Common Fixed Point Theorems of Altman Integral Type Mappings in G-Metric Spaces

and Applied Analysis 3 Proposition 1.7 see 6 . Let X,G be a G-metric space. Then the followings are equivalent. 1 The sequence {xn} is G-Cauchy; 2 For every > 0, there exists k ∈ N such that G xn, xm, xm < , for all n,m ≥ k. Proposition 1.8 see 6 . Let X,G be a G-metric space. Then the function G x, y, z is jointly continuous in all three of its variables. Definition 1.9 see 6 . Let X,G and X′, G′ be G-metric space, and f : X,G → X′, G′ be a function. Then f is said to be G-continuous at a point a ∈ X if and only if for every ε > 0, there is δ > 0 such that x, y ∈ X, and G a, x, y < δ imply G′ f a , f x , f y < ε. A function f is G-continuous at X if and only if it is G-continuous at all a ∈ X. Proposition 1.10 see 6 . Let X,G and X′, G′ be G-metric space. Then f : X → X′ is Gcontinuous at x ∈ X if and only if it is G-sequentially continuous at x, that is, whenever {xn} is G-convergent to x, {f xn } is G-convergent to f x . Proposition 1.11 see 6 . Let X,G be a G-metric space. Then, for any x, y, z, a in 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 ≤ 2G 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 . Definition 1.12 see 8 . Self-mappings f and g of a G-metric space X,G are said to be compatible if limn→∞G fgxn, gfxn, gfxn 0 and limn→∞G gfxn, fgxn, fgxn 0, whenever {xn} is a sequence in X such that limn→∞fxn limn→∞gxn t, for some t ∈ X. In 2010, Manro et al. 9 introduced the concept of weakly commuting mappings, Rweakly commuting mappings into G-metric space as follows. Definition 1.13 see 9 . A pair of self-mappings f, g of aG-metric space is said to be weakly commuting if G ( fgx, gfx, gfx ) ≤ Gfx, gx, gx, ∀x ∈ X. 1.1 Definition 1.14 see 9 . A pair of self-mappings f, g of a G-metric space is said to be Rweakly commuting, if there exists some positive real number R such that G ( fgx, gfx, gfx ) ≤ RGfx, gx, gx, ∀x ∈ X. 1.2 Remark 1.15. If R ≤ 1, then R-weakly commuting mappings are weakly commuting. Now we introduce the new concept of φ-weakly commuting mappings as follow. 4 Abstract and Applied Analysis Definition 1.16. A pair of self-mappings f, g of a G-metric space is said to be φ-weakly commuting, if there exists a continuous function φ : 0,∞ → 0,∞ , φ 0 0, such that G ( fgx, gfx, gfx ) ≤ φGfx, gx, gx, ∀x ∈ X. 1.3 Remark 1.17. Commuting mappings are weakly commuting mappings, but the reverse is not true. For example: let X 0, 1/2 , G x, y, z |x − y| |y − z| |z − x|, for all x, y, z ∈ X, define f x x/2, g x x2/2, through a straightforward calculation, we have: fgx x2/4, gfx x2/8, G fgx, gfx, gfx G x2/4, x2/8, x2/8 x2/4, but G fx, gx, gx |x − x2| x − x2, hence, G fgx, gfx, gfx ≤ G fx, gx, gx , but fgx / gfx. Remark 1.18. Weakly commuting mappings are R-weakly commuting mappings, but the reverse is not true. For example: let X −1, 1 , define G x, y, z |x − y| |y − z| |z − x|, for all x, y, z ∈ X, f x |x|, g x |x| − 1, then gfx |x| − 1, fgx 1 − |x|, |fx − gx| 1, |fgx − gfx| 2 1 − |x| , G fgx, gfx, gfx 2|fgx − gfx| 4 1 − |x| ≤ 4 4|fx − gx| G fx, gx, gx , when R 2, we get that f and g are R-weakly commuting mappings, but not weakly commuting mappings. Remark 1.19. R-weakly commuting mappings are φ-weakly commuting mappings but the reverse is not true. For example: let X 0, ∞ , G x, y, z |x − y| |y − z| |z − x|, for all x, y, z ∈ X, f x x2/4, g x x2, thus, we have fgx x4/4, gfx x4/16, G fgx, gfx, gfx 3/8 x4, G fx, gx, gx 3/2 x2. Let φ x 1/2 x2, then G ( fgx, gfx, gfx ) 3 8 x4 ≤ 9 8 x4 1 2 ( 3 2 x2 )2 φ ( 3 2 x2 ) φ ( G ( fx, gx, gx )) . 1.4 For any given R > 0, since limx→ ∞ 1/4 x2 ∞, there exists x ∈ X such that 1/4 x2 > R, so we getG fgx, gfx, gfx 1/4 x2G fx, gx, gx > RG fx, gx, gx . Therefore, f and g are φ-weakly commuting mappings, but not R-weakly commuting mappings. Lemma 1.20. Let δ t be Lebesgue integrable, and δ t > 0, for all t > 0, let F x ∫x 0 δ t dt, then F x is an increasing function in 0, ∞ . Definition 1.21. Let f and g be self-mappings of a set X. If w fx gx for some x in X, then x is called a coincidence point of f and g, and w is called s point of coincidence of f and g. 2. Main Results In this paper, we denote φ : 0, ∞ → 0, ∞ the function satisfying 0 < φ t < t, for all t > 0. Theorem 2.1. Let X,G be a completeG-metric space and let S, T , R, f , g, and h be six mappings of X into itself. If there exists an increasing function Q : 0, ∞ → 0, ∞ satisfying the conditions (i)∼(iii) and the following conditions: iv S X ⊆ g X , T X ⊆ h X , R X ⊆ f X , v ∫G Sx,Ty,Rz 0 δ t dt ≤ φ ∫Q G fx,gy,hz 0 δ t dt , for all x, y, z ∈ X, Abstract and Applied Analysis 5 where δ t is a Lebesgue integrable function which is summable nonnegative such that ∫and Applied Analysis 5 where δ t is a Lebesgue integrable function which is summable nonnegative such that ∫ 0 δ t dt > 0, ∀ > 0. 2.1 Then, a one of the pairs S, f , T, g , and R, h has a coincidence point in X, b if S, f , T, g , and R, h are three pairs of continuous φ-weakly commuting mappings, then the mappings S, T , R, f , g, and h have a unique common fixed point in X. Proof. Let x0 be an arbitrary point inX, from the condition iv , there exist x1, x2, x3 ∈ X such that y1 Sx0 gx1, y2 Tx1 hx2, y3 Rx2 fx3. 2.2 By induction, there exist two sequences {xn}, {yn} in X, such that y3n 1 Sx3n gx3n 1, y3n 2 Tx3n 1 hx3n 2, y3n 3 Rx3n 2 fx3n 3, n ∈ N. 2.3 If yn yn 1 for some n, with n 3m, then p x3m 1 is a coincidence point of the pair S, f ; if yn 1 yn 2 for some n, with n 3m, then p x3m 2 is a coincidence point of the pair T, g ; if yn 2 yn 3 for some n, with n 3m, then p x3m 3 is a coincidence point of the pair R, h . On the other hand, if there exists n0 ∈ N such that yn0 yn0 1 yn0 2, then yn yn0 for any n ≥ n0. This implies that {yn} is a G-Cauchy sequence. In fact, if there exists p ∈ N such that y3p y3p 1 y3p 2, then applying the contractive condition v with x y3p, y y3p 1, and z y3p 2, and the property of φ, we get ∫G y3p 1,y3p 2,y3p 3 0 δ t dt ∫G Sx3p,Tx3p 1,Rx3p 2