The Discussion Of Some Nonlinear Operators And Applications To Boundary Value Problems For Differential Equations

The purpose of this paper is to discuss two classes of nonlinear problems, one of which is nonlinear operator equations and the other is some applications of nonlinear operator theory to boundary value problems for differential equations.The methods employed are mainly partial ordering method and iterative techniques and so on.This paper includes three chapters.In chapter 1, we provide a research summary of several classes of nonlinear operators (α—homogeneous operators, a class of operators of which fixed points are obtained by iterative technique, set-valued operators) since the beginning when these operators' definitions were introduced, and the general thoughts in which we can deal with some integral equations,boundary value problems for differential equations by using nonlinear theory.In chapter 2, we present some fixed point theorems for several classes of nonlinear operators.In §2.1, sufficient conditions for the existence and uniqueness of fixed points to α—homogeneous operators and some mean value theorems of homogeneous operators are given under much weaker conditions, such results and methods have seldom been seen in literature available.In §2.2, some existence theorems of fixed pionts for a class of operators of which fixed points are obtained by iterative technique are given, and an application to a class of integral equations is considered.In §2.3, we obtain some new fixed point theorems for set-valued maps in ordered Banach spaces, and the sufficient and necessary conditions for existence of fixed points to certain operators similar to α—concave operators.In addition, we utilize the results to study the existence and uniqueness of positive fixed points for a class of α—convex operators. The idea of studying α—concave operators via set-valued maps has not been seen in literature available.In §2.4, a new nonzero fixed point theorem for completely continuous operators in C[0,1] is given by using the properties of fixed point index. Its applications improve and generalize previous results of three-point,two-point and m-point boundary valueproblems for differential equations.In chapter 3,the existence and multiplicity of positive solutions for several classes of nonlinear boundary value problems for differential equations are discussed by using nonlinear operator theory. In some degree, we develop the methods of studying boundary value problems for differential equations.In §3.1, by using the fixed point theorem of cone expansion and compression, the results in §2.4 and multi- solution theorems in cones, we establish the existence and multiplicity of positive solutions to several classes of three-point boundary value problems as follows: (I) Semi-positone three-point BVPu"(t) + Xf(t,u(t)) = 0, i€(0,l),u(0) = 0, au(ri) = u(l).where 0 < 77 < 1, 0 < a < ^, f(t, u) > —M, here M is a positive constant. (II)Three-point BVP for differential equations with an advanced argumentu"(t) + a(t)f(u(h(t))) = 0, ie(0,l),u,(0) = 0, au(rj) = u(l).where 0 < r) < 1,0 < a < ±, h e C((0,1), (0,1]) satisfies t < h(t) < 1, t € (0,1). (III)Three-point BVP with the parameter in closed intervalu(0) = 0,au(T)) = m(1).where 0 Ao > 0.In §3.2. by using Leggett-Williams fixed point theorem, we establish the existence ofat least two positive solutions to the nonlinear semi-positone m-point boundary valueproblems t,u) = 0,t€(0,l),m-2 m-2u'(0) = ￡>?'(&), u(l) = J ?=1 t=l...