of Positive Solutions For Integral Boundary Value Problems Of Nonlinear Fractional Differential Equations

Compared to the integer calculus in describing natural phenomena and objective laws,fractional calculus is more accurate,widely used in many fields,and obtained better achievements in the research topic especially in the subjects of physics,chemistry,engineering and so on.Among them,the study of the existence of positive solutions for the boundary value problem of the nonlinear fractional differential equation is an important part of the fractional calculus theory,which has important theoretical significance.In this paper,three kinds of integral boundary value problems for nonlinear fractional differential equations are studied.By using the cone theory,the fixed point theorem and related theories,a series of sufficient conditions for judging the existence and uniqueness of positive solution are given.The main contents are as follows:Firstly,a class of nonlinear boundary value problem for impulsive fractional differential equations is studied,in which the impulsive condition contains fractional state variables and boundary value conditions contain integral condition of bounded variation function.The sufficient conditions for the uniqueness of the positive solution and the multiplicity of the positive solutions are obtained respectively by using the fixed point theorem of the operator and the fixed point theorem of Avery-Peterson.Then,a class of integral boundary value problem for fractional differential equations with infinite number of impulse points and fractional state variables in Banach space is discussed.The sufficient conditions for the existence of positive solutions are obtained respectively by using the fixed point theorem of Schauder and the fixed point theorem of GuoKrasnosel’ski?.Finally,a class of integral boundary value problem for fractional differential equations on the infinite intervals is considered,and the mixed fractional derivative of Caputo and Riemann-Liouville is included in the equations.The sufficient conditions for the existence and uniqueness of the positive solution are obtained respectively by using the fixed point theorem of Schauder and the principle of Banach compression mapping.