Research On The Existence Of Positive Solutions For Some Class Of Boundary Value Problems Of Differential Equations

Since the20th century, with the development of the functional theory, as well asthe need to solve practical problems, boundary value problems has been developingrapidly in nearly half of this century. Especially, the study on the existence of thepositive solutions for the boundary value problems had already aroused great concernof experts and scholars at home and abroad. Therefore, these factors are making thisareas to be one of the hot points of mathematical research workers recently. Focusingon the existence of positive solutions for several types of boundary value problems inthis article by using some different methods to obtain several criteria for the existenceof positive solution of boundary value problems is the mainly purpose here.This paper consists of the following four parts.The first part is introduction. Mainly introduce the origin and development of theboundary value problems of the differential equations and briefly present its bothresearch status and research significance;In the second part, we introduced some fixed point theorems and discussed theexistence of positive solution of second order N-point boundary value problem fornonlinear ordinary differential equations. We got some theorems of the existence forpositive solutions under the differential conditions;In the third part of this article, we studied the existence of the positive solution ofthe nonlinear second-order four points integral boundary value problem of theordinary differential equations and established the criteria for the existence of positivesolution. And then we gave two examples to prove the correctness of our main results;In the fourth part of this article, we introduced briefly the definition and simplenature of fractional differential equations and studied the existence of positivesolutions fractional differential equations with integral boundary by using the properties of the Green’s function. Finally, we illustrated the validity of theconclusions of our main results by using two examples.