Some Fixed-Point Results for a G-Weak Contraction in G-Metric Spaces

and Applied Analysis 3 Proposition 2.5. Let X,G be a G-metric space. Then the following statements are equivalent. 1 The sequence {xn} is G-Cauchy. 2 For every ε > 0, there is k ∈ N such that G xn, xm, xm < ε, for all n,m ≥ k. Definition 2.6. A G-metric space X,G is called G-complete if every G-Cauchy sequence in X,G is G-convergent in X,G . Proposition 2.7. Let X,G be a G-metric space. Then, for any x, y, z, a ∈ X it follows that: i if G x, y, z 0, then x y z; ii G x, y, z ≤ G x, x, y G x, x, z ; iii G x, y, y ≤ 2 G y, x, x ; iv G x, y, z ≤ G x, a, z G a, y, z ; v G x, y, z ≤ 2/3 G x, y, a G x, a, z G a, y, z ; vi G x, y, z ≤ G x, a, a G y, a, a G z, a, a . 3. Main Results We start with the following definition. Definition 3.1 see 37 . The function φ : 0, ∞ → 0, ∞ is called an altering distance function, if the following properties are satisfied. 1 φ is continuous and nondecreasing. 2 φ t 0 if and only if t 0. Let X,G be a G-metric space and F : X → X be a mapping. We set M ( x, y, y ) max { G ( x, y, y ) , G x, Fx, Fx , G ( y, Fy, Fy ) , G ( x, Fy, Fy ) G ( y, Fy, Fy ) G ( y, Fx, Fx ) 3 } , M1 ( x, y, y ) max { G ( x, y, y ) , G ( y, Fy, Fy )} , N ( x, y, y ) min { G x, Fx, Fx , G ( y, Fy, Fy ) , G ( y, Fx, Fx )} . 3.1 With this setting, we introduce the following definitions. Definition 3.2. Let X,G be a G-metric space. A mapping F : X → X is called a G-weak contraction of type A if and only if there exist two constants a ∈ 0, 1 and L ≥ 0 such that G ( Fx, Fy, Fy ) ≤ aMx, y, y LNx, y, y, 3.2