Common Coupled Fixed Point Theorems of Single-Valued Mapping forc-Distance in Cone Metric Spaces

Abstract

The existence and uniqueness of the common coupled fixed point in cone metric spaces have been studied by considering different types of contractive conditions. A new concept of thec-distance in cone metric space has been recently introduced in 2011. Then, coupled fixed point results for contraction-type mappings in ordered cone metric spaces and cone metric spaces have been considered. In this paper, some common coupled fixed point results onc-distance in cone metric spaces are obtained. Some supporting examples are given.

Some New Theorems on c-Distance without Continuity in Cone Metric Spaces over Banach Algebras

“…Our main results improve and generalize some important known results in the literature [6,7,[9][10][11][12][13][14][15][16][17]. In addition, we give an example to show that the main results are indeed real improvements and generalizations of the corresponding results in the literature.…”

“…Comparing the main results in the literature [6,7,[9][10][11][13][14][15][16][17], we see that many conclusions in the literature relied strongly on either the assumption of the continuity of the mappings (Theorems 3.1 and 3.3 in [6]; Theorems 3.5 and 3.9 in [9]; Theorems 3.1 and 3.3 in [10]; Theorems 2-5 in [11]; Theorem 3.3 in [14]; Theorem 2.4 and Corollaries 2.5-2.7 in [16]; Theorems 3.1, 3.2, 3.5, and 3.6 in [17]); or the normality of the cone on -distance in cone metric space (Theorems 3.2 and 3.4 in [6]; Theorems 3.2 and 3.5 in [10] [7]). They must have one of the two.…”

“…In 2011, Cho et al [6] and Wang and Guo [7] defined the concept of -distance which is a cone version ofdistance of Kada et al [8]. After that, lots of fixed point results on -distance in cone metric spaces and in tvs-cone metric spaces were introduced in [9][10][11][12][13][14][15][16][17]. However, the conditions relied strongly on the assumptions that the underlying cone is normal or the mappings are continuous.…”

T-Reich contraction and fixed point results in cone metric spaces with c-distance

“…In 2011, Cho et al [1] introduced the concept of the c-distance in cone metric spaces. Then, many authors studied the existence and uniqueness of the fixed point, common fixed point, coupled fixed point and common coupled fixed point problems using this distance in cone metric spaces and ordered cone metric spaces; see for example [5][6][7][8][9][10][11][12][13][14].…”

“…Example 2.6. ( [8,9]) Let E = R 2 and P = {(x, y) ∈ E : x, y ≥ 0}. Let X = [0, 1] and define a mapping d :…”