Existence Of Multi-positive Solutions For Multi-point Boundary Value Problem For Fractional Differential Equations

Now, the scope of application of the fractional differential equations is very extensive, including genetic mechanics, description of molecular diffusion theory, rheological properties of rock, viscoelastic fractal theory, control system etc. Fractional differential equations have become the focus of the International Mathematics Research. However, research on the differential equation with fractional derivative nonlinear terms in the boundary value problem is little. And so far, this kind of article mostly adopts the degree theory to study the existence of solution. The research methods and research contents are in need of a breakthrough. It is based on this situation, this article was born. In the article, the main research is the fractional differential equations of fractional order derivative with conditions in the nonlinear term and the boundary value.The first chapter introduces the background knowledge of fractional differential equations and some research results.The second chapter presents some basic definitions and theorems, including the fractional differential equations of spaces, the cone theory basic knowledge and the fixed point theorem.The third chapter proves the existence of positive solutions of boundary value problems of fractional differential equation by using the Krasnosel’skii fixed point theorem. Finally, a specific example explains that the conclusion is practical.The fourth chapter discusses that the fractional order differential equation boundary value problem has at least 3 positive solutions by using the Leggett and Williams fixed point theorem. And then, the existence criteria for at least three or arbitrary n or 2n-1 positive solutions are established.The last chapter summarizes the whole framework of this article, and explains the research direction of the next step.