Solvability Of An Operator And Operator Equations In Banach Space

In this paper, we make use of partial order method to study the existence of fixed points of non-linear operators in Banach space E with a standard basic sequence and the solvability of two classes of operator equations. More over we set up some theorems of fixed point, the existence and uniqueness of solution of two classes of operator equations. The main results are given as follows:In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we define a partial order which depends on the standard basic sequence in Banach space E . Use the partial order, we construct a cone in E and discusses properties of the cone and the partial order. Based on the properties above, we proves some fixed point theorems and discusses the existence of the Hammerstein integral equation's solutions in the finite dimension space.In Chapter 3, we use partial order theory to study the solvability of two classes of operator equations of A( x,y)=Bx and A( x,y)= B(x,y) in Banach space, and have the existence and uniqueness theorem of fixed point without compact or continuous operators.In Chapter 4, we discuss fixed points of some mixed monotone operators on the Pf cone in the linear normal space through partial order theory, and get the existence and uniqueness theorem of fixed point on condition that the operators are no compact or no continuous. Finally we apply the new result to study the solvability of the Hammerstein integral equation.In Chapter 5, we use the Mountain Pass Lemma to obtain the existence theorem of weak solutions of P ( x)?Laplaceequations in the bounded domains.