Quartet fixed point for nonlinear contraction

QUAR TET FIXED POINT THEOREMS F OR NONLINEAR CONTRA CTIONS IN P AR TIALL Y ORDERED METRIC SP ACES ERDAL KARAPINAR Abstract. The n otion of coupled fixed p oint is int ro duced in by Bhask ar and Lakshmik an tham in [2]. V er y recen tly , the concept of tr i pled fixe d p oin t is int ro duced b y Ber inde and Borcut [1]. In this man uscript, by using the mixed g monoton e mapping, some new quartet fixed point theorems are obt ained. W e also give some examples to s upport our results. 1. Introduction and Preliminaries In 2006, B ha sk a r and Lakshmik an tham [2] intro duced the notion of coupled fixed po in t and proved some fixed point theorem under certain c o ndition. Later, Lakshmik an tham and ´ Ciri´ c in [8] ex tended these results by defining of g -monotone prop ert y . After that man y results appeared o n coupled fixed p oin t theor y (see e.g. [3, 4, 6, 5, 11, 10]). V ery recently , Ber inde and Borcut [1] int ro duced the co ncept of tripled fixed po in t and proved some related theorems . In this man uscript, the qua rtet fixed po in t is considered and b y using the mixed g -monotone mapping, existence and uniqueness o f qua rtet fixed p oint are obtained. First we recall the bas ic definitions and res ults from which quartet fixed p oint is inspired. Let ( X , d ) b e a metr ic spa ce and X 2 := X × X . Then the mapping ρ : X 2 × X 2 → [0 , ∞ ) such that ρ (( x 1 , y 1 ) , ( x 2 , y 2 )) := d ( x 1 , x 2 ) + d ( y 1 , y 2 ) forms a metric o n X 2 . A s equence ( { x n } , { y n } ) ∈ X 2 is said to b e a double s equence of X . Definition 1. (Se e [2] ) L et ( X , ≤ ) b e p artial ly or der e d set and F : X × X → X . F is said to have mixe d monotone pr op ert y if F ( x, y ) is monotone nonde c r e asing in x and is monotone non-incr e asing in y , that is, for any x, y ∈ X , x 1 ≤ x 2 ⇒ F ( x 1 , y ) ≤ F ( x 2 , y ) , for x 1 , x 2 ∈ X , and y 1 ≤ y 2 ⇒ F ( x, y 2 ) ≤ F ( x, y 1 ) , for y 1 , y 2 ∈ X . Definition 2. ( s e e [2] ) A n element ( x , y ) ∈ X × X is said to b e a c ouple d fixe d p oint of the mapping F : X × X → X if F ( x , y ) = x and F ( y, x ) = y . 2000 Mathematics Subje c t Classific ation. 47H10,54H25,46J10, 46J15. Key wor ds and phr ases. Fixed point theorems, Nonlinear con traction, Partially or dered, Quar- tet Fixed Poin t, mixed g monotone. 1 2 E. KARAPINAR Throughout this paper, let ( X, ≤ ) be pa r tially order ed set and d b e a metric on X such that ( X , d ) is a complete metric spa ce. F urther, the pr oduct spaces X × X satisfy the following: ( u, v ) ≤ ( x, y ) ⇔ u ≤ x, y ≤ v ; for a ll ( x, y ) , ( u, v ) ∈ X × X . (1.1) The following tw o results of Bhask a r and Lakshmik antham in [2 ] were extended to clas s o f co ne metr ic spaces in [5]: Theorem 3. L et F : X × X → X b e a c ontinuous mapping having t he mixe d monotone pr op erty on X . Assum e that ther e exists a k ∈ [0 , 1) with d ( F ( x, y ) , F ( u, v )) ≤ k 2 [ d ( x, u ) + d ( y , v )] , for al l u ≤ x, y ≤ v . (1.2) If ther e exist x 0 , y 0 ∈ X su ch that x 0 ≤ F ( x 0 , y 0 ) and F ( y 0 , x 0 ) ≤ y 0 , then, ther e exist x, y ∈ X su ch t ha t x = F ( x , y ) and y = F ( y , x ) . Theorem 4. L et F : X × X → X b e a mappi ng having the mixe d monotone pr op erty on X . Supp ose that X has the following pr op erties: ( i ) if a non-de cr e asing se qu en c e { x n } → x , then x n ≤ x, ∀ n ; ( i ) if a non-incr e asing se quenc e { y n } → y , then y ≤ y n , ∀ n. Assume that ther e exists a k ∈ [0 , 1) with d ( F ( x, y ) , F ( u, v )) ≤ k 2 [ d ( x, u ) + d ( y , v )] , for al l u ≤ x, y ≤ v . (1.3) If ther e exist x 0 , y 0 ∈ X su ch that x 0 ≤ F ( x 0 , y 0 ) and F ( y 0 , x 0 ) ≤ y 0 , then, ther e exist x, y ∈ X su ch t ha t x = F ( x , y ) and y = F ( y , x ) . Inspired by Definition 1, the follo wing co ncept of a g -mixed monotone mapping int ro duced by V. Lakshmik antham and L. ´ Ciri´ c [8]. Definition 5. L et ( X , ≤ ) b e p artial ly or der e d set and F : X × X → X and g : X → X . F is said t o have mixe d g - m onotone pr op ert y if F ( x, y ) is m onotone g -non- de cr e asing in x and is monotone g -non-incr e asing in y , that is, for any x, y ∈ X , g ( x 1 ) ≤ g ( x 2 ) ⇒ F ( x 1 , y ) ≤ F ( x 2 , y ) , for x 1 , x 2 ∈ X , and (1.4) g ( y 1 ) ≤ g ( y 2 ) ⇒ F ( x, y 2 ) ≤ F ( x, y 1 ) , for y 1 , y 2 ∈ X . (1.5) It is clear that Definition 13 reduces to Definition 9 when g is the identit y . Definition 6. An element ( x , y ) ∈ X × X is c al le d a c ouple p oint of a mapping F : X × X → X and g : X → X if F ( x , y ) = g ( x ) , F ( y, x ) = g ( y ) . Definition 7. L et F : X × X → X and g : X → X wher e X 6 = ∅ . The mappings F and g ar e said to c ommu t e if g ( F ( x, y )) = F ( g ( x ) , g ( y )) , for al l x, y ∈ X . QUAR TET FIXED POINT THEOREMS FOR NONLINEAR CONTRA CTIONS IN P AR TIALL Y ORDERED SETS 3 Theorem 8. L et ( X , ≤ ) b e p artial ly or der e d set and ( X, d ) b e a c omplete metric sp ac e and also F : X × X → X and g : X → X wher e X 6 = ∅ . Supp ose that F has the mixe d g -monotone pr op erty and t ha t ther e exist s a k ∈ [0 , 1 ) with d ( F ( x, y ) , F ( u, v )) ≤ k 2  d ( g ( x ) , g ( u )) + d ( g ( y ) , g ( v )) 2  (1.6) for al l x, y , u, v ∈ X for which g ( x ) ≤ g ( u ) and g ( v ) ≤ g ( y ) . Supp ose F ( X × X ) ⊂ g ( X ) , g is se quential ly c ont inuous and c ommutes w ith F and also supp ose either F is c ontinuous or X has the fol lowi ng pr op erty: if a non-de cr e asing se quenc e { x n } → x, then x n ≤ x, for al l n, (1.7) if a non-incr e asing se quenc e { y n } → y , then y ≤ y n , for al l n. (1.8) If ther e exist x 0 , y 0 ∈ X su ch t ha t g ( x 0 ) ≤ F ( x 0 , y 0 ) and g ( y 0 ) ≤ F ( y 0 , x 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 c oincidenc e. Berinde and Bor cut [1] intro duced the following partial order o n the pro duct space X 3 = X × X × X : ( u, v , w ) ≤ ( x, y , z ) if and o nly if x ≥ u, y ≤ v , z ≥ w, (1.9) where ( u, v , w ) , ( x, y , z ) ∈ X 3 . Regar ding this partial o rder, we state the definition of the follo wing mapping. Definition 9. (Se e [1] ) L et ( X, ≤ ) b e p artial ly or der e d set and F : X 3 → X . We say t ha t F has the mixe d m onotone pr op erty if F ( x, y , z ) is monotone non- de cr e asing in x and z , and it is monotone n on-incr e asing in y , t ha t is, for any x, y , z ∈ X x 1 , x 2 ∈ X , x 1 ≤ x 2 ⇒ F ( x 1 , y , z ) ≤ F ( x 2 , y , z ) , y 1 , y 2 ∈ X , y 1 ≤ y 2 ⇒ F ( x, y 1 , z ) ≥ F ( x, y 2 , z ) , z 1 , z 2 ∈ X , z 1 ≤ z 2 ⇒ F ( x, y , z 1 ) ≤ F ( x, y , z 2 ) . (1.10) Theorem 10 . (Se e [1 ] ) L et ( X , ≤ ) b e p artial ly or der e d set and ( X , d ) b e a c omplete metric sp ac e. L et F : X × X × X → X b e a mapping having the mixe d monotone pr op erty on X . Assume that ther e exist c onstants a , b, c ∈ [0 , 1) such that a + b + c < 1 for whi ch d ( F ( x, y , z ) , F ( u, v , w )) ≤ ad ( x, u ) + bd ( y , v ) + cd ( z , w ) (1.11) for al l x ≥ u, y ≤ v , z ≥ w . Assum e that X has the fol lowing pr op ert ies: ( i ) if non-de cr e asing se quenc e x n → x , t hen x n ≤ x for al l n, ( ii ) if non-incr e asing se qu en c e y n → y , then y n ≥ y f or al l n , If ther e exist x 0 , y 0 , z 0 ∈ X such that x 0 ≤ F ( x 0 , y 0 , z 0 ) , y 0 ≥ F ( y 0 , x 0 , y 0 ) , z 0 ≤ F ( x 0 , y 0 , z 0 ) then ther e exist x, y , z ∈ X such that F ( x , y , z ) = x and F ( y , x, y ) = y and F ( z , y , x ) = z The aim of this pap er is in tro duce the concept o f quartet fixed p oin t and prov e the r elated fixed p oint theorems. 4 E. KARAPINAR 2. Quar tet Fixed Point Theorems Let ( X, ≤ ) be pa rtially ordered set and ( X, d ) b e a complete metric space. W e state the definition of the following mapping. Throughout the ma n us c ript we deno te X × X × X × X by X 4 . Definition 11. (Se e [7 ] ) L et ( X, ≤ ) b e p artial ly or der e d set and F : X 4 → X . We say that F has the mixe d monotone pr op erty if F ( x, y , z , w ) is monotone non- de cr e asing in x and z , a nd it is monotone n on-incr e asing in y and w , that is, for any x, y , z , w ∈ X x 1 , x 2 ∈ X , x 1 ≤ x 2 ⇒ F ( x 1 , y , z , w ) ≤ F ( x 2 , y , z , w ) , y 1 , y 2 ∈ X , y 1 ≤ y 2 ⇒ F ( x, y 1 , z , w ) ≥ F ( x, y 2 , z , w ) , z 1 , z 2 ∈ X , z 1 ≤ z 2 ⇒ F ( x, y , z 1 , w ) ≤ F ( x, y , z 2 , w ) , w 1 , w 2 ∈ X , w 1 ≤ w 2 ⇒ F ( x, y , z , w 1 ) ≥ F ( x, y , z , w 2 ) . (2.1) Definition 12. (Se e [7] ) An element ( x, y , z , w ) ∈ X 4 is c al le d a quartet fixe d p oint of F : X × X × X × X → X if F ( x , y , z , w ) = x, F ( x , w , z , y ) = y , F ( z , y , x, w ) = z , F ( z , w, x, y ) = w. (2.2) Definition 13. L et ( X, ≤ ) b e p artial ly or der e d set and F : X 4 → X . We say that F has t he mixe d g -monotone pr op ert y if F ( x , y , z , w ) is monotone g -non- de cr e asing in x and z , and it is monotone g -non-incr e asing in y and w , that is, for any x, y , z , w ∈ X x 1 , x 2 ∈ X , g ( x 1 ) ≤ g ( x 2 ) ⇒ F ( x 1 , y , z , w ) ≤ F ( x 2 , y , z , w ) , y 1 , y 2 ∈ X , g ( y 1 ) ≤ g ( y 2 ) ⇒ F ( x, y 1 , z , w ) ≥ F ( x, y 2 , z , w ) , z 1 , z 2 ∈ X , g ( z 1 ) ≤ g ( z 2 ) ⇒ F ( x, y , z 1 , w ) ≤ F ( x, y , z 2 , w ) , w 1 , w 2 ∈ X , g ( w 1 ) ≤ g ( w 2 ) ⇒ F ( x, y , z , w 1 ) ≥ F ( x, y , z , w 2 ) . (2.3) Definition 14. An element ( x, y , z , w ) ∈ X 4 is c al le d a quartet c oincidenc e p oint of F : X 4 → X and g : X → X if F ( x , y , z , w ) = g ( x ) , F ( y , z , w, x ) = g ( y ) , F ( z , w, x, y ) = g ( z ) , F ( w, x, y , z ) = g ( w ) . (2.4) Notice that if g is iden tit y ma pping, then Definition 1 3 and Definition14 r educe to Definition 11 and Definit ion12 , resp ectively . Definition 15. Le t F : X 4 → X and g : X → X . F and g ar e c al le d c ommutative if g ( F ( x, y , z , w )) = F ( g ( x ) , g ( y ) , g ( z ) , g ( w )) , for al l x, y , z , w ∈ X . (2.5) F or a metric space ( X , d ), the function ρ : X 4 × X 4 → [0 , ∞ ), given by , ρ (( x, y , z , w ) , ( u, v , r, t )) := d ( x, u ) + d ( y , v ) + d ( z , r ) + d ( w, t ) forms a metric space on X 4 , tha t is, ( X 4 , ρ ) is a metric induced by ( X , d ). Let Φ denote the all functions φ : [0 , ∞ ) → [0 , ∞ ) which is contin uous and satisfy that ( i ) φ ( t ) < t QUAR TET FIXED POINT THEOREMS FOR NONLINEAR CONTRA CTIONS IN P AR TIALL Y ORDERED SETS 5 ( i ) lim r → t + φ ( r ) < t for ea ch r > 0 . The aim o f this pa per is to prov e the f ollowing theorem. Theorem 16. L et ( X , ≤ ) b e p artial ly or der e d s et and ( X, d ) b e a c omplete metric sp ac e. Supp ose F : X 4 → X and ther e ex ist s φ ∈ Φ such that F has the mixe d g -monotone pr op erty and d ( F ( x, y , z , w ) , F ( u, v, r , t )) ≤ φ  d ( x, u ) + d ( y , v ) + d ( z , r ) + d ( w, t ) 4  (2.6) for al l x, u, y , v , z , r , w , t for which g ( x ) ≤ g ( u ) , g ( y ) ≥ g ( v ) , g ( z ) ≤ g ( r ) and g ( w ) ≥ g ( t ) . S u pp ose ther e exist x 0 , y 0 , z 0 , w 0 ∈ X such that g ( x 0 ) ≤ F ( x 0 , y 0 , z 0 , w 0 ) , g ( y 0 ) ≥ F ( x 0 , w 0 , z 0 , y 0 ) , g ( z 0 ) ≤ F ( z 0 , y 0 , x 0 , w 0 ) , g ( w 0 ) ≥ F ( z 0 , w 0 , x 0 , y 0 ) . (2.7) Assume also that F ( X 4 ) ⊂ g ( X ) and g c ommutes with F . Supp ose either ( a ) F is c ontinu ou s , or ( b ) X has the fol lowi ng pr op erty: ( i ) if non-de cr e asing se quenc e x n → x , then x n ≤ x for al l n, ( ii ) if non-incr e asing se qu en c e y n → y , then y n ≥ y f or al l n , then ther e exist x, y , z , w ∈ X such that F ( x , y , z , w ) = g ( x ) , F ( x, w, z , y ) = g ( y ) , F ( z , y , x, w ) = g ( z ) , F ( z , w, x, y ) = g ( w ) . that is, F and g have a c ommon c oincidenc e p oint. Pr o of. Let x 0 , y 0 , z 0 , w 0 ∈ X be such that (2 .7) . W e constr uct the sequences { x n } , { y n } , { z n } and { w n } as follows g ( x n ) = F ( x n − 1 , y n − 1 , z n − 1 , w n − 1 ) , g ( y n ) = F ( x n − 1 , w n − 1 , z n − 1 , y n − 1 ) , g ( z n ) = F ( z n − 1 , y n − 1 , x n − 1 , w n − 1 ) , g ( w n ) = F ( z n − 1 , w n − 1 , x n − 1 , y n − 1 ) . (2.8) for n = 1 , 2 , 3 , .... . W e claim that g ( x n − 1 ) ≤ g ( x n ) , g ( y n − 1 ) ≥ g ( y n ) , g ( z n − 1 ) ≤ g ( z n ) , g ( w n − 1 ) ≥ g ( w n ) , for a ll n ≥ 1 . (2.9) Indeed, we sha ll use mathematical induction to pro ve (2.9). Due to (2 .7), we have g ( x 0 ) ≤ F ( x 0 , y 0 , z 0 , w 0 ) = g ( x 1 ) , g ( y 0 ) ≥ F ( x , w 0 , z 0 , y 0 ) = g ( y 1 ) , g ( z 0 ) ≤ F ( z 0 , y 0 , x 0 , w 0 ) = g ( z 1 ) , g ( w 0 ) ≥ F ( z 0 , w 0 , x 0 , y 0 ) = g ( w 1 ) . Thu s, the ineq ualities in (2.9) hold for n = 1 . Supp ose now that the inequalities in (2.9) hold for some n ≥ 1 . By mixed g -monotone prop erty o f F , together with (2.8) a nd (2.3) we hav e g ( x n ) = F ( x n − 1 , y n − 1 , z n − 1 , w n − 1 ) ≤ F ( x n , y n , z n , w n ) = g ( x n +1 ) , g ( y n ) = F ( x n − 1 , w n − 1 , z n − 1 , y n − 1 ) ≥ F ( x n , w n , z n , y n ) = g ( y n +1 ) , g ( z n ) = F ( z n − 1 , y n − 1 , x n − 1 , w n − 1 ) ≤ F ( z n , y n , x n , w n ) = g ( z n +1 ) , g ( w n ) = F ( z n − 1 , w n − 1 , x n − 1 , y n − 1 ) ≥ F ( z n − 1 , w n − 1 , x n − 1 , y n − 1 ) = g ( w n +1 ) , (2.10) 6 E. KARAPINAR Thu s, (2.9) holds for all n ≥ 1. Hence, w e have · · · g ( x n ) ≥ g ( x n − 1 ) ≥ · · · ≥ g ( x 1 ) ≥ g ( x 0 ) , · · · g ( y n ) ≤ g ( y n − 1 ) ≤ · · · ≤ g ( y 1 ) ≤ g ( y 0 ) , · · · g ( z n ) ≥ g ( z n − 1 ) ≥ · · · ≥ g ( z 1 ) ≥ g ( z 0 ) , · · · g ( w n ) ≤ g ( w n − 1 ) ≤ · · · ≤ g ( w 1 ) ≤ g ( w 0 ) , (2.11) Set δ n = d ( g ( x n ) , g ( x n +1 )) + d ( g ( y n ) , g ( y n +1 )) + d ( g ( z n ) , g ( z n +1 )) + d ( g ( w n ) , g ( w n +1 )) W e shall show that δ n +1 ≤ 4 φ ( δ n 4 ) . (2.12) Due to (2.6), (2.8 ) and (2 .11), we have d ( g ( x n +1 ) , g ( x n +2 )) = d ( F ( x n , y n , z n , w n ) , F ( x n +1 , y n +1 , z n +1 , w n +1 )) φ  d ( g ( x n ) ,g ( x n +1 ))+ d ( g ( y n ) ,g ( y n +1 ))+ d ( g ( z n ) ,g ( z n +1 ))+ d ( g ( w n ) ,g ( w n +1 )) 4  ≤ φ ( δ n 4 ) (2.13) d ( g ( y n +1 ) , g ( y n +2 )) = d ( F ( y n , z n , w n , x n ) , F ( y n +1 , z n +1 , w n +1 , x n +1 )) ≤ φ  d ( g ( y n ) ,g ( y n +1 ))+ d ( g ( z n ) ,g ( z n +1 ))+ d ( g ( w n ) ,g ( w n +1 ))+ d ( g ( x n ) ,g ( x n +1 )) 4  ≤ φ ( δ n 4 ) (2.14) d ( g ( z n +1 ) , g ( z n +2 )) = d ( F ( z n , w n , x n , y n ) , F ( z n +1 , w n +1 , x n +1 , y n +1 )) ≤ φ  d ( g ( z n ) ,g ( z n +1 ))+ d ( g ( w n ) ,g ( w n +1 ))+ d ( g ( x n ) ,g ( x n +1 ))+ d ( g ( y n ) ,g ( y n +1 )) 4  ≤ φ ( δ n 4 ) (2.15) d ( g ( w n +1 ) , g ( w n +2 )) = d ( F ( w n , x n , y n , z n ) , F ( w n +1 , x n +1 , y n +1 , z n +1 )) φ  d ( g ( w n ) ,g ( w n +1 ))+ d ( g ( x n ) ,g ( x n +1 ))+ d ( g ( y n ) ,g ( y n +1 ))+ d ( g ( z n ) ,g ( z n +1 )) 4  ≤ φ ( δ n 4 ) (2.16) Due to (2.13)-(2.16), we conclude that d ( x n +1 , x n +2 ) + d ( y n +1 , y n +2 ) + d ( z n +1 , z n +2 ) + d ( w n +1 , w n +2 ) ≤ 4 φ ( δ n 4 ) (2.17) Hence we have (2.12). Since φ ( t ) < t for a ll t > 0 , then δ n +1 ≤ δ n for all n . Hence { δ n } is a no n- increasing s equence. Since it is b ounded b elo w, there is some δ ≥ 0 such that lim n →∞ δ n = δ + . (2.18) W e shall show that δ = 0. Supp ose, to the c o n trary , that δ > 0. T aking the limit as δ n → δ + of b oth s ides of (2.12) and having in mind that we supp ose lim t → r φ ( r ) < t for a ll t > 0, w e ha ve δ = lim n →∞ δ n +1 ≤ lim n →∞ 4 φ ( δ n 4 ) = lim δ n → δ + 4 φ ( δ n 4 ) < 4 δ 4 < δ (2.19) QUAR TET FIXED POINT THEOREMS FOR NONLINEAR CONTRA CTIONS IN P AR TIALL Y ORDERED SETS 7 which is a contradiction. Th us, δ = 0, that is, lim n →∞ [ d ( x n , x n − 1 ) + d ( y n , y n − 1 ) + d ( z n , z n − 1 ) + d ( w n , w n − 1 )] = 0 . (2.20) Now, w e sha ll pro ve tha t { g ( x n ) } , { g ( y n ) } , { g ( z n ) } and { g ( w n ) } ar e Ca uc hy se- quences. Suppo se, to the contrary , that at least one of { g ( x n ) } , { g ( y n ) } , { g ( z n ) } and { g ( w n ) } is not Cauch y . So , there exists an ε > 0 for which we can find sub- sequences { g ( x n ( k ) ) } , { g ( x n ( k ) ) } of { g ( x n ) } and { g ( y n ( k ) ) } , { g ( y n ( k ) ) } of { g ( y n ) } and { g ( z n ( k ) ) } , { g ( z n ( k ) ) } of { g ( z n ) } and { g ( w n ( k ) ) } , { g ( w n ( k ) ) } of { g ( w n ) } with n ( k ) > m ( k ) ≥ k such that d ( g ( x n ( k ) ) , g ( x m ( k ) )) + d ( g ( y n ( k ) ) , g ( y m ( k ) )) + d ( g ( z n ( k ) ) , g ( z m ( k ) )) + d ( g ( w n ( k ) ) , g ( w m ( k ) )) ≥ ε. (2.21) Additionally , corr esponding to m ( k ), we may choose n ( k ) s uc h that it is the smallest int eger satisfying (2.21) and n ( k ) > m ( k ) ≥ k . Th us, d ( g ( x n ( k ) − 1 ) , g ( x m ( k ) )) + d ( g ( y n ( k ) − 1 ) , g ( y m ( k ) )) + d ( g ( z n ( k ) − 1 ) , g ( z m ( k ) )) + d ( g ( w n ( k ) − 1 ) , g ( w m ( k ) )) < ε. (2.22) By using triangle inequa lit y and having (2.21),(2.22) in mind ε ≤ t k =: d ( g ( x n ( k ) ) , g ( x m ( k ) )) + d ( g ( y n ( k ) ) , g ( y m ( k ) )) + d ( g ( z n ( k ) ) , g ( z m ( k ) )) + d ( g ( w n ( k ) ) , g ( w m ( k ) )) ≤ d ( g ( x n ( k ) ) , g ( x n ( k ) − 1 )) + d ( g ( x n ( k ) − 1 ) , g ( x m ( k ) )) + d ( g ( y n ( k )) , g ( y n ( k ) − 1 )) + d ( g ( y n ( k ) − 1 ) , g ( y m ( k ) )) + d ( g ( z n ( k ) ) , g ( z n ( k ) − 1 )) + d ( g ( z n ( k ) − 1 ) , g ( z m ( k ) )) + d ( g ( w n ( k ) ) , g ( w n ( k ) − 1 )) + d ( g ( w n ( k ) − 1 ) , g ( w m ( k ) )) < d ( g ( x n ( k ) ) , g ( x n ( k ) − 1 )) + d ( g ( y n ( k ) ) , g ( y n ( k ) − 1 ))+ d ( g ( z n ( k ) ) , g ( z n ( k ) − 1 )) + d ( g ( w n ( k ) ) , g ( w n ( k ) − 1 )) + ε. (2.23) Letting k → ∞ in (2.23) and us ing (2.20) lim k →∞ t k = lim k →∞  d ( g ( x n ( k ) ) , g ( x m ( k ) )) + d ( g ( y n ( k ) ) , g ( y m ( k ) )) + d ( g ( z n ( k ) ) , g ( z m ( k ) )) + d ( g ( w n ( k ) ) , g ( w m ( k ) ))  = ε + (2.24) Again by triang le inequalit y , t k = d ( g ( x n ( k ) ) , g ( x m ( k ) )) + d ( g ( y n ( k ) ) , g ( y m ( k ) )) + d ( g ( z n ( k ) ) , g ( z m ( k ) )) + d ( g ( w n ( k ) ) , g ( w m ( k ) )) ≤ d ( g ( x n ( k ) ) , g ( x n ( k )+1 )) + d ( g ( x n ( k )+1 ) , g ( x m ( k )+1 )) + d ( g ( x m ( k )+1 ) , g ( x m ( k ) )) + d ( g ( y n ( k ) ) , g ( y n ( k )+1 )) + d ( g ( y n ( k )+1 ) , g ( y m ( k )+1 )) + d ( g ( y m ( k )+1 ) , g ( y m ( k ) )) + d ( g ( z n ( k ) ) , g ( z n ( k )+1 )) + d ( g ( z n ( k )+1 ) , g ( z m ( k )+1 )) + d ( g ( z m ( k )+1 ) , g ( z m ( k ) )) + d ( g ( w n ( k ) ) , g ( w n ( k )+1 )) + d ( g ( w n ( k )+1 ) , g ( w m ( k )+1 )) + d ( g ( w m ( k )+1 ) , g ( w m ( k ) )) ≤ δ n ( k )+1 + δ m ( k )+1 + d ( g ( x n ( k )+1 ) , g ( x m ( k )+1 )) + d ( g ( y n ( k )+1 ) , g ( y m ( k )+1 )) + d ( g ( z n ( k )+1 ) , g ( z m ( k )+1 )) + d ( g ( w n ( k )+1 ) , g ( w m ( k )+1 )) (2.25) Since n ( k ) > m ( k ), t hen g ( x n ( k ) ) ≥ g ( x m ( k ) ) a nd g ( y n ( k ) ) ≤ g ( y m ( k ) ) , g ( z n ( k ) ) ≥ g ( z m ( k ) ) a nd g ( w n ( k ) ) ≤ g ( w m ( k ) ) . (2.26) Hence from (2.26), (2.8) and (2.6), w e have, 8 E. KARAPINAR d ( g ( x n ( k )+1 ) , g ( x m ( k )+1 )) = d ( F ( x n ( k ) , y n ( k ) , z n ( k ) , w n ( k ) ) , F ( x m ( k ) , y m ( k ) , z m ( k ) , w m ( k ) )) ≤ φ  1 4 [ d ( g ( x n ( k ) ) , g ( x m ( k ) )) + d ( g ( y n ( k ) ) , g ( y m ( k ) )) + d ( g ( z n ( k ) ) , g ( z m ( k ) )) + d ( g ( w n ( k ) ) , g ( w m ( k ) ))]  (2.27) d ( g ( y n ( k )+1 ) , g ( y m ( k )+1 )) = d ( F ( y n ( k ) , z n ( k ) , w n ( k ) , x n ( k ) ) , F ( y m ( k ) , z m ( k ) , w m ( k ) , x m ( k ) )) ≤ φ  1 4 [ d ( g ( y n ( k ) ) , g ( y m ( k ) )) + d ( g ( z n ( k ) ) , g ( z m ( k ) )) + d ( g ( w n ( k ) ) , g ( w m ( k ) )) + d ( g ( x n ( k ) , x m ( k ) ))]  (2.28) d ( g ( z n ( k )+1 ) , g ( z m ( k )+1 )) = d ( F ( z n ( k ) , w n ( k ) , x n ( k ) , y n ( k ) ) , F ( z m ( k ) , w m ( k ) , x m ( k ) , y m ( k ) )) ≤ φ  1 4 [ d ( g ( z n ( k ) ) , g ( z m ( k ) )) + d ( g ( w n ( k ) ) , g ( w m ( k ) )) + d ( g ( x n ( k ) ) , g ( x m ( k ) )) + dg (( y n ( k ) ) , g ( y m ( k ) ))]  (2.29) d ( g ( w n ( k )+1 ) , g ( w m ( k )+1 ) = d ( F ( w n ( k ) , x n ( k ) , y n ( k ) , z n ( k ) ) , F ( w m ( k ) , x m ( k ) , y m ( k ) , z m ( k ) ) ≤ φ  1 4 [ d ( g ( w n ( k ) ) , g ( w m ( k ) )) + d ( g ( x n ( k ) ) , g ( x m ( k ) )) + d ( g ( y n ( k ) ) , g ( y m ( k ) )) + d ( g ( z n ( k ) ) , g ( z m ( k ) ))]  (2.30) Combining (2 .25) with (2.27)-(2.30), we o btain tha t t k ≤ δ n ( k )+1 + δ m ( k )+1 + d ( g ( x n ( k )+1 ) , g ( x m ( k )+1 ) + d ( g ( y n ( k )+1 ) , g ( y m ( k )+1 )) + d ( g ( z n ( k )+1 ) , g ( z m ( k )+1 )) + d ( g ( w n ( k )+1 ) , g ( w m ( k )+1 ))) ≤ δ n ( k )+1 + δ m ( k )+1 + t k + 4 φ  t k 4  < δ n ( k )+1 + δ m ( k )+1 + t k + 4 t k 4 (2.31) Letting k → ∞ , w e get a contradiction. This sho ws tha t { g ( x n ) } , { g ( y n ) } , { g ( z n ) } and { g ( w n ) } are Ca uchy se quences. Since X is co mplete metr ic space, there exists x, y , z , w ∈ X suc h that lim n →∞ g ( x n ) = x and lim n →∞ g ( y n ) = y , lim n →∞ g ( z n ) = z and lim n →∞ g ( w n ) = w . (2.32) Since g is contin uous, (2.32) implies that lim n →∞ g ( g ( x n )) = g ( x ) and lim n →∞ g ( g ( y n )) = g ( y ) , lim n →∞ g ( g ( z n )) = g ( z ) and lim n →∞ g ( g ( w n )) = g ( w ) . (2.33) F rom (2.10) and b y reg arding commutativit y of F and g , g ( g ( x n +1 )) = g ( F ( x n , y n , z n , w n )) = F ( g ( x n ) , g ( y n ) , g ( z n ) , g ( w n )) , g ( g ( y n +1 )) = g ( F ( x n , w n , z n , y n )) = F ( g ( x n ) , g ( w n ) , g ( z n ) , g ( y n )) , g ( g ( z n +1 )) = g ( F ( z n , y n , x n , w n )) = F ( g ( z n ) , g ( y n ) , g ( x n ) , g ( w n )) , g ( g ( w n +1 )) = g ( F ( z n , w n , x n , y n )) = F ( g ( z n ) , g ( w n ) , g ( x n ) , g ( y n )) , (2.34) W e shall show that F ( x , y , z , w ) = g ( x ) , F ( x , w , z , y ) = g ( y ) , F ( z , y , x, w ) = g ( z ) , F ( z , w, x, y ) = g ( w ) . QUAR TET FIXED POINT THEOREMS FOR NONLINEAR CONTRA CTIONS IN P AR TIALL Y ORDERED SETS 9 Suppo se now ( a ) holds. Then b y (2.8),(2.34 ) and (2.32), w e have g ( x ) = lim n →∞ g ( g ( x n +1 )) = lim n →∞ g ( F ( x n , y n , z n , w n )) = lim n →∞ F ( g ( x n ) , g ( y n ) , g ( z n ) , g ( w n )) = F ( lim n →∞ g ( x n ) , lim n →∞ g ( y n ) , lim n →∞ g ( z n ) , lim n →∞ g ( w n )) = F ( x, y , z , w ) (2.35) Analogously , we a lso o bserv e that g ( y ) = lim n →∞ g ( g ( y n +1 )) = lim n →∞ g ( F ( x n , w n , z n , y n ) = lim n →∞ F ( g ( x n ) , g ( w n ) , g ( z n ) , g ( y n )) = F ( lim n →∞ g ( x n ) , lim n →∞ g ( w n ) , lim n →∞ g ( z n ) , lim n →∞ g ( y n )) = F ( x, w, z , y ) (2.36) g ( z ) = lim n →∞ g ( g ( z n +1 )) = lim n →∞ g ( F ( z n , y n , x n , w n )) = lim n →∞ F ( g ( z n ) , g ( y n ) , g ( x n ) , g ( w n )) = F ( lim n →∞ g ( z n ) , lim n →∞ g ( y n ) , lim n →∞ g ( x n ) , lim n →∞ g ( w n )) = F ( z , y , x, w ) (2.37) g ( w ) = lim n →∞ g ( g ( w n +1 )) = lim n →∞ g ( F ( z n , w n , x n , y n )) = lim n →∞ F ( g ( z n ) , g ( w n ) , g ( x n ) , g ( y n )) = F ( lim n →∞ g ( z n ) , lim n →∞ g ( w n ) , lim n →∞ g ( x n ) , lim n →∞ g ( y n )) = F ( z , w , x, y ) (2.38) Thu s, w e have F ( x , y , z , w ) = g ( x ) , F ( y, z , w, x ) = g ( y ) , F ( z , , w, x, y ) = g ( z ) , F ( w, x, y , z ) = g ( w ) . Suppo se now the ass umption ( b ) ho lds. Since { g ( x n ) } , { g ( z n ) } is non- dec reasing and g ( x n ) → x, g ( z n ) → z and a lso { g ( y n ) } , { g ( w n ) } is no n-increasing and g ( y n ) → y , g ( w n ) → , then b y assumption ( b ) w e ha ve g ( x n ) ≥ x, g ( y n ) ≤ y , g ( z n ) ≥ z , g ( w n ) ≤ w (2.39) for a ll n . Thus, by tria ng le ine q ualit y and (2.34) d ( g ( x ) , F ( x, y , z , w )) ≤ d ( g ( x ) , g ( g ( x n +1 ))) + d ( g ( g ( x n +1 )) , F ( x, y , z , w )) ≤ d ( g ( x ) , g ( g ( x n +1 ))) + φ  1 4  d ( g ( g ( x n ) , g ( x ))) + d ( g ( g ( y n ) , g ( y ))) + d ( g ( g ( z n ) , g ( z ))) + d ( g ( g ( w n ) , g ( w )))  (2.40) Letting n → ∞ imp lies that d ( g ( x ) , F ( x, y , z , w )) ≤ 0 . Hence, g ( x ) = F ( x, y , z , w ). Analogously w e can get that F ( y, z , w, x ) = g ( y ) , F ( z , w, x, y ) = g ( z ) and F ( w, x, y , z ) = g ( w ) . Thu s, w e pr o ved that F and g hav e a quartet coincidence p oin t.  10 E. KARAPINAR References [1] V. Berinde and M. Borcut, T ri pl ed fixed p oint theorems for con tractive ty p e mappings in partially ordered metric s pac es, Nonline ar Analysis , 7 4 (15 ), 4889–4897 (2011). [2] Bhask ar, T.G., Lakshmik ant ham, V.: Fixed Poin t Theory i n partiall y ordered metric spac es and applications Nonline ar Analysis , 65 , 1379–139 3 (2006). [3] N.V. Luong and N.X. Thuan, Coupled fixed p oin ts in partiall y ordered m et ric spaces and application, Nonline ar Analysis , 74 , 983-992(2011) . [4] B. Samet, Coupled fixed p oin t theorems for a generalized MeirKeeler contract ion in partiall y ordered metric spaces, Nonline ar Analysis , 7 4 (12 ), 45084517(201 0). [5] E. Karapınar, Couple Fixed Poin t on Cone Metric Spaces, Gazi Univ ersity Journal of Scienc e , 24 (1),51-58(2011 ). [6] E. Karapınar, Coupled fixed poi n t theorems for nonlinear con tractions i n cone metric spaces, Comput. Math. Appl. , 59 (12), 3656-3668(2010) . [7] E. Karapınar, N.V.Luong, Quartet Fixed Poin t Theorems f or nonlinear con tractions, submi t- ted. [8] Lakshmik an tham, V., ´ Ciri´ c, L.: : Couple Fixed Poin t Theorems for nonlinear con tractions i n partially ordered metric s pac es Nonline ar Analysis , 70 , 4341-4349 (2009). [9] Nieto, J. J., Rodri gue z-L´ opez, R.: Cont ractive mapping theorems in partially ordered sets and applications to ordinary di fferential equations. Or der 2005(22), 3 , 223–239 (2006). [10] Bina ya k S. Choudhu ry , N . Metiy a and A. Kundu, Cou pled coincidence p oin t theorems i n ordered metric spaces, Ann. Univ. F err ar a , 57 , 1-16(2011). [11] B.S. Choudh ury , A. Kundu : A coupled coincidence p oin t r esult i n partiall y or dered metric spaces for compatible m appings. Nonline ar Anal. TMA 73 , 25242531 (2010) erdal karapınar, Dep ar tment of Ma thema tics, A tilim University 0 6836, ˙ Incek, An kara, Turkey E-mail addr e ss : erdalkarapin ar@yahoo.com E-mail addr e ss : ekarapinar@a tilim.edu.tr