Some Fixed Point Theorems in Ordered G-Metric Spaces and Applications

and Applied Analysis 3 The following are examples of G-metric spaces. Example 1.8 see 7 . Let R, d be the usual metric space. Define Gs by Gs ( x, y, z ) d ( x, y ) d ( y, z ) d x, z , 1.4 for all x, y, z ∈ R. Then it is clear that R, Gs is a G-metric space. Example 1.9 see 7 . Let X {a, b}. Define G on X ×X ×X by G a, a, a G b, b, b 0, G a, a, b 1, G a, b, b 2 1.5 and extend G to X ×X ×X by using the symmetry in the variables. Then it is clear that X,G is a G-metric space. Definition 1.10 see 7 . A G-metric space X,G is called G-complete if every G-Cauchy sequence in X,G is G-convergent in X,G . The notion of weakly increasing mappings was introduced in by Altun and Simsek 16 . Definition 1.11 see 16 . Let X, be a partially ordered set. Two mappings F,G : X → X are said to be weakly increasing if Fx GFx and Gx FGx, for all x ∈ X. Two weakly increasing mappings need not be nondecreasing. Example 1.12 see 16 . Let X R, endowed with the usual ordering. Let F,G : X → X defined by Fx ⎧ ⎨ ⎩ x, 0 ≤ x ≤ 1, 0, 1 < x < ∞, gx ⎧ ⎨ ⎩ √ x, 0 ≤ x ≤ 1, 0, 1 < x < ∞. 1.6 Then F and G are weakly increasing mappings. Note that F and G are not nondecreasing. The aim of this paper is to study a number of fixed point results for two weakly increasing mappings f and g with respect to partial ordering relation in a generalized metric space. 4 Abstract and Applied Analysis 2. Main Results Theorem 2.1. Let X, be a partially ordered set and suppose that there exists G-metric in X such that X,G is G-complete. Let f, g : X → X be two weakly increasing mappings with respect to . Suppose there exist nonnegative real numbers a, b, and c with a 2b 2c < 1 such that G ( fx, gy, gy ) ≤ aGx, y, y bGx, fx, fx Gy, gy, gy c [ G ( x, gy, gy ) G ( y, fx, fx )] , 2.1 G ( gx, fy, fy ) ≤ aGx, y, y bGx, gx, gx Gy, fy, fy c [ G ( x, fy, fy ) G ( y, gx, gx )] , 2.2 for all comparative x, y ∈ X. If f or g is continuous, then f and g have a common fixed point u ∈ X. Proof. By inequality 2.2 , we have G ( gy, fx, fx ) ≤ aGy, x, x bGy, gy, gy Gx, fx, fx c [ G ( y, fx, fx ) G ( x, gy, gy )] . 2.3 If X is a symmetric G-metric space, then by adding inequalities 2.1 and 2.3 , we obtain G ( fx, gy, gy ) G ( gy, fx, fx ) ≤ aGx, y, y Gy, x, x 2bGx, fx, fx Gy, gy, gy 2c [ G ( x, gy, gy ) G ( y, fx, fx )] , 2.4 which further implies that dG ( fx, fy ) ≤ adG ( x, y ) b [ dG ( x, fx ) dG ( y, gy )] c [ dG ( x, gy ) dG ( y, fx )] , 2.5 for all x, y ∈ X with 0 ≤ a 2b 2c < 1 and the fixed point of f and g follows from 2 . Now if X is not a symmetric G-metric space. Then by the definition of metric X, dG and inequalities 2.1 and 2.3 , we obtain dG ( fx, gy ) G ( fx, gy, gy ) G ( gy, fx, fx ) ≤ aGx, y, y Gx, x, y 2bGx, fx, fx Gy, gy, gy 2c [ G ( x, gy, gy ) G ( y, fx, fx )] Abstract and Applied Analysis 5 ≤ adG ( x, y ) 2b [ 2 3 dG ( x, fx ) 2 3 dG ( y, gy )]and Applied Analysis 5 ≤ adG ( x, y ) 2b [ 2 3 dG ( x, fx ) 2 3 dG ( y, gy )] 2c [ 2 3 dG ( x, gy ) 2 3 dG ( y, fx ) ] adG ( x, y ) 4 3 b [ dG ( x, fx ) dG ( y, gy )] 4 3 c [ dG ( x, gy ) dG ( y, fx )] , 2.6 for all x ∈ X. Here, the contractivity factor a 8/3 b 8/3 c may not be less than 1. Therefore metric gives no information. In this case, for given x0 ∈ X, choose x1 ∈ X such that x1 fx0. Again choose x2 ∈ X such that gx1 x2. Also, we choose x3 ∈ X such that x3 fx2. Continuing as above process, we can construct a sequence {xn} in X such that x2n 1 fx2n, n ∈ N∪ {0} and x2n 2 gx2n 1, n ∈ N∪ {0}. Since f and g are weakly increasing with respect to , we have x1 fx0 g ( fx0 ) gx1 x2 f ( gx1 ) fx2 x3 g ( fx2 ) gx3 x4 · · · . 2.7 Thus from 2.1 , we have G x2n 1, x2n 2, x2n 2 G ( fx2n, gx2n 1, gx2n 1 ) ≤ aG x2n, x2n 1, x2n 1 b [ G ( x2n, fx2n, fx2n ) G ( x2n 1, gx2n 1, gx2n 1 )] c [ G ( x2n, gx2n 1, gx2n 1 ) G ( x2n 1, fx2n, fx2n )] aG x2n, x2n 1, x2n 1 b G x2n, x2n 1, x2n 1 G x2n 1, x2n 2, x2n 2 c [ G ( x2n, x2n 2, gx2n 2 ) G x2n 1, x2n 1, x2n 1 ] a b G x2n, x2n 1, x2n 1 bG x2n 1, x2n 2, x2n 2 cG x2n, x2n 2, x2n 2 . 2.8 By G5 , we have G x2n 1, x2n 2, x2n 2 ≤ a b c 1 − b − c x2n, x2n 1, x2n 1 . 2.9