Research On Positive Solutions For Boundary Value Problems Of Fractional Differential Equations With P-Laplacian Operator

In this passage,we for several kinds of higher order fractional differential equations(systems)with P-Laplacian operators carried on the thorough analysis and discuss,and testified the existence and uniqueness of its positive solutions.Some of the existing papers are improved and perfected to some extent,and some significant results are obtained.The full text is mainly made up of the following four chapters.In chapter 1,we briefly described the origin,development,application value and research significance of fractional calculus theory.In addition to all that,with regard to the solvability of positive solutions of boundary value problems of fractional differential equations(systems)with P-Laplacian operator,the corresponding research actuality at home and abroad is briefly expounded,and at the same time,this chapter introduces the relevant concepts,lemmas and theorems that must be used in the research of this paper.In chapter 2,The uniqueness of positive solutions of integral boundary value problem of a class of fractional P-Laplacian differential equations with fractional derivatives terms and linear operators in nonlinear terms is analyzed and investigated.The boundary conditions of the equation are Riemann-Stieltjes integral with fractional derivatives term.In combination with the properties of mixed monotone operators on cones,we established and verified the uniqueness of positive solutions of the boundary value problem according to the fixed point theorem of related operator equations.In the end,a numerical value example is given as an application to reflect the correctness and feasibility of the main conclusions.In chapter 3,we are focused on analyzing the sufficient conditions for the unique positive solutions of mixed fractional differential systems with P-Laplacian operators.At the same time,the boundary conditions in the system contain parameters and have certain dependence on parameters.Based on the properties of Green functions,and using the fixed point theorem of the sum of two operators in semi-ordered Banach spaces,the uniqueness of positive solution of the system is proved through a series of derivations.Apart from that,an iterative sequence that approximates the unique positive solution is constructed.Finally,it is reasonable and correct to use two concrete examples as applications to illustrate the main results.In chapter 4,the semipositone system of fractional differential equations with P-Laplacian operator is analyzed and investigated.The nonlinear term in the system is singular and contains fractional integral term,and the boundary conditions of the system include fractional derivatives terms,Riemann-Stieltjes integrals and infinite point boundary condition of the unknown functions.Based on the properties of Green function obtained from the formula derivation and the fixed point index theorem on cone,the sufficient condition for the existence of at least one positive solution for the system is established when the parameters are in a suitable interval.In the end,a numerical example is used as an application to show that the main results are valid and practical.