Research On The Properties Of Solutions Of Several Types Of Fractional Differential Equations

Nonlinear functional analysis is an important research field of mathematics with a wide-ranging application.The establishment of this field aims to translate various phenomena appearing in the real world into nonlinear mathematical problems,and then create a com-plete theories and methods,which can be used to deal with nonlinear problems.The main contents and methods of nonlinear functional analysis include analytical methods,partial order methods,topology degree theory,critical point theory,and monotone mapping theo-ry,and so on.These important methods and theoretical results can be widely applied to the study of nonlinear integral equations,ordinary differential equations,partial differential equations and other various types of equations,and there boundary value problems.Boundary value problem for fractional differential equations is an important branch of differential equations.With the development of the theory of fractional calculus,this theory and its applications have received widespread attention.In fact,it has been found that fractional boundary value problems can be widely used in fluid mechanics,viscoelastic mechanics,the electrical conduction in biological systems,and characterization of the frac-tional regression model,and so on.Due to the practicality and accuracy of the fractional differential equation model,more and more scholars have made a relatively systematic and in-depth study of the properties of the solutions for fractional differential equations.Howev-er,as is well-known,there are many results and methods that are the same as the classical calculus equations.These works can be regarded as an extension of the classical calculus theory.Therefore,a further research on fractional differential equations is necessary.Based on the above reasons,the research on the solutions of fractional boundary value problems is meaningful.Finally,based on previous research,further research and discussion on fractional boundary value problems are viewed.This paper makes full use of a series of method:topology degree theory,the spectral radius of completely positive linear operator and positive eigenvalue theory,fixed point theorem of mixed monotone operator,Schauder fixed point theorem,Banach contraction mapping principle,the monotonic iterative method of increasing operators and partial order topological methods,and so on.We then obtain the existence,uniqueness,nonexistence,upper and lower bound estimates,and dependence on parameters of the solutions for several types of nonlinear higher-order(singular)fractional differential equations,which are novel and meaningful.In addition,notice that there is little theoretical knowledge about cones in Holder space,which needs a further study.Therefore,we want to define a new cone with normality,regularity and other good properties are constructed.And on this basis,we consider some problems of fractional differential equations.The paper is divided into eight chapters.In Chapter I,the theoretical background,some preliminary definitions and properties of nonlinear functional analysis and fractional calculus are given.Several lemmas about fixed point theorem are then given,which play an important role in this paper.In Chapter ?,we consider a class of singular fractional differential equations with p-Laplacian operator,in which the uniqueness,the convergence,the continuous dependence and monotonicity on parameters of positive solutions are nu-merically analyzed.The results are then shown in graphs and tables.In Chapter ?,by using the reduced method of fractional derivative,the fixed point index theory and the Ba-nach contraction mapping principle,we introduce a class of nonlinear fractional differential equations with mixed-type boundary value conditions,in which the nonlinear term contains the fractional derivatives.It then holds that the existence and the uniqueness of positive solutions.In Chapter IV,we consider a class of singular nonlinear higher order fractional boundary value problems supplemented with sum of Riemann-Stieltjes integral type and nonlocal infinite-point discrete type boundary conditions.The fractional derivative of differ-ent orders is involved in the nonlinear term and boundary conditions.The uniqueness and continuous dependence on parameters of iterative positive solution are established by using the fixed point theorem of mixed monotone operator.In Chapter V,we introduce a class of differential equations,which derived from physics.This problem aims to describe the me-chanics phenomenon of turbulent flow in a porous medium.On this basis,we consider a class of singular nonlinear boundary value problems involving two types of fractional derivatives with p-Lapla.cian operator.Based on the fixed point theorem of mixed monotone operator,the uniqueness and continuous dependence on parameters of iterative positive solutions are obtained.Finally,two numerical examples are presented.We then analyze the data through graphs and tables,which illustrate the availability of our main results.In Chapter ?,we consider a class of higher-order fractional boundary value problems.Based on the Schaud-er fixed point theorem,the existence of solutions is obtained with the hypothesis that the nonlinear term satisfies the Caratheodory conditions.It then holds that the uniqueness of solutions based on the Banach contraction mapping principle.Finally,based on the theory of spectral radius,the uniqueness and nonexistence of positive solutions are obtained.In Chapter ?,we consider a class of boundary value problems with p-Laplacian operator in-volving two kinds of fractional differential derivatives.By using the Schauder fixed point theorem,the existence of solutions is proved.The numerical estimation of upper and lower bound of the unique solution is analyzed by using the Banach contraction mapping princi-ple.In Chapter ?,we define a new pone in the Holder space,and consider its normality and regularity,and so on.On this basis,we consider its applications for the problems of fractional differential equations.