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A generalization of Banach’s contraction principle for some non-obviously contractive operators in a cone metric space

This paper investigates the fixed points for self-maps of a closed set in a space of abstract continuous functions. Our main results essentially extend and generalize some fixed point theorems in cone metric spaces. An application to differential equations is given. | Turk J Math 36 (2012) , 297 – 304. ¨ ITAK ˙ c TUB doi:10.3906/mat-1005-267 A generalization of Banach’s contraction principle for some non-obviously contractive operators in a cone metric space Yingxin Guo Abstract This paper investigates the fixed points for self-maps of a closed set in a space of abstract continuous functions. Our main results essentially extend and generalize some fixed point theorems in cone metric spaces. An application to differential equations is given. Key Words: Cone metric space, fixed point, Ordered Banach space, self-maps of a closed set, iterative sequence 1. Introduction Fixed point theory is a mixture of analysis, topology and geometry. The theory of existence of fixed points of maps has been revealed as a very powerful and important tool in the study of nonlinear phenomena [3, 5–9, 15–23, 27–29]. If a topological space is a metric space, or a linear topological space, then the fixed point theory in such spaces is very abundant. Cone metric spaces were introduced in [10]. The authors there described convergence in cone metric spaces and introduced completeness, then they proved some fixed point theorems of contractive mappings on cone metric spaces. Recently, in [1, 2, 4, 11–14, 16, 21, 24–25] some fixed point theorems were proved for maps on cone metric spaces. In particular, Du [26] showed that from each cone metric one can get the usual metric by using a scalarization function. Hence the results of Huang-Zhang [10] and the results of many other authors are obtained trivially by Du’s method. But there is a paper, [15], on cone metric spaces in which Du’s method is not applicable. In this work we prove some fixed point theorems in cone metric spaces, including results which generalize those from Huang and Zhang’s work. Given the fact that, in a cone, one has only a partial ordering, it is doubtful that their Theorem 2.1 can be further generalized. The organization of this paper is as follows. In Section 2, problem formulation and .