aa r X i v : . [ m a t h . GN ] J un QUARTET FIXED POINT THEOREMS FOR NONLINEARCONTRACTIONS IN PARTIALLY ORDERED METRIC SPACES
ERDAL KARAPINAR
Abstract.
The notion of coupled fixed point is introduced in by Bhaskar andLakshmikantham in [2]. Very recently, the concept of tripled fixed point isintroduced by Berinde and Borcut [1]. In this manuscript, by using the mixed g monotone mapping, some new quartet fixed point theorems are obtained.We also give some examples to support our results. Introduction and Preliminaries
In 2006, Bhaskar and Lakshmikantham [2] introduced the notion of coupledfixed point and proved some fixed point theorem under certain condition. Later,Lakshmikantham and ´Ciri´c in [8] extended these results by defining of g -monotoneproperty. After that many results appeared on coupled fixed point theory (see e.g.[3, 4, 6, 5, 11, 10]).Very recently, Berinde and Borcut [1] introduced the concept of tripled fixedpoint and proved some related theorems. In this manuscript, the quartet fixedpoint is considered and by using the mixed g -monotone mapping, existence anduniqueness of quartet fixed point are obtained.First we recall the basic definitions and results from which quartet fixed pointis inspired. Let ( X, d ) be a metric space and X := X × X . Then the mapping ρ : X × X → [0 , ∞ ) such that ρ (( x , y ) , ( x , y )) := d ( x , x ) + d ( y , y ) forms ametric on X . A sequence ( { x n } , { y n } ) ∈ X is said to be a double sequence of X . Definition 1. (See [2] ) Let ( X, ≤ ) be partially ordered set and F : X × X → X . F is said to have mixed monotone property if F ( x, y ) is monotone nondecreasingin x and is monotone non-increasing in y , that is, for any x, y ∈ X , x ≤ x ⇒ F ( x , y ) ≤ F ( x , y ) , for x , x ∈ X, and y ≤ y ⇒ F ( x, y ) ≤ F ( x, y ) , for y , y ∈ X. Definition 2. (see [2] ) An element ( x, y ) ∈ X × X is said to be a coupled fixedpoint of the mapping F : X × X → X if F ( x, y ) = x and F ( y, x ) = y. Mathematics Subject Classification.
Key words and phrases.
Fixed point theorems, Nonlinear contraction, Partially ordered, Quar-tet Fixed Point, mixed g monotone.
Throughout this paper, let ( X, ≤ ) be partially ordered set and d be a metric on X such that ( X, d ) is a complete metric space. Further, the product spaces X × X satisfy the following:( u, v ) ≤ ( x, y ) ⇔ u ≤ x, y ≤ v ; for all ( x, y ) , ( u, v ) ∈ X × X. (1.1)The following two results of Bhaskar and Lakshmikantham in [2] were extendedto class of cone metric spaces in [5]: Theorem 3.
Let F : X × X → X be a continuous mapping having the mixedmonotone property on X . Assume that there exists a k ∈ [0 , with d ( F ( x, y ) , F ( u, v )) ≤ k d ( x, u ) + d ( y, v )] , for all u ≤ x, y ≤ v. (1.2) If there exist x , y ∈ X such that x ≤ F ( x , y ) and F ( y , x ) ≤ y , then, thereexist x, y ∈ X such that x = F ( x, y ) and y = F ( y, x ) . Theorem 4.
Let F : X × X → X be a mapping having the mixed monotoneproperty on X . Suppose that X has the following properties: ( i ) if a non-decreasing sequence { x n } → x , then x n ≤ x, ∀ n ;( i ) if a non-increasing sequence { y n } → y , then y ≤ y n , ∀ n. Assume that there exists a k ∈ [0 , with d ( F ( x, y ) , F ( u, v )) ≤ k d ( x, u ) + d ( y, v )] , for all u ≤ x, y ≤ v. (1.3) If there exist x , y ∈ X such that x ≤ F ( x , y ) and F ( y , x ) ≤ y , then, thereexist x, y ∈ X such that x = F ( x, y ) and y = F ( y, x ) . Inspired by Definition 1, the following concept of a g -mixed monotone mappingintroduced by V. Lakshmikantham and L. ´Ciri´c [8]. Definition 5.
Let ( X, ≤ ) be partially ordered set and F : X × X → X and g : X → X . F is said to have mixed g -monotone property if F ( x, y ) is monotone g -non-decreasing in x and is monotone g -non-increasing in y , that is, for any x, y ∈ X , g ( x ) ≤ g ( x ) ⇒ F ( x , y ) ≤ F ( x , y ) , for x , x ∈ X, and (1.4) g ( y ) ≤ g ( y ) ⇒ F ( x, y ) ≤ F ( x, y ) , for y , y ∈ X. (1.5)It is clear that Definition 13 reduces to Definition 9 when g is the identity. Definition 6.
An element ( x, y ) ∈ X × X is called a couple point of a mapping F : X × X → X and g : X → X if F ( x, y ) = g ( x ) , F ( y, x ) = g ( y ) . Definition 7.
Let F : X × X → X and g : X → X where X = ∅ . The mappings F and g are said to commute if g ( F ( x, y )) = F ( g ( x ) , g ( y )) , for all x, y ∈ X. UARTET FIXED POINT THEOREMS FOR NONLINEAR CONTRACTIONS IN PARTIALLY ORDERED SETS3
Theorem 8.
Let ( X, ≤ ) be partially ordered set and ( X, d ) be a complete metricspace and also F : X × X → X and g : X → X where X = ∅ . Suppose that F hasthe mixed g -monotone property and that there exists a k ∈ [0 , with d ( F ( x, y ) , F ( u, v )) ≤ k (cid:20) d ( g ( x ) , g ( u )) + d ( g ( y ) , g ( v ))2 (cid:21) (1.6) for all x, y, u, v ∈ X for which g ( x ) ≤ g ( u ) and g ( v ) ≤ g ( y ) . Suppose F ( X × X ) ⊂ g ( X ) , g is sequentially continuous and commutes with F and also suppose either F is continuous or X has the following property:if a non-decreasing sequence { x n } → x, then x n ≤ x, for all n, (1.7) if a non-increasing sequence { y n } → y, then y ≤ y n , for all n. (1.8) If there exist x , y ∈ X such that g ( x ) ≤ F ( x , y ) and g ( y ) ≤ F ( y , x ) , thenthere exist x, y ∈ X such that g ( x ) = F ( x, y ) and g ( y ) = F ( y, x ) , that is, F and g have a couple coincidence. Berinde and Borcut [1] introduced the following partial order on the productspace X = X × X × X :( u, v, w ) ≤ ( x, y, z ) if and only if x ≥ u, y ≤ v, z ≥ w, (1.9)where ( u, v, w ) , ( x, y, z ) ∈ X . Regarding this partial order, we state the definitionof the following mapping. Definition 9. (See [1] ) Let ( X, ≤ ) be partially ordered set and F : X → X .We say that F has the mixed monotone property if F ( x, y, z ) is monotone non-decreasing in x and z , and it is monotone non-increasing in y , that is, for any x, y, z ∈ X x , x ∈ X, x ≤ x ⇒ F ( x , y, z ) ≤ F ( x , y, z ) ,y , y ∈ X, y ≤ y ⇒ F ( x, y , z ) ≥ F ( x, y , z ) ,z , z ∈ X, z ≤ z ⇒ F ( x, y, z ) ≤ F ( x, y, z ) . (1.10) Theorem 10. (See [1] ) Let ( X, ≤ ) be partially ordered set and ( X, d ) be a completemetric space. Let F : X × X × X → X be a mapping having the mixed monotoneproperty on X . Assume that there exist constants a, b, c ∈ [0 , such that a + b + c < for which d ( F ( x, y, z ) , F ( u, v, w )) ≤ ad ( x, u ) + bd ( y, v ) + cd ( z, w ) (1.11) for all x ≥ u, y ≤ v, z ≥ w . Assume that X has the following properties: ( i ) if non-decreasing sequence x n → x , then x n ≤ x for all n, ( ii ) if non-increasing sequence y n → y , then y n ≥ y for all n ,If there exist x , y , z ∈ X such that x ≤ F ( x , y , z ) , y ≥ F ( y , x , y ) , z ≤ F ( x , y , z ) then there 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 paper is introduce the concept of quartet fixed point and provethe related fixed point theorems.
E. KARAPINAR Quartet Fixed Point Theorems
Let ( X, ≤ ) be partially ordered set and ( X, d ) be a complete metric space. Westate the definition of the following mapping. Throughout the manuscript we denote X × X × X × X by X . Definition 11. (See [7] ) Let ( X, ≤ ) be partially ordered set and F : X → X .We say that F has the mixed monotone property if F ( x, y, z, w ) is monotone non-decreasing in x and z , and it is monotone non-increasing in y and w , that is, forany x, y, z, w ∈ Xx , x ∈ X, x ≤ x ⇒ F ( x , y, z, w ) ≤ F ( x , y, z, w ) ,y , y ∈ X, y ≤ y ⇒ F ( x, y , z, w ) ≥ F ( x, y , z, w ) ,z , z ∈ X, z ≤ z ⇒ F ( x, y, z , w ) ≤ F ( x, y, z , w ) ,w , w ∈ X, w ≤ w ⇒ F ( x, y, z, w ) ≥ F ( x, y, z, w ) . (2.1) Definition 12. (See [7] ) An element ( x, y, z, w ) ∈ X is called a quartet fixed pointof 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.
Let ( X, ≤ ) be partially ordered set and F : X → X . We say that F has the mixed g -monotone property if F ( x, y, z, w ) is monotone g -non-decreasingin x and z , and it is monotone g -non-increasing in y and w , that is, for any x, y, z, w ∈ Xx , x ∈ X, g ( x ) ≤ g ( x ) ⇒ F ( x , y, z, w ) ≤ F ( x , y, z, w ) ,y , y ∈ X, g ( y ) ≤ g ( y ) ⇒ F ( x, y , z, w ) ≥ F ( x, y , z, w ) ,z , z ∈ X, g ( z ) ≤ g ( z ) ⇒ F ( x, y, z , w ) ≤ F ( x, y, z , w ) ,w , w ∈ X, g ( w ) ≤ g ( w ) ⇒ F ( x, y, z, w ) ≥ F ( x, y, z, w ) . (2.3) Definition 14.
An element ( x, y, z, w ) ∈ X is called a quartet coincidence pointof F : X → 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 identity mapping, then Definition 13 and Definition14 reduceto Definition 11 and Definition12, respectively. Definition 15.
Let F : X → X and g : X → X . F and g are called commutativeif g ( F ( x, y, z, w )) = F ( g ( x ) , g ( y ) , g ( z ) , g ( w )) , for all x, y, z, w ∈ X. (2.5)For a metric space ( X, d ), the function ρ : X × X → [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 , that is, ( X , ρ ) is a metric induced by ( X, d ).Let Φ denote the all functions φ : [0 , ∞ ) → [0 , ∞ ) which is continuous and satisfythat( i ) φ ( t ) < t UARTET FIXED POINT THEOREMS FOR NONLINEAR CONTRACTIONS IN PARTIALLY ORDERED SETS5 ( i ) lim r → t + φ ( r ) < t for each r > Theorem 16.
Let ( X, ≤ ) be partially ordered set and ( X, d ) be a complete metricspace. Suppose F : X → X and there exists φ ∈ Φ such that F has the mixed g -monotone property and d ( F ( x, y, z, w ) , F ( u, v, r, t )) ≤ φ (cid:18) d ( x, u ) + d ( y, v ) + d ( z, r ) + d ( w, t )4 (cid:19) (2.6) for all 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 ) . Suppose there exist x , y , z , w ∈ X such that g ( x ) ≤ F ( x , y , z , w ) , g ( y ) ≥ F ( x , w , z , y ) ,g ( z ) ≤ F ( z , y , x , w ) , g ( w ) ≥ F ( z , w , x , y ) . (2.7) Assume also that F ( X ) ⊂ g ( X ) and g commutes with F . Suppose either ( a ) F is continuous, or ( b ) X has the following property: ( i ) if non-decreasing sequence x n → x , then x n ≤ x for all n, ( ii ) if non-increasing sequence y n → y , then y n ≥ y for all n ,then there 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 common coincidence point.Proof. Let x , y , z , w ∈ X be such that (2.7). We construct the sequences { x n } , { y n } , { z n } and { w n } as follows g ( x n ) = F ( x n − , y n − , z n − , w n − ) ,g ( y n ) = F ( x n − , w n − , z n − , y n − ) ,g ( z n ) = F ( z n − , y n − , x n − , w n − ) ,g ( w n ) = F ( z n − , w n − , x n − , y n − ) . (2.8)for n = 1 , , , .... .We claim that g ( x n − ) ≤ g ( x n ) , g ( y n − ) ≥ g ( y n ) ,g ( z n − ) ≤ g ( z n ) , g ( w n − ) ≥ g ( w n ) , for all n ≥ . (2.9)Indeed, we shall use mathematical induction to prove (2.9). Due to (2.7), we have g ( x ) ≤ F ( x , y , z , w ) = g ( x ) , g ( y ) ≥ F ( x , w , z , y ) = g ( y ) ,g ( z ) ≤ F ( z , y , x , w ) = g ( z ) , g ( w ) ≥ F ( z , w , x , y ) = g ( w ) . Thus, the inequalities in (2.9) hold for n = 1. Suppose now that the inequalitiesin (2.9) hold for some n ≥
1. By mixed g -monotone property of F , together with(2.8) and (2.3) we have g ( x n ) = F ( x n − , y n − , z n − , w n − ) ≤ F ( x n , y n , z n , w n ) = g ( x n +1 ) ,g ( y n ) = F ( x n − , w n − , z n − , y n − ) ≥ F ( x n , w n , z n , y n ) = g ( y n +1 ) ,g ( z n ) = F ( z n − , y n − , x n − , w n − ) ≤ F ( z n , y n , x n , w n ) = g ( z n +1 ) ,g ( w n ) = F ( z n − , w n − , x n − , y n − ) ≥ F ( z n − , w n − , x n − , y n − ) = g ( w n +1 ) , (2.10) E. KARAPINAR
Thus, (2.9) holds for all n ≥
1. Hence, we have · · · g ( x n ) ≥ g ( x n − ) ≥ · · · ≥ g ( x ) ≥ g ( x ) , · · · g ( y n ) ≤ g ( y n − ) ≤ · · · ≤ g ( y ) ≤ g ( y ) , · · · g ( z n ) ≥ g ( z n − ) ≥ · · · ≥ g ( z ) ≥ g ( z ) , · · · g ( w n ) ≤ g ( w n − ) ≤ · · · ≤ g ( w ) ≤ g ( w ) , (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 ))We shall show that δ n +1 ≤ φ ( δ n . (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 (cid:16) 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 (cid:17) ≤ φ ( δ n ) (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 (cid:16) 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 (cid:17) ≤ φ ( δ n ) (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 (cid:16) 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 (cid:17) ≤ φ ( δ n ) (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 (cid:16) 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 (cid:17) ≤ φ ( δ n ) (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 ) ≤ φ ( δ n φ ( t ) < t for all t >
0, then δ n +1 ≤ δ n for all n . Hence { δ n } is a non-increasing sequence. Since it is bounded below, there is some δ ≥ n →∞ δ n = δ + . (2.18)We shall show that δ = 0. Suppose, to the contrary, that δ >
0. Taking the limit as δ n → δ + of both sides of (2.12) and having in mind that we suppose lim t → r φ ( r ) < t for all t >
0, we have δ = lim n →∞ δ n +1 ≤ lim n →∞ φ ( δ n δ n → δ + φ ( δ n < δ < δ (2.19) UARTET FIXED POINT THEOREMS FOR NONLINEAR CONTRACTIONS IN PARTIALLY ORDERED SETS7 which is a contradiction. Thus, δ = 0, that is,lim n →∞ [ d ( x n , x n − ) + d ( y n , y n − ) + d ( z n , z n − ) + d ( w n , w n − )] = 0 . (2.20)Now, we shall prove that { g ( x n ) } , { g ( y n ) } , { g ( z n ) } and { g ( w n ) } are Cauchy se-quences. Suppose, to the contrary, that at least one of { g ( x n ) } , { g ( y n ) } , { g ( z n ) } and { g ( w n ) } is not Cauchy. So, there exists an ε > { 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, corresponding to m ( k ), we may choose n ( k ) such that it is the smallestinteger satisfying (2.21) and n ( k ) > m ( k ) ≥ k . Thus, 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. (2.22)By using triangle inequality 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 ) − )) + d ( g ( x n ( k ) − ) , g ( x m ( k ) ))+ d ( g ( y n ( k )) , g ( y n ( k ) − )) + d ( g ( y n ( k ) − ) , g ( y m ( k ) ))+ d ( g ( z n ( k ) ) , g ( z n ( k ) − )) + d ( g ( z n ( k ) − ) , g ( z m ( k ) ))+ d ( g ( w n ( k ) ) , g ( w n ( k ) − )) + d ( g ( w n ( k ) − ) , g ( w m ( k ) )) < d ( g ( x n ( k ) ) , g ( x n ( k ) − )) + d ( g ( y n ( k ) ) , g ( y n ( k ) − ))+ d ( g ( z n ( k ) ) , g ( z n ( k ) − )) + d ( g ( w n ( k ) ) , g ( w n ( k ) − )) + ε. (2.23)Letting k → ∞ in (2.23) and using (2.20)lim k →∞ t k = lim k →∞ (cid:20) 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 ) )) (cid:21) = ε + (2.24)Again by triangle inequality, 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 ), then g ( x n ( k ) ) ≥ g ( x m ( k ) ) and g ( y n ( k ) ) ≤ g ( y m ( k ) ) ,g ( z n ( k ) ) ≥ g ( z m ( k ) ) and g ( w n ( k ) ) ≤ g ( w m ( k ) ) . (2.26)Hence from (2.26), (2.8) and (2.6), we have, 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 (cid:18) [ 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 ) ))] (cid:19) (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 (cid:18) [ 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 ) ))] (cid:19) (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 (cid:18) [ 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 ) ))] (cid:19) (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 (cid:18) [ 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 ) ))] (cid:19) (2.30)Combining (2.25) with (2.27)-(2.30), we obtain that 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 (cid:0) t k (cid:1) < δ n ( k )+1 + δ m ( k )+1 + t k + 4 t k (2.31)Letting k → ∞ , we get a contradiction. This shows that { g ( x n ) } , { g ( y n ) } , { g ( z n ) } and { g ( w n ) } are Cauchy sequences. Since X is complete metric space, there exists x, y, z, w ∈ X such thatlim 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 continuous, (2.32) implies thatlim 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)From (2.10) and by regarding commutativity 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)We 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 ) . UARTET FIXED POINT THEOREMS FOR NONLINEAR CONTRACTIONS IN PARTIALLY ORDERED SETS9
Suppose now ( a ) holds. Then by (2.8),(2.34) and (2.32), we 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 also observe 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)Thus, we 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 ) . Suppose now the assumption ( b ) holds. Since { g ( x n ) } , { g ( z n ) } is non-decreasingand g ( x n ) → x, g ( z n ) → z and also { g ( y n ) } , { g ( w n ) } is non-increasing and g ( y n ) → y, g ( w n ) → , then by assumption ( b ) we have g ( x n ) ≥ x, g ( y n ) ≤ y, g ( z n ) ≥ z, g ( w n ) ≤ w (2.39)for all n . Thus, by triangle inequality 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 (cid:18) (cid:20) 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 ))) (cid:21)(cid:19) (2.40)Letting n → ∞ implies that d ( g ( x ) , F ( x, y, z, w )) ≤
0. Hence, g ( x ) = F ( x, y, z, w ).Analogously we 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 ) . Thus, we proved that F and g have a quartet coincidence point. (cid:3) References [1] V. Berinde and M. Borcut, Tripled fixed point theorems for contractive type mappings inpartially ordered metric spaces,
Nonlinear Analysis , (15), 4889–4897 (2011).[2] Bhaskar, T.G., Lakshmikantham, V.: Fixed Point Theory in partially ordered metric spacesand applications Nonlinear Analysis , , 1379–1393 (2006).[3] N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces andapplication, Nonlinear Analysis , , 983-992(2011).[4] B. Samet, Coupled fixed point theorems for a generalized MeirKeeler contraction in partiallyordered metric spaces, Nonlinear Analysis , (12), 45084517(2010).[5] E. Karapınar, Couple Fixed Point on Cone Metric Spaces, Gazi University Journal of Science , (1),51-58(2011).[6] E. Karapınar, Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. , (12), 3656-3668(2010).[7] E. Karapınar, N.V.Luong, Quartet Fixed Point Theorems for nonlinear contractions, submit-ted.[8] Lakshmikantham, V., ´Ciri´c, L.: : Couple Fixed Point Theorems for nonlinear contractions inpartially ordered metric spaces Nonlinear Analysis , , 4341-4349 (2009).[9] Nieto, J. J., Rodriguez-L´opez, R.: Contractive mapping theorems in partially ordered sets andapplications to ordinary differential equations. Order , 223–239 (2006).[10] Binayak S. Choudhury, N. Metiya and A. Kundu, Coupled coincidence point theorems inordered metric spaces, Ann. Univ. Ferrara , , 1-16(2011).[11] B.S. Choudhury, A. Kundu : A coupled coincidence point result in partially ordered metricspaces for compatible mappings. Nonlinear Anal.
TMA , 25242531 (2010) erdal karapınar,Department of Mathematics, Atilim University 06836, ˙Incek, Ankara, Turkey E-mail address : [email protected] E-mail address ::