Positive Solutions For Multi-point Boundry Value Problems Of Some Fractional Differential Equations

In recent years, along with a wide range of applications in various fields of physics, biology, chemistry, and so on. The study of fractional differential equations has attracted the interest of many scholars, there has been a lot of research solutions for fractional order differential equations and multiple solutions of the article. Most of the research is through the fractional initial value problem into an equivalent integral equation, and then use the fixed point theory to obtain the initial value problem of fractional existence of solutions.In this paper, by using u0-bounded operator, the fixed point index theory, a nonlinear alternative of Leray-Schauder type, Krasnoselskii’s fixed point theorem in a cone and the properties of the Green’s function, we obtain some existence results of positive solutions for a class of fractional differential equation with infinity-point boundary value problem. This thesis contains four chapters:In chapter1, we present here the necessary definitions, lemmas and theorems from fractional calculus theory and then give a number of results about fixed point index theory and fixed point theorem for the existence of solution.In chapter2, we establish the existence of solutions for the following nonlinear fractional differential equations by means of the properties of the Green’s function, u0-positive function and the fixed point index theory under some conditions concerning the first Eigen value with respect to the relevant linear operator [Dα0+u(t)+h(t)f(t,u(t))=0,0<t<1,n-1<α≤n, where i(≥0) are integers, i≤n-2, ηj≥0(j=1,2,…,m-2),0<ξ1<ξ2<…<ξm-2<1, In chapter3, In this paper, by using Krasnoselskii fixed point theorem and the properties of the Green’s function, we obtain some existence results of positive solutions for a class of fractional differential equations with multi-point boundary value problem. where n-1<α,β≤n, i is a fixed constant i∈N,i≤n-2,α,β≥2.f,g:[0,1]×[0,∞)â†'R are two given continuous function, Dα0+v(t),Dβ0+(t) are the standard Riemann-Liouville derivative, In chapter4, we consider the existence results of positive solutions for the following fractional differential equations with infinity-point boundary value problem by using the properties of the Green’s function and alternative theorem. wheren-1<α,β≥n, i is a fixed constant, i∈N,i≤n-2,α,β≥2.f:[0,1]×[0,∞)â†'R...