The Existence Research Of Positive Solutions For Boundary Value Problem Of Differential Equations At Resonance

Differential equations originated from the practical problems, such as gas dynamics, chemistry, fluid dynamics, economics and so on. It is because the study of practical problems ultimately attributable to the study of differential equations that differential equations is the key to solve practical problems in proper ways. Many physics, engineering practical problems related to derivative can be attributed to differential equations boundary value problems. As a branch of mathematics research, differential equations boundary value problems have attracted wide attention by more and more scholars. With further research on practical problems, it appears a lot kinds of differential equations boundary value problems, it can be divided into, roughly speaking, such as resonance and non-resonance differential equations boundary value problems; integer order and fractional differential equations boundary value problems. Compared to the non-resonance boundary value problems, it is more difficult to study the existence of solutions for boundary value problems of differential equations at resonance. Therefore, it is more difficult to get positive solutions of boundary value problems for differential equations at resonance. This paper will focus on the existence of positive solutions of three kinds of boundary value problems for differential equations at resonance.By selecting the appropriate spaces, defining the proper operators, using the Leggett-Williams norm-type theorem due to O’Regan and Zima, and endowing reasonable assumptions for the nonlinear term, the existence theorem of positive solutions for three kinds of boundary value problem at resonance were obtained. It is proved the existence of positive solutions for boundary value problem with integral boundary condition at resonance; the existence of positive solutions for a coupled system of differential equations with multi-point boundary value conditions at resonance is obtained; the existence of positive solutions for multi-point boundary value problem at resonance on the half-line is proved.