Solvability Of The Second-order Differential System With Sign-changing Weight Function

Based on cone theory in Banach space and Guo-Krasnosel'skii fixed point theorem,fixed point index theorem and properties of the fixed point index,the existence and multiplicity of positive solutions for three kinds of second-order differential systems sign-changing weight functions are discussed in this master's thesis.The full text is divided into four parts,as follows:Chapter 1 introduces the research background and development status of second-order differential systems,p-Laplacian impulsive differential systems and differential systems with indefinite weight functions,and summarizes the main work of this master's thesis.In the last section,the basic knowledge of definitions and theorems used in this master's thesis is given.Chapter 2 studies the existence of positive solutions for a class of second-order n-dimensional differential systems with sign-changed weight functions.According to Guo-Krasnosel'skii fixed point theorem on cone and combining the characteristics of sign-changed weight function in the systems,it is concluded that there exists at least one positive solution for n-dimensional system when the non-linear term satisfies the appropriate conditions.Chapter 3 discusses the existence of multiple positive solutions for a class of second-order differential systems with indefinite weight functions and multiple parameter.According to the finite additivity of integral interval and combining the characteristics of sign-changed weight function,the difficulty of parameter selection due to the increase of linear term in the system is overcome.According to the different values of the two parameters ? ?,,using the fixed point index theory in cones,two positive solutions and three positive solutions are obtained for this kind of problem.Finally,two examples are given to verify the correctness of the main results.Chapter 4 investigates the existence of two positive solutions for a class of impulsive differential systems with p-Laplace operators and sign-changed weight function.By constructing suitable norm and cone,combining the properties of p-Laplace operators,and the fixed point theorem in cones,the optimal intervals of parameters are obtained when there are two positive solutions.Finally,an example is given to illustrate the rationality of the theorem conditions.In Chapter 5,the main conclusions of this master's thesis are summarized,the difficulties and innovations of the research are summarized,and the future research work is prospected.