Some new theorems of expanding mappings without continuity.pdf

Some new theorems of expanding mappings without continuity in cone metric spaces Yan Han Shaoyuan Xu? School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, P.R.China Abstract. In this paper, the fixed point theorems for one mapping and the common fixed point theorems for two mappings satisfying certain expansive conditions are obtained. The mappings are not necessary continuous and the cone is not normal. Example is given to illustrate the results. Keywords. Cone metric space; Expanding mapping; Common fixed point 2010MR Subject Classification. 54H25; 47H10 1 Introduction and preliminaries Recently, Huang and Zhang [1] introduced the concept of cone metric space as a generalization of metric space. They proved the properties of sequences in cone metric spaces and obtained various fixed point theorems for contractive mappings. Afterwards, Abbas and Jungck [3] established the common fixed points for two mappings without exploiting the notion of continuity. Since then, common fixed point theorems in cone metric spaces were proved for mappings satisfying different contractive conditions by many authors (see [4–6, 8–10]). But there are a few results about expanding mappings. Chintaman and Jagannath [7] introduced several meaningful fixed point theorems for expanding mapping. However, the mapping depended strongly on continuous at the fixed points. In this paper, we delete the continuity of the mappings and obtain some fixed points theorems for one expanding mapping and common fixed points theorems for two expanding mappings, which satisfy generalized expansive conditions, in nonnormal cone metric spaces. These results improve and generalize some important known results in [7]. We recall some definitions of cone metric spaces and some of their properties [1]. Let E be a real Banach space and P be a subset of E, θ denotes the zero element of E and intP denotes the interior of P . The subset P is called a cone if and only if: (i) P is closed, none