Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces

and Applied Analysis 3 Proposition 2.2 see 2 . Let X,G be a G-metric space and xn a sequence in X. Then, for all x ∈ X, the following statements are equivalent: i xn is G-convergent to x, ii G xn, xn, x → 0 as n → ∞, iii G xn, x, x → 0 as n → ∞, iv G xn, xm, x → 0 as n,m → ∞. Proposition 2.3 see 2 . Let X,G be a G-metric space and xn a sequence in X. Then, the following statements are equivalent: i xn is G-Cauchy, ii For every > 0 there existsN ∈ N such that G xn, xm, xm < , for all n,m ≥ N. Lemma 2.4 see 2 . If X,G is a G-metric space then G x, y, y ≤ 2G y, x, x for all x, y ∈ X. Let X,G be a G-metric space and F : X × X → X a mapping. Then, a map F is said to be continuous 10 in X,G if for every G-convergent sequences xn → x and yn → y, F xn, yn is G-convergent to F x, y . Quite recently, Bhaskar and Lakshmikantham 14 defined and studied the concepts of mixed monotone property and coupled fixed point in partially ordered metric space. Let X,≤ be a partially ordered set and F : X × X → X a mapping. Then, a map F is said to have mixed monotone property if F x, y is monotone nondecreasing in x and is monotone nonincreasing in y; that is, for any x, y ∈ X, x1, x2 ∈ X, x1 ≤ x2 implies F ( x1, y ) ≤ Fx2, y ) , y1, y2 ∈ X, y1 ≤ y2 implies F ( x, y1 ) ≥ Fx, y2 ) . 2.3 An element x, y ∈ X×X is said to be a coupled fixed point of the mapping F : X×X → X if F ( x, y ) x, F ( y, x ) y. 2.4 The following class of functions are considered in 25 . Denote with Φ the set of all functions φ : 0,∞ → 0,∞ which satisfy that i φ is continuous and nondecreasing, ii φ t 0 if and only if t 0, iii φ t s ≤ φ t φ s , for all t, s ∈ 0,∞ . ByΨwe denote the set of all functions ψ : 0,∞ → 0,∞ which satisfy limt→ r ψ t > 0, for all r > 0 and limt→ 0 ψ t 0. 4 Abstract and Applied Analysis For example, functions φ1, φ2, φ3, φ3 ∈ Φ, where φ1 t kt k > 0 , φ2 t t/t 1, φ3 t ln t 1 , and φ4 t min{t, 1}, and the functions ψ1, ψ2, ψ3 ∈ Ψ, where ψ1 t kt, ψ2 t ln 2t 1 /2, and ψ3 t ⎧ ⎪ ⎪ ⎪⎨ ⎪ ⎪ ⎪⎩ 1, if t 0, t 1, t t 1 , if 0 < t < 1,