The Existence Of Positive Solutions For Boundary Value Problems Of Fractional Differential Equations

In recent decades, fractional calculus have been widely used in electromagnetics, chemical, control science and mechanics, and other disciplines. Research shows that the introduction of fractional calculus perhaps have new findings and conclusions in traditional method powerless and provides a powerful tool to solve nonlinear problem, and it has become a hotspot in the research and engineering application.This paper of master degree chiefly studies the existence and uniqueness of posi-tive solutions for four-point boundary value problem of nonlinear fractional differential equations, the differential operator we discussed is defined in Caputo sense.2Here what different from other literatures is positive solution that we have discussed, and the con-ditions which the nonlinear term satisfies are discussed fully.First, we give the historical background, current situation of the problem we studied. Second, we transform the differential equations of boundary value conditions under Ca-puto definition into Fredholm integral equations by using relevant properties. Then we deduce its green's function and fully exploit the corresponding properties. The solution of integral equation is equivalent to fixed point of operator equations and proof its com-pletely continuous. Finally, Wc use Banach fixed point theorem and Lipschitz condition to obtain the existence and uniqueness of positive solutions; using the Leray-Schauder nonlinear choices and linear growth conditions to get at least a positive solutions; using Solution theorem to gain at least three positive solutions; using the cone tensile com-pression theorem and nonlinear term meeting super linear conditions and times linear conditions acquire boundary value problems at least one, two or three positive solutions, the limit of nonlinear term is generalized to normal number which is one of the paper's important work. We also give some simple relative examples to illustrate what we said above, thus we make previous research results more plentiful.