QUASICONTRACTION NONSELF-MAPPINGS ON CONVEX METRIC SPACES AND COMMON FIXED POINT THEOREMS

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1 QUASICONTRACTION NONSELF-MAPPINGS ON CONVEX METRIC SPACES AND COMMON FIXED POINT THEOREMS LJILJANA GAJIĆ AND VLADIMIR RAKOČEVIĆ Received 29 September 2004 and in revised form 24 January 2005 We consider quasicontraction nonself-mappings on Takahashi convex metric spaces and common fixed point theorems for a pair of maps. Results generalizing and unifying fixed point theorems of Ivanov, Jungck, Das and Naik, and Ćirić are established. 1. Introduction and preliminaries Let X be a complete metric space. A map T : X X such that for some constant λ 0,1 and for every x, y X dtx,ty λ max { dx, y,dx,tx,dy,ty,dx,ty,dy,tx } 1.1 is called quasicontraction. Let us remark that Ćirić[1] introduced and studied quasicontraction as one of the most general contractive type map. The well known Ćirić s result see, e.g., [1, 6, 11] is that quasicontraction T possesses a unique fixed point. Fortheconvenienceofthereaderwe recallthefollowingrecentćirić s result. Theorem 1.1 [2, Theorem 2.1]. Let X be a Banach space, C a nonempty closed subset of X, and C the boundary of C.LetT : C X be a nonself mapping such that for some constant λ 0,1 and for every x, y C Suppose that dtx,ty λ max { dx, y,dx,tx,dy,ty,dx,ty,dy,tx }. 1.2 Then T has a unique fixed point in C. T C C. 1.3 Following Ćirić [3], let us remark thatproblem to extend the known fixed point theorem for self mappings T : C C,definedby1.1, to corresponding nonself mappings T : C X, C X, was open more than 20 years. In 1970, Takahashi [15] introduced the definition of convexity in metric space and generalized same important fixed point theorems previously proved for Banach spaces. In Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005: DOI: /FPTA

2 366 Quasicontraction nonself-mappings this paper we consider quasicontraction nonself-mappings on Takahashi convex metric spaces and common fixed point theorems for a pair of maps. Results generalizing and unifying fixed point theorems of Ivanov [7], Jungck [8], Das and Naik [3], Cirić [2], Gajić[5]andRakočević[12] are established. Let us recall that see Jungck [9] the self maps f and g on a metric space X,d are said to be a compatible pair if lim d gfx n, fgx n = n whenever {x n } is a sequence in X such that lim n gx n = lim n fx n = x 1.5 for some x in X. Following Sessa [14]wewillsaythat f,g : X X are weakly commuting if d fgx,gfx d fx,gx foreveryx X. 1.6 Clearly weak commutativity of f and g is a generalization of the conventional commutativity of f and g, and the concept of compatibility of two mappings includes weakly commuting mappings as a proper subclass. We recall the following definition of a convex metric space see [15]. Definition 1.2. Let X be a metric space and I = [0,1] the closed unit interval. A Takahashi convex structure on X is a function W : X X I X which has the property that for every x, y X and λ I d z,wx, y,λ λdz,x+1 λdz, y 1.7 for every z X. IfX,d is equipped with a Takahashi convex structure, then X is called a Takahashi convex metric space. If X,d is a Takahashi convex metric space, then for x, y X we set seg[x, y] = { Wx, y,λ:λ [0,1] }. 1.8 Let us remark that any convex subset of normed space is a convex metric space with Wx, y,λ = λx +1 λy.

3 2. Main results The next theorem is our main result. L. GajićandV.Rakočević 367 Theorem 2.1. Let X,d be a complete Takahashi convex metric space with convex structure W whichiscontinuousinthethirdvariable,c a nonempty closed subset of X and C the boundary of C. Letg : C X, f : X X and f : C C. Supposethat C, f is continuous, and let us assume that f and g satisfy the following conditions. i For every x, y C where dgx,gy M ω x, y, 2.1 M ω x, y = max { ω [ d fx, fy ],ω [ d fx,gx ],ω [ d fy,gy ], ω [ d fx,gy ],ω [ d fy,gx ]}, 2.2 ω :[0,+ [0,+ is a nondecreasing semicontinuous function from the right, such that ωr <r,forr>0,andlim r [r ωr] = +. ii f and g are a compatible pair on C, that is, whenever {x n } isasequenceinc such that for some x in X. iii iv v Then f and g haveauniquecommonfixedpointz in C. lim d gfx n, fgx n = n lim n gx n = lim n fx n = x 2.4 gc C f C. 2.5 g C C. 2.6 f C C. 2.7 Proof. Starting with an arbitrary x 0 C, weconstructasequence{x n } of points in C as follows. By 2.6 gx 0 C. Hence,2.5 implies that there is x 1 C such that f x 1 = gx 0. Let us consider gx 1. If gx 1 C, againby2.5 there is x 2 C such that f x 2 = gx 1. Suppose that gx 1 C. Now, because W is continuous in the third

4 368 Quasicontraction nonself-mappings variable, there exists λ 11 [0,1] such that W f x 1,g x1,λ11 C seg [ f x1,g x1 ]. 2.8 By 2.7 there is x 2 C such that f x 2 = W f x 1,gx 1,λ 11. Hence, by induction we construct a sequence {x n } of points in C as follows. If gx n C, thanby2.5 f x n+1 = gx n forsomex n+1 C; ifgx n C, then there exists λ nn [0,1] such that Now, by 2.7pickx n+1 C such that W f x n,g,λnn C seg [ f,g ]. 2.9 f x n+1 = W f,g,λnn Let us remark see [6] that for every x, y X and every λ [0,1] dx, y = d x,wx, y,λ + d Wx, y,λ, y Furthermore, if u X and z = Wx, y,λ seg[x, y]then First let us prove that du,z = d u,wx, y,λ max { du,x,du, y } f x n+1 g = f = g Suppose the contrary that f x n gx n 1. Then x n C.Now,by2.5 gx n C,hence f x n+1 = gx n, a contradiction. Thus we prove We will prove that gx n and f x n are Cauchy sequences. First we will prove that these sequences are bounded, that is that the set is bounded. For each n 1set We will prove that { } { A = f xi gxi } 2.14 A n = i=0 n 1 i=0 { } n 1 { } f xi g xi, i=0 i= a n = diam A n. a n = max { d f x 0,g xi :0 i n 1 }. 2.16

5 L. GajićandV.Rakočević 369 If a n = 0, then f x 0 = gx 0. We will prove that gx 0 is a common fixed point for f and g.by2.3 it follows that Now we obtain fg x 0 = gf x0 = gg x d gg x 0,g x0 Mω gx0,x 0 = ω d ggx0,g x0, 2.18 and hence ggx 0 = gx 0. From 2.17, we conclude that gx 0 = z is also a fixed point of f. To prove the uniqueness of the common fixed point, let us suppose that fu= gu = u for some u C.Now,by2.1wehave dz,u = dgz,gu M ω z,u = ω dz,u, 2.19 and so, z = u. Suppose that a n > 0. To prove 2.16 we have to consider three cases. Case 1. Suppose that a n = d fx i,gx j forsome0 i, j n 1. 1i Now, if i 1and fx i = gx i 1,wehave a n = d fx i,gx j = d gxi 1,gx j Mω xi 1,x j ω an <an and we get a contradiction. Hence i = 0. 1ii If i 1and fx i gx i 1,wehavei 2, and fx i 1 = gx i 2.Hence fx i seg [ g x i 2,g xi 1 ], 2.21 we have a n = d fx i,gx j max { d gxi 2,gx j,d gxi 1,gx j } max { M ω xi 2,x j,mω xi 1,x j } ω an <a n 2.22 and we get a contradiction. Case 2. Suppose that a n = d fx i, fx j forsome0 i, j n 1. 2i If fx j = gx j 1, then Case 2i reduces to Case 1i. 2ii If fx j gx j 1, then as in the Case 1ii we have j 2, fx j 1 = gx j 2,and fx j C seg [ gx j 2,gx j 1 ] Hence a n = d fx i, fx j max { d fxi,gx j 2,d fxi,gx j 1 } 2.24 and Case 2ii reduces to Case 1i.

6 370 Quasicontraction nonself-mappings Case 3. The remaining case a n = dgx i,gx j forsome0 i, j n 1, is not possible see Case 1i. Hence we proved Now a n = d fx 0,gx i d fx0,gx 0 + d gx0,gx i d fx0,gx 0 + ωan, 2.25 By i there is r 0 [0,+ suchthat Thus, by 2.26 and clearly a n ω a n d fx0,gx r ωr >d fx 0,gy 0, forr>r a n r 0, n = 1,2,..., 2.28 a = lim a n = diama r n Hence we proved that gx n and fx n are bounded sequences. To prove that gx n and fx n are Cauchy sequences, let us consider the set { } { } B n = fxi gxi, n = 2,3, By 2.16wehave i=n i=n b n diam B n = sup d fx n,gx j, n = 1,2, j n If fx n = gx n 1,thenasinCase1iforeachj n b n = d fx n,gx j = d g 1,gx j ω bn 1, n = 1,2, If fx n gx n 1, then as in Case 1ii for each n 1andj n b n = d fx n,gx j max { d g 2,gx j,d g 1,gx j } ω bn By 2.32and2.33weget b n ω b n 2, n = 2,3, Clearly, b n b n+1 for each n, and set lim n b n = b. Wewillprovethatb = 0. If b>0, then 2.34 and i imply b ωb <b, and we get a contradiction. It follows that both fx n and gx n are Cauchy sequences. Since fx n C and C is a closed subset of a complete metric space X we conclude that lim n fx n = y C.Furthermore, d f x n,g 0, n, 2.35

7 L. GajićandV.Rakočević 371 implies limgx n = y.hence, limg x n = lim f = y C By continuity of f lim f g x n = lim f f = f y C Now, by 2.3, we have d gf x n, f y d gf x n, fg + d fg, f y 0, n, 2.38 that is limgf x n = f y Now, M ω f, y ω d fy,gy n, d gfx n,gy M ω f, y n, 2.40 implies d fy,gy ω d fy,gy Hence, f y = gy, and gy is a common fixed point of f and g see In the special case, when ωr = λ r where 0 <λ<1, we obtain the following result. Theorem 2.2. Let X,d be a complete Takahashi convex metric space with convex structure W whichiscontinuousinthethirdvariable,c a nonempty closed subset of X and C the boundary of C. Letg : C X, f : X X and f : C C. Supposethat C, f is continuous, and let us assume that f and g satisfy the following conditions. i There exists a constant λ 0,1 such that for every x, y C where dgx,gy λ Mx, y, 2.42 Mx, y = max { d fx, fy,d fx,gx,d fy,gy,d fx,gy,d fy,gx } Suppose that the conditions ii v in Theorem 2.1 are satisfied. Then f and g have a unique common fixed point z in C and g is continuous at z.moreover,ifz n C, n = 1,2,..., then limd fz n,gz n = 0 iff limn z n = z Proof. By Theorem 2.1 we know that f and g have a unique common fixed point z in C. Now, we show that g is continuous at z. Let{y n } be a sequence in C such that y n z.

8 372 Quasicontraction nonself-mappings Now we have that is d gy n,gz λ M y n,z = λ max { d fy n, fz,d } fy n,gy n,d fz,gyn = λ max { d fy n, fz,d } fy n,gy n λ d fy n, fz + d fz,gy n, 2.45 d gy n,gz 1 λ 1 λ d fy n, fz Therefore, we have gy n gz and so g is continuous at z. Toprove2.44, let us suppose that w C. Now, since fz= gz = z,wehave d fw,gw d fw, fz+dgw,gz d fw, fz+λ Mw,z d fw, fz+λ max { d fw, fz,d fw,gw,d fz,gw } 2.47 d fw, fz+λ d fw, fz+d fw,gw, that is 1 λd fw,gw 1 + λd fw, fz Let us remark that d fw, fz d fw,gw+dgw,gz d fw,gw+λ Mw,z d fw,gw+λ max { d fw, fz,d fw,gw,d fz,gw } 2.49 d fw,gw+λ d fw, fz+d fw,gw, that is By 2.48and2.50weobtain 1 λd fw, fz 1 + λd fw,gw Clearly 2.51 implies λd fw,gw 1 + λd fw, fz 1 λ λ 2 d fw,gw Remark 2.3. Let K,ρ be a bounded metric space. It is said that the fixed point problem for a mapping A : K K is well posed if there exists a unique x A K such that Ax A = x A and the following property holds: If {x n } K and ρx n,ax n 0asn, then ρx n,x A 0asn. Let us remark that condition 2.44 is related to the notion

9 L. GajićandV.Rakočević 373 of well posed fixed point problem, and the notion of well-posedness is of central importance in many areas of Mathematics and its applications [4, 10, 13]. Remark 2.4. If in Theorem 2.1 we let f be the identity map on X and ωr = λ r where 0 <λ<1, we get Ćirić s Theorem 1.1 Gajić s theorem [5] stated for a Banach convex complete metric space X. Remark 2.5. If in Theorem 2.1 we let f be the identity map on X and C = X, weget Ivanov s result [6, 7]stated for a Banach space X. Remark 2.6. Let us recall that the first part of Theorem 2.2, that is the existence of the unique common fixed point of f and g was proved by Rakočević[12]. By the proof of Theorem 2.1 we can recover some results of Das and Naik [3] and Jungck [8]. Corollary 2.7 [3, Theorem 2.1]. Let X be a complete metric space. Let f be a continuous self-map on X and g be any self-map on X that commutes with f. Further let f and g satisfy and there exists a constant λ 0,1 such that for every x, y X where gx f X 2.52 dgx,gy λ Mx, y, 2.53 Mx, y = max { d fx, fy,d fx,gx,d fy,gy,d fx,gy,d fy,gx } Then f and g haveauniquefixedpoint. Proof. We follow the proof of Theorem 2.1. Let us remark that the condition 2.52 implies that starting with an arbitrary x 0 X, weconstructasequence{x n } of points in X such that f x n+1 = gx n, n = 0,1,2,... The rest of the proof follows by the proof of Theorem 2.1. Corollary 2.8 [3, Theorem 3.1]. Let X be a complete metric space. Let f 2 be a continuous self-map on X and g be any self-map on X that commutes with f. Further let f and g satisfy gfx f 2 X 2.55 and f gx = g f x whenever both sides are defined. Further, let there exist a constant λ 0,1 such that for every x, y f X where dgx,gy λ Mx, y, 2.56 Mx, y = max { d fx, fy,d fx,gx,d fy,gy,d fx,gy,d fy,gx } Then f and g haveauniquecommonfixedpoint.

10 374 Quasicontraction nonself-mappings Proof. Again, we follow the proof of Theorem 2.1. By2.55 starting with an arbitrary x 0 f X, we construct a sequence {x n } of points in f X suchthat f x n+1 = gx n = y n, n = 0,1,2,...Now f y n = f gx n = g f x n = gy n 1 = z n, n = 1,2,...,andfrom the proof of Theorem 2.1 we conclude that {z n } is a Cauchy sequence in X and hence convergent to some z X.Now,foreachn 1 d f 2 g x n,gfz = d gf 2 x n,gfz λ M f 2 x n, f z = λ max { d f 2 f x n, f 2 z,d f 2 f x n, f 2 g x n, Now, by continuity of f 2 d f 2 z,gfz,d f 2 f x n,gfz,d f 2 z, f 2 g x n } d f 2 z,gfz λ d f 2 z,gfz Whence, f 2 z = gfz, and gfzisauniquecommonfixedof f and g. Let us remark that from Theorem 2.1 and the proof of Corollary 2.7, wegetthefollowing. Corollary 2.9. Let X be a complete metric space. Let f be a continuous self-map on X and g be any self-map on X that weakly commutes with f. Further let f and g satisfy 2.52and Then f and g haveauniquecommonfixedpoint. Now as a corollary we get the following result of Jungck [8]. Corollary Let X be a complete metric space. Let f be a continuous self-map on X and g be any self-map on X that commutes with f. Further let f and g satisfy 2.52 and there exists a constant λ 0,1 such that for every x, y X Then f and g haveauniquecommonfixedpoint. dgx,gy λ d fx, fy Corollary Let X be a convex complete metric space, C a nonempty compact subset of X,and C the boundary of C.Letg : C X, f : X X and f : C C.Supposethatg and f are continuous, f and g satisfy the conditions ii v in Theorem 2.1,andforallx, y C, x y where dgx,gy <Mx, y, 2.61 Mx, y = max { d fx, fy,d fx,gx,d fy,gy,d fx,gy,d fy,gx } Then f and g haveauniquecommonfixedpointinc. Proof. By Theorem 2.2 andtheproofof[12,theorem4].

11 Acknowledgment L. GajićandV.Rakočević 375 The authors are grateful to the referees for some helpful comments and suggestions. References [1] Lj. B. Ćirić, A generalization of Banach s contraction principle, Proc. Amer. Math. Soc , [2], Quasi contraction non-self mappings on Banach spaces, Bull.Cl.Sci.Math.Nat.Sci. Math , [3] K.M.DasandK.V.Naik,Common fixed-point theorems for commuting maps on a metric space, Proc. Amer. Math.Soc , no. 3, [4] F.S.DeBlasiandJ.Myjak,Sur la porosité de l ensemble des contractions sans point fixe [On the porosity of the set of contractions without fixed points],c.r.acad.sci.parissér. I Math , no. 2, French. [5] Lj. Gajić, Quasi-contractive nonself mappings on Takahashi convex metric spaces, NoviSadJ. Math , no. 3, [6] O. Hadžić, Foundations of Fixed Point Theory, Institut za Matematiku, Novi Sad, [7] A.A.Ivanov, Fixed points of mappings of metric spaces. Studies in topology, II,Zap.Naučn. Sem. Leningrad. Otdel. Mat. Inst.Steklov. LOMI , 5 102, 207 Russian. [8] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly , no. 4, [9], Compatible mappings and common fixed points, Int. J. Math. Math. Sci , no. 4, [10] W. A. Kirk and L. M. Saliga, Some results on existence and approximation in metric fixed point theory, J. Comput. Appl. Math , no. 1-2, [11] V.Rakočević, Funkcionalna analiza,naučna knjiga, Beograd,1994. [12], Quasi contraction nonself mappings on Banach spaces and common fixed point theorems, Publ. Math. Debrecen , no. 3, [13] S. Reich and A. J. Zaslavski, Well-posedness of fixed point problems,fareastj.math.sci.fjms, 2001, Special Volume, Part III, [14] S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. Beograd N.S , [15] W. Takahashi, A convexity in metric space and nonexpansive mappings. I, Kōdai Math. Sem. Rep , Ljiljana Gajić: Institute of Mathematics, Faculty of Science, University of Novi Sad, Trg D. Obradovića 4, Novi Sad, Serbia and Montenegro address: gajic@im.ns.ac.yu Vladimir Rakočević: Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, Višegradska33, Niš, Serbia and Montenegro address: vrakoc@bankerinter.net