Some Research On The Fixed Point Theorems Of Nonlinear Operators

In this thesis, we discuss the fixed point theorems of nonlinear operators andtheir applications.In Chapter1, we introduce the backgrounds of the problem and the main resultsof this thesis.In Chapter2, we investigate the existence of periodic solution for a class ofnonlinear functional integral equation. We first prove a fixed point theorem in aBanach algebra and with its help, an existence theorem about periodic solution tothe addressed functional integral equation is presented. In addition, an example isgiven to illustrate our result.In Chapter3, we investigate mixed g-monotone mappings in partially orderedmetric spaces. We establish several coupled coincidence and coupled common fixedpoint theorems, which generalize some known results. Especially, our main resultsextend and complement some recent results due to the article of Lakshmikantham.In Chapter4, we introduce a notion of weakly increasing mappings with two vari-ables. Several coupled common fixed point theorems for weakly increasing mappingsin ordered metric spaces are established. Then, by using a scalarization method,we obtain two coupled common fixed point theorems in ordered cone metric spaces,which extend some earlier results about weakly increasing mappings with one vari-able.In Chapter5, we discuss the common fixed point theorem of two multivaluedgeneralized-weak contractive mappings in a complete metric space. This theoremis a generalization of the common fixed point theorem in [41].