A coincidence point theorem for two pairs of mappings is proved

B. Fisher

D. Türkoğlu

O. Özer

Türkoğlu, D.

Özer, O.

Fisher, B.

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Mathematical Communications 7(2002), 39-44 39A coincidence point theorem for multi-valuedcontractions∗Duran Tu¨rkog˘lu†, Orhan O¨zer‡ and Brian Fisher§Abstract. A coincidence point theorem for two pairs of mappingsis proved.Key words: coincidence point, multi-valued mapping, weakly com-muting mappings, compatible mappingsAMS subject classifications: 54H25Received February 6, 2002 Accepted May 15, 20021. Introduction and preliminariesLet (X, d) be a metric space and let f and g be mappings from X into itself. In [5],S. Sessa defined f and g to be weakly commuting ifd(gfx, fgx) ≤ d(gx, fx)for all x ∈ X . It can be seen that two commuting mappings are weakly commuting,but the converse is false as shown in the example of [5].Recently, Jungck [1] extended the concept of weak commutativity in the follow-ing way:Definition 1. Let f and g be mappings from a metric space (X, d) into itself.The mappings f and g are said to be compatible iflimn→∞(fgxn, gfxn) = 0whenever {xn} is a sequence in X such that limn→∞ fxn = limn→∞ gxn = z forsome z in X.It is obvious that two weakly commuting mappings are compatible, but theconverse is not true, see the examples in [1].∗This research was supported by TUBITAK project No.TBAG-1742 (1999).†Department of Mathematics, Kırıkkale University, Kırıkkale, Turkey, e-mail:turkoglu@sinanoglu.kku.edu.tr‡Department of Mathematics, Anadolu University, Eskisehir, Turkey, e-mail:oozer@vm.baum.anadolu.edu.tr§Department of Mathematics and Computer Science, Leicester University, Leicester, LE1 7RH,England, e-mail: fbr@le.ac.uk40 D.Tu¨rkog˘lu, O. O¨zer and B. FisherRecently, Kaneko [2] and Singh et al. [6] extended the concepts of weak com-mutativity and compatibility, see Kaneko et al. [3], for single-valued mappings tothe setting of single-valued and multi-valued mappings, respectively.Now let (X, d) be a metric space and let CB(X) denote the family of all non-empty closed and bounded subsets of X . Let H be the Hausdorff metric on CB(X)induced by the metric d, i.e.,H(A,B) = max{supx∈Ad(x,B), supy∈Bd(y,A)}for A,B ∈ CB(X), where d(x,A) = infy∈A d(x, y).It is well-known that (CB(X), H) is a metric space, and if (X, d) is complete,then (CB(X), H) is also complete.The following lemma was proved in Nadler [4].Lemma 1. Let A,B ∈ CB(X) and k > 1. Then for each a ∈ A, there exists apoint b ∈ B such that d(a, b) ≤ kH(A,B).Definition 2. Let (X, d) be a metric space and let f : X → X and S : X →CB(X) be single-valued and multi-valued mappings, respectively. The mappings fand S are said to be weakly commuting if for all x ∈ X, fSx ∈ CB(X) andH(Sfx, fSx) ≤ d(fx, Sx),where H is the Hausdorff metric defined on CB(X).Definition 3. The mappings f and S are said to be compatible iflimn→∞ d(fyn, Sfxn) = 0whenever {xn} and {yn} are sequences in X such that limn→∞ fxn = limn→∞ yn =z for some z ∈ X, where yn ∈ Sxn for n = 1, 2, . . . .Remark 1.(i) Definition 3 is slightly different from Kaneko’s definition [2].(ii) If S is a single-valued mapping on X in Definitions 2 and 3, then Defini-tions 2 and 3 become the definitions of weak commutativity and compatibilityfor single-valued mappings.(iii) If the mappings f and S are weakly commuting, then they are compatible, butthe converse is not true.In fact, suppose that f and S are weakly commuting and let {xn} and {yn}be two sequences in X such that yn ∈ Sxn for n = 1, 2, . . . and limn→∞ fxn =limn→∞ yn = z for some z ∈ X. From d(fxn, Sxn) ≤ d(fxn, yn), it follows thatlimn→∞ d(fxn, Sxn) = 0. Thus, since f and g are weakly commuting, we havelimn→∞H(Sfxn, fSxn) = 0.On the other hand, since d(fyn, Sfxn) ≤ H(fSxn, Sfxn), we havelimn→∞ d(fyn, Sfxn) = 0,A coincidence point theorem for multi-valued contractions 41which means that f and S are compatible.Example 1. Let X = [1,∞) be set with the Euclidean metric d and definefx = 2x4 − 1 and Sx = [1, x2] for all x ≥ 1. Note that f and S are continuous andS(X) = f(X) = X. Let {xn} and {yn} be sequences in X defined by xn = yn = 1for n = 1, 2, . . . . Then we havelimn→∞ fxn = limn→∞ yn = 1 ∈ X, where yn ∈ Sxn.On the other hand, we can show that H(fSxn, Sfxn) = 2(x4n − 1)2 → 0 if andonly xn → 1 as n→∞ and so, since d(fyn, Sfxn) ≤ H(fSxn, Sfxn), we havelimn→∞ d(fyn, Sfxn) = 0.Therefore, f and T are compatible, but f and T are not weakly commuting at x = 2.2. Main resultsTheorem 1. Let (X, d) be a complete metric space. Let f, g : X → X be continuousmappings and S, T : X → CB(X) be H-continuous mappings such that T (X) ⊆f(X) and S(X) ⊆ g(X), the pair S and g are compatible mappings andHp(Sx, T y) ≤ max{ad(fx, gy)dp−1(fx, Sx), ad(fx, gy)dp−1(gy, T y),ad(fx, Sx)dp−1(gy, T y), cdp−1(fx, T y)d(gy, Sx)} (1)for all x, y ∈ X, where p ≥ 2 is an integer, 0 < a < 1 and c ≥ 0. Then there existsa point z ∈ X such that fz ∈ Sz and gz ∈ Tz, i.e., z is a coincidence point of f, Sand of g, T . Further, z is unique when 0 < c < 1.Proof. Let x0 be an arbirary point in X . Since Sx0 ⊆ g(X), there exists apoint x1 ∈ X such that gx1 ∈ Sxo and so there exists a point y ∈ Tx1d(gx1, y) ≤ kH(Sx0, Tx1),where k = a−1/2 > 1, which is possible by Lemma1. Since Tx1 ⊆ f(X), thereexists a point x2 ∈ X such that y = fx2 and so we haved(gx1, fx2) ≤ kH(Sx0, Tx1).Similarly, there exists a point x3 ∈ X such that gx3 ∈ Sx2 andd(gx3, fx2) ≤ kH(Sx2, Tx1).Inductively, we can obtain a sequence {xn} in X such thatfx2n ∈ Tx2n, n ∈ N,gx2n+1 ∈ Sx2n, n ∈ N0 = N ∪ {0},d(gx2n+1, fx2n) ≤ kH(Sx2n, Tx2n−1), n ∈ N,d(gx2n+1, fx2n) ≤ kH(Sx2n, Tx2n+1), n ∈ N0,42 D.Tu¨rkog˘lu, O. O¨zer and B. Fisherwhere N denotes the set of positive integers. Then, by (1), we havedp(gx2n+1, fx2n+2) ≤ kpHp(Sx2n, Tx2n+1)≤ a−p/2 max{ad(fx2n, gx2n+1)dp−1(fx2n, Sx2n),ad(fx2n, gx2n+1)dp−1(gx2n+1, Tx2n+1),ad(fx2n, Sx2n)dp−1(gx2n+1, Tx2n+1),cdp−1(fx2n, Tx2n+1)dp−1(gx2n+1, fx2n+2)}≤ a−p/2 max{ad(fx2n, gx2n+1)dp−1(fx2n, gx2n+1),ad(fx2n, gx2n+1)dp−1(gx2n+1, fx2n+2),ad(fx2n, Sx2n)dp−1(gx2n+1, fx2n+2),cdp−1(fx2n, fx2n+2)dp−1(gx2n+1, gx2n+1)}.Putting a−p/2 = β, we havedp(gx2n+1, fx2n+2) ≤ βmax{ad(fx2n, gx2n+1)dp−1(fx2n, gx2n+1),ad(fx2n, gx2n+1)dp−1(gx2n+1, fx2n+2),ad(fx2n, Sx2n)dp−1(gx2n+1, fx2n+2),cdp−1(fx2n, fx2n+2)dp−1(gx2n+1, gx2n+1)},≤ βad(fx2n, gx2n+1)dp−1(fx2n, gx2n+1),dp(gx2n+1, fx2n+2) ≤ βnand(x0, gx1).Since 0 < β < 1, it follows that{gx1, fx2, gx3, fx4, . . . , gx2n−1, fx2n, gx2n+1, . . . }is a Cauchy sequence in X . Since (X, d) is a complete metric space, letlimn→∞ gx2n+1 = limn→∞ fx2n = z.Now, we will prove that z is a coincidence point of f and S. For every n ∈ N ,we haved(fgx2n+1, Sz) ≤ d(fgx2n+1, Sfx2n) +H(Sfx2n, Sz). (2)It follows from the H-continuity of S thatlimn→∞H(Sfx2n, Sz) = 0, (3)since fx2n → z as n→∞. Since f and S are compatible mappings andlimn→∞ fx2n = limn→∞ yn = z,where yn = gx2n+1 ∈ Sx2n and zn = x2n, we havelimn→∞ d(fyn, Sfzn) = limn→∞ d(fgx2n+1, Sfx2n) = 0. (4)A coincidence point theorem for multi-valued contractions 43Thus, from equations (2), (3) and (4), we havelimn→∞ d(fgx2n+1, Sz) = 0and sod(fz, Sz) ≤ d(fz, fgx2n+1) + d(fgx2n+1, Sz).Letting n tend to infinity, it follows that d(fz, Sz) = 0. This implies that fz ∈ Sz,since Sz is a closed subset of X . Thus z is a coincidence point of f and S. Similarly,we can prove that z is a coincidence point of g and T . This completes the proof ofthe theorem. ✷Letting f = g be the identity mapping onX, in Theorem 1, we have the followingcorollary:Corollary 1. Let (X, d) be a complete metric space and let S, T : X → CB(X)be H-continuous multi-valued mappings such thatHp(Sx, T y) ≤ max{ad(x, y)dp−1(y, T y), ad(x, y)dp−1(y, T y),ad(x, Sx)dp−1(y, T y), cdp−1(x, T y)d(y, Sx)}for all x, y ∈ X, where p ≥ 2, 0 < a < 1, c > 0. Then S and T have a commonfixed point z in X.Putting f = g and S = T in Theorem 1, we have the following corollary:Corollary 2. Let (X, d) be a complete metric space, let f : X → X be acontinuous mapping and let S : X → CB(X) be an H-continuous mapping suchthat S(X) ⊂ g(X), andHp(Sx, Sy) ≤ max{ad(fx, fy)dp−1(fx, Sx), ad(fx, fy)dp−1(gy, Sy),cdp−1(fx, Sy)d(fy, Sx)}for all x, y ∈ X, where p ≥ 2 is an integer, 0 < a < 1 and c ≥ 0. Then there existsa coincidence point z of f and S.References[1] G. Jungck, Compatible mappings and common fixed points, Internat. J. Math.Math. Sci. 9(1986), 771-773.[2] H.Kaneko, A common fixed point of weakly commuting multi-valued map-pings, Math. Japon. 33(1988), 741-744.[3] H.Kaneko, S. Sessa, Fixed point theorems for compatible multi-valued andsingle-valued mappings, Internat. J. Math. Math. Sci. 12(1989), 257-262.[4] S. B.Nadler, Multivalued contraction mappings, Pacific J. Math. 30(1969),475-488.44 D.Tu¨rkog˘lu, O. O¨zer and B. Fisher[5] S. Sessa,On a weak commutativity condition of mappings in fixed point con-siderations, Publ. Inst. Math. (Beograd) 32(1982), 146-153.[6] S. L. Singh, K. S.Ha, Y. J. Cho, Coincidence and fixed points of nonliearhybrid contractions, Internat. J. Math. Math. Sci. 12(1979), 247-256.

A coincidence point theorem for multi-valued contractions

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