Coupled coincidence point theorems for mixed (G, S)-monotone operators on partially ordered metric spaces and applications

Couple d coincide nce p oint theorems for mixed ( G, S ) -monoton e op erators on partially ordered metri c spaces and applicatio ns Habib Y azidi Abstract In this pap er, we in tro duce the concept of mixed ( G, S )-monotone mapp ings and pro v e coupled co incidence and coupled common fixed p oin t theorems for suc h mappings satisfying a nonlinear con traction in v olving altering distance f u nctions. Presen ted theorems extend , impr o v e and generalize the ve ry recen t results of Harj an i, L´ op ez and Sadarangani [J. Harjan i, B. L´ op ez and K. Sad aran gani, Fixed p oin t theorems for mixed mon otone op erators and applications to in tegral equations, Nonlinear Analysis (2010 ), doi:10.101 6/j.na.2010 .10.047] and other existing results in the literature. Some applications to p erio dic b oundary v a lue problems are also considered. Key w ords: Coincidence p oint, coupled common fixed p oin t, ( G, S )-monotone mapping, ordered set. 1 In tro du ction and preliminaries Fixed p oint problems of co nt ractiv e mappings in metric spaces endow e d w ith a partially order ha v e b een studied b y man y authors (see [1 ]-[17]). Bhask ar and Lakshmik an tham [3] in tro duced the concept of a coupled fi x ed p oin t and stud ied th e problems of a u niqueness of a coupled fi xed p oin t in partially ord ered metric s paces an d app lied their theo rems to p roblems of the existence of solution for a p erio dic boun dary v alue pr oblem. In [8], Lakshmik antham and ´ Ciri ´ c established some coincidence and common coupled fi xed p oin t theorems un d er nonlinear contract ions in partially ordered metric sp aces. V ery recently , Harjani, L´ op ez and S adarangani [7] obtained some coupled fi xed p oin t theorems for a mixed monotone op erator in a complete metric sp ace endo w ed with a partial ord er b y using altering distance f unctions. T h ey applied th eir results to the study of the existence and uniquen ess of a nonlinear int egral equation. No w , we briefly recall v arious b asic d efinitions and f acts. Definition 1.1 (se e Bhaskar and L akshmikantham [3]). L et ( X,  ) b e a p artial ly or der e d set and F : X × X → X . Then the ma p F is said to have mixe d monotone pr op erty if F ( x, y ) is mon otone non-de cr e asing in x and is monotone non-incr e asing in y , that is, for any x, y ∈ X , x 1  x 2 implies F ( x 1 , y )  F ( x 2 , y ) for all y ∈ X 1 and y 1  y 2 implies F ( x, y 2 )  F ( x, y 1 ) for all x ∈ X. The main result obtained by Bhask ar and L aksh mik ant ham [3] is the follo wing. Theorem 1.1 (se e Bhaskar and L akshmikantham [3]). L et ( X ,  ) b e a p artial ly or der e d set and supp ose ther e is a metric d on X such that ( X , d ) is a c omplete metric sp ac e . L et F : X × X → X b e a mapping having the mixe d monotone pr op erty on X . Assume that ther e exists k ∈ [0 , 1) such tha t d ( F ( x, y ) , F ( u, v )) ≤ k 2 [ d ( x, u ) + d ( y , v )] for e ach u  x and y  v. Supp ose either F is c ontinuous or X has the fol lowing pr op erties: (i) if a non-de c r e asing se quenc e x n → x , then x n  x for al l n , (ii) if a non-incr e asing se quenc e x n → x , then x  x n for al l n . If ther e exi st x 0 , y 0 ∈ X such that x 0  F ( x 0 , y 0 ) and F ( y 0 , x 0 )  y 0 , then F has a c ouple d fixe d p oint. Inspired by Definition 1.1, Lakshmik antham and ´ Ciri ´ c in [8] introd uced the concept of a g -mixed monotone mapp ing. Definition 1.2 (se e L akshmikantham and ´ Ciri´ c [8]). L et ( X ,  ) b e a p artial ly or der e d set, F : X × X → X a nd g : X → X . Then the map F is said to have mixe d g -monotone pr op e rty i f F ( x, y ) is monotone g -non-de cr e asing in x and is monotone g -non-inc r e asing in y , that is, for any x, y ∈ X , g x 1  g x 2 implies F ( x 1 , y )  F ( x 2 , y ) for all y ∈ X and g y 1  g y 2 implies F ( x, y 2 )  F ( x, y 1 ) for all x ∈ X . Definition 1.3 (se e L akshmikantham and ´ Ciri´ c [8]). L et X b e a non-empty set, and let F : X × X → X , g : X → X b e given mappings. An element ( x, y ) ∈ X × X is c al le d a c ouple d c oincidenc e p oint of the mappings F an d g if F ( x, y ) = g x and F ( y , x ) = gy . Definition 1.4 (se e L akshmikantham and ´ Ciri´ c [8]). L et X b e a non-empty set. Then we say th at the mappings F : X × X → X a nd g : X → X ar e c ommutative if g ( F ( x, y )) = F ( g x, g y ) . The main result of Lakshmik antham and ´ Ciri ´ c [8] is the f ollo wing. 2 Theorem 1.2 (se e L akshmikantham and ´ Ciri´ c [8]). L et ( X,  ) b e a p artial ly or der e d set and supp ose ther e is a metric d on X such that ( X, d ) is a c omplete metric sp ac e. Assume ther e is a function φ : [0 , + ∞ ) → [0 , + ∞ ) with φ ( t ) < t and lim r → t + φ ( r ) < t for e ach t > 0 and also supp ose F : X × X → X and g : X → X ar e su ch that F has the mixe d g -monotone pr op erty and d ( F ( x, y ) , F ( u, v )) ≤ φ  d ( g x, g u ) + d ( g y , g v ) 2  for al l x, y , u, v ∈ X with g x  g u a nd g v  g y . Assume that F ( X × X ) ⊆ g ( X ) , g is c ontinuous and c ommutes with F and also supp ose either F i s c ontinuous or X has the fol lowing pr op erties: (i) if a non-de c r e asing se quenc e x n → x , then x n  x for al l n , (ii) if a non-incr e asing se quenc e x n → x , then x  x n for al l n . If ther e exist x 0 , y 0 ∈ X such that gx 0  F ( x 0 , y 0 ) a nd F ( y 0 , x 0 )  g y 0 then ther e exist x, y ∈ X such that g x = F ( x, y ) and g y = F ( y, x ) , that is, F and g have a c ouple d c oincidenc e p oint. V ery recen tly , Harjani, L´ op ez and Sadarangani [7] established coupled fixed p oint the- orems for a mixed monotone op erator satisfying con tractio n in v olving altering distance functions in a complete partially ord er ed m etric space. Denote b y F the set of fu nctions ϕ : [0 , + ∞ ) → [0 , + ∞ ) satisfying the follo wing p rop erties: (a) ϕ is cont in uous and non-d ecreasing, (b) ϕ ( t ) = 0 if and only if t = 0. Theorem 1.3 (Harjani, L´ op ez and Sadar angani [ 7]). L et ( X ,  ) b e a p artial ly or der e d set and d b e a metric on X su c h th at ( X, d ) is a c omplete metric sp ac e. L et F : X × X → X b e a mapping having the mixe d monotone pr op erty on X an d satisfying ϕ ( d ( F ( x, y ) , F ( u, v )) ≤ ϕ ( max { d ( x, u ) , d ( y , v ) } ) − Φ(max { d ( x, u ) , d ( y , v ) } ) for al l x, y , u, v ∈ X with u  x and y  v , wher e ϕ, ψ ∈ F . Supp ose either F is c ontinuous or X has the fol lowing pr op erties: (i) if a non-de c r e asing se quenc e x n → x , then x n  x for al l n , (ii) if a non-incr e asing se quenc e x n → x , then x  x n for al l n . If ther e exist x 0 , y 0 ∈ X such that x 0  F ( x 0 , y 0 ) and F ( y 0 , x 0 )  y 0 then F has a c ouple d fixe d p oint. In this pap er, w e in tro duce the conce pt of mixed ( G, S )-monotone mapp ings and pro v e coupled coincidence and coupled common fixed p oin t theorems for suc h mappings satisfying a nonlinear cont raction in volving altering d istance fun ctions. Presente d theorems extend, impro v e and generalize th e results of Harjani, L´ op ez and Sadarangani [7]. As app lications of our obtained resu lts, w e study the existence and uniqueness of solution to p erio dic b ound ary v alue pr ob lem. 3 2 Main Res ults No w , we in tro duce the concept of mixed ( G, S )-monotone prop ert y . Definition 2.1 L et X b e a non-empty set endowe d with a p artial or der  . Consider the mappings F : X × X → X and G, S : X → X . We say that F has the mixe d ( G, S ) - monotone pr op erty on X i f f or al l x, y ∈ X , x 1 , x 2 ∈ X , G ( x 1 )  S ( x 2 ) ⇒ F ( x 1 , y )  F ( x 2 , y ) , x 1 , x 2 ∈ X , G ( x 1 )  S ( x 2 ) ⇒ F ( x 1 , y )  F ( x 2 , y ) , y 1 , y 2 ∈ X , G ( y 1 )  S ( y 2 ) ⇒ F ( x, y 1 )  F ( x, y 2 ) , y 1 , y 2 ∈ X, G ( y 1 )  S ( y 2 ) ⇒ F ( x, y 1 )  F ( x, y 2 ) . Remark 1 If we take G = S , then F has the mixe d ( G, S ) -monotone pr op erty implies that F has the mixe d G -mono tone pr op e rty. No w , we state and pr o v e our firs t resu lt. Theorem 2.1 L et ( X ,  ) b e a p artial ly or der e d set and supp ose that ther e exists a metric d on X such tha t ( X, d ) i s a c omplete metric sp ac e. L et G, S : X → X and F : X × X → X b e a mapping having the mixe d ( G, S ) -monotone pr op erty on X . Supp ose that ϕ ( d ( F ( x, y ) , F ( u, v ))) ≤ ϕ (max { d ( Gx, S u ) , d ( S y , Gv ) } ) − φ (max { d ( Gx, S u ) , d ( S y , Gv ) } ) , (1) for al l x, y , u, v ∈ X with G ( x )  S ( u ) or G ( x )  S ( u ) and S ( y )  G ( v ) or S ( y )  G ( v ) , wher e ϕ, φ ∈ F . A ssume that F ( X × X ) ⊆ G ( X ) ∩ S ( X ) and assume also that G, S and F satisfy the fol lowing hyp otheses: (I) F , G and S ar e c ontinuous, (II) F c ommutes r esp e ctively with G and S . If ther e exi st x 0 , y 0 , x 1 and y 1 such tha t  G ( x 0 )  S ( x 1 )  F ( x 0 , y 0 ); G ( y 0 )  S ( y 1 )  F ( y 0 , x 0 ) , then th er e exist x, y ∈ X such that G ( x ) = S ( x ) = F ( x, y ) and G ( y ) = S ( y ) = F ( y , x ) , that is, G, S and F have a c ouple d c oincidenc e p oint ( x, y ) ∈ X × X . Pro of. Let x 0 , y 0 , x 1 , y 1 ∈ X such that G ( x 0 )  S ( x 1 )  F ( x 0 , y 0 ) and G ( y 0 )  S ( y 1 )  F ( y 0 , x 0 ) . Since F ( X × X ) ⊆ G ( X ) ∩ S ( X ), we can c ho ose x 2 , y 2 , x 3 , y 3 ∈ X s uc h that  G ( x 2 ) = F ( x 0 , y 0 ) G ( y 2 ) = F ( y 0 , x 0 ) and  S ( x 3 ) = F ( x 1 , y 1 ) S ( y 3 ) = F ( y 1 , x 1 ) · 4 Con tin uing this pro cess we can construct sequences { x n } and { y n } in X suc h th at  G ( x 2 n +2 ) = F ( x 2 n , y 2 n ) G ( y 2 n +2 ) = F ( y 2 n , x 2 n ) ;  S ( x 2 n +3 ) = F ( x 2 n +1 , y 2 n +1 ) S ( y 2 n +3 ) = F ( y 2 n +1 , x 2 n +1 ) for all n ≥ 0. (2) W e shall show that for all n ≥ 0, G ( x 2 n )  S ( x 2 n +1 )  G ( x 2 n +2 ) (3) and G ( y 2 n ) ≥ S ( y 2 n +1 ) ≥ G ( y 2 n +2 ) . (4) As G ( x 0 )  S ( x 1 )  F ( x 0 , y 0 ) = G ( x 2 ) and G ( y 0 )  S ( y 1 )  F ( y 0 , x 0 ) = G ( y 2 ), our claim is satisfied for n = 0. Supp ose that (3) and (4) hold for some fi xed n ≥ 0. Since G ( x 2 n )  S ( x 2 n +1 )  G ( x 2 n +2 ) and G ( y 2 n )  S ( y 2 n +1 )  G ( y 2 n +2 ), and as F h as the mixed ( G, S )-monotone p rop erty , w e ha v e G ( x 2 n +2 ) = F ( x 2 n , y 2 n )  F ( x 2 n +1 , y 2 n )  F ( x 2 n +1 , y 2 n +1 )  F ( x 2 n +2 , y 2 n +1 )  F ( x 2 n +2 , y 2 n +2 ) , then G ( x 2 n +2 )  S ( x 2 n +3 )  G ( x 2 n +4 ) . On the other hand , G ( y 2 n +2 ) = F ( y 2 n , x 2 n )  F ( y 2 n +1 , x 2 n )  F ( y 2 n +1 , x 2 n +1 )  F ( y 2 n +2 , x 2 n +1 )  F ( y 2 n +2 , x 2 n +2 ) , then G ( y 2 n +2 )  S ( y 2 n +3 )  G ( y 2 n +4 ) . Th us b y induction, we pr ov ed that (3) an d (4) h old for all n ≥ 0. W e complete the pr o of in the f ollo wing steps Step 1: W e w ill pr o v e that lim n → + ∞ d ( F ( x n , y n ) , F ( x n +1 , y n +1 )) = lim n → + ∞ d ( F ( y n , x n ) , F ( y n +1 , x n +1 )) = 0 . (5) F rom (3), (4) and (1), we hav e ϕ ( d ( F ( x 2 n , y 2 n ) , F ( x 2 n +1 , y 2 n +1 ))) (6) ≤ ϕ (max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } ) − φ (max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } ) ≤ ϕ (max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } ) . (7) Since ϕ is a non-decreasing fu n ction, we get that d ( F ( x 2 n , y 2 n ) , F ( x 2 n +1 , y 2 n +1 )) ≤ max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } . Therefore d ( Gx 2 n +2 , S x 2 n +3 ) ≤ max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } . (8) 5 Again, u sing (3), (4) and (1), we hav e ϕ ( d ( F ( y 2 n , x 2 n ) , F ( y 2 n +1 , x 2 n +1 ))) ≤ ϕ (max { d ( Gy 2 n , S y 2 n +1 ) , d ( Gx 2 n , S x 2 n +1 ) } ) − φ (max { d ( Gy 2 n , S y 2 n +1 ) , d ( Gx 2 n , S x 2 n +1 ) } ) ≤ ϕ (max { d ( Gy 2 n , S y 2 n +1 ) , d ( Gx 2 n , S x 2 n +1 ) } ) . (9) Since ϕ is non-decreasing, we ha v e d ( F ( y 2 n , x 2 n ) , F ( y 2 n +1 , x 2 n +1 )) ≤ max { d ( Gy 2 n , S y 2 n +1 ) , d ( Gx 2 n , S x 2 n +1 } . Therefore d ( Gy 2 n +2 , S y 2 n +3 ) ≤ max { d ( Gy 2 n , S y 2 n +1 ) , d ( Gx 2 n , S x 2 n +1 ) } . (10) Com bining (8) and (10), w e obtain max { d ( Gx 2 n +2 , S x 2 n +3 ) , d ( Gy 2 n +2 , S y 2 n +3 ) } ≤ max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } . Then  max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) }  is a p ositiv e decreasing sequence. Hence there exists r ≥ 0 su c h that lim n → + ∞ max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } = r. Com bining (7) and (9), we obtain max { ϕ ( d ( G x 2 n +2 , S x 2 n +3 )) , ϕ ( d ( Gy 2 n +2 , S y 2 n +3 )) } ≤ ϕ (max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } ) − φ (max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } ) . Since ϕ is non-decreasing, we get ϕ (max { d ( Gx 2 n +2 , S x 2 n +3 ) , d ( Gy 2 n +2 , S y 2 n +3 ) } ) ≤ ϕ (max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } ) − φ (max { d ( Gx 2 n , S x 2 n +1 ) , d ( Gy 2 n , S y 2 n +1 ) } ) . Letting n → + ∞ in the ab ov e inequalit y , w e get ϕ ( r ) ≤ ϕ ( r ) − φ ( r ) , whic h implies that φ ( r ) = 0 and then, since φ is a n altering distance function, r = 0. Consequent ly lim n → + ∞ max { d ( F ( x 2 n , y 2 n ) , F ( x 2 n +1 , y 2 n +1 )) , d ( F ( y 2 n , x 2 n ) , F ( y 2 n +1 , x 2 n +1 )) } = 0 . (11) By the same w a y , we obtain lim n → + ∞ max { d ( F ( x 2 n +1 , y 2 n +1 ) , F ( x 2 n +2 , y 2 n +2 )) , d ( F ( y 2 n +1 , x 2 n +1 ) , F ( y 2 n +2 , x 2 n +2 )) } = 0 . (12) Finally , (11) and (12) give the d esir ed r esult, that is, (5) holds. 6 Step 2: W e w ill pr o v e that F ( x n , y n ) and F ( y n , x n ) are C auc h y sequences. F rom (5), it is su fficien t to sho w th at F ( x 2 n , y 2 n ) and F ( y 2 n , x 2 n ) are Cauc hy sequences. W e pro ceed b y negation and s u pp ose that at least one of the sequ ences F ( x 2 n , y 2 n ) or F ( y 2 n , x 2 n ) is not a Cauc h y sequence. This implies th at d ( F ( x 2 n , y 2 n ) , F ( x 2 m , y 2 m )) 9 0 or d ( F ( y 2 n , x 2 n ) , F ( y 2 m , x 2 m )) 9 0 as n, m → + ∞ . Consequent ly max { d ( F ( x 2 n , y 2 n ) , F ( x 2 m , y 2 m )) , d ( F ( y 2 n , x 2 n ) , F ( y 2 m , x 2 m )) } 9 0 , as n, m → + ∞ . Then there exists ε > 0 for wh ic h w e can find tw o sub s equences of p ositiv e in tegers { m ( i ) } and { n ( i ) } s u c h th at n ( i ) is the smallest ind ex f or w hic h n ( i ) > m ( i ) > i , max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } ≥ ε. (13) This means that max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i ) − 2 , y 2 n ( i ) − 2 )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i ) − 2 , x 2 n ( i ) − 2 )) } < ε. (14) F rom (2.1), (14) and us ing the triangular inequalit y , w e get ε ≤ max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } ≤ max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i ) − 2 , y 2 n ( i ) − 2 )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i ) − 2 , x 2 n ( i ) − 2 )) } + max { d ( F ( x 2 n ( i ) − 2 , y 2 n ( i ) − 2 ) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 )) , d ( F ( y 2 n ( i ) − 2 , x 2 n ( i ) − 2 ) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 )) } + max { d ( F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , ( F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } < ε + max { d ( F ( x 2 n ( i ) − 2 , y 2 n ( i ) − 2 ) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 )) , d ( F ( y 2 n ( i ) − 2 , x 2 n ( i ) − 2 ) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 )) } + max { ( F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } . Letting i → + ∞ in ab o v e inequ ality and usin g (5), we obtain that lim i → + ∞ max( d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i ) , x 2 n ( i ) ))) = ε. (15) Also, we h a v e ε ≤ max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } ≤ max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( y 2 m ( i ) − 1 , x 2 m ( i ) − 1 )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 m ( i ) − 1 , x 2 m ( i ) − 1 )) } + max { d ( F ( x 2 m ( i ) − 1 , y 2 m ( i ) − 1 ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 m ( i ) − 1 , x 2 m ( i ) − 1 ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } ≤ 2 max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 m ( i ) − 1 , y 2 m ( i ) − 1 )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 m ( i ) − 1 , x 2 m ( i ) − 1 )) } + max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } . Using (5), (1 5) and letting i → + ∞ in the ab o ve inequalit y , we obtain lim i → + ∞ max { d ( F ( x 2 m ( i ) − 1 , y 2 m ( i ) − 1 ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 m ( i ) − 1 , x 2 m ( i ) − 1 ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } = ε. (16) 7 On other hand, we ha v e max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } ≤ max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i )+1 , y 2 n ( i )+1 )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i )+1 , x 2 n ( i )+1 )) } + max { d ( F ( x 2 n ( i )+1 , y 2 n ( i )+1 ) , F ( x 2 n ( i ) , y 2 n ( i ) )) , d ( F ( y 2 n ( i )+1 , x 2 n ( i )+1 ) , F ( y 2 n ( i ) , x 2 n ( i ) )) } . Since ϕ is a con tin uous non-d ecreasing f u nction, it foll o ws f rom the ab o v e inequalit y that ϕ ( ε ) ≤ (17) lim sup i → + ∞ ϕ (max { d ( F ( x 2 m ( i ) , y 2 m ( i ) ) , F ( x 2 n ( i )+1 , y 2 n ( i )+1 )) , d ( F ( y 2 m ( i ) , x 2 m ( i ) ) , F ( y 2 n ( i )+1 , x 2 n ( i )+1 )) } ) . Using the co nt ractiv e condition, on one h an d we hav e ϕ ( d ( F ( x 2 m ( i ) , y 2 m ( i ) )) , F ( x 2 n ( i )+1 , y 2 n ( i )+1 )) ≤ ϕ (max { d ( Gx 2 m ( i ) , S x 2 n ( i )+1 ) , d ( Gy 2 m ( i ) , S y 2 n ( i )+1 ) } ) − φ (max { d ( Gx 2 m ( i ) , S x 2 n ( i )+1 ) , d ( Gy 2 m ( i ) , S y 2 n ( i )+1 ) } ) ≤ ϕ (max { d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 )) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 ) , d ( F ( y 2 m ( i ) − 2 , x 2 m ( i ) − 2 )) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 ) } ) − φ (max { d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 )) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 ) , d ( F ( y 2 m ( i ) − 2 , x 2 m ( i ) − 2 )) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 ) } ) . On the other hand we hav e ϕ ( d ( F ( y 2 m ( i ) , x 2 m ( i ) )) , F ( y 2 n ( i )+1 , x 2 n ( i )+1 )) ≤ ϕ (max { d ( Gy 2 m ( i ) , S y 2 n ( i )+1 ) , d ( Gx 2 m ( i ) , S x 2 n ( i )+1 ) } ) − φ (max { d ( Gy 2 m ( i ) , S y 2 n ( i )+1 ) , d ( Gx 2 m ( i ) , S x 2 n ( i )+1 ) } ) ≤ ϕ (max { d ( F ( y 2 m ( i ) − 2 , x 2 m ( i ) − 2 )) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 ) , d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 )) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 )) } ) − φ (max { d ( F ( y 2 m ( i ) − 2 , x 2 m ( i ) − 2 )) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 ) , d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 )) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 ) } ) . Therefore max { ϕ ( d ( F ( x 2 m ( i ) , y 2 m ( i ) )) , F ( x 2 n ( i )+1 , y 2 n ( i )+1 )) , ϕ ( d ( F ( y 2 m ( i ) , x 2 m ( i ) )) , F ( y 2 n ( i )+1 , x 2 n ( i )+1 )) } ≤ ϕ (max { d ( Gx 2 m ( i ) , S x 2 n ( i )+1 ) , d ( Gy 2 m ( i ) , S y 2 n ( i )+1 ) } ) − φ (max { d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 )) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 ) , d ( F ( y 2 m ( i ) − 2 , x 2 m ( i ) − 2 )) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 ) } ) . (18) W e claim that max { d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 )) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 ) , d ( F ( y 2 m ( i ) − 2 , x 2 m ( i ) − 2 )) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 ) } ) → ε as i → + ∞ . (19) In fact, using the triangular inequalit y , w e hav e d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 ) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 )) ≤ d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 ) , F ( x 2 m ( i ) − 1 , y 2 m ( i ) − 1 )) + d ( F ( x 2 m ( i ) − 1 , y 2 m ( i ) − 1 ) , F ( x 2 n ( i ) , y 2 n ( i ) )) + d ( F ( x 2 n ( i ) , y 2 n ( i ) ) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 )) . Letting i → + ∞ in the ab ov e inequality and using (5) and (16), we obtain lim i → + ∞ d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 ) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 )) ≤ ε. (20) 8 On the other hand , we hav e d ( F ( x 2 m ( i ) − 1 , y 2 m ( i ) − 1 ) , F ( x 2 n ( i ) , y 2 n ( i ) )) ≤ d ( F ( x 2 m ( i ) − 1 , y 2 m ( i ) − 1 ) , F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 )) + d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 ) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 )) + d ( F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 ) , F ( x 2 n ( i ) , y 2 n ( i ) )) . Letting i → + ∞ in the ab ov e inequality and using (5) and (16), we obtain ε ≤ lim i → + ∞ d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 ) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 )) . (21) Com bining (20) and (21), we get lim i → + ∞ d ( F ( x 2 m ( i ) − 2 , y 2 m ( i ) − 2 ) , F ( x 2 n ( i ) − 1 , y 2 n ( i ) − 1 )) = ε. By the same w a y , we obtain lim i → + ∞ d ( F ( y 2 m ( i ) − 2 , x 2 m ( i ) − 2 ) , F ( y 2 n ( i ) − 1 , x 2 n ( i ) − 1 )) = ε. Th us w e pr o v ed (19). Finally , letti ng i → + ∞ in (18), u sing (17), (19) and the con tin uit y of ϕ and φ , w e get ϕ ( ε ) ≤ ϕ ( ε ) − φ ( ε ), wh ic h implies that φ ( ε ) = 0, that is, ε = 0, a con tradiction. Th us ( F ( x 2 n , y 2 n )) and ( F ( y 2 n , x 2 n )) are C auc h y sequences in X , whic h giv es us that ( F ( x n , y n )) and ( F ( y n , x n )) are also Cauc h y sequences. Step 3: Existence of a coupled coincidence p oin t. Since( F ( x n , y n )) and (( F ( y n , x n ))) are Cauch y sequences in the complete met ric space ( X, d ), th ere exist α, α ′ ∈ X s uc h that: lim n → + ∞ F ( x n , y n ) = α and lim n → + ∞ F ( y n , x n ) = α ′ . Therefore, lim n → + ∞ G ( x 2 n +2 ) = α , lim n → + ∞ G ( y 2 n +2 ) = α ′ , lim n → + ∞ S ( x 2 n +3 ) = α and lim n → + ∞ S ( y 2 n +3 ) = α ′ . using the con tin u it y and the comm utativit y of F and G , we hav e G ( G ( x 2 n +2 )) = G ( F ( x 2 n , y 2 n )) = F ( Gx 2 n , Gy 2 n ) and G ( G ( y 2 n +2 )) = G ( F ( y 2 n , x 2 n )) = F ( Gy 2 n , Gx 2 n ) . Letting n → + ∞ , we get G ( α ) = F ( α, α ′ ) and G ( α ′ ) = F ( α ′ , α ) . Using also the co n tin uit y and the commutati vit y of F and S , by the same w a y , we obtain S ( α ) = F ( α, α ′ ) and S ( α ′ ) = F ( α ′ , α ). Therefore G ( α ) = F ( α, α ′ ) = S ( α ) and G ( α ′ ) = F ( α ′ , α ) = S ( α ′ ) . Th us w e pro v ed that ( α, α ′ ) is a coupled coincidence p oin t of G, S and F .  In the next result, we pro v e that the previous th eorem is s till v alid if we replace th e con tin u it y of F by some conditions. Theorem 2.2 If we r eplac e the c ontinuity hyp othesis of F in The or em 2.1 by the fol lowing c onditions: 9 (i) if ( x n ) is a non-de cr e asing se que nc es with x n → x then x n ≤ x for e ach n ∈ N , (ii) if ( y n ) is a non-incr e asing se quenc es with y n → y then y ≤ y n for e ach n ∈ N , (iii) x, y ∈ X , x  y ⇒ Gx  S y , (iv) x, y ∈ X , x  y ⇒ Gx  S y . Then G, S a nd F have a c ouple d c oincidenc e p oint. Pro of. F ollo wing the pro of of Theorem 2.1, w e ha ve that F ( x n , y n ) a nd F ( y n , x n ) are Cauc h y sequences in the complete m etric s pace ( X, d ), there exist α , α ′ ∈ X s uc h that lim n → + ∞ F ( x n , y n ) = α and lim n → + ∞ F ( y n , x n ) = α ′ . Therefore lim n → + ∞ F ( x 2 n , y 2 n ) = α and lim n → + ∞ F ( y 2 n , x 2 n ) = α ′ . Hence lim n → + ∞ G ( x 2 n +2 ) = α , lim n → + ∞ G ( y 2 n +2 ) = α ′ , lim n → + ∞ S ( x 2 n +3 ) = α and lim n → + ∞ S ( y 2 n +3 ) = α ′ . Using the co mmuta - tivit y of F and G and of F and S and the con tractiv e condition, it follo ws fr om co nditions (iii)-(iv) that ϕ ( d ( G ( F ( x 2 n , y 2 n )) , S ( F ( x 2 n +1 , y 2 n +1 )))) = ϕ ( d ( F ( Gx 2 n , Gy 2 n ) , F ( S x 2 n +1 , S y 2 n +1 ))) ≤ ϕ (max { d ( G ( Gx 2 n ) , S ( S x 2 n +1 )) , d ( G ( Gy 2 n ) , S ( S y 2 n +1 )) } ) (22) − φ (max { d ( G ( Gx 2 n ) , S ( S x 2 n +1 )) , d ( G ( Gy 2 n ) , S ( S y 2 n +1 )) } ) . (23) Similarly , we ha v e ϕ ( d ( G ( F ( y 2 n , x 2 n )) , S ( F ( y 2 n +1 , x 2 n +1 )))) = ϕ ( d ( F ( Gy 2 n , Gx 2 n ) , F ( S y 2 n +1 , S x 2 n +1 ))) ≤ ϕ (max { d ( G ( Gy 2 n ) , S ( S y 2 n +1 )) , d ( G ( Gx 2 n ) , S ( S x 2 n +1 )) } ) (24) − φ (max { d ( G ( Gy 2 n ) , S ( S y 2 n +1 )) , d ( G ( Gx 2 n ) , S ( S x 2 n +1 )) } ) . (25) Com bining (22), (24) an d the fact that max { ϕ ( a ) , ϕ ( b ) } = ϕ ( max { a, b } ) for a, b ∈ [0 , + ∞ ), from (iii)-(iv), we obtain ϕ (max { d ( G ( F ( x 2 n , y 2 n )) , S ( F ( x 2 n +1 , y 2 n +1 ))) , d ( G ( F ( y 2 n , x 2 n )) , S ( F ( y 2 n +1 , x 2 n +1 ))) } ) ≤ ϕ (max { d ( G ( Gx 2 n ) , S ( S x 2 n +1 )) , d ( G ( Gy 2 n ) , S ( S y 2 n +1 )) } ) − φ (max { d ( G ( Gx 2 n ) , S ( S x 2 n +1 )) , d ( G ( Gy 2 n ) , S ( S y 2 n +1 )) } ) . Letting n → + ∞ in the last expression, using the contin uity of G and S , we get ϕ (max { d ( G ( α ) , S ( α )) , d ( G ( α ′ ) , S ( α ′ )) } ) ≤ ϕ (max { d ( G ( α ) , S ( α )) , d ( G ( α ′ ) , S ( α ′ )) } ) − φ (max { d ( G ( α ) , S ( α )) , d ( G ( α ′ ) , S ( α ′ )) } ) . This implies that φ (max { d ( G ( α ) , S ( α )) , d ( G ( α ′ ) , S ( α ′ )) } ) = 0 and, since φ is an altering distance function, then max { d ( G ( α ) , S ( α )) , d ( G ( α ′ ) , S ( α ′ )) } = 0 . 10 Consequent ly G ( α ) = S ( α ) an d G ( α ′ ) = S ( α ′ ) . (26) T o finish the pro of, we claim that F ( α, α ′ ) = G ( α ) = S ( α ) and F ( α ′ , α ) = G ( α ′ ) = S ( α ′ ). Indeed, using the con tractiv e condition, it follo ws fr om (i)-(iv) th at ϕ ( d ( F ( Gx 2 n , Gy 2 n ) , F ( α, α ′ ))) ≤ ϕ (max { d ( G ( Gx 2 n ) , S ( α )) , d ( G ( Gy 2 n ) , S ( α ′ )) } ) − φ (max { d ( G ( G x 2 n ) , S ( α )) , d ( G ( Gy 2 n ) , S ( α ′ )) } ) ≤ ϕ (max { d ( G ( Gx 2 n ) , S ( α )) , d ( G ( Gy 2 n ) , S ( α ′ )) } ) . Using the fact that ϕ is non-decreasing, w e get d ( F ( Gx 2 n , Gy 2 n ) , F ( α, α ′ )) ≤ max { d ( G ( Gx 2 n ) , S ( α )) , d ( G ( Gy 2 n ) , S ( α ′ )) } . (27) Similarly , we ha v e ϕ ( d ( F ( Gy 2 n , Gx 2 n ) , F ( α ′ , α ))) ≤ ϕ (max { d ( G ( Gy 2 n ) , S ( α ′ )) , d ( G ( Gx 2 n ) , S ( α )) } ) − φ (max { d ( G ( G y 2 n ) , S ( α ′ )) , d ( G ( Gx 2 n ) , S ( α )) } ≤ ϕ (max { d ( G ( Gy 2 n ) , S ( α ′ )) , d ( G ( Gx 2 n ) , S ( α )) } ) . Using the fact that ϕ is non-decreasing, w e see that d ( F ( Gy 2 n , Gx 2 n ) , F ( α ′ , α )) ≤ max { d ( G ( Gy 2 n ) , S ( α ′ )) , d ( G ( Gx 2 n ) , S ( α )) } . (28) Com bining (27) and (28), we get max { d ( F ( Gx 2 n , Gy 2 n ) , F ( α, α ′ )) , d ( F ( Gy 2 n , Gx 2 n ) , F ( α ′ , α ))) ≤ max { d ( G ( Gy 2 n ) , S ( α ′ )) , d ( G ( Gx 2 n ) , S ( α )) } . Using the co mmuta tivit y o f F and G , we w r ite max { d ( G ( F ( x 2 n , y 2 n ))) , F ( α, α ′ )) , d ( G ( F ( y 2 n , x 2 n )) , F ( α ′ , α )) } ≤ max { d ( G ( Gy 2 n ) , S ( α ′ )) , d ( G ( Gx 2 n ) , S ( α )) } . Letting n → + ∞ , u sing the con tin uit y of G , we obtain max { d ( G ( α ) , F ( α, α ′ )) , d ( G ( α ′ ) , F ( α ′ , α )) } ≤ max { d ( G ( α ) , S ( α )) , d ( G ( α ′ ) , S ( α ′ )) } . Lo oking at (26), we deduce that max { d ( G ( α ) , F ( α, α ′ )) , d ( G ( α ′ ) , F ( α ′ , α )) } = 0 . Therefore, d ( G ( α ) , F ( α, α ′ )) = 0 and d ( G ( α ′ ) , F ( α ′ , α )) = 0 . Consequent ly G ( α ) = F ( α, α ′ ) and G ( α ′ ) = F ( α ′ , α ) . (29) By the same w a y , we get S ( α ) = F ( α, α ′ ) and S ( α ′ ) = F ( α ′ , α ) . (30) 11 Finally , com bining (26), (29) and (30), we dedu ce t hat ( α, α ′ ) is a coupled coincidence p oint of F , G and S .  No w , we giv e a su fficien t condition for the existence and the un iqueness of the coup led common fixed p oint . Not ice that if ( X,  ) is a partially ordered set, w e endow X × X with the follo wing partial order relation: for ( x, y ) , ( u, v ) ∈ X × X , ( x, y )  ( u, v ) ⇔ x  u and y  v . Theorem 2.3 In addition to the hyp otheses of The or em 2.1 (r esp. The or em 2.2 ), sup- p ose that for every ( x, y ) , ( x ∗ , y ∗ ) ∈ X × X ther e exists a ( u, v ) ∈ X × X such that ( F ( u, v ) , F ( v , u )) is c omp ar able to ( F ( x, y ) , F ( y , x )) and ( F ( x ∗ , y ∗ ) , F ( y ∗ , x ∗ )) . Then F , G and S have a unique c ouple d c ommon fixe d p oint, tha t is, th er e exist a unique ( x, y ) ∈ X × X such that x = G ( x ) = F ( x, y ) = S ( x ) and y = G ( y ) = F ( y , x ) = S ( y ) . Pro of. W e kno w, from Theorem 2.1 (r esp. Th eorem 2.2), that exists a coupled coincidence p oint. W e s upp ose that exist ( x, y ) and ( x ∗ , y ∗ ) tw o coupled coincidence p oin ts, that is, G ( x ) = F ( x, y ) = S ( x ), G ( y ) = F ( y , x ) = S ( y ), G ( x ∗ ) = F ( x ∗ , y ∗ ) = S ( x ∗ ) and G ( y ∗ ) = F ( y ∗ , x ∗ ) = S ( y ∗ ). W e claim that G ( x ) = G ( x ∗ ) = S ( x ∗ ) = S ( x ) and G ( y ) = G ( y ∗ ) = S ( y ∗ ) = S ( y ) . (31) If ( F ( x, y ) , F ( y , x )) is comparable to ( F ( x ∗ , y ∗ ) , F ( y ∗ , x ∗ )), it is easy to reac h the result, then w e supp ose the general case. By assumption there is ( u, v ) ∈ X × X suc h that ( F ( u, v ) , F ( v , u )) is comparable to ( F ( x, y ) F ( y , x )) and ( F ( x ∗ , y ∗ ) F ( y ∗ , x ∗ )). W e distinguish tw o cases: First case: W e assu me that ( F ( x, y ) , F ( y , x ))  ( F ( u, v ) , F ( v , u )) and ( F ( x ∗ , y ∗ ) , F ( y ∗ , x ∗ ))  ( F ( u, v ) , F ( v , u )). Put u 0 = u and v 0 = v and w e choose u 1 and v 1 suc h that G ( u 0 )  S ( u 1 )  F ( u 0 , v 0 ), G ( v 0 )  S ( v 1 )  F ( v 0 , u 0 ). Similarly as in the pr o of of Th eorem 2.1, we can constru ct sequences { u n } and { v n } in X suc h that  G ( u 2 n +2 ) = F ( u 2 n , v 2 n ) G ( v 2 n +2 ) = F ( v 2 n , u 2 n ) and  S ( u 2 n +3 ) = F ( u 2 n +1 , v 2 n +1 ) S ( v 2 n +3 ) = F ( v 2 n +1 , u 2 n +1 ) for all n ≥ 0. Lo oking at the p ro of of Theorem 2.1, precisely at (3), we see that { G ( u 2 n ) } is a non- decreasing sequence, G ( u 2 n ) ≤ S ( u 2 n +1 ), and { G ( v 2 n ) } is a non-increasing sequen ce, G ( v 2 n )  S ( v 2 n +1 ). Therefore, w e ha v e G ( x ) = F ( x, y ) ≤ F ( u 0 , v 0 ) = G ( u 2 )  G ( u 2 n )  S ( u 2 n +1 ) and (32) G ( y ) = F ( y , x )  F ( v 0 , u 0 ) = G ( v 2 )  G ( v 2 n )  S ( v 2 n +1 ) . 12 Similarly , we ha v e G ( x ∗ ) = F ( x ∗ , y ∗ )  F ( u 0 , v 0 ) = G ( u 2 )  G ( u 2 n )  S ( u 2 n +1 ) and (33) G ( y ∗ ) = F ( y ∗ , x ∗ )  F ( v 0 , u 0 ) = G ( v 2 )  G ( v 2 n )  S ( v 2 n +1 ) . Using (32 ) and the cont ractiv e condition, we w r ite ϕ ( d ( F ( x, y ) , F ( u 2 n +1 , v 2 n +1 ))) ≤ ϕ (max { d ( Gx, S u 2 n +1 ) , d ( Gy , S v 2 n +1 ) } ) − φ (max { d ( Gx, S u 2 n +1 ) , d ( Gy , S v 2 n +1 ) } ) and ϕ ( d ( F ( y , x ) , F ( v 2 n +1 , u 2 n +1 ))) ≤ ϕ (max { d ( Gy , S v 2 n +1 ) , d ( Gx, S u 2 n +1 ) } ) − φ (max { d ( Gy, S v 2 n +1 ) , d ( Gx, S u 2 n +1 ) } ) . Therefore ϕ (max { d ( F ( x, y ) , F ( u 2 n +1 , v 2 n +1 )) , d ( F ( y , x ) , F ( v 2 n +1 , u 2 n +1 )) } ) ≤ ϕ (max { d ( Gx, S u 2 n +1 ) , d ( Gy , S v 2 n +1 ) } ) − φ (max { d ( Gx, S u 2 n +1 ) , d ( Gy , S v 2 n +1 ) } ) . Therefore ϕ (max( d ( G ( x ) , S u 2 n +3 ) , d ( Gy , S v 2 n +3 ))) ≤ ϕ (max( d ( G ( x ) , S u 2 n +1 ) , d ( Gy , S v 2 n +1 ))) (34) − φ (max( d ( G ( x ) , S u 2 n +1 ) , d ( Gy , S v 2 n +1 ))) . W e see that ϕ (max { d ( Gx, S u 2 n +3 ) , d ( Gy , S v 2 n +3 ) } ) ≤ ϕ (max { d ( Gx, S u 2 n +1 ) , d ( Gy , S v 2 n +1 ) } ) . Using the non-decreasing pr op ert y of ϕ , w e get max { d ( Gx, S u 2 n +3 ) , d ( Gy , S v 2 n +3 ) } ≤ max { d ( Gx, S u 2 n +1 ) , d ( Gy , S v 2 n +1 ) } . This imp lies that max { d ( Gx, S u 2 n +1 ) , d ( Gy , S v 2 n +1 ) } is a non-increasing sequence. Hence, th ere exists r ≥ 0 such that lim n → + ∞ max { d ( Gx, S u 2 n +1 ) , d ( Gy , S v 2 n +1 ) } = r. P assing to limit in (34) as n → + ∞ , we obtain ϕ ( r ) ≤ ϕ ( r ) − φ ( r ) , whic h implies that φ ( r ) = 0 and th en , sin ce φ is an altering distance f u nction, r = 0. W e deduce that lim n → + ∞ max { d ( Gx, S u 2 n +1 ) , d ( Gy , S v 2 n +1 ) } = 0 . (35) 13 Similarly , one can p ro v e th at lim n → + ∞ max { d ( Gx ∗ , S u 2 n +1 ) , d ( Gy ∗ , S v 2 n +1 ) } = 0 . (36) By the triangle inequalit y , (35) and (36), d ( Gx, Gx ∗ ) ≤ d ( Gx, S u 2 n +1 ) + d ( G ( x ∗ ) , S u 2 n +1 ) → 0 as n → + ∞ , (37) d ( Gy , Gy ∗ ) ≤ d ( Gy , S v 2 n +1 ) + d ( G ( y ∗ ) , S v 2 n +1 ) → 0 as n → + ∞ . (38) Hence G ( x ) = G ( x ∗ ) and G ( y ) = G ( y ∗ ) . (39) This p ro v e the claim (31) in this case. Second case: W e assum e that ( F ( x, y ) , F ( y , x ))  ( F ( u, v ) , F ( v , u )) an d ( F ( x ∗ , y ∗ ) , F ( y ∗ , x ∗ ))  ( F ( u, v ) , F ( v , u )). Put u 0 = u and v 0 = v and w e choose u 1 and v 1 suc h that G ( u 0 )  S ( u 1 )  F ( u 0 , v 0 ), G ( v 0 )  S ( v 1 )  F ( v 0 , u 0 ). Similarly as in the pr o of of Th eorem 2.1, we can constru ct sequences { u n } and { v n } in X suc h that  G ( u 2 n +2 ) = F ( u 2 n , v 2 n ) G ( v 2 n +2 ) = F ( v 2 n , u 2 n ) and  S ( u 2 n +3 ) = F ( u 2 n +1 , v 2 n +1 ) S ( v 2 n +3 ) = F ( v 2 n +1 , u 2 n +1 ) for all n ≥ 0. Lo oking at the p ro of of Theorem 2.1, precisely at (3), we see that { G ( u 2 n ) } is a non- increasing sequence, G ( u 2 n )  S ( u 2 n +1 ), and { G ( v 2 n ) } is a non-decreasing sequence, G ( v 2 n )  S ( v 2 n +1 ). Therefore, w e ha v e G ( x ) = F ( x, y )  F ( u 0 , v 0 ) = G ( u 2 )  G ( u 2 n )  S ( u 2 n +1 ) and G ( y ) = F ( y , x )  F ( v 0 , u 0 ) = G ( v 2 )  G ( v 2 n )  S ( v 2 n +1 ) . Similarly , we ha v e G ( x ∗ ) = F ( x ∗ , y ∗ )  F ( u 0 , v 0 ) = G ( u 2 )  G ( u 2 n )  S ( u 2 n +1 ) and G ( y ∗ ) = F ( y ∗ , x ∗ )  F ( v 0 , u 0 ) = G ( v 2 )  G ( v 2 n )  S ( v 2 n +1 ) . F rom this, we complete the p ro of iden tically as in the fi rst case and w e obtain the cla im (31) in this case. Since G ( x ) = F ( x, y ) = S ( x ) and G ( y ) = F ( y , x ) = S ( y ), by th e comm utativit y of F , G and F , S , we hav e  G ( G ( x )) = G ( F ( x, y )) = F ( Gx, Gy ) G ( G ( y )) = G ( F ( y , x )) = F ( Gy , Gx ) and  S ( S ( x )) = S ( F ( x, y )) = F ( S ( x ) , S ( y )) S ( S ( y )) = S ( F ( y , x )) = F ( S ( y ) , S ( x )) . (40) Set G ( x ) = a = S ( x ), G ( y ) = b = S ( y ). Then from (40), G ( a ) = F ( a, b ) = S ( a ) and G ( b ) = F ( b, a ) = S ( b ) . (41) 14 Th us ( a, b ) is a coupled coincidence p oin t. Then from (3 1) with x ∗ = a and y ∗ = b it follo ws that G ( a ) = G ( x ) = S ( a ) and G ( b ) = G ( y ) = S ( b ). Therefore G ( a ) = a = S ( a ) and G ( b ) = b = S ( b ) . ( 42) W e deduce th at ( a, b ) is a coup led common fixed p oint. T o pro v e the uniqueness, assume that ( c, d ) is another coupled common fixed p oin t. Then b y (31) and (42) we h a v e c = G ( c ) = G ( a ) = a and d = G ( d ) = G ( b ) = b .  Remark 2 T aking G = S = I X (the identity mapping of X ) in The or em 2.1, we obtain [7, The or em 2]. T aking G = S = I X in The or em 2.2, we obtain [7, The or em 3]. T aking S = G in T heorem 2.3 , w e obtain the follo wing resu lt. Corollary 2.1 L et ( X ,  ) b e a p artial ly or der e d set and supp ose that ther e exists a metric d on X such that ( X, d ) is a c omplete metr ic sp ac e. L et G : X → X b e two mappings and F : X × X → X b e a map ping with the mixe d G -monotone pr op erty and satisfying ϕ ( d ( F ( x, y ) , F ( u, v ))) ≤ ϕ ( max { d ( Gx, Gu ) , d ( Gy , Gv ) } ) − φ (max { Gx, Gu ) , d ( Gy , Gv ) } ) , for al l x, y , u, v ∈ X with G ( x )  G ( u ) or G ( x )  G ( u ) and G ( y )  G ( v ) or G ( y )  G ( v ) , wher e ϕ and φ ar e altering distanc e functions. Assume that F ( X × X ) ⊆ G ( X ) and assume also the fol lowing hyp otheses: 1. G is c ontinuous, 2. F is c ontinuous or G is non-de cr e asing mapping and X satisfies the fol lowing pr op- erties: • i f ( x n ) is a non-de cr e asing se qu e nc es with x n → x then x n  x for e ach n ∈ N , • i f ( y n ) is a non-incr e asing se quenc es with y n → y then y  y n for e ach n ∈ N ; 3. for every ( x, y ) , ( x ∗ , y ∗ ) ∈ X × X ther e exists a ( u, v ) ∈ X × X such that ( F ( u, v ) , F ( v , u )) is c omp ar able to ( F ( x, y ) , F ( y , x )) and ( F ( x ∗ , y ∗ ) , F ( y ∗ , x ∗ )) , 4. F c ommutes with G . If ther e exi st x 0 , y 0 ∈ X such that  G ( x 0 )  F ( x 0 , y 0 ) G ( y 0 ))  F ( y 0 , x 0 ) then th er e exists a unique ( x, y ) ∈ X × X such that x = G ( x ) = F ( x, y ) and y = G ( y ) = F ( y , x ) , that is, G and F have a unique c ouple d c ommon fixe d p oint. 15 3 Applicatio ns to p erio dic b oundary v alue prob- lems In this sect ion, w e stud y the existence and un iqueness of solution to a p erio dic b oundary v alue problem, as an app lication to the fi xed p oint theorem giv en by Corollary 2.1. Let C ([0 , T ] , R ) be th e s et of all con tinuous fun ctions u : [0 , T ] → R and consid er a mapping G : C ([0 , T ] , R ) → C ([0 , T ] , R ). Consider th e p erio d ic b oundary v alue p roblem u ′ = f ( t, u ) + h ( t, u ) , t ∈ (0 , T ) (43) u (0) = u ( T ) , (44) where f , h are t w o con tin uous fu nctions s atisfying the f ollo win g conditions: There exist p ositiv e constan ts λ 1 , λ 2 , µ 1 and µ 2 , suc h that for all u, v ∈ ( C ([0 , T ] , R ), Gv ( t ) ≤ Gu ( t ), 0 ≤ ( f ( t, u ( t )) + λ 1 u ( t )) − ( f ( t, v ( t )) + λ 1 v ( t )) ≤ µ 1 ln[( Gu ( t ) − Gv ( t )) 2 + 1] ( 45) − µ 2 ln[( Gu ( t ) − Gv ( t )) 2 + 1] ≤ ( h ( t, u ( t )) + λ 2 u ( t )) − ( h ( t, v ( t )) + λ 2 v ( t )) ≤ 0 (46) with 2 m ax { µ 1 , µ 2 } λ 1 + λ 2 < 1 . (47) W e fir stly stu d y the existence of a solution of the follo wing p erio dic system: u ′ + λ 1 u − λ 2 v = f ( t, u ) + h ( t, v ) + λ 1 u − λ 2 v v ′ + λ 1 v − λ 2 u = f ( t, v ) + h ( t, u ) + λ 1 v − λ 2 u, (48) with the p erio d icit y condition u (0) = u ( T ) and v (0) = v ( T ) . (49) This p roblem is equiv alent to the integ ral equations: u ( t ) = Z T 0 k 1 ( t, s )[ f ( s, u ) + h ( s, v ) + λ 1 u − λ 2 v ] + Z T 0 k 2 ( t, s )[ f ( s, v ) + h ( s, u ) + λ 1 v − λ 2 u ] ds v ( t ) = Z T 0 k 1 ( t, s )[ f ( s, v ) + h ( s, u ) + λ 1 v − λ 2 u ] + Z T 0 k 2 ( t, s )[ f ( s, u ) + h ( s, v ) + λ 1 u − λ 2 v ] ds where k 1 ( t, s ) =            1 2 " e σ 1 ( t − s ) 1 − e σ 1 T + e σ 2 ( t − s ) 1 − e σ 2 T # 0 ≤ s < t ≤ T 1 2 " e σ 1 ( t + T − s ) 1 − e σ 1 T + e σ 2 ( t + T − s ) 1 − e σ 2 T # 0 ≤ t < s ≤ T k 2 ( t, s ) =            1 2 " e σ 2 ( t − s ) 1 − e σ 2 T + e σ 1 ( t − s ) 1 − e σ 1 T # 0 ≤ s < t ≤ T 1 2 " e σ 2 ( t + T − s ) 1 − e σ 2 T + e σ 1 ( t + T − s ) 1 − e σ 1 T # 0 ≤ t < s ≤ T . 16 Here, σ 1 = − ( λ 1 + λ 2 ) and σ 2 = ( λ 2 − λ 1 ). F rom [3, Lemma 3.2], we ha v e k 1 ( t, s ) ≥ 0 , 0 ≤ t, s ≤ T and k 2 ( t, s ) ≤ 0 , 0 ≤ t, s ≤ T . (50) W e assume that there exist α, β ∈ C ([0 , T ]) su ch that G ( α ( t )) ≤ Z 1 0 k 1 ( t, s )( f ( s, α ( s )) + h ( s, β ( s )) + λ 1 α ( s ) − λ 2 β ( s )) ds + Z 1 0 k 2 ( t, s )( f ( s, β ( s )) + h ( s, α ( s )) + λ 1 β ( s ) − λ 2 α ( s )) ds (51) and G ( β ( t )) ≥ Z 1 0 k 1 ( t, s )( f ( s, β ( s )) + h ( s, α ( s )) + λ 1 β ( s ) − λ 2 α ( s )) ds + Z 1 0 k 2 ( t, s )( f ( s, α ( s )) + h ( s, β ( s )) + λ 1 α ( s ) − λ 2 β ( s )) ds. (52) W e endow X = C ([0 , T ] , R ) w ith the metric d ( u, v ) = max t ∈ [0 ,T ] | u ( t ) − v ( t ) | for u, v ∈ X . This sp ace can b e equip p ed w ith a partial order giv en by x, y ∈ C ([0 , T ]) , x  y ⇔ x ( t ) ≤ y ( t ) , for any t ∈ [0 , T ] . In X × X we defin e the follo wing p artial order ( x, y ) , ( u, v ) ∈ X × X, ( x, y )  ( u, v ) ⇔ x  u and y  v . Since for an y x, y ∈ X w e ha ve that max( x, y ) and min( x, y ) ∈ X , assumption 3 of Corollary 2.1 is satisfied for ( X,  ). Moreo v er in [10] it is pr o v ed that ( X,  ) satisfies assumption 2 of Corollary 2.1. No w , we shall prov e the follo wing r esult. Theorem 3.1 Supp ose that G : X → X is a non-de cr e asing c ontinuous mapping. Supp ose also that (45)-(47) and (51)-(52) hold. Then (48)-(49) has a u ni q ue solution. Ther efor e (43)-(44) h as also a unique solution. Pro of. W e introd uce the operator F : X × X → X defined by F ( u, v )( t ) = Z T 0 k 1 ( t, s )[ f ( s, u ) + h ( s, v ) + λ 1 u − λ 2 v ] ds + Z T 0 k 2 ( t, s )[ f ( s, v ) + h ( s, u ) + λ 1 v − λ 2 u ] ds for all u, v ∈ X and t ∈ [0 , T ]. 17 W e claim that F has the mixed G -monotone prop ert y . In fact, for Gx 1 ≤ Gx 2 and t ∈ [0 , T ], w e h a v e F ( x 1 , y )( t ) − F ( x 2 , y )( t ) = Z T 0 k 1 ( t, s )( f ( s, x 1 ( s )) − f ( s, x 2 ) + λ 1 ( x 1 ( s ) − x 2 ( s )) ds + Z T 0 k 2 ( t, s )( h ( s, x 1 ( s )) − h ( s, x 2 ) − λ 2 ( x 1 − x 2 )) ds. F rom (45), (46) and (50), for all t ∈ [0 , T ], w e ha ve F ( x 1 , y )( t ) − F ( x 2 , y )( t ) ≤ 0 . This imp lies that F ( x 1 , y )  F ( x 2 , y ) . Also, f or Gy 1  Gy 2 and t ∈ [0 , T ], we h a v e F ( x, y 1 )( t ) − F ( x, y 1 )( t ) = Z T 0 k 1 ( t, s )( h ( s, y 1 ( s )) − h ( s, y 2 ) − λ 2 ( y 1 ( s ) − y 2 ( s )) ds + Z T 0 k 2 ( t, s )( f ( s, y 1 ( s )) − f ( s, y 2 ) + λ 1 ( y 1 − y 2 )) ds. Lo oking at (45), (46) and (50), for all t ∈ [0 , T ], w e ha ve F ( x, y 1 )( t ) − F ( x, y 2 )( t ) ≥ 0 , that is, F ( x, y 1 ) ≥ F ( x, y 2 ) . Th us, we p ro v ed that F has the mixed G -monotone p rop erty . F or G ( x )  G ( u ) and G ( y )  G ( v ), we h av e F ( x, y )  F ( u, v ) and d ( F ( x, y ) , F ( u, v )) = max t ∈ [0 ,T ] | F ( x, y )( t ) − F ( u, v )( t ) | = max t ∈ [0 ,T ] ( F ( x, y )( t ) − F ( u, v )( t )) = max t ∈ [0 ,T ] Z T 0 k 1 ( t, s )[( f ( s, x ( s )) − f ( s, u ( s )) + λ 1 ( x − u )) − ( h ( s, v ( s )) − h ( s, y ( s )) − λ 2 ( y − v ))] ds − Z T 0 k 2 ( t, s )[( f ( s, v ( s )) − f ( s, y ( s )) + λ 1 ( v − y )) − ( h ( s, u ( s )) − h ( s, x ( s )) − λ 2 ( u − x ))] ds. Using (45 ) and (4 6) we get d ( F ( x, y ) , F ( u, v )) ≤ max t ∈ [0 ,T ] Z T 0 k 1 ( t, s )  µ 1 ln[( Gx ( s ) − Gu ( s )) 2 + 1] + µ 2 ln[( Gy ( s ) − Gv ( s )) 2 + 1]  ds + Z T 0 ( − k 2 ( t, s ))  µ 1 ln[( Gv ( s ) − Gy ( s )) 2 + 1] + µ 2 ln[( Gx ( s ) − Gu ( s )) 2 + 1]  ds ≤ max( µ 1 , µ 2 ) max t ∈ [0 ,T ] Z T 0 ( k 1 ( t, s ) − k 2 ( t, s )) ln[( Gx ( s ) − Gu ( s )) 2 + 1] ds + Z T 0 ( k 1 ( t, s ) − k 2 ( t, s )) ln[( Gy ( s ) − Gv ( s )) 2 + 1] ds. 18 An easy computation yields d ( F ( x, y ) , F ( u, v )) ≤  max t ∈ [0 ,T ] Z T 0 ( k 1 ( t, s ) − k 2 ( t, s )) ds  max( µ 1 , µ 2 )  ln[( d ( Gx, Gu )) 2 + 1] + ln[( d ( Gy , Gv )) 2 + 1]  ≤ 2  max t ∈ [0 ,T ] Z T 0 ( k 1 ( t, s ) − k 2 ( t, s )) ds  max( µ 1 , µ 2 ) ln[(max( d ( Gx, Gu ) , d ( Gy , Gv ))) 2 + 1] ≤ 2 max( µ 1 , µ 2 ) max t ∈ [0 ,T ]      Z t 0 e σ 1 ( t − s ) 1 − e σ 1 T ds + Z T t e σ 1 ( t + T − s ) 1 − e σ 1 T ds      ln[(max( d ( Gx, Gu ) , d ( Gy , Gv ))) 2 + 1] . After int egrating, we get d ( F ( x, y ) , F ( u, v )) ≤ 2 m ax( µ 1 , µ 2 ) λ 1 + λ 2 ln[(max( d ( Gx, Gu ) , d ( Gy , Gv ))) 2 + 1] . F rom (47), we obtain d ( F ( x, y ) , F ( u, v )) ≤ ln [(max( d ( Gx, Gu ) , d ( Gy , Gv ))) 2 + 1] whic h implies that ( d ( F ( x, y ) , F ( u, v ))) 2 ≤ (ln[(max( d ( Gx, Gu ) , d ( Gy , Gv ))) 2 + 1]) 2 . Then, ( d ( F ( x, y ) , F ( u, v ))) 2 ≤ (max( d ( Gx, Gu ) , d ( Gy , Gv ))) 2 −  (max( d ( Gx, Gu ) , d ( Gy , Gv ))) 2 − (ln[(max( d ( Gx, Gu ) , d ( Gy , Gv ))) 2 + 1]) 2  . Set ϕ ( t ) = t 2 and φ ( t ) = t 2 − ln( t 2 + 1). C learly ϕ and φ are altering distance fu nctions and from the ab o v e inequality , we obtain ϕ ( d ( F ( x, y ) , F ( u, v ))) ≤ ϕ ( max { d ( Gx, Gu ) , d ( Gy , Gv ) } ) − φ ((max { d ( Gx, Gu ) , d ( Gy , Gv ) } )) for all x , y , u, v ∈ X suc h that G ( x )  G ( u ) and G ( y )  G ( v ). 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Reurings, A fixed p oint theorem in partially ord ered sets and some applications to metrix equ ations, Proc. Amer. Math. So c. 132 (5)(2004)1 435- 1443. [15] M-D. Rus, Fixed p oin t theorems for generalize d con tractions in p artially or- dered metric spaces with semi-monotone metric, Nonlinear Analysis (2010) doi:10.10 16/j.na.201 0.10.053. [16] B. Samet, Coupled fi x ed p oin t theorems for a ge neralized Meir-Keeler con traction in partially ordered metric spaces, Nonlinear Anal. 72 (2010) 4 508-451 7. [17] W. Shatana w i, Pa rtially ordered cone metric sp aces and coupled fixed p oin t r esu lts, Computers and Mathematics with App lications 60 (2010) 2508-2515. Habib Y azidi UNIVERSIT ´ E DE TUNIS, DEP AR TMENT O F MA T HEMA T I CS, TUNIS COLLE GE OF SCIENCES AND TECHNIQ UES, 5 A VENUE T AHA HUSSEIN, BP , 59 , BAB MANARA, TUNIS. E-mail add r ess: habib .y azidi@gmail.c om 20