The Research Of Existence Positive Solutions For Some Nonlinear Fractional Differential Equations

In the recent years, since it could be better to describe and description many phenomena of physical, biological, mechanical and aeronautical engineering, researchers are more and more focus about fractional differential equation. Such aspects closely linked our life and job as weather and climate researching, medical image and analysis of seismic singularity can also through the establishment of the fractional differential equation to solve some relevant problems. This paper mainly use cone fixed point theorem, monotone iterative technique to discuss classes of nonlinear fractional differential equation and the existence of positive solution. Our results extend and improve the previous result, meanwhile, obtain the sufficient conditions for the study of the nontrivial solution.The structure arrangement of the thesis is as follows:Chapter one we mainly make an overview about the contents of our questions, background status and the development trend of the research.In chapter two we list the necessary basic definition, property, theorem and lemma of the special function, fractional calculus, cone and partial ordering.Chapter three research the existence of the type of Riemann-Liouville derivative nonlinear fractional differential equation positive solution, and obtain the necessary condition for nontrivial solution under some conditions. Firstly, we present the expression of integral solution of fractional differential equation(3.1)product, the accurately expression of Green function and some properties of Green function. For better to prove it, we also delimiting some necessary space and lemma. Secondly, on the basis of the Theorem 2.2.1, we arrival at the main results of this chapter. Finally, some examples are given to verify the feasibility of this chapter work.Chapter four research the existence of the nonlinear fractional differential equation with integral boundary condition positive solution, and obtain the necessary condition for nontrivial solution under some conditions. Firstly, we present the expression of integral solution of the nonlinear fractional differential equation(4.1), some properties of Green function and the accurately expression of Green function. To prove the conclusion of this chapter, we also delimiting some necessary space and lemma. Secondly, according to the Theorem 2.2.1, we arrival at the main results of this chapter. Finally, some examples are given to verify the feasibility of this chapter work.In chapter five. We bring the paper summarizes in this chapter, moreover, definite the direction for future research.