Study Of The Theorys And Applications For Some Nonlinear Operators

There are mainly two works in this paper. On the one hand, we discuss two classes of nonlinear operator equations, including nonlinear mixed monotone operator equations in Bnanch space and the set- valued mapping equation in an ordered locally convex topological linear space. The methods employed are mainly partial ordering method and iterative techniques and so on. On the other hand, we will use more precise theory of nonlinear operator to discuss the existence problems of solutions for differential equations.This paper includes three chapters.In Chapter 1, we provide a research summary of several classes of nonlinear operators ( including u₀—concave operators, mixed monotone operators, set-valued (multivalued ) operators,etc). At the some time, we introduce briefly the main works in our papers.In Chapter 2, we present some fixed point theorems for several classes of nonlinear operators.In §2.1 and §2.2, we obtain new fixed points theorems of the existence and uniqueness for t — a(t) model concace convex or t — a(t,u,v) model concave convex mixed monotone operators (see Theorem 2.1.1 and Theorem 2.2.1). The resulting conclusion essentially improves many related conditions and results to date. In fact, similar results that associted to these theorems are also valid to u₀ concace increasing operators. Moreover,as an application of our results, we prove the existence and uniqueness of positive solution of a class of ninlinear integral equation.In §2.3, we introduced a class of w— concace convex mixed monotone operator, and give both necessary and sufficient conditions for the existence and uniqueness of fixed point of this kind of mixed monotone operator (see Theorem 2.3.1 etc );In §2.4, we discuss the existence of fixed point about the sum of mixed monotone operator (see Theorem 2.4.3 etc ). We improve and generalize some related results.In §2.5, by using the obtained results in §2.1 and §2.2, we discuss the existence of solutions of nonlinear evolution equations in Banach space (see Theorem 2.5.2 and Theorem 2.5.3 etc ), improve and generalize some related results in references.In §2.6, firstly, we discuss several fixed point theorems about set-valued mapping in an ordered locally convex topological linear space (see Theorem 2.6.3 and Theorem 2.6.8 etc ), generalize some related results in reference. Secondly, By using the obtained results, we discuss the minimum problem with constraint condition in superior theoreyxeG(x), w(x,x)= min w(x, y),GG()where ui is continuous function, and 6* is set-valued mapping. We give its sufficient conditione for existence of solution (see Theorem 2.6.9 ).In §2.7, by using fixed point theorems of set -valued mappings and Schauder , we discuss the differential equations x'(t) + g(t, x(t)) = 0. The existence and properties of solutions of this equation is obtained (see Theorem 2.7.2 ). To the knowledge of the author, this discussion method are very few in reference.In Chapter 3, we study the existence,nonexistence,multiplicity of periodic positive solutions for functional differential equation and the existence multiplicity of positive solutions for a kind of mixed boundary value problem by employing the fixed-point theorem of Krasnoselskii's cone, the fixed point index in cones and cone expansion or compression type fixed point theorem, etc.In §3.1, we will use more precise theory of the fixed point index in cones to discuss the existence of positive periodic solutions for the functional differential equations with parametery'(t) = -a{t)f(y(t))y(t) + Xg(t,y(t - r(t))),where A > 0 is a parameter.In §3.2, we study the existence,multiplicity and nonexistence of positive u— periodic solutions for a kind of second order functional differential equation with parameterU"(t) + a(t)u(t) = Xf(t, U(t - T0(t)), U(t - Tl(t)), â–  â– ;U{t - Tn{t)))by employing the fixed point theorem of cone expansion or compression , where A > 0 is a parameter.As far as I know, in §3.1, §3.2, our conclusions (see Theorem 3.1.5- 3.1.7 and Theorem 3.2.1- 3.2.3 )are new and original.In §3.3, by using the fixed point theorem of cone expansion and compression and fixed point index theory, we establish the existence and multiplicity of positive solutions for a class of mixed boundary value problems as follows:(?(?'))' + f{t,u) = 0, 0 < i < 1,a