| Nonlinear functional analysis is a research field of mathematics with profoundtheories and extensive applications. It constructs many general theories and methodsto deal with nonlinear problems on the basis of the study of the nonlinear problemswhich appeared in mathematics and the natural sciences. Its rich theories and ad-vanced methods are widely used in studies of solving many kinds of nonlinear difer-ential equations, nonlinear integral equations and some other types of equations, andhandling many nonlinear problems in computational mathematics, cybernetics, opti-mized theory, dynamic system, economical mathematics, etc. At present, nonlinearfunctional analysis mainly covers topology degree theory, critical point theory, partialorder method, analysis method, monotone mapping theory, and so on.The boundary value problems of nonlinear integer diferential equations are im-portant subjects in the theory of diferential equations. Owing to the importance inboth theory and application, boundary value problems for integer diferential equa-tions have attracted many researchers’ attention, and a large number of results havebeen obtained. In recent years, fractional diferential equations have been widely usedin difusion and transport theory, chaos and turbulence, viscoelastic mechanics, non-newtonian fluid mechanics etc. As one of the hottest issues in the international researchfield, it has received highly attention of the domestic and foreign mathematics and nat-ural science field.The theory and method of nonlinear functional analysis has been employed inthe present paper, such as cone theory, fixed point theory, fixed point index theory,Krasnoselskii fixed point theorem and monotone iterative technique, to investigate theexistence, multiplicity of positive solutions to several kinds of (singular) boundary valueproblems of nonlinear integer (fractional) diferential equations (system). Besides, theexistence of solutions and multiple solutions for nonlinear second order impulsive difer-ential equation has been investigated. Having studied thoroughly, some new interestingresults under weaker conditions have been obtained, most of which have been publishedin Commun. Nonlinear Sci. Numer. Simul.(SCI), Abstr. Appl. Anal.(SCI),Discrete Dyn. Nat. Soc.(SCI), Adv. Diference Equ.(SCI), Bull. Malays.Math. Sci. Soc.(SCI) and Electron. J. Qual. Theory Difer. Equ.(SCI), ect. The dissertation is divided into six chapters. In Chapter I, the background ofnonlinear functional analysis and some basic concepts and theorems have been intro-duced. In Chapter II, the existence results for positive solutions are derived to twokinds of nonlocal boundary value problem of integer diferential systems. In§2.2, theexistence and multiplicity of symmetric positive solutions for a class of singular second-order diferential system with coupled boundary condition is established. In§2.3, theexistence results for positive solutions are considered as a class of singular p-Laplacianfourth-order diferential systems with integral boundary conditions. In Chapter III, theexistence of positive solutions for two kinds of semipositone boundary value problemsof singular integer diferential equation has been studied.§3.2, the existence of at leastone or two positive solutions for a class of singular integral boundary value problemhas been discussed. In§3.3, the existence results for positive solutions to a class ofintegral boundary value problem have been obtained, where the nonlinear term areboth singular at two variables. In Chapter IV, the existence of at least one, or threepositive solutions for a class of singular second order impulsive diferential equationwith integral boundary condition has been dealt with. In Chapter V, the existence andmultiplicity results for positive solutions have been studied to two kinds of higher orderfractional diferential equations. In§5.2, we established the uniqueness of positive so-lution and the dependence of solution on the parameter to a kind of m-point boundaryvalue problem, and the properties of positive solution are given. In§5.3, we obtain theuniqueness, the existence and multiplicity of positive solutions for singular fractionaldiferential equation. In Chapter VI, the study of two kinds of semipositone fractionaldiferential systems has been focused on. In§6.2, the existence of positive solutionsfor singular fractional diferential system has been discussed in a sign-changing nonlin-ear term and integral boundary condition. In§6.3, the existence of positive solutionsto a singular fractional diferential system with coupled boundary condition has beeninvestigated. |
