tailieunhanh - Note on common fixed point for noncommuting mappings without continuity in cone metric spaces
In this work, we prove a common fixed point theorem by using the generalized distance in a cone metric space. Note on common fixed point for noncommuting mappings without continuity in cone metric spaces | Note on common fixed point for noncommuting mappings without continuity in cone metric spaces Nguyen Duc Lang University of Science, Thainguyen University, Vietnam E-mail: nguyenduclang2002@ Abstract: In this work, we prove a common fixed point theorem by using the generalized distance in a cone metric space. Keywords: Cone metric space, Common fixed point, Fixed point. Mathematics Subject Classification: 47H10, 54H25. 1 Introduction In 2007, Huang and Zhang [8] introduced the concept of the cone metric space, replacing the set of real numbers by an ordered Banach space, and proved some fixed point theorems of contractive type mappings in cone metric spaces. Afterward, several fixed and common fixed point results in cone metric spaces were introduced in [1, 2, 10, 11, 14] and the references contained therein. Also, the existence of fixed and common fixed points in partially ordered cone metric spaces was studied in [3, 4, 5, 12]. The aim of this paper is to generalize and unify the common fixed point theorems of Abbas and Jungck [1], Hardy and Rogers [7], Huang and Zhang [8], Abbas et al. [2], Song et al. [14], Wang and Guo [15] and Cho et al. [6] on c-distance in a cone metric space. 2 Preliminaries Lemma . ([4, 9]). Let E be a real Banach space with a cone P in E. Then, for all u, v, w, c ∈ E, the following hold: (p1 ) If u v and v w, then u w. (p2 ) If 0 u c for each c ∈ intP , then u = 0. (p3 ) If u λu where u ∈ P and 0 n0 . Lemma . ([6, 13, 15]). Let (X, d) be a cone metric space and let q be a c-distance on X. Also, let {xn } and {yn } be sequences in X and x, y, z ∈ X. Suppose that {un } and {vn } are two sequences in P converging to 0. Then the following hold: (qp1 ) If q(xn , y) un and q(xn , z) vn for n ∈ N, then y = z. Specifically, if q(x, y) = 0 and q(x, z) = 0, then y = z. (qp2 ) If q(xn , yn ) un and q(xn , z) vn for n ∈ N, then {yn } converges to z. (qp3 ) If q(xn , xm ) un for m > n, then {xn } is a .
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