Applied Mathematical Sciences, Vol. 7, 2013, no. 75, 3703-3713 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.35273 Common Fixed Point Theorems for Generalized (ψ, φ)-type Contactive Mappings on Metric Spaces Seong-Hoon Cho Department of Mathematics Hanseo University, Seosan Chungnam, 356-706, South Korea Jong-Sook Bae Department of Mathematics Myongji University, Yongin Gyeonggi, 449-728, South Korea Copyright c 2013 Seong-Hoon Cho and Jong-Sook Bae. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Some new fixed point theorems for generalized (ψ,φ)-type contactive mappings are established. Mathematics Subject Classification:47H10, 54H25 Keywords: Common fixed points, Compatible mappings, Weakly compatible mappings, Generalized (ψ, φ)-type contactive mappings, Metric spaces 1 Introduction and preliminaries Let (X, d) be a metric space. A self mapping S on X is said to be contraction if there exists k (0, 1) such that d(sx,sy) kd(x, y) for all x, y X. (1.1)
3704 Seong-Hoon Cho and Jong-Sook Bae According to the Banach contraction principle, a mapping S satisfying (1.1) in complete metric space X has a unique fixed point. A study on generalizations of the Banach contraction principle has been a very active field of research during recent years. For example, a quasi cotraction mapping has been dealt with in [3, 4, 5, 7]. Recectly, the author [10] proved some common fixd point theorems which generalize the fixed point theorems of Banach, Kannan [6], Jungck [8], Das and Naik [5] and Berinde [2, 3]. In this paper, we give some new common fixed point theorems, which are generalizations of results of [2, 10]. Let (X, d) be a metric space, and let S and T be self mappings on X. S and T are said to be compatible [7] if lim n d(stx n,tsx n ) = 0, whenever {x n } is a sequence of points in X such that lim n Tx n = t and lim n Sx n = t for some t X. S and T are said to be weakly compatible [8] if STp = TSp, whenever Tp = Sp. It is known that compatible mappings are weakly compatible mappings, but the reverse is not true. S and T are said to be pointwise R-weak commuting [9] if given x X, there exists R>0 such that d(st x, T Sx) Rd(Tx,Sx). Note that S and T are weakly compatible if and only if they are pointwise R-weak commuting (see [9]). Suppose that S(X) T (X). (1.2) If x 0 X, then Sx 0 T (X). There exists x 1 X such that Sx 0 = Tx 1. Again, since Sx 1 T (X), there exists x 2 X such that Sx 1 = Tx 2. Inductively, we construct two sequences {x n } and {y n } of points in X such that y n = Sx n = Tx n+1,n=0, 1, 2, (1.3) Define the sets O(y 0 ; n) ={y 0,y 1,y 2,,y n } and O(y 0 ; ) ={y 0,y 1,y 2,,y n, }. For A X, we denote by δ(a) the diameter of A. From now on, let ψ :[0, ) [0, ) be a nondecreasing and continuous function such that ψ(t) t for all t>0, and let φ :[0, ) [0, ) bea function such that (φ1) φ is nondecreasing; (φ2) φ is upper semi-continuous; (φ3) φ(t) <tfor all t>0.
Fixed points 3705 Lemma 1.1. [1, 2] If φ :[0, ) [0, ) satisfies (φ1) and (φ2), then the following hold. (1) φ(0) = 0; (2) lim n φ n (t) =0if and only if φ(t) <tfor all t>0. Throughout this paper, let K :[0, ) 5 [0, ) be a continuous and nondecreasing function in each coordinate variable such that K(a, a, a, a, a) a for a 0. A function f : (X, d) R is called upper semi-continuous if, for any {x n } X and x X, lim n f(x n ) f(x) whenever lim n x n = x. A function ψ :[0, ) [0, ) is called subadditive if ψ(s+t) ψ(t)+ψ(s) for all s, t 0. 2 Common fixed point theorems Let (X, d) be a metric space, and let S and T be self mappings on X. Then, we say that S and T satisfy generalized (ψ, φ)-type contractive condition whenever ψ(d(sx,sy)) φ(ψ(m(x, y))) for all x, y X, where m(x, y) =K(d(Tx,Ty),d(Tx,Sx),d(Ty,Sy),d(Tx,Sy),d(Ty,Sx)). Proposition 2.1. Let (X, d) be a metric space. Suppose that S and T are self mappings on X satisfying generalized (ψ, φ)-type contractive condition and S(X) T (X). Suppose that ψ is subadditive and there exists n 0 N such that lim (t t φn 0 (t)) =. Then O(y 0 ; ) is bounded for x 0 X. Proof. Suppose that there exists n 0 N such that lim t (t φ n 0 (t)) =. Let x 0 X be a fixed. If n 0 <i<j n, then we have ψ(d(y i,y j )) φ(ψ(k(d(y i 1,y j 1 ),d(y i 1,y i ),d(y j 1,y j ),d(y i 1,y j )),d(y j 1,y i ))) φ(ψ(k(v 1,v 1,v 1,v 1,v 1 ))) φ(ψ(v 1 )), where v 1 = max{d(y i 1,y j 1 ),d(y i 1,y i ),d(y j 1,y j ),d(y i 1,y j ),d(y j 1,y i )} δ(o(y 0 ; n)).
3706 Seong-Hoon Cho and Jong-Sook Bae For this v 1, there exists v 2 δ(o(y 0 ; n)) such that ψ(v 1 ) φ(ψ(v 2 )). Hence ψ(d(y i,y j )) φ 2 (ψ(v 2 )), and hence ψ(d(y i,y j )) φ n 0 (ψ(v n0 )) for some v n0 δ(o(y 0 ; n)). Thus, ψ(d(y i,y j )) φ n 0 (ψ(δ(o(y 0 ; n)))). (2.1) So, we have ψ(d(y i,y j )) <ψ(δ(o(y 0 ; n))). Hence we have d(y i,y j ) <δ(o(y 0 ; n)). Therefore, for all n>n 0, we have δ(o(y 0 ; n)) = max{d(y 0,y k ),d(y i,y j ):1 k n, 1 i, j n 0,i n 0 <j<n}. We consider the following four cases. 1 :Ifδ(O(y 0 ; n)) = d(y 0,y k ) for some k such that 1 k n 0, then δ(o(y 0 ; n)) = max{d(y 0,y l ):1 l n 0 }. 2 : If δ(o(y 0 ; n)) = d(y 0,y k ) for some k such that n 0 k n, then d(y 0,y k ) d(y 0,y n0 +1)+d(y n0 +1,y k ). By (2.1), we obtain ψ(δ(o(y 0 ; n))) ψ(d(y 0,y n0 +1))+d(y n0 +1,y k )) ψ(d(y 0,y n0 +1))+ψ(d(y n0 +1,y k )) ψ(d(y 0,y n0 +1)) + φ n 0 (ψ(δ(o(y 0 ; n)))). Since ψ(δ(o(y 0 ; n))) is nondecreasing, lim n ψ(δ(o(y 0 ; n))) exists. Suppose that lim n ψ(δ(o(y 0 ; n))) =. Then, we have lim (t t φn 0 (t)) = lim (ψ(δ(o(y 0 ; n))) φ n 0 (ψ(δ(o(y 0 ; n))))) n ψ(d(y 0,y n0 +1)) <, which is a contradiction. Thus, ψ(δ(o(y 0 ; ))) = lim n ψ(δ(o(y 0 ; n))) <, and so δ(o(y 0 ; )) <. 3 :Ifδ(O(y 0 ; n)) = d(y i,y j ) for some i, j such that 1 i, j n 0, then we have ψ(d(y i,y j )) φ(ψ(k(d(y i 1,y j 1 ),d(y i 1,y i ),d(y j 1,y j ),d(y i 1,y j )),d(y j 1,y i ))) φ(ψ(k(u, u, u, u, u))) φ(ψ(u)) φ(ψ(δ(o(y 0 ; n 0 )))) <ψ(δ(o(y 0 ; n 0 ))),
Fixed points 3707 where u = max{d(y i 1,y j 1 ),d(y i 1,y i ),d(y j 1,y j ),d(y i 1,y j ),d(y j 1,y i )}. Hence, δ(o(y 0 ; n)) <δ(o(y 0 ; n 0 ). 4 : If δ(o(y 0 ; n)) = d(y i,y j ) for some i, j such that i n 0 < j < n, then we have ψ(δ(o(y 0 ; n))) = ψ(d(y i,y j )) ψ(d(y i,y n0 +1) +d(y n0 +1,y j )) ψ(d(y i,y n0 +1)) + ψ(d(y n0 +1,y j )) ψ(d(y i,y n0 +1)) + φ n 0 (ψ(δ(o(y 0 ; n)))). As in case 2, we have δ(o(y 0 ; )) <. In all cases, δ(o(y 0 ; )) <. Let (X, d) be a metric space, x X, and let A X. We denote by d(x, A) = sup{d(x, y) :y A}. Proposition 2.2. Let (X, d) be a metric space. Suppose that S and T are self mappings on X satisfying generalized (ψ, φ)-type contractive condition and S(X) T (X). Suppose that ψ is subadditive and inf x X ψ(d(sx,s(t 1 (Sx)))) < lim t (t φ(t)). Then there exists x 0 X such that O(y 0 ; ) is bounded. Proof. Suppose that inf x X ψ(d(sx,s(t 1 (Sx)))) < lim t (t φ(t)). Then there exists x 0 X such that ψ(d(sx 0,S(T 1 (Sx 0 ))) < lim t (t φ(t)). Then there exists x 1 T 1 (Sx 0 ) such that ψ(d(sx 0,Sx 1 )) < lim t (t φ(t)). Then Sx 0 = Tx 1 = y 0. Next, one can choose x 2,x 3, X such that (1.3) is satisfied. Then ψ(d(y 0,y 1 )) < lim t (t φ(t)). Hence there exists M>0 such that for all t>m ψ(d(y 0,y 1 )) <t φ(t) (2.2) Let i, j N such that 1 i, j n. If d(y i,y j ) > 0, then we have ψ(d(y i,y j )) d(y i,y j ) > 0. Thus we obtain ψ(d(y i,y j )) φ(ψ(k(d(y i 1,y j 1 ),d(y i 1,y i ),d(y j 1,y j ),d(y i 1,y j ),d(y j 1,y i )))) φ(ψ(k(δ(o(y 0 ; n)),δ(o(y 0 ; n)),δ(o(y 0 ; n)),δ(o(y 0 ; n)),δ(o(y 0 ; n))))) φ(ψ(δ(o(y 0 ; n)))) <ψ(δ(o(y 0 ; n))). Since ψ is nondecreasing, d(y i,y j ) <δ(o(y 0 ; n)). Thus we have δ(o(y 0 ; n)) = max{d(y 0,y k ):1 k n}.
3708 Seong-Hoon Cho and Jong-Sook Bae Let δ(o(y 0 ; n)) = d(y 0,y l ) for some l N with 1 l n. From the following inequality ψ(d(y 0,y l )) ψ(d(y 0,y 1 )) + ψ(d(y 1,y l ))+ ψ(d(y 0,y 1 )) + φ(ψ(δ(o(y 0 ; n)))), we obtain ψ(δ(o(y 0 ; n))) φ(ψ(δ(o(y 0 ; n)))) ψ(d(y 0,y 1 )). (2.3) From (2.2) and (2.3), we have ψ(δ(o(y 0 ; n))) M for all n N. Thus, if δ(o(y 0 ; n)) > 0, then δ(o(y 0 ; n)) M for all n N. Hence, O(y 0 ; ) is bounded. Lemma 2.3. Let (X, d) be a metric space. Suppose that S and T are self mappings on X satisfying generalized (ψ, φ)-type contractive condition and S(X) T (X). If there exists x 0 X such that O(y 0 ; ) is bounded, then {y n } is Cauchy sequence in X. Proof. Let x 0 X be such that O(y 0 ; ) is bounded. Let B n = {y i : i n}, for n =0, 1, 2,. Then B = δ(b 0 ) <. We claim that, for n =0, 1, 2,, ψ(δ(b n )) φ n (ψ(b)). (2.4) If n = 0, then, obviousely (2.4) hold. Suppose that (2.4) hold when n = k, i.e. ψ(δ(b k )) φ k (ψ(b)). Let y i,y j B k+1 for any i, j k + 1. Then ψ(d(y i,y j )) = ψ(d(sx i,sx j )) φ(ψ(k(d(tx i,tx j ),d(tx i,sx i ),d(tx j,sx j ),d(tx i,sx j ),d(tx j,sx i )))) =φ(ψ(k(d(y i 1,y j 1 ),d(y i 1,y i ),d(y j 1,y j ),d(y i 1,y j ),d(y j 1,y i )))) φ(ψ(k(δ(b k ),δ(b k ),δ(b k ),δ(b k ),δ(b k )))) φ(ψ(δ(b k ))) φ(φ k (ψ(b))) =φ k+1 (ψ(b)). Hence, ψ(δ(b k+1 )) φ k+1 (ψ(b)). Therefore, (2.4) is true, for n =0, 1, 2,. Since lim n φ n (ψ(b)) = 0, for any ɛ>0, there exists n 0 N such that φ n 0 (ψ(b)) <ɛ.form, n n 0, we have d(y m,y n ) δ(b n0 ) ψ(δ(b n0 )) φ n 0 (ψ(b)) <ɛ. Hence {y n } is a Cauchy sequence in X.
Fixed points 3709 Theorem 2.4. Let (X, d) be a complete metric space.suppose that S and T are weakly compatible self mappings on X satisfying generalized (ψ, φ)-type contractive condition and S(X) T (X). Assume that T (X) is closed. If there exists x 0 X such that O(y 0 ; ) is bounded, then S and T have a unique common fixed point in X. Proof. Let x 0 X be such that O(y 0 ; ) is bounded. By Lemma 2.2, {y n } is Cauchy sequence in X, and so there exists u X such that lim n y n = u. Since T (X) is closed, there exists z X such that u = Tz. Hence lim Tx n = lim Sx n = Tz. n n We show that Tz = Sz. We have ψ(d(sx n,sz)) φ(ψ(k(d(tx n,tz),d(tx n,sx n ),d(tz,sz),d(tx n,sz),d(tz,sx n )))). Thus, we obtain ψ(d(tz,sz)) = lim sup ψ(d(sx n,sz)) lim sup φ(ψ(k(d(tx n,tz),d(tx n,sx n ),d(tz,sz),d(tx n,sz),d(tz,sx n )))) φ(ψ(k(0, 0,d(Tz,Sz),d(Tz,Sz), 0))) φ(ψ(k(d(tz,sz),d(tz,sz),d(tz,sz),d(tz,sz),d(tz,sz)))) φ(ψ(d(tz,sz))). If Tz Sz, then 0 <d(tz,sz) ψ(d(tz,sz)). Hence, we have ψ(d(tz,sz)) φ(ψ(d(tz,sz))) <ψ(d(tz,sz)), which is a contradiction. Thus, Tz = Sz = w. Since S and T are weakly compatible, Sw = Tw. We show that w is a fixed point of S. We have ψ(d(w, Sw)) = ψ(d(sz,sw)) φ(ψ(k(d(tz,tw),d(tz,sz),d(tw,sw),d(tz,sw),d(tw,sz)))) =φ(ψ(k(d(w, Sw),d(w, w),d(sw,sw),d(sw,w),d(w, Sw)))) φ(ψ(k(d(w, Sw),d(w, Sw),d(w, Sw),d(Sw,w),d(w, Sw)))) φ(ψ(d(w, Sw))).
3710 Seong-Hoon Cho and Jong-Sook Bae If w Sw, then 0 <d(w, Sw) ψ(d(w, Sw). Thus we have ψ(d(w, Sw) φ(ψ(d(w, Sw))) < ψ(d(w, Sw)), which is a contradiction. Hence w = Sw, and hence w = Sw = Tw. Let q be another common fixed point of S and T. We obtain ψ(d(q, w)) = ψ(d(sq,sw)) φ(ψ(k(d(tq,tw),d(tq,sq),d(tw,sw),d(tq,sw),d(tw,sq)))) φ(ψ(k(d(q, w),d(q, q),d(w, w),d(q, w),d(w, q)))) φ(ψ(k(d(q, w),d(q, w),d(q, w),d(q, w),d(w, q)))) φ(ψ(d(q, w))). If w q, then 0 <d(q, w) ψ(d(q, w)). Thus, we have ψ(d(q, w)) φ(ψ(d(q, w))) <ψ(d(q, w)), which is a contradiction. Hence w = q. Hence, S and T have a unique common fixed point in X. Remark 2.1. In Theorem 2.4, if we have K(t 1,t 2,t 3,t 4,t 5 ) = max{t 1,t 2,t 3, t 4,t 5 }, then we obtain Theorem 2.2 of [10]. Theorem 2.5. Let (X, d) be a complete metric space.suppose that S and T are weakly compatible self mappings on X satisfying generalized (ψ, φ)-type contractive condition and S(X) T (X). Assume that T is continuous. If there exists x 0 X such that O(y 0 ; ) is bounded, then S and T have a unique common fixed point in X. Proof. Let x 0 X be such that O(y 0 ; ) is bounded. By Lemma 2.2, {y n } is Cauchy sequence in X. Let lim n y n = p X. Since T is continuous, lim Ty n = Tp. n We have d(sy n,tp) d(sy n,ty n+1 )+d(ty n+1,tp)=d(stx n+1,tsx n+1 )+d(ty n+1,tp) Sine S and T are weakly compatible, they are pointwise R-weak commuting. Hence, for x n+1 X, there exists R>0 such that d(stx n+1,tsx n+1 ) Rd(Tx n+1,sx n+1 ). Thus we have d(sy n,tp) Rd(Tx n+1,sx n+1 )+d(ty n+1,tp) =Rd(y n,y n+1 )+d(ty n+1,tp).
Fixed points 3711 Hence, We now show that Sp = Tp. We have Thus, we get lim n Sy n = Tp. ψ(d(sy n,sp)) φ(ψ(k(d(ty n,tp),d(ty n,sy n ), d(tp,sp),d(ty n,sp),d(tp,sy n )))). ψ(d(tp,sp)) = lim sup ψ(d(sy n,sp)) φ(ψ(k(0, 0,d(Tp,Sp),d(Tp,Sp), 0))) φ(ψ(k(d(tp,sp),d(tp,sp),d(tp,sp),d(tp,sp),d(tp,sp)))) φ(ψ(d(tp,sp))) If Sp Tp, then we have ψ(d(tp,sp)) φ(ψ(d(tp,sp))) <ψ(d(tp,sp)), which is a contradiction. Hence Sp = Tp = u. Since S and T are weakly compatible, Su = Tu.As in the proof of Theorem 2.4, we have u is a unique common fixed point of S and T. Remark 2.2. (1) In Theorem 2.5, if we have K(t 1,t 2,t 3,t 4,t 5 ) = max{t 1,t 2, t 3,t 4,t 5 }, then we obtain Theorem 2.3 of [10]. (2) In Theorem 2.5, if we have K(t 1,t 2,t 3,t 4,t 5 ) = max{t 1,t 2,t 3,t 4,t 5 } and ψ(t) =t for all t 0, then we obtain a generalization of Theorem 2 of [2] with weak compatibility. We now give an example to support Theorem 2.4. Example 2.1. Let X = { 1 : n =0, 1, 2 } {0} and d(x, y) = x y 2 n for all x, y X, and φ(t) = 1 t for all t 0. Let ψ(t) =t for all t 0 and 2 K(t 1,t 2,t 3,t 4,t 5 ) = max{t 1,t 2,t 3,t 4,t 5 }. Let S and T be self mappings on X defined by { { 1 (x = 1,n=0, 1, 2 ) 1 (x = 1,n=0, 1, 2 ) 2 Sx = n+2 2 n 2 and Tx = n+1 2 n 0 (x =0) 0 (x =0).
3712 Seong-Hoon Cho and Jong-Sook Bae Then T (X) is closed, and S(X) T (X), and S and T are weakly compatible. Also, for x 0 =0,O(y 0 ; ) is bounded. We now show that S and T satisfy generalized (ψ, φ)-type contractive condition. For x = 1 and y = 1 (m>n), we have 2 n 2 m ψ(d(sx,sy)) = d(sx,sy)=d(s 1 2,S 1 n 2 )= 1 m 2 1 n+2 2 m+2 1 2 ( 1 2 1 n+1 2 )=1 m+1 2 d(t 1 2,T 1 )=φ(d(tx,ty)) φ(ψ(m(x, y))). n 2m For x = 0 and y = 1 2 n, we have ψ(d(sx,sy)) = d(sx,sy)=d(s0,s 1 2 n)= 1 2 n+2 1 1 = φ(d(tx,ty)) φ(ψ(m(x, y))). 2 2n+1 Thus S and T satisfy generalized (ψ, φ)-type contractive condition. Therefore, all conditions of Theorem 2.4 are satisfied, and S and T have a unique common fixed point in X. References [1] R.P. Agarwal, M.A. El-gebeily, D. O regan, Generalized contractions in partially ordered metric spaces, Applicable Analysis 87(2008), 109-116. [2] V. Berinde, A common fixed point theorem for compatible quasi contractive self mappings in metric spaces, Applied Mathematics and Computation 213(2009), 348-354. [3] V. Berinde, A common fixed point theorem for quasi contractive type mappings, Ann. Univ. Sci. Budapest 46(2003), 81-90. [4] Lj. B. Ćirić, A generalization of Banach s contraction principle, Proc. Amer. Math. Soc.45(1974), 267-273. [5] K. M. Das, K. V. Naik, Common fixed point theorems for commuting maps on metric spaces, Proc. Amer. Math. Soc. 77(1979), 369-373. [6] R. Kannan, Some results on fixed points, Bull. Clcultta Math. Soc, 60(1968), 71-76. [7] G. Jungck, Commuting maps and fixed points, Amer. Math Monthly 83(1976), 261-263.
Fixed points 3713 [8] G. Jungck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci. 4(1996), 199-215. [9] R.P. Pant, Common fixed points for four mapppings, Bull. Calcutta Math, Soc. 9(1998), 281-287. [10] Sh. Rezapour, N. Shahzad, Common fixed points of (ψ, φ)-type contractive maps, Applied Mathematics Letters 25(2012), 959-962. Received: May 21, 2013