Coincidence and Common Fixed Point Results for Contraction Type Maps in Partially Ordered Metric Spaces

COINCIDENCE AND COMMO N FIX ED POINT R E SUL TS F OR CONTRA CTION TYPE MAPS IN P AR TIALL Y ORDER ED METRIC SP A CES HASSEN A YDI Abstract. W e presen t coincidenc e and c ommon fixed point results of self- mappings satisfying a con trac tion type i n partially ordered metric spaces. As an application, we give a n existence theorem for a common solution of integr al equations. Key W or ds and Phras es: partially ordered, coincidence point, common fixed po int , weakly incr easing ma ppings, co mpatible pair o f ma pping s , integral equa tions. 1. Introduction Existence of fixed or common fixed results in partially o rdered sets has fasci- nated many resear chers in the few recent years wher e some applications to matrix equation, o rdinary differential equations and integral equations a re presented, we cite for example [1, 3, 4, 9, 13, 14]. Like in [9 ], we denote S the class of the functions β : [0 , ∞ [ → [0 , 1[ which satisfies the condition (1.1) β ( t n ) → 1 implies t n → 0 . A. Amini-Harandi and H. Emami [9] proved recently in the con test o f partially ordered sets the following r esult Theorem 1.1. L et ( X,  ) b e a p artia l ly or der e d set and supp ose that ther e ex ists a metric d in X such that ( X, d ) is a c omplete metric sp ac e. L et T : X → X b e a non-de cr e asing mapping s u ch t hat (1.2) d ( T x, T y ) ≤ β ( d ( x, y )) d ( x, y ) , for x, y ∈ X with x  y , wher e β ∈ S . Assume that either T is c ontinuous or x satisfies the fol lowing c ondition : If { x n } is a nonde cr e asing se quenc e in X such that x n → x , t hen x n  x ∀ n ∈ N . Besides, supp ose that for e ach x, y ∈ X , ther e exists z ∈ X which is c omp ar able to x and y . If ther e exists x 0 ∈ X with x 0  T x 0 , then T has a unique fixe d p oint. This previous theorem is an extension o f Geraght y’s result [7] wher e the contest of metric complete s paces is consider e d. This paper is again an ex tensio n of theorem 1.1. F or pa rtially ordered metric spac e s, we take three self-mappings satisfying the following co nt raction d ( f x, g y ) ≤ β ( d ( H x, H y )) d ( H x, H y ) . W e pre s ent coincidence and co mmon fixed point results of the self-mapping s f , g and H . These results will be giv en in section 2. As an applicatio n of the ab ov e results, in section 3, we give in a particular case an existence theor em for a commo n solution of integral equations. 2000 Mathematics Subje c t Classific ation. Primary 47J10; Secondary 34J15. 1 2 H. A YDI 2. Main resul ts W e start some definitions, which we need in the sequel Definition 2.1. L et X b e a non-empty set, N is a natu r a l numb er such that N ≥ 2 and T 1 , T 2 , · · · , T N : X → X ar e given self-mapp ings on X . If w = T 1 x = T 2 x = · · · = T N x for some x ∈ X , then x is c al le d a c oincide nc e p oi nt of T 1 , T 2 , · · · , T N − 1 and T N , and w is c al le d a p oint of c oincidenc e of T 1 , T 2 , · · · , T N − 1 and T N . If w = x , t hen x is c al le d a c ommon fixe d p oint of T 1 , T 2 , · · · , T N − 1 and T N . Definition 2.2. [8] L et ( X , d ) b e a metric sp ac e and f , g : X → X ar e given self- mappings on X . The p ai r { f , g } is said to b e c omp atible if lim n → + ∞ d ( f g x n , g f x n ) = 0 , whenever { x n } is a se quenc e in X such that lim n → + ∞ f x n = lim n → + ∞ g x n = t for some t in X . Let X b e a non-empty set and R : X → X be a given mapping. F or ev ery x ∈ X , we deno te by R − 1 ( x ) the subset o f X defined by: R − 1 ( x ) := { u ∈ X | Ru = x } . Definition 2.3. L et ( X ,  ) b e a p artial ly or d er e d set and T , S, R : X → X ar e given mappings such that T X ⊆ RX and S X ⊆ R X . W e say t hat S and T ar e we akly incr e asing with r esp e ct to R if and only if for al l x ∈ X , we have: T x  S y , ∀ y ∈ R − 1 ( T x ) and S x  T y , ∀ y ∈ R − 1 ( S x ) . Remark 2.4. If R : X → X is the identity mapp ing ( Rx = x for al l x ∈ X ), then S and T ar e we akly incr e asing with r esp e ct t o R implies that S and T ar e we akly incr e asing mappi ngs. Note that the notion of we ak ly incr e asing mappings was intr o duc e d in [2] (also se e [5 , 6] ). Our firs t ma in res ult is an extension of theorem 1.1, and it is the following Theorem 2.5. L et ( X ,  ) b e a p artial ly or der e d set. Supp ose that ther e exists a metric d on X such t hat ( X , d ) is c omplete. L et f , g , H : X → X b e given mappings satisfying (a) f X ⊆ H X , g X ⊆ H X , (b) f , g and H ar e c ontinuous, (c) t he p airs { f , H } and { g , H } ar e c omp atible, (g) f and g ar e we akly incr e asing with r esp e ct to H . Supp ose t hat for every ( x, y ) ∈ X × X such t hat H x and H y ar e c omp ar ab le, we have (2.1) d ( f x, g y ) ≤ β ( d ( H x, H y )) d ( H x, H y ) , wher e β ∈ S . Then, f , g and H have a c oincidenc e p oi nt u ∈ X , that is, f u = g u = H u . Pr o of. Let x 0 be an arbitra ry p oint in X . Since f X ⊆ H X , there exists x 1 ∈ X such that H x 1 = f x 0 . Since g X ⊆ H X , there exists x 2 ∈ X s uch that H x 2 = g x 1 . Contin uin g this pro c e s s, we can construct sequences { x n } and { y n } in X defined b y (2.2) H x 2 n +1 = f x 2 n = y 2 n , H x 2 n +2 = g x 2 n +1 = y 2 n +1 , ∀ n ∈ N COINCIDENCE AND COMMON FIXED P OINT RESUL TS 3 By co ns truction, we have x 1 ∈ H − 1 ( f x 0 ) and x 2 ∈ H − 1 ( g x 1 ) , then using the fact that f and g ar e weakly incre asing with res pect to H , w e obtain H x 1 = f x 0  g x 1 = H x 2  f x 2 = H x 3 . W e co nt in ue the pro ce s s to get (2.3) H x 1  H x 2  ...  H x 2 n +1  H x 2 n +2  ... W e ca n then write (2.4) y 0  y 1  ...  y 2 n  y 2 n +1  ... First ca se . If there exists n ∈ N ∗ such that y 2 n − 1 = y 2 n , then b y construction of the sequence { y m } and the contraction (2.1) with x = x 2 n , y = x 2 n +1 . d ( y 2 n , y 2 n +1 ) = d ( f x 2 n , g x 2 n +1 ) ≤ β ( d ( H x 2 n , H x 2 n +1 )) d ( H x 2 n , H x 2 n +1 ) = β ( d ( y 2 n − 1 , y 2 n )) d ( y 2 n − 1 , y 2 n ) = 0 , which implies y 2 n = y 2 n +1 . This le a ds to y m = y 2 n − 1 for a ny m ≥ 2 n . Hence for every m ≥ 2 n we have H x m = H x 2 n . This implies that { H x n } is a Ca uch y sequence. The s ame conclusion holds if y 2 n = y 2 n +1 . Second c a se . Suppose that y n 6 = y n +1 for any in teger n . First, we will show tha t (2.5) lim n → + ∞ d ( y n +1 , y n ) = 0 . Thanks to (2.3), H x 2 n and H x 2 n +1 are comparable, then us ing (2.2) and taking x = x 2 n +2 and y = x 2 n +1 in (2.1), we g et d ( y 2 n +2 , y 2 n +1 ) = d ( H x 2 n +3 , H x 2 n +2 ) = d ( f x 2 n +2 , g x 2 n +1 ) ≤ β ( d ( H x 2 n +2 , H x 2 n +1 )) d ( H x 2 n +2 , H x 2 n +1 ) = β ( d ( y 2 n +1 , y 2 n )) d ( y 2 n +1 , y 2 n ) . (2.6) Using 0 ≤ β < 1 , w e deduce then (2.7) d ( y 2 n +2 , y 2 n +1 ) ≤ d ( y 2 n +1 , y 2 n ) . Similarly to this, one can find for x = x 2 n and y = x 2 n +1 in (2.1) that (2.8) d ( y 2 n +1 , y 2 n ) ≤ d ( y 2 n , y 2 n − 1 ) . Thu s co mbi ning (2 .7) tog ether with (2.8 ) lea ds that for any n ∈ N (2.9) d ( y n +2 , y n +1 ) ≤ d ( y n +1 , y n ) . It follows that the se q uence { d ( y n +1 , y n ) } is mo no tonic decreasing. Hence, there exists r ≥ 0 such that (2.10) lim n → + ∞ d ( y n +1 , y n ) → r . F r o m (2.6 ), we hav e d ( y 2 n +2 , y 2 n +1 ) d ( y 2 n +1 , y 2 n ) ≤ β ( d ( y 2 n +1 , y 2 n )) < 1 . Letting n → + ∞ in the a bove inequa lit y , then thanks to (2.10), we obtain lim n → + ∞ β ( d ( y 2 n +1 , y 2 n )) = 1 , and s ince β ∈ S , this implies that r = 0 . Hence, (2 .5 ) holds. W e need now to c hec k that { H x n } is a Cauc h y sequence. F ollowing (2.2), it suffices to prov e that { H x 2 n } is a Cauch y sequence. T o do this, we pro ceed by contradic- tion. Suppo se that { H x 2 n } is not a Ca uch y sequence . Then for a n y ε > 0 , fo r 4 H. A YDI which there exist tw o sequence s of p ositive integers integers { m ( k ) } and { n ( k ) } such tha t for a ll p ositive integers k , (2.11) n ( k ) > m ( k ) > k , d ( H x 2 m ( k ) , H x 2 n ( k ) ) > ε, d ( H x 2 m ( k ) , H x 2 n ( k ) − 2 ) ≤ ε Therefore, we use (2.11) a nd the triangula r inequality to get ε < d ( H x 2 m ( k ) , H x 2 n ( k ) ) ≤ d ( H x 2 m ( k ) , H x 2 n ( k ) − 2 ) + d ( H x 2 n ( k ) − 2 , H x 2 n ( k ) − 1 ) + d ( H x 2 n ( k ) − 1 , H x 2 n ( k ) ) ≤ ε + d ( H x 2 n ( k ) − 2 , H x 2 n ( k ) − 1 ) + d ( H x 2 n ( k ) − 1 , H x 2 n ( k ) ) . Letting k → + ∞ in the ab ov e inequa lit y a nd using (2.5), we find (2.12) lim k → + ∞ d ( H x 2 m ( k ) , H x 2 n ( k ) ) = ε. Again, using the tria ngular inequa lit y we have | d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 ) − d ( H x 2 n ( k ) , H x 2 m ( k ) ) |≤ d ( H x 2 m ( k ) , H x 2 m ( k ) − 1 ) . Letting again k → + ∞ in the a bove inequa lity and using (2.5 )-(2 .12) we find (2.13) lim k → + ∞ d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 ) = ε. On the o ther ha nd, w e have d ( H x 2 n ( k ) , H x 2 m ( k ) ) ≤ d ( H x 2 n ( k ) , H x 2 n ( k )+1 ) + d ( H x 2 n ( k )+1 , H x 2 m ( k ) ) = d ( H x 2 n ( k ) , H x 2 n ( k )+1 ) + d ( f x 2 n ( k ) , g x 2 m ( k ) − 1 ) . Thanks to (2.5 )-(2.12), then letting k → + ∞ , w e have from the ab ov e inequality (2.14) ε ≤ lim k → + ∞ d ( f x 2 n ( k ) , g x 2 m ( k ) − 1 ) . W e take now x = x 2 n ( k ) and y = x 2 m ( k ) − 1 in (2.1). Hence, d ( f x 2 n ( k ) , g x 2 m ( k ) − 1 ) ≤ β ( d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 )) d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 ) < d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 ) Letting again k − → + ∞ in the a bove inequa lity and using (2.5)-(2.13 ), we obtain (2.15) lim k → + ∞ d ( f x 2 n ( k ) , g x 2 m ( k ) − 1 ) ≤ ε. Combining (2.14) to (2 .15) yields lim k → + ∞ d ( f x 2 n ( k ) , g x 2 m ( k ) − 1 ) = ε, Therefore, s ince we are in the cas e H x 2 n ( k ) 6 = H x 2 m ( k ) − 1 , then writing d ( f x 2 n ( k ) , g x 2 m ( k ) − 1 ) d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 ) ≤ β ( d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 )) < 1 , and using the fact that ε = lim k → + ∞ d ( f x 2 n ( k ) , g x 2 m ( k ) − 1 ) = lim k → + ∞ d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 ) , we g et lim k → + ∞ β ( d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 )) = 1 . W e know tha t β ∈ S , hence lim k → + ∞ d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 ) = 0 , which is a cont radiction with (2.13 ), tha t is lim k − → + ∞ d ( H x 2 n ( k ) , H x 2 m ( k ) − 1 ) = ε > 0 . W e deduce then { H x n } is a C a uch y sequence. Let us now prove the existence of a coincidence point. First, we know that { H x n } COINCIDENCE AND COMMON FIXED P OINT RESUL TS 5 is a Ca uch y sequence, then since ( X, d ) is a complete metric spa c e , there exists u ∈ X such that (2.16) lim n − → + ∞ H x n = u. F r o m (2.1 6) a nd the contin uit y of H , w e ge t (2.17) lim n → + ∞ H ( H x n ) = H u . The triangular inequa lit y yields (2.18) d ( H u, f u ) ≤ d ( H u, H ( H x 2 n +1 )) + d ( H ( f x 2 n , f ( H x 2 n )) + d ( f ( H x 2 n ) , f u ) . Thanks to (2.2 ) and (2 .1 6) (2.19) H x 2 n → u, f x 2 n → u The pair { f , H } is compatible, then (2.20) d ( H ( f x 2 n ) , f ( H x 2 n )) → 0 Using the co nt in uit y of f and (2.16), we hav e (2.21) d ( f ( H x 2 n ) , f u ) → 0 Combining (2.1 7 )-(2.20) together with (2.21) and letting n → + ∞ in (2.1 8), w e obtain d ( H u, f u ) ≤ 0 , which mea ns that H u = f u . Again by the tria ngular inequality (2.22) d ( H u, g u ) ≤ d ( H u, H ( H x 2 n +2 )) + d ( H ( g x 2 n +1 ) , g ( Gx 2 n +1 )) + d ( g ( Gx 2 n +1 ) , g u ) . On the o ther ha nd (2.23) H x 2 n +1 → u, g x 2 n +1 → u The pair { g , H } is co mpatible, then (2.24) d ( H ( g x 2 n +1 ) , g ( Gx 2 n +1 )) → 0 The contin uit y of g together with (2.16) gives us (2.25) d ( g ( H x 2 n +1 ) , g u ) → 0 Combining (2.17)-(2.2 4) tog ether with (2.25) and letting n − → + ∞ in (2.22), we obtain d ( H u, g u ) ≤ 0 , which means tha t H u = g u . W e finish then by finding f u = g u = H u , that is, u is a coincidence p oint of f , g and H . The pr o of o f theore m 2.5 is prov ed.  Now, w e o mit in the pro o f of theor e m 2.5 , the contin uit y of f , g and H , and the compatibility of the pairs { f , H } and { g , H } , and we replace them b y other conditions in order to find the sa me result. This will be the purp ose of the next theorem Theorem 2.6. L et ( X ,  ) b e a p artial ly or der e d set. Supp ose that ther e exists a c omplete met ric d on X such that X is r e gular (i.e, if { z n } is a non-de cr e asing se qu enc e in X with r esp e ct to  such t hat z n → z as n → + ∞ , then z n  z for al l n ∈ N ). L et f , g , H : X → X b e given mapp ings satisfying (a) f X ⊆ H X , g X ⊆ H X , (b) H X is a close d subsp a c e of ( X, d ) , (c) f and g ar e we akly incr e asing with r esp e ct to H . Supp ose t hat for every ( x, y ) ∈ X × X such t hat H x and H y ar e c omp ar ab le, we have (2.1) holds. 6 H. A YDI Then, f , g and H have a c oi ncidenc e p oint u ∈ X , that is, f u = g u = H u . Pr o of. W e take the same seq uence s { x n } and { y n } as in the pro of of theorem 2 .5. In particular { H x n } is a Cauch y sequence in the clo sed subspace H X , then there exists v = H u „ u ∈ X such that (2.26) lim n → + ∞ H x n = v = H u. Thanks to (2.4), { H x n } is non-decrea sing sequence, then since it conv erges to v = H u , we g et H x n  H u, ∀ n ∈ N , so the terms H x n and H u ar e comparable. Putting now x = x 2 n and y = u in (2.1 ) and using (2.2 ) one ca n wr ite d ( H x 2 n +1 , g u ) = d ( T x 2 n , g u ) ≤ β ( d ( H x 2 n , H u )) d ( H x 2 n , H u ) ≤ d ( H x 2 n , H u ) . Letting n → + ∞ in the a bove inequa lit y , using (2.26), we obtain d ( H u, g u ) = 0 . This means that f u = H u . W e use now the same strateg y , one can wr ite for x = u and y = x 2 n +1 in (2.1) d ( f u, H x 2 n +2 ) = d ( f u, g x 2 n +1 ) ≤ β ( d ( H u, H x 2 n +1 )) d ( H u, H x 2 n +1 ) ≤ d ( H u, H x 2 n +1 ) . Similarly , we let n − → + ∞ in the ab ov e inequality and we obtain f u = H u. W e co nclude that u is a co incidence p oint of H , f a nd g , and then the pro of of theorem 2.6 is co mpleted.  Now, we sha ll prov e the existence a nd uniqueness theor em of a co mmon fixed po in t. Theorem 2. 7. In addition to the hyp otheses of the or em 2.5, supp ose that for any ( x, y ) ∈ X × X , ther e exists u ∈ X such that f x  f u and f y  f u . Then, f , g and H have a un ique c ommon fix e d p oi nt, that is ther e exists a unique z ∈ X such that z = H z = f z = g z . Pr o of. Referring to theo rem 2.5, the set of coincidence p oints is no n-empt y . W e shall show that if x ∗ and y ∗ are coincidence po in ts, that is, H x ∗ = f x ∗ = g x ∗ and H y ∗ = f y ∗ = g y ∗ , then (2.27) H x ∗ = H y ∗ . By a ssumption, ther e exists u 0 ∈ X such that (2.28) f x ∗  f u 0 , f y ∗  f u 0 . Now, we pro ceed similar to the pro o f o f theorem 2.5, we can immediately define the sequence { H u n } as fo llows (2.29) H u 2 n +1 = f u 2 n , H u 2 n +2 = g u 2 n +1 , ∀ n ∈ N Again, w e have (2.30) f x ∗ = H x ∗  H u n , f y ∗ = H y ∗  H u n , ∀ n ∈ N ∗ COINCIDENCE AND COMMON FIXED P OINT RESUL TS 7 Putting x = u 2 n and y = x ∗ in (2.1) a nd using β < 1 and (2.30), we get d ( H u 2 n +1 , H x ∗ ) = d ( f u 2 n , g x ∗ )) ≤ β ( d ( H u 2 n , H x ∗ )) d ( H u 2 n , H x ∗ ) ≤ d ( H u 2 n , H x ∗ ) . This g ives us (2.31) d ( H u 2 n +1 , H x ∗ ) ≤ d ( H u 2 n , H x ∗ ) Putting x = x ∗ and y = u 2 n in (2.1), then similar ly to the ab ove, o ne ca n find (2.32) d ( H u 2 n +2 , H x ∗ ) ≤ d ( H u 2 n +1 , H x ∗ ) . Here w e have used H x ∗ = g x ∗ . W e combine (2.31 ) to (2 .3 2) to rema rk that (2.33) d ( H u n +1 , H x ∗ ) ≤ d ( H u n , H x ∗ ) . Then the s equence { d ( H u n , H x ∗ ) } is no n-increasing, so there exists r ≥ 0 such that d ( H u n , H x ∗ ) → r a s n → + ∞ . W e know tha t d ( H u 2 n +1 , H x ∗ ) ≤ β ( d ( H u 2 n , H x ∗ )) d ( H u 2 n , H x ∗ ) , hence d ( H u 2 n +1 , H x ∗ ) d ( H u 2 n , H x ∗ ) ≤ β ( d ( H u 2 n , H x ∗ )) < 1 . Letting n → + ∞ in the a bove inequa lit y , then we obtain lim n → + ∞ β ( d ( H u 2 n , H x ∗ )) = 1 , and s ince β ∈ S , this implies that d ( H u 2 n , H x ∗ ) → r = 0 . W e then wr ite (2.34) d ( H u n , H x ∗ ) → 0 as n → + ∞ . The same idea y ields (2.35) d ( H u n , H y ∗ ) → 0 as n → + ∞ . (2.34)-(2.35 ) together with the fac t that the limit is unique a llows that (2.27) holds. Now, thanks to (2.29)-(2.34), we ca n wr ite (2.36) lim n → + ∞ f u 2 n = H x ∗ = H y ∗ , lim n → + ∞ g u 2 n +1 = H x ∗ = H y ∗ . F r o m the compatibility of the pairs { f , H } and { g , H } , we obtain using (2.34 )-(2.36) (2.37) lim n → + ∞ d ( H ( f u 2 n ) , f ( H u 2 n )) = 0 , lim n → + ∞ d ( H ( g u 2 n +1 ) , g ( H u 2 n +1 )) = 0 . Let us deno te z = H x ∗ . By the tr ia ngular ineq ua lit y , we hav e d ( H z , f z ) ≤ d ( H z , H ( f u 2 n )) + d ( H ( f u 2 n ) , f ( H u 2 n )) + d ( f ( H u 2 n ) , f z ) , Using (2.36)-(2.37) and the con tin uit y of f and letting n − → + ∞ in the ab ov e inequality , we g e t d ( H z , f z ) ≤ 0 , that is, H z = f z and z is a coincidence point of H a nd f . Again the triangula r inequality gives us d ( H z , g z ) ≤ d ( H z , H ( g u 2 n +1 )) + d ( H ( g u 2 n +1 ) , f ( H u 2 n +1 )) + d ( f ( H u 2 n +1 ) , g z ) , Using (2.36)-(2.37) and the contin uit y of g and letting n − → + ∞ in the ab ove inequality , we g e t d ( H z , g z ) ≤ 0 , that is, H z = g z and z is a coincidence p oint of g and H . F rom (2.27), we hav e z = H x ∗ = H z = f z = g z . 8 H. A YDI This pro ves that z is a common fixed po in t o f the mappings H , g and f . Now our purp ose is to chec k that s uc h a p oint is unique. Suppose there is an another common fixed po in t p , that is p = H p = f p = g p. This implies that p is a co incidence p oint o f H , f and g . F rom (2.27), this implies that H p = H z . Hence, we get p = H p = H z = z , which yields the uniqueness of the common fixe d po in t. The pro of of theorem 2 .7 is co mpleted. The next result is an immediate consequence of theorems 2.5 a nd 2 .7 by taking H = I d X .  Corollary 2.8. L et ( X,  ) b e a p artial ly or der e d set. Supp ose that ther e exists a c omplete metric d on X . L et f , g : X → X b e given mappi ngs satisfying (a) f , g ar e c ont inuous, (b) f and g ar e we akly incr e asing. Supp ose that for every ( x, y ) ∈ X × X such that x and y ar e c omp ar able, we have (2.38) d ( f x, g y ) ≤ β ( d ( x, y )) d ( x, y ) , wher e β ∈ S . Su pp ose again that for e a ch x, y ∈ X , ther e exists z ∈ X which is c omp ar able to x and y . Then, f , g have a un ique c ommon fixe d p oint. Remark 2.9. T aking g = f in c or ol lary 2.8, we find t he r esult given in the or em 1.1. 3. Applica tion : existence f or a common solution of integral equa tion s Consider the integral equatio ns (3.1) ( u ( t ) = R T 0 K 1 ( t, s, u ( s )) ds + h ( t ) , t ∈ [0 , T ] u ( t ) = R T 0 K 2 ( t, s, u ( s )) ds + h ( t ) , t ∈ [0 , T ] where T > 0 . The purpose of this section is to giv e an existence theorem for common solution of (3.1 ), using Corolla ry 2.8. This application is inspired in [2], [10]. Let consider the space X = C ( I ) ( I = [0 , T ] ) o f contin uous functions defined on I . Obviously , this s pace with the metric g iven by d ( x, y ) = sup t ∈ I | x ( t ) − y ( t ) | , ∀ x, y ∈ X , is a complete metric spa ce. X = C ( I ) can also be equipp ed w ith the partial o rder  given b y ∀ x, y ∈ X, x  y ⇐ ⇒ x ( t ) ≤ y ( t ) , ∀ t ∈ I . Moreov er, in [12], it is prov ed that ( C ( I ) ,  ) is regular (the definition is given in theorem 2.6). No w we will state the following theorem which we find it in [11] Theorem 3.1. Supp ose that the fol lowing hyp otheses hold: i) K 1 , K 2 : I × I × R → R and h : R → R ar e c ontinuous, ii) for al l t , s ∈ I , we have K 1 ( t, s, u ( t )) ≤ K 2 t, s, Z T 0 K 1 ( s, τ , u ( τ )) dτ + h ( s ) ! . COINCIDENCE AND COMMON FIXED P OINT RESUL TS 9 K 2 ( t, s, u ( t )) ≤ K 1 t, s, Z T 0 K 2 ( s, τ , u ( τ )) dτ + h ( s ) ! . iii) t her e exists a c ontinuous function G : I × I → R + such that | K 1 ( t, s, x ) − K 2 ( t, s, y ) | ≤ G ( t, s ) p log[( x − y ) 2 + 1] , for al l t , s ∈ I and x, y ∈ R such that y  x , iv) su p t ∈ I R T 0 G 2 ( t, s ) ds ≤ 1 T . Then the inte gr al e quations (3.1) have a solution u ∗ ∈ C ( I ) . Pr o of. As mentioned ab ov e, the pro of we find it in [11], with o f cour se a leg e r mo dification a t the end. W e need to bring the quantities f x ( t ) = Z T 0 K 1 ( t, s, u ( s )) ds + h ( t ) , t ∈ I . g x ( t ) = Z T 0 K 2 ( t, s, u ( s )) ds + h ( t ) , t ∈ I . Again, the authors [11] found that f and g ar e weakly increasing . Mor eov er, they obtained d ( f x, g y ) ≤ p log[ d 2 ( x, y ) + 1] . Now, the leger mo dification w ith r esp e ct to [1 1] is to choose the function β a s β ( t ) = p log[ t 2 + 1] t . It is clea r that with this choice, β ∈ S . One can then write d ( f x, g y ) ≤ β ( d ( x, y )) d ( x, y ) . Now, a ll the hypo theses of Corollary 2.8 are satisfied. Then, there exists u ∗ ∈ C ( I ) , a common fixe d p o int of f and g , that is u ∗ is a so lution to (3.1 ).  References [1] R. P . Agar wa l, M. A. El-Gebeily , D. O ’Regan, Ge neralized con t ractions in partially orde red metric spaces, App. Anal. 87 (2008), 109-116. [2] I. Altun, H. Simsek, some fixed p oint theorems on ordere d metric spaces and applications. Fixed Poin t theo ry and Applications, 2010, article ID 621492, 17 pages, 2010. [3] T. Gnana. Bhask er, V. Lakshmik a n tham, fixed p oint theorems in partially ordere d metric spaces and applications, Nonli near analysis, 6, (2006), 1379-1393. [4] J. Caballero, J. 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