Some Coupled Fixed Point Results on Partial Metric Spaces

SOME COUPLED FIXED POINT R ESUL TS ON P AR TIAL METRIC SP A CES HASSEN A YDI Abstract. In this paper we give some coupled fixed point r esults for mappings satisfying different con tr active conditions on complete partial metric spaces. 2000 Mathematics Sub ject Cl ass ification. 47H10, 5 4H25. Key W ords and Phrases : Coupled fixed p oint, partial metric spac e, contractive condition 1. INTR ODUCTION AND PRELIMINARIES F or a given partially ordered s et X , Bha sk ar and Lak shmik antham in [3] in tro- duced the concept of coupled fixed p oint of a mapping F : X × X → X . L ater in [4], Lakshmik a nth am and C ´ ır ´ ıc inv estigated some more coupled fixed p oint theo rems in partially order ed sets. The following is the corr esp onding definition o f a coupled fixed p oint. Definition 1.1. [3] An element ( x, y ) ∈ X × X is said to b e a coupled fixed p o in t of the mapping F : X × X → X if F ( x, y ) = x and F ( y , x ) = y . F. Sabetg hadam e t a l. [12] obtained the following Theorem 1.2. Le t ( X, d ) b e a c omplete c one m etric sp ac e. Supp ose t hat the mapping F : X × X → X satisfies the fol lo wing c ontr active c ondition for al l x , y , u, v ∈ X d ( F ( x, y ) , F ( u, v )) ≤ k d ( x, u ) + ld ( y , v ) wher e k , l ar e nonne gative c onstants with k + l < 1 . Then F has a unique c ouple d fixe d p oint. In this pap er, we give the analo gous of this result (and some others in [12]) on partial metric spa ces, a nd we establish some coupled fixed p oint r esults. The concept of partial metric s pa ce ( X , p ) w as introduced by Matthews in 199 4. In such spa ces, the distance o f a p oint in the self may not be zero. First, we start with some preliminaries definitions o n the partial metric spaces [1, 2, 5, 6, 7, 8, 9, 10, 1 1, 13] Definition 1.3. ([5, 6, 7]) A partial metric on a nonempt y set X is a function p : X × X − → R + such that for all x, y , z ∈ X : (p1) x = y ⇐ ⇒ p ( x, x ) = p ( x, y ) = p ( y , y ), (p2) p ( x, x ) ≤ p ( x, y ), (p3) p ( x, y ) = p ( y , x ) , (p4) p ( x, y ) ≤ p ( x, z ) + p ( z , y ) − p ( z , z ). 1 2 H. A YDI A pa r tial metric space is a pair ( X , p ) s uch that X is a nonempty set and p is a partial metric o n X . Remark 1.4 . It is clear that, if p ( x, y ) = 0, then from (p1), (p2) and (p3), x = y . But if x = y , p ( x, y ) may not b e 0. If p is a partial metric on X , then the function p s : X × X − → R + given by p s ( x, y ) = 2 p ( x, y ) − p ( x, x ) − p ( y , y ) , is a metric on X . Definition 1.5. ([5, 6, 7]) Let ( X, p ) be a partial metric space. Then: (i) a se q uence { x n } in a par tial metric spa ce ( X , p ) conv erges to a point x ∈ X if and o nly if p ( x, x ) = lim n − → + ∞ p ( x, x n ); (ii) A sequence { x n } in a partial metric space ( X , p ) is called a Cauch y sequence if there ex ists (and is finite) lim n,m − → + ∞ p ( x n , x m ). (iii) A partial metr ic s pa ce ( X , p ) is said to b e complete if every Cauch y s equence { x n } in X co nv erges to a p oint x ∈ X , that is p ( x, x ) = lim n,m − → + ∞ p ( x n , x m ). Lemma 1.6. ([5, 6, 8]) Let ( X , p ) be a partia l metric space. (a) { x n } is a Ca uch y sequence in ( X , p ) if a nd only if it is a Cauch y sequence in the metric spa ce ( X , p s ). (b) A partial metric space ( X , p ) is complete if and only if the metric spa ce ( X, p s ) is co mplete. F ur thermore, lim n − → + ∞ p s ( x n , x ) = 0 if and only if p ( x, x ) = lim n − → + ∞ p ( x n , x ) = lim n,m − → + ∞ p ( x n , x m ) . 2. MAIN R E SUL TS Our first main re s ult is the following Theorem 2.1. L et ( X , p ) b e a c omplete p artial metric sp ac e. Supp ose that the mapping F : X × X → X satisfies the fol lowing c ontr active c ondition for al l x , y , u, v ∈ X (2.1) p ( F ( x, y ) , F ( u, v )) ≤ k p ( x, u ) + l p ( y , v ) wher e k , l ar e nonne gative c onstants with k + l < 1 . Then F has a unique c ouple d fixe d p oint. Pro of. Cho ose x 0 , y 0 ∈ X and set x 1 = F ( x 0 , y 0 ) and y 1 = F ( y 0 , x 0 ). Repeating this pro c e ss, s e t x n +1 = F ( x n , y n ) a nd y n +1 = F ( y n , x n ). Then by (2.1), we have p ( x n , x n +1 ) = p ( F ( x n − 1 , y n − 1 ) , F ( x n , y n )) ≤ k p ( x n − 1 , x n ) + lp ( y n − 1 , y n ) , (2.2) and simila rly p ( y n , y n +1 ) = p ( F ( y n − 1 , x n − 1 ) , F ( y n , x n )) ≤ k p ( y n − 1 , y n ) + l p ( x n − 1 , x n ) . (2.3) Therefore, by letting (2.4) d n = p ( x n , x n +1 ) + p ( y n , y n +1 ) , COUPLED FIXED POINT RESUL TS 3 we have d n = p ( x n , x n +1 ) + p ( y n , y n +1 ) ≤ k p ( x n − 1 , x n ) + lp ( y n − 1 , y n ) + k p ( y n − 1 , y n ) + l p ( x n − 1 , x n ) =( k + l )[ p ( y n − 1 , y n ) + p ( x n − 1 , x n )] =( k + l ) d n − 1 . (2.5) Consequently , if we set δ = k + l then for each n ∈ N we hav e (2.6) d n ≤ δ d n − 1 ≤ δ 2 d n − 2 ≤ ... ≤ δ n d 0 . If d 0 = 0 then p ( x 0 , x 1 ) + p ( y 0 , y 1 ) = 0. Hence, fro m Remar k 1.4, we get x 0 = x 1 = F ( x 0 , y 0 ) and y 0 = y 1 = F ( y 0 , x 0 ), meaning that ( x 0 , y 0 ) is a coupled fixed p oint of F . Now, le t d 0 > 0. F o r e a ch n ≥ m we hav e in view of the co ndition ( p 4 ) p ( x n , x m ) ≤ p ( x n , x n − 1 ) + p ( x n − 1 , x n − 2 ) − p ( x n − 1 , x n − 1 ) + p ( x n − 2 , x n − 3 ) + p ( x n − 3 , x n − 4 ) − p ( x n − 3 , x n − 3 )+ + ... + p ( x m +2 , x m +1 ) + p ( x m +1 , x m ) − p ( x m +1 , x m +1 ) ≤ p ( x n , x n − 1 ) + p ( x n − 1 , x n − 2 ) + ... + p ( x m +1 , x m ) . Similarly , we hav e p ( y n , y m ) ≤ p ( y n , y n − 1 ) + p ( y n − 1 , y n − 2 ) + ... + p ( y m +1 , y m ) . Thu s, p ( x n , x m ) + p ( y n , y m ) ≤ d n − 1 + d n − 2 + ... + d m ≤ ( δ n − 1 + δ n − 2 + ... + δ m ) d 0 ≤ δ m 1 − δ d 0 . (2.7) By definition of p s , we have p s ( x, y ) ≤ 2 p ( x, y ), s o for any n ≥ m (2.8) p s ( x n , x m ) + p s ( y n , y m ) ≤ 2 p ( x n , x m ) + 2 p ( y n , y m ) ≤ 2 δ m 1 − δ d 0 . which implies that { x n } and { y n } ar e Cauch y sequenc e s in ( X , p s ) b ecause o f 0 ≤ δ = k + l < 1. Since the partial metric s pace ( X , p ) is complete, hence thanks to Lemma 1.6, the metric space ( X, p s ) is complete, so there exist u ∗ , v ∗ ∈ X such that (2.9) lim n → + ∞ p s ( x n , u ∗ ) = lim n → + ∞ p s ( y n , v ∗ ) = 0 . Again, fr o m Lemma 1.6, we get p ( u ∗ , u ∗ ) = lim n → + ∞ p ( x n , u ∗ ) = lim n → + ∞ p ( x n , x n ) , and p ( v ∗ , v ∗ ) = lim n → + ∞ p ( y n , v ∗ ) = lim n → + ∞ p ( y n , y n ) . But, from co nditio n ( p 2) and (2.6), p ( x n , x n ) ≤ p ( x n , x n +1 ) ≤ d n ≤ δ n d 0 , so since δ ∈ [0 , 1 [, hence letting n → + ∞ , we get lim n → + ∞ p ( x n , x n ) = 0. It follows that (2.10) p ( u ∗ , u ∗ ) = lim n → + ∞ p ( x n , u ∗ ) = lim n → + ∞ p ( x n , x n ) = 0 . 4 H. A YDI Similarly , we get (2.11) p ( v ∗ , v ∗ ) = lim n → + ∞ p ( y n , v ∗ ) = lim n → + ∞ p ( y n , y n ) = 0 . Therefore, we have us ing (2.1) p ( F ( u ∗ , v ∗ ) , u ∗ ) ≤ p ( F ( u ∗ , v ∗ ) , x n +1 ) + p ( x n +1 , u ∗ ) − p ( x n +1 , x n +1 ) , By (p4 ) ≤ p ( F ( u ∗ , v ∗ ) , F ( x n , y n )) + p ( x n +1 , u ∗ ) ≤ k p ( x n , u ∗ ) + l p ( y n , v ∗ ) + p ( x n +1 , u ∗ ) , and letting n → + ∞ , then from (2.1 0) and (2.11), we obtain p ( F ( u ∗ , v ∗ ) , u ∗ )) = 0, so F ( u ∗ , v ∗ ) = u ∗ . Similarly , we hav e F ( v ∗ , u ∗ ) = v ∗ , meaning that ( u ∗ , v ∗ ) is a coupled fixe d p oint of F . Now, if ( u ′ , v ′ ) is a nother coupled fixed p oint of F , then p ( u ′ , u ∗ ) = p ( F ( u ′ , v ′ ) , F ( u ∗ , v ∗ )) ≤ k p ( u ′ , u ∗ ) + l p ( v ′ , v ∗ ) p ( v ′ , v ∗ ) = p ( F ( v ′ , u ′ ) , F ( v ∗ , u ∗ )) ≤ k p ( v ′ , v ∗ ) + l p ( u ′ , u ∗ ) . It follows that p ( u ′ , u ∗ ) + p ( v ′ , v ∗ ) ≤ ( k + l )[ p ( u ′ , u ∗ ) + p ( v ′ , v ∗ )] . In view of k + l < 1, this implies that p ( u ′ , u ∗ ) + p ( v ′ , v ∗ ) = 0, so u ∗ = u ′ and v ∗ = v ′ . The pro of of Theo rem 2 .1 is completed. It is worth noting tha t when the constants in Theorem 2.1 ar e equal we ha ve the following Co rollar y Corollary 2.2. L et ( X , p ) b e a c omplete p artial metr ic s p ac e. Supp ose t hat the mapping F : X × X → X satisfies the fol lowing c ontr active c ondition for al l x , y , u, v ∈ X (2.12) p ( F ( x, y ) , F ( u, v )) ≤ k 2 ( p ( x, u ) + p ( y , v )) wher e 0 ≤ k < 1 . Then, F has a unique c ouple d fixe d p oint. Example 2.3 . Let X = [0 , + ∞ [ endowed with the usual pa rtial metric p defined by p : X × X → [0 , + ∞ [ with p ( x, y ) = max { x, y } . The partial metric s pace ( X , p ) is co mplete beca use ( X , p s ) is complete. Indeed, for a ny x, y ∈ X , p s ( x, y ) = 2 p ( x, y ) − p ( x, x ) − p ( y , y ) =2 max { x, y } − ( x + y ) = | x − y | , Thu s, ( X , p s ) is the Euclidean metric space which is complete. Consider the map- ping F : X × X → X defined by F ( x, y ) = x + y 6 . F or a ny x, y , u, v ∈ X , we have p ( F ( x, y ) , F ( u, v )) = 1 6 max { x + y , u + v } ≤ 1 6 [max { x, u } +max { y , v } ] = 1 6 [ p ( x, u )+ p ( y , v )] , which is the contractiv e condition (2.12) for k = 1 3 . Therefore, by Coro llary 2.2 , F has a unique coupled fixe d p oint, which is (0 , 0). Note that if the mapping F : X × X → X is given by F ( x, y ) = x + y 2 , then F satisfies the co ntractiv e condition (2 .12) fo r k = 1, that is, p ( F ( x, y ) , F ( u, v )) = 1 2 max { x + y , u + v } ≤ 1 2 [max { x, u } +max { y , v } ] = 1 2 [ p ( x, u )+ p ( y , v )] , COUPLED FIXED POINT RESUL TS 5 In this ca s e, (0 , 0) a nd (1 , 1) are b oth coupled fixed p oints o f F and hence the coupled fixed p oint of F is not unique. This shows that the condition k < 1 in Corollary 2.2, and hence k + l < 1 in Theorem 2.1 can not b e o mitted in the statement of the aforesa id results. Theorem 2.4. L et ( X , p ) b e a c omplete p artial metric sp ac e. Supp ose that the mapping F : X × X → X satisfies the fol lowing c ontr active c ondition for al l x , y , u, v ∈ X (2.13) p ( F ( x, y ) , F ( u, v )) ≤ k p ( F ( x, y ) , x ) + l p ( F ( u, v ) , u ) wher e k , l ar e nonne gative c onstants with k + l < 1 . Then F has a unique c ouple d fixe d p oint. Pro of. W e ta ke the s a me seq uences { x n } and { y n } g iven in the pro o f of Theor em 2.1 by x n +1 = F ( x n , y n ) , y n +1 = F ( y n , x n ) for any n ∈ N . Applying (2.1 3), we get (2.14) p ( x n , x n +1 ) ≤ δ p ( x n − 1 , x n ) (2.15) p ( y n , y n +1 ) ≤ δ p ( y n − 1 , y n ) , where δ = k 1 − l . By definition of p s , we have (2.16) p s ( x n , x n +1 ) ≤ 2 p ( x n , x n +1 ) ≤ 2 δ n p ( x 1 , x 0 ) (2.17) p s ( y n , y n +1 ) ≤ 2 p ( y n , y n +1 ) ≤ 2 δ n p ( y 1 , y 0 ) . Since k + l < 1, hence δ < 1, so the sequences { x n } and { y n } are Ca uch y sequences in the metric space ( X , p s ). The partial metric spa ce ( X , p ) is complete, hence from Lemma 1 .6, ( X , p s ) is complete, so there exist u ∗ , v ∗ ∈ X such tha t (2.18) lim n → + ∞ p s ( x n , u ∗ ) = lim n → + ∞ p s ( y n , v ∗ ) = 0 . F ro m Lemma 1.6, we g et p ( u ∗ , u ∗ ) = lim n → + ∞ p ( x n , u ∗ ) = lim n → + ∞ p ( x n , x n ) , and p ( v ∗ , v ∗ ) = lim n → + ∞ p ( y n , v ∗ ) = lim n → + ∞ p ( y n , y n ) . By the condition (p2) and (2.14), we hav e p ( x n , x n ) ≤ p ( x n , x n +1 ) ≤ δ n p ( x 1 , x 0 ) , so lim n → + ∞ p ( x n , x n ) = 0. It follows that (2.19) p ( u ∗ , u ∗ ) = lim n → + ∞ p ( x n , u ∗ ) = lim n → + ∞ p ( x n , x n ) = 0 . Similarly , we find (2.20) p ( v ∗ , v ∗ ) = lim n → + ∞ p ( y n , v ∗ ) = lim n → + ∞ p ( y n , y n ) = 0 . 6 H. A YDI Therefore, by (2.1 3) p ( F ( u ∗ , v ∗ ) , u ∗ ) ≤ p ( F ( u ∗ , v ∗ ) , x n +1 ) + p ( x n +1 , u ∗ ) = p ( F ( u ∗ , v ∗ ) , F ( x n , y n )) + p ( x n +1 , u ∗ ) ≤ k p ( F ( u ∗ , v ∗ ) , u ∗ ) + lp ( F ( x n , y n ) , x n ) + p ( x n +1 , u ∗ ) = k p ( F ( u ∗ , v ∗ ) , u ∗ ) + lp ( x n +1 , x n ) + p ( x n +1 , u ∗ ) and letting n → + ∞ , then from (2.16)-(2.19), we obtain p ( F ( u ∗ , v ∗ ) , u ∗ ) ≤ k p ( F ( u ∗ , v ∗ ) , u ∗ ) . F ro m the pre c eding inequality we can deduce a co n tradiction if we a ssume that p ( F ( u ∗ , v ∗ ) , u ∗ ) 6 = 0, b ecause in that case we conclude that 1 ≤ k and now this inequality is , in fact, a contradiction, so p ( F ( u ∗ , v ∗ ) , u ∗ ) = 0, that is, F ( u ∗ , v ∗ ) = u ∗ . Similarly , we have F ( v ∗ , u ∗ ) = v ∗ , meaning that ( u ∗ , v ∗ ) is a coupled fixed p oint of F . Now, if ( u ′ , v ′ ) is a nother coupled fixed p oint of F , then in view of (2 .1 3) p ( u ′ , u ∗ ) = p ( F ( u ′ , v ′ ) , F ( u ∗ , v ∗ )) ≤ k p ( F ( u ′ , v ′ ) , u ′ ) + l p ( F ( u ∗ , v ∗ ) , u ∗ ) = k p ( u ′ , u ′ ) + l p ( u ∗ , u ∗ ) ≤ k p ( u ′ , u ∗ ) + l p ( u ′ , u ∗ ) = ( k + l ) p ( u ′ , u ∗ ) , using (p2 ) that is p ( u ′ , u ∗ ) = 0 since ( k + l ) < 1. It follows that u ∗ = u ′ . Similar ly , we ca n hav e v ∗ = v ′ , and the pro o f of Theo rem 2 .4 is completed. Theorem 2.5. L et ( X , p ) b e a c omplete p artial metric sp ac e. Supp ose that the mapping F : X × X → X satisfies the fol lowing c ontr active c ondition for al l x , y , u, v ∈ X (2.21) p ( F ( x, y ) , F ( u, v )) ≤ k p ( F ( x, y ) , u ) + l p ( F ( u, v ) , x ) wher e k , l ar e nonne gative c onstant s with k + 2 l < 1 . Then F has a unique c ouple d fixe d p oint. Pro of. Since, k + 2 l < 1, he nce k + l < 1, and as a conseque nc e the pro of of the uniqueness in this Theore m is as trivial a s in the other results. T o prov e the existence of the fixed p oint, choose the sequences { x n } and { y n } lik e in the pr o of of Theor em 2 .1, that is x n +1 = F ( x n , y n ) , y n +1 = F ( y n , x n ) for any n ∈ N . Applying ag ain (2 .21), we have p ( x n , x n +1 ) = p ( F ( x n − 1 , y n − 1 ) , F ( x n , y n )) ≤ k p ( F ( x n − 1 , y n − 1 ) , x n ) + l p ( F ( x n , y n ) , x n − 1 ) = k p ( x n , x n ) + l p ( x n +1 , x n − 1 ) ≤ k p ( x n +1 , x n ) + l p ( x n +1 , x n − 1 )] , by (p2 ) ≤ k p ( x n +1 , x n ) + l p ( x n +1 , x n ) + l p ( x n , x n − 1 ) − lp ( x n , x n ) , using (p4 ) ≤ ( k + l ) p ( x n , x n +1 ) + l p ( x n − 1 , x n ) . It follows that for any n ∈ N ∗ p ( x n , x n +1 ) ≤ l 1 − l − k p ( x n − 1 , x n ) . COUPLED FIXED POINT RESUL TS 7 Let us take δ = l 1 − l − k . Hence, we deduce (2.22) p s ( x n , x n +1 ) ≤ 2 p ( x n , x n +1 ) ≤ 2 δ n p ( x 0 , x 1 ) . Under the condition 0 ≤ k + 2 l < 1, we g e t 0 ≤ δ < 1. F r om this fact w e immediately obtain that { x n } is Cauch y in the complete metric space ( X , p s ). O f course , similar arguments apply to the case of the s e quence { y n } in or der to prove that (2.23) p s ( y n , y n +1 ) ≤ 2 p ( y n , y n +1 ) ≤ 2 δ n p ( y 0 , y 1 ) , and, thus, that the sequence { y n } is Cauchy in ( X , p s ). Therefore, there exist u ∗ , v ∗ ∈ X such that (2.24) lim n → + ∞ p s ( x n , u ∗ ) = lim n → + ∞ p s ( y n , v ∗ ) = 0 . Thanks to Lemma 1.6, we have lim n → + ∞ p ( x n , u ∗ ) = lim n → + ∞ p ( x n , x n ) = p ( u ∗ , u ∗ ) , and lim n → + ∞ p ( y n , v ∗ ) = lim n → + ∞ p ( y n , y n ) = p ( v ∗ , v ∗ ) . The conditio n ( p 2 ) together with (2.22) yield that p ( x n , x n ) ≤ p ( x n , x n +1 ) ≤ δ n p ( x 0 , x 1 ) , hence letting n → + ∞ , we get lim n → + ∞ p ( x n , x n ) = 0. It follows that (2.25) p ( u ∗ , u ∗ ) = lim n → + ∞ p ( x n , u ∗ ) = lim n → + ∞ p ( x n , x n ) = 0 . Similarly , we hav e (2.26) p ( v ∗ , v ∗ ) = lim n → + ∞ p ( y n , v ∗ ) = lim n → + ∞ p ( y n , y n ) = 0 . Therefore, we have us ing (2.21) p ( F ( u ∗ , v ∗ ) , u ∗ ) ≤ p ( F ( u ∗ , v ∗ ) , x n +1 ) + p ( x n +1 , u ∗ ) = p ( F ( u ∗ , v ∗ ) , F ( x n , y n )) + p ( x n +1 , u ∗ ) ≤ k p ( F ( u ∗ , v ∗ ) , x n ) + lp ( F ( x n , y n ) , u ∗ ) + p ( x n +1 , u ∗ ) = k p ( F ( u ∗ , v ∗ ) , x n ) + lp ( x n +1 , u ∗ ) + p ( x n +1 , u ∗ ) ≤ k p ( F ( u ∗ , v ∗ ) , u ∗ ) + k p ( u ∗ , x n ) + l p ( x n +1 , u ∗ ) + p ( x n +1 , u ∗ ) , using p(4 ) . Letting n → + ∞ yields, using (2.25) p ( F ( u ∗ , v ∗ ) , u ∗ ) ≤ k p ( F ( u ∗ , v ∗ ) , u ∗ ) , and since k < 1, we hav e p ( F ( u ∗ , v ∗ ) , u ∗ ) = 0, tha t is F ( u ∗ , v ∗ ) = u ∗ . Sim ilarly , thanks to (2.26), w e get F ( v ∗ , u ∗ ) = v ∗ , and hence ( u ∗ , v ∗ ) is a coupled fixed po int of F . When the consta nts in Theo rems 2 .4 and 2.5 ar e equal, w e get the following corolla r ies Corollary 2.6. L et ( X , p ) b e a c omplete p artial metr ic s p ac e. Supp ose t hat the mapping F : X × X → X satisfies the fol lowing c ontr active c ondition for al l x , y , u, v ∈ X (2.27) p ( F ( x, y ) , F ( u, v )) ≤ k 2 ( p ( F ( x, y ) , x ) + p ( F ( u, v ) , u )) 8 H. A YDI wher e 0 ≤ k < 1 . Then, F has a unique c ouple d fixe d p oint. Corollary 2.7. L et ( X , p ) b e a c omplete p artial metr ic s p ac e. Supp ose t hat the mapping F : X × X → X satisfies the fol lowing c ontr active c ondition for al l x , y , u, v ∈ X (2.28) p ( F ( x, y ) , F ( u, v )) ≤ k 2 ( p ( F ( x, y ) , u ) + p ( F ( u, v ) , x )) wher e 0 ≤ k < 2 3 . Then, F has a un ique c ou ple d fi xe d p oint. Pro of. The co ndition 0 ≤ k < 2 3 follows from the hypothesis on k and l given in Theorem 2.5. Remark 2.8. • Theorem 2.1 extends the Theor em 2 .2 of [12] on the class o f partial metric spaces. • Theorem 2.4 extends the Theor em 2.5 of [12] on the class of partial metric spaces. Remark 2.9 . Note that in Theorem 2.4, if the mapping F : X × X → X satisfies the c ontractiv e condition (2.13) for all x , y , u , v ∈ X , then F also satisfies the following co ntractiv e condition p ( F ( x, y ) , F ( u, v )) = p ( F ( u, v ) , F ( x, y )) ≤ k p ( F ( u, v ) , u ) + lp ( F ( x, y ) , x ) (2.29) Consequently , by adding (2.13) and (2.29), F also s atisfies the following: (2.30) p ( F ( x, y ) , F ( u, v )) ≤ k + l 2 p ( F ( u, v ) , u ) + k + l 2 p ( F ( x, y ) , x ) which is a contractiv e co ndition of the t yp e (2.27) in Co rollar y 2 .6 with equal constants. Therefore, o ne ca n also reduce the pr o of o f genera l case (2.13) in Theor em 2.4 to the sp ecial ca se of equal constants. A similar argument is v alid for the contractiv e co nditions (2.21) in Theor em 2.5 and (2.28) in C o rollar y 2.7. Ac kno wl edgment. The author thanks the editor and the referee s fo r their k ind comments and sugg estions to improve this pap er. References [1] I. Altun, F. Sola, H. Simsek, Gener alize d co ntr actions on p artial metric sp ac e s , T op ology and its Applications, 157 (18) (2010) 2778-2785. [2] H. Aydi, Some fixe d p oint r esults in or der e d p artial metric sp ac es , Accepted in J. Nonlinear Sci. Appl, (2011). [3] T. Gnana Bhask ar, V. Lakshmik antham, Fixe d p oint the or ems in p artial ly or der e d metric sp ac es and applic ations , Nonlinear Analysi s . 65 (2006) 1379-1393. [4] L. B. C ´ ır ´ ıc, V . Lakshmik an tham, Couple d fixed p oint the or ems for nonline ar c ontr acti ons in p artial ly or der e d metric sp ac es , Nonlinear Analysis: Theory , Methods Applications, vol. 70, no. 12, pp. 4341-4349, 2009. [5] S.G. 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V alero, A quantitative c omputational mo del for c ompl ete p artialmetric sp ac es via formal b al ls , Mathe matical Structures in Computer Science, 19 (3) (2009) 541-563. [11] M. P . Sc hellek ens, The c orr esp ondenc e b etwe en p artial metrics and semivaluations , Theoret. Comput. Sci. 315 (2004) 135-149. [12] F. Sabetghadam, H. P . Masiha, and A. H. Sanatpour, Some Coupled Fixed Poin t Theorems in Cone Metric Spaces, Fi xed P oint Theory Appl. V olume 2009, Article ID 125426, 8 pages doi:10.1155/2009 /125426. [13] O. V alero, On Banach fixe d p oint the or ems for p artial metric sp ac es , Appl. Gen. T opol, 6 (2) (2005) 229-240. Hassen Aydi: Univ ersit´ e de Mo nastir. Institut Sup´ erieur d’Informatique de Mahdia . Route de R´ ejiche, Km 4, BP 35, Mahdia 5 121, T unisie. Email-addr e s s: hassen.aydi@isima.rnu.tn