Functional Boundary Value Problems Of Several Classes Of Ordinary Differential Equations

The concepts and methods of functional analysis have penetrated into many branches of modern mathematics,and their influence on boundary value problems of ordinary differential equations has reached a very deep level.Since it is studied in the form of pure algebra and topological structure,many theorems and methods developed in the process of dealing with boundary value problems of ordinary differential equations also appear more extensive and profound.The use of Mawhin’s degree theory in this thesis is an application and extension of Brouwer’s degree theory of operators in functional analysis.Then the research of functional boundary value problem has its important practical value and theoretical significance;Moreover,considering that functional boundary conditions can abstract specific boundary value conditions to more general conditions for research,using Mawhin’s coincidence degree theory and its extension,this thesis studies the existence of solutions of several kinds of ordinary differential equations at resonance under functional boundary conditions.1.We briefly introduce the research background and significance of this topic,as well as the current situation of this topic at home and abroad,and expound the main structure and arrangement of this thesis.Subsequently,we mainly introduce some relevant preliminary knowledge about fractional operators and related properties,the properties of the p-Laplacian operator,and the related definitions and theorems used in this thesis.2.We investigate three kinds of linear operator differential equations according to Mawhin’s coincidence degree theory-the existence of solutions for n order functional boundary value problems at resonance on the half-line,the solvability of impulsive problems for fractional differential equations with functional boundary value conditions at resonance and resonant functional problems involving mixed fractional derivatives.Firstly,we construct suitable Banach spaces and define projection operators that satisfy the appropriate conditions.Secondly,according to the Mawhin’s continuation theorem,we establish lemmas,theorems,and give proofs.Finally,the necessary examples are used to illustrate the conclusions.3.We discuss the functional boundary value problems of three types of nonlinear operator fractional differential equations with p-Laplacian operator.We explore separately functional p-Laplacian boundary value problems with dimKerM=1 on[0,1],p-Laplacian problems with dimKerM=2 on[0,1]under functional boundary value conditions,and higher-order p-Laplacian problems under boundary value conditions on unbounded domains with dimKerM=2.Since the operator M is nonlinear,these three problems are studied by applying Mawhin’s coincidence degree theory and its extension.It is well known that the higher the dimension of the kernel space of the corresponding homogeneous boundary value problem,the more difficult and skillful the operator’s construction will be.In the process of constructing the operator Q,we bypass the rigid requirement that Q is the projection operator in the theorem,and give a new operator Q to develop the proof of the theorem,and finally give related examples to prove the applicability of main results.