Nonlinear Operators, Fixed Point Theory And Its Applications

Nonlinear functional analysis is an important branch of mathematics, and it can explain several kinds of natural phenomena. So, the mathematical world and the natural science world attach importance to the nonlinear functional analysis. They have obtained some new results for the nonlinear functional analysis and its applications.It is well known that, the fixed point theorem for the nonlinear operators, including increasing operators, decreasing operators, and mixed monotone operators,and so on, is the advantageous tool to research the nonlinear differentialequations, and nonlinear integral equations. People obtain the existence and uniqueness theorem of nonlinear differential equations, and nonlinear integralequations, the existence of the maximal or minimal solution, and the iterative methods, by using the existence and uniqueness of nonlinear operators and the contract rate of iterative sequence.In recently years, it has obtained some new results of the fixed point theorem and applications for nonlinear operators. People investigate the existenceand uniqueness of fixed point, the existence and the scope of existence of multi-point by using the partial order theory and topological degree theory. In the past, they research the operators with continuous or compact conditions.But now, we do not require the operator to be compact or continuous, we also obtain some new theories, which have extensive and impotent application.The fixed point of mixed monotone operators has been investigated with concrete and profoundly. But the results about the fixed point of singular operators, multivalued operators are still few, even there are some theories, they always require strong conditions. By the arouse of documents [1-29], the present paper does not require strong conditions, and obtains some new fixed point theorems. The results obtained are essentially generalize and improve the previous relevant ones by using weakly conditions.The thesis is divided into three chapters according to contents:In Chapter 1, by a fixed point theorem for mixed monotone operators, the existence and uniqueness is established for second-order two-point boundary value problem of singular differential equationsFinally, we apply the conclusion to nonlinear boundary value problem of singulardifferential equation.In Chapter 2, we obtain the existence and uniqueness theorem of positive fixed point for more general operators, precisely, for generalized e-concave and generalized e-convex monotone operators with perturbation, by using the partial order theory and monotone iterative technique. That is, we obtain some new existence and uniqueness theorems for positive fixed point of C = A + B, where A is a monotone operator with generalized e-concavity and generalized e-convexity, B is a sublinear operator, and does not require continuous or compact condition, which extend the corresponding results in [10-14]. Moreover, some applications to nonlinear integral equations are given.In Chapter 3, by the arouse of documents [15-17, 20-25, 27], the existence of fixed points for a class of t -α(t) mixed monotone multivalued model operators is discussed, by means of the properties of cone, the monotonicity of multivalued operators, and get some correlated conclusions. We also present some new fixed point theorems of increasing multivalued operators and decreasing multivalued operators. Our conclusions improve the relevant results obtained by Wu and others. The mixed monotonicity of operator in Chapter 3 is different from operator in Chapter 1 and 2. In Chapter 1 and 2, the mixed monotone operator is general operator, but in Chapter 3, the operator is t -α(t) mixed monotone multivalued model operator.