Existence Of Solutions Of Several Kinds Of Differential Equations

The existence of solutions of differential equations is very important for solving many practical problems.In this paper,we consider the existence of solutions of several kinds of integer and fractional differential equations.Using the fixed point theorem of mixed monotone operator and a new iterative method,we obtain the existence of positive solutions of the problem.According to the content,this paper is divided into six chapters.The first chapter introduced the research background,current situation and significance of several kinds of integer and fractional differential equations,the main work of the full text and innovation points.The second chapter,we consider the existence of solutions of fractional differential equations with Riemann-Stieltjes integral boundary conditions.Using fixed point theorem and some properties of mixed monotone operator theory,we obtain the existence and uniqueness of solutions of the equation.At the end of this chapter,we give two examples to verify our results.The third chapter,we study the existence of solutions for a class of fourth-order equations.Fourth-order equations are widely used in many practical fields.Different from the previous studying,we construct a closed ball in the Banach space,using a new iterative method,we obtain the existence and uniqueness solutions of equations by the means of the contraction mapping theorem.Then the iteration sequence of the problem is given and the form of the iteration solution is obtained.Finally,two numerical examples are given to verify our results.The fourth chapter and the fifth chapter,we obtain the existence of solutions for fractional differential equations with p-Laplace operators and 2n differential equations with higher derivatives.Using the iterative method mentioned in the third chapter,we obtain the existence and uniqueness of the solution of the equation.Finally,two examples are given to verify our results.This chapter can be seen as the application of the iterative method in fractional and higher order equations.The sixth chapter summarized the main contents of the full-text research and explained the practical significance of the corresponding conclusions and looks forward to the future work.