FIXED POINT THEOREMS FOR A FAMILY OF HYBRID PAIRS OF MAPPINGS IN METRICALLY CONVEX SPACES M. IMDAD AND LADLAY KHAN Received 30 December 2004 and in revised form 24 March 2005 The present paper establishes some coincidence and common fixed point theorems for a sequence of hybrid-type nonself-mappings defined on a closed subset of a metrically convex metric space. Our results generalize some earlier results due to Khan et al. (2000), Itoh (1977), Khan (1981), Ahmad and Imdad (1992 and 1998), and several others. Some related results are also discussed. 1. Introduction In recent years several fixed point theorems for hybrid pairs of mappings are proved and by now there exists considerable literature in this direction. To mention a few, one can cite Imdad and Ahmad [10], Pathak [19], Popa [20] and references cited therein. On the other hand Assad and Kirk [4]gaveasufficient condition enunciating fixed point of set-valued mappings enjoying specific b oundary condition in metrically convex metric spaces. In the current years the work due to Assad and Kirk [4] has inspired extensive activities whichincludesItoh[12], Khan [14],AhmadandImdad[1, 2], Imdad et al. [11]and some others. Most recently, Huang and Cho [9]andDhageetal.[6] proved some fixed point theo- rems for a sequence of set-valued mappings which generalize several results due to Itoh [12], Khan [14], Ahmad and Khan [3] and others. The purpose of this paper is to prove some coincidence and common fixed point theorems for a sequence of hybrid type non- self mappings satisfying certain contraction type condition which is essentially patterned after Khan et al. [15]. Our results either partially or completely generalize earlier results duetoKhanetal.[15], Itoh [12], Khan [14], Ahmad and Imdad [1, 2], Ahmad and Khan [3] and several others. 2. Preliminaries Before proving our results, we collect the relevant definitions and results for our future use. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 281–294 DOI: 10.1155/FPTA.2005.281 282 Hybrid fixed point theorems in metrically convex spaces Let (X,d) be a metric space. Then following Nadler [17], we recall (i) CB(X) = {A : A is nonempty closed and bounded subset of X}. (ii) C(X) = {A : A is nonempt y compact subset of X}. (iii) For nonempty subsets A, B of X and x ∈ X, d(x,A) = inf  d(x,a):a ∈ A  , H(A,B) = max  supd(a,B):a ∈ A  ,  supd(A,b):b ∈ B  . (2.1) It is well known (cf. Kuratowski [16]) that CB(X) is a metric space with the distance H which is known as Hausdorff-Pompeiu metric on X. The following definitions and lemmas will be frequently used in the sequel. Definit ion 2.1. Let K be a nonempty subset of a metric space (X, d), T : K → X and F : K → CB(X). The pair (F,T) is said to be pointwise R-weakly commuting on K if for given x ∈ K and Tx ∈ K, there exists some R = R(x) > 0suchthat d(Ty,FTx) ≤ R · d(Tx,Fx)foreachy ∈ K ∩ Fx. (2.2) Moreover, the pair (F,T)willbecalledR-weakly commuting on K if (2.2)holdsfor each x ∈ K, Tx ∈ K with some R>0. If R = 1, we get the definition of weak commutativity of (F, T)onK due to Hadzic and Gajic [8]. For K = X Definition 2.1 reduces to “pointwise R-weak commutativity and R-weak commutativity” for single valued self mappings due to Pant [18]. Definit ion 2.2 [7, 8]. Let K beanonemptysubsetofametricspace(X,d), T : K → X and F : K → CB(X). The pair (F,T)issaidtobeweaklycommuting(cf.[7]) if for every x, y ∈ K with x ∈ Fyand Ty∈ K,wehave d(Tx,FTy) ≤ d(Ty,Fy), (2.3) whereas the pair (F,T) is said to be compatible (cf. [8]) if for every sequence {x n }⊂K, from the relation lim n→∞ d  Fx n ,Tx n  = 0 (2.4) and Tx n ∈ K (for every n ∈ N) it follows that lim n→∞ d(Ty n ,FTx n ) = 0, for every se- quence {y n }⊂K such that y n ∈ Fx n , n ∈ N. For hybrid pairs of self type mappings these definitions were introduced by Kaneko and Sessa [13]. Definit ion 2.3 [11]. Let K beanonemptysubsetofametricspace(X,d), T : K → X and F : K → CB(X). The pair (F,T) is said to be quasi-coincidentally commuting if for all coincidence points “x”of(T,F), TFx ⊂ FTx whenever Fx ⊂ K and Tx ∈ K for all x ∈ K. M. Imdad and L. Khan 283 Definit ion 2.4 [11]. A mapping T : K → X is said to be coincidentally idempotent w.r.t mapping F : K → CB(X), if T is idempotent at the coincidence points of the pair (F,T). Definit ion 2.5 [4]. A metric space (X,d)issaidtobemetricallyconvexifforanyx, y ∈ X with x = y there exists a point z ∈ X, x = z = y such that d(x,z)+d(z, y) = d(x, y). (2.5) Lemma 2.6 [4]. Let K be a nonempty clos e d subset of a metrically convex metric space (X,d).Ifx ∈ K and y/∈ K then there exists a point z ∈ δK (the boundary of K) such that d(x,z)+d(z, y) = d(x, y). Lemma 2.7 [17]. Let A,B ∈ CB(X) and a ∈ A,thenforanypositivenumberq<1 there exists b = b(a) in B such that q · d(a,b) ≤ H( A, B). 3. Main results Our main result runs as follows. Theorem 3.1. Let (X, d) be a complete metrically convex metric space and K anonempty closed subset of X.Let{F n } ∞ n=1 : K → CB(X) and S,T : K → X satisfying (iv) δK ⊆ SK ∩ TK, F i (K) ∩ K ⊆ SK, F j (K) ∩ K ⊆ TK, (v) Tx ∈ δK ⇒ F i (x) ⊆ K, Sx ∈ δK ⇒ F j (x) ⊆ K,and H  F i (x), F j (y)  ≤ a · max  1 2 d(Tx,Sy),d  Tx,F i (x)  ,d  Sy,F j (y)   + b  d  Tx,F j (y)  + d  Sy,F i (x)  , (3.1) where i = 2n − 1, j = 2n, (n ∈ N), i = j for all x, y ∈ K with x = y, a,b ≥ 0,and 2b<a, 2a +3b<q<1, (vi) (F i ,T) and (F j ,S) are compatible pairs, (vii) {F n }, S and T are continuous on K. Then (F i ,T) as well as (F j ,S) has a point of coincidence. Proof. Firstly, we proceed to construct two sequences {x n } and {y n } in the following way. Let x ∈ δK.Then(duetoδK ⊆ TK) there exists a point x 0 ∈ K such that x = Tx 0 . From the i mplication Tx ∈ δK which implies F 1 (x 0 ) ⊆ F 1 (K) ∩ K ⊆ SK,letx 1 ∈ K be such that y 1 = Sx 1 ∈ F 1 (x 0 ) ⊆ K.Sincey 1 ∈ F 1 (x 0 ), there exists a point y 2 ∈ F 2 (x 1 )such that q · d  y 1 , y 2  ≤ H  F 1  x 0  ,F 2  x 1  . (3.2) Suppose y 2 ∈ K.Theny 2 ∈ F 2 (K) ∩ K ⊆ TK implies that there exists a point x 2 ∈ K such that y 2 = Tx 2 . Otherwise, if y 2 /∈ K, then there exists a point p ∈ δK such that d  Sx 1 , p  + d  p, y 2  = d  Sx 1 , y 2  . (3.3) 284 Hybrid fixed point theorems in metrically convex spaces Since p ∈ δK ⊆ TK, there exists a point x 2 ∈ K with p = Tx 2 so that d  Sx 1 ,Tx 2  + d  Tx 2 , y 2  = d  Sx 1 , y 2  . (3.4) Let y 3 ∈ F 3 (x 2 )besuchthatq · d(y 2 , y 3 ) ≤ H(F 2 (x 1 ),F 3 (x 2 )). Thus, repeating the foregoing arguments, we obtain two sequences {x n } and {y n } such that (viii) y 2n ∈ F 2n (x 2n−1 ), y 2n+1 ∈ F 2n+1 (x 2n ), (ix) y 2n ∈ K ⇒ y 2n = Tx 2n or y 2n /∈ K ⇒ Tx 2n ∈ δK and d  Sx 2n−1 ,Tx 2n  + d  Tx 2n , y 2n  = d  Sx 2n−1 , y 2n  , (3.5) (x) y 2n+1 ∈ K ⇒ y 2n+1 = Sx 2n+1 or y 2n+1 /∈ K ⇒ Sx 2n+1 ∈ δK and d  Tx 2n ,Sx 2n+1  + d  Sx 2n+1 , y 2n+1  = d  Tx 2n , y 2n+1  . (3.6) We denote P ◦ =  Tx 2i ∈  Tx 2n  : Tx 2i = y 2i  , P 1 =  Tx 2i ∈  Tx 2n  : Tx 2i = y 2i  , Q ◦ =  Sx 2i+1 ∈  Sx 2n+1  : Sx 2i+1 = y 2i+1  , Q 1 =  Sx 2i+1 ∈  Sx 2n+1  : Sx 2i+1 = y 2i+1  . (3.7) One can note that (Tx 2n ,Sx 2n+1 ) ∈ P 1 × Q 1 and (Sx 2n−1 ,Tx 2n ) ∈ Q 1 × P 1 .  Now, we distinguish the following three cases. Case 1. If (Tx 2n ,Sx 2n+1 ) ∈ P ◦ × Q ◦ ,then q · d  Tx 2n ,Sx 2n+1  ≤ H  F 2n+1  x 2n  ,F 2n  x 2n−1  ≤ a · max  1 2 d  Tx 2n ,Sx 2n−1  ,d  Tx 2n ,F 2n+1  x 2n  ,d  Sx 2n−1 ,F 2n  x 2n−1   + b ·  d  Tx 2n ,F 2n  x 2n−1  + d  Sx 2n−1 ,F 2n+1  x 2n  ≤ a · max  1 2 d  y 2n , y 2n−1  ,d  y 2n , y 2n+1  ,d  y 2n−1 , y 2n   + b ·  d  y 2n−1 , y 2n  + d  y 2n , y 2n+1  , (3.8) M. Imdad and L. Khan 285 which in turn yields d  Tx 2n ,Sx 2n+1  ≤                 a + b q − b  d  Sx 2n−1 ,Tx 2n  ,ifd  y 2n−1 , y 2n  ≥ d  y 2n+1 , y 2n   b q − b −a  d  Sx 2n−1 ,Tx 2n  ,ifd  y 2n−1 , y 2n  ≤ d  y 2n+1 , y 2n  , (3.9) or d  Tx 2n ,Sx 2n+1  ≤ h · d  Sx 2n−1 ,Tx 2n  , (3.10) where h = max{((a + b)/(q − b)),(b/(q − b − a))} < 1, since 2a +3b<1. Similarly if (Sx 2n−1 ,Tx 2n ) ∈ Q ◦ × P ◦ ,then d  Sx 2n−1 ,Tx 2n  ≤                 a + b q − b  d(Sx 2n−1 ,Tx 2n−2 ), if d  y 2n−2 , y 2n−1  ≥ d  y 2n−1 , y 2n   b q − b −a  d  Sx 2n−1 ,Tx 2n−2  ,ifd  y 2n−2 , y 2n−1  ≤ d  y 2n−1 , y 2n  , (3.11) or d  Sx 2n−1 ,Tx 2n  ≤ h · d  Sx 2n−1 ,Tx 2n−2  , (3.12) where h = max{((a + b)/(q − b)),(b/(q − b − a))} < 1, since 2a +3b<1. Case 2. If (Tx 2n ,Sx 2n+1 ) ∈ P ◦ × Q 1 ,then d  Tx 2n ,Sx 2n+1  + d  Sx 2n+1 , y 2n+1  = d  Tx 2n , y 2n+1  , (3.13) which in turn yields d  Tx 2n ,Sx 2n+1  ≤ d  Tx 2n , y 2n+1  = d  y 2n , y 2n+1  , (3.14) and hence q · d  Tx 2n ,Sx 2n+1  ≤ q · d  y 2n , y 2n+1  ≤ H  F 2n+1  x 2n  ,F 2n  x 2n−1  . (3.15) 286 Hybrid fixed point theorems in metrically convex spaces Now, proceeding as in Case 1,wehave d  Tx 2n ,Sx 2n+1  ≤                 a + b q − b  d  Sx 2n−1 ,Tx 2n ), if d  y 2n−1 , y 2n  ≥ d  y 2n+1 , y 2n   b q − b −a  d  Sx 2n−1 ,Tx 2n  ,ifd  y 2n−1 , y 2n  ≤ d  y 2n+1 , y 2n  , (3.16) or d  Tx 2n ,Sx 2n+1  ≤ h · d  Sx 2n−1 ,Tx 2n  . (3.17) In case (Sx 2n−1 ,Tx 2n ) ∈ Q 1 × P ◦ , then as earlier, one also obtains d  Sx 2n−1 ,Tx 2n  ≤                 a + b q − b  d  Sx 2n−1 ,Tx 2n−2  ,ifd  y 2n−2 , y 2n−1  ≥ d  y 2n−1 , y 2n   b q − b −a  d  Sx 2n−1 ,Tx 2n−2  ,ifd  y 2n−2 , y 2n−1  ≤ d  y 2n−1 , y 2n  , (3.18) or d  Sx 2n−1 ,Tx 2n  ≤ h · d  Sx 2n−1 ,Tx 2n−2  , (3.19) where h = max{((a + b)/(q − b)),(b/(q − b − a))} < 1, since 2a +3b<1. Case 3. If (Tx 2n ,Sx 2n+1 ) ∈ P 1 × Q ◦ ,thenSx 2n−1 = y 2n−1 . Proceeding as in Case 1, one gets q · d  Tx 2n ,Sx 2n+1  = q · d  Tx 2n , y 2n+1  ≤ q · d  Tx 2n , y 2n  + q · d  y 2n , y 2n+1  ≤ q · d  Sx 2n−1 , y 2n  + H  F 2n+1 (x 2n  ,F 2n  x 2n−1  ≤ q · d  Sx 2n−1 , y 2n  + a · max  1 2 d  y 2n , y 2n−1  ,d  y 2n , y 2n+1  ,d  y 2n−1 , y 2n   + b  d  y 2n , y 2n  + d  y 2n−1 , y 2n+1  , (3.20) which in turn yields d  Tx 2n ,Sx 2n+1  ≤                 q + b q − a− b  d  Sx 2n−1 , y 2n  ,ifd  y 2n−1 , y 2n  ≤ d  y 2n+1 , y 2n   q + a+ b q − b  d  Sx 2n−1 , y 2n  ,ifd  y 2n−1 , y 2n  ≥ d  y 2n+1 , y 2n  . (3.21) M. Imdad and L. Khan 287 Now, proceeding as earlier, one also obtains d  Sx 2n−1 , y 2n  ≤                 a + b q − b  d  Sx 2n−1 ,Tx 2n−2  ,ifd  y 2n−2 , y 2n−1  ≥ d  y 2n−1 , y 2n   b q − a− b  d  Sx 2n−1 ,Tx 2n−2  ,ifd  y 2n−2 , y 2n−1  ≤ d  y 2n−1 , y 2n  . (3.22) Therefore combining above inequalities, we have d  Tx 2n ,Sx 2n+1  ≤ k · d  Sx 2n−1 ,Tx 2n−2  , (3.23) where k = max  a + b q − b  q + b q − a− b  ,  a + b q − b  q + a+ b q − b  ,  b q − a− b  q + b q − a− b  ,  b q − a− b  q + a+ b q − b  < 1, (3.24) since 2a +3b<1. To substantiate that, the inequality 2a +3b<q<1 implies all foregoing inequalities, onemaynotethat 2a +3b<q =⇒ 2aq +3bq < q 2 , (3.25) or aq + ab + bq + b 2 + aq +2bq − ab − b 2 <q 2 , (3.26) or aq + ab + bq + b 2 <q 2 − aq − 2bq + ab + b 2 , (3.27) or  a + b q − b  q + b q − a − b  < 1, (3.28) and 2a +3b<q=⇒ a +3b<q, (3.29) or aq +3bq < q 2 =⇒ aq + bq + bq + bq < q 2 , (3.30) 288 Hybrid fixed point theorems in metrically convex spaces or bq + ab + b 2 <q 2 − bq − aq + ab − bq + b 2 , (3.31) or  b q − a − b  q + a + b q − b  < 1. (3.32) Similarly one can establish the other inequalities as well. Thus in all the cases, we have d  Tx 2n ,Sx 2n+1  ≤ k · max  d  Sx 2n−1 ,Tx 2n  ,d  Tx 2n−2 ,Sx 2n−1  (3.33) whereas d  Sx 2n+1 ,Tx 2n+2  ≤ k · max  d  Sx 2n−1 ,Tx 2n  ,d  Tx 2n ,Sx 2n+1  . (3.34) Now on the lines of Assad and Kirk [4], it can be shown by induction that for n ≥ 1, we have d  Tx 2n ,Sx 2n+1  <k n · δ, d  Sx 2n+1 ,Tx 2n+2  <k n+(1/2) · δ (3.35) whereas δ = k −1/2 max  d  Tx 0 ,Sx 1  ,d  Sx 1 ,Tx 2  . (3.36) Thus the sequence {Tx 0 ,Sx 1 ,Tx 2 ,Sx 3 , ,Sx 2n−1 ,Tx 2n ,Sx 2n+1 , } is Cauchy and hence converges to the point z in X. Then as noted in [7] there exists at least one subsequence {Tx 2n k } or {Sx 2n k +1 } which is contained in P ◦ or Q ◦ respectively. Suppose that the sub- sequence {Tx 2n k } contained in P ◦ for e ach k ∈ N converges to z. Using compatibility of (F j ,S), we have lim k→∞ d  Sx 2n k −1 ,F j  x 2n k −1  = 0foranyevenintegerj ∈ N, (3.37) which implies that lim k→∞ d(STx 2n k ,F j (Sx 2n k −1 )) = 0. Using the continuity of S and F j , one obtains Sz ∈ F j (z), for any even integer j ∈ N. Similarly the continuit y of T and F i implies Tz ∈ F i (z), for any odd integer i ∈ N.Now q · d(Tz,Sz) ≤ H  F i (z), F j (z)  ≤ a · max  1 2 d(Tz,Sz),d  Tz,F i (z)  ,d  Sz, F j (z)   + b  d  Tz,F j (z)  + d  Sz, F i (z)  ≤ a · max  1 2 d(Tz,Sz),0,0  + b  d(Tz,Sz)+d(Tz,Sz)  ≤  a 2 +2b  · d(Tz,Sz), (3.38) M. Imdad and L. Khan 289 yielding thereby Tz = Sz which shows that z is a common coincidence point of the maps {F n }, S and T. Remark 3.2. By setting F i = F (for any odd integer i ∈ N)andF j = G (for any even integer j ∈ N)inTheorem 3.1, one deduces a rectified and sharpened form of a result due to Ahmad and Imdad [2]. Remark 3.3. By setting F i = F (for any odd integer i ∈ N), F j = G (for any even integer j ∈ N)andS = T in Theorem 3.1, one deduces a rectified and improved version of a result due to Ahmad and Imdad [1]. In an attempt to prove Theorem 3.1 for pointwise R-weakly commuting mappings, we have the following. Theorem 3.4. Let (X, d) be a complete metrically convex metric space and K anonempty closed subset of X.Let{F n } ∞ n=1 : K → CB(X) and S,T : K → X satisfying (3.1), (iv), (v) and (vii). Suppose that (xi) (F i ,T) and (F j ,S) are pointwise R-weakly commuting pairs. Then (F i ,T) as well as (F j ,S) has a point of coincidence. Proof. On the lines of the proof of Theorem 3.1, one can show that the sequence {Tx 2n } converges to a point z ∈ X. Now we assume that there exists a subsequence {Tx 2n k } of {Tx 2n } which is contained in P ◦ . Further subsequence {Tx 2n k } and {Sx 2n k +1 } both converge to z ∈ K as K is a closed subset of the complete metric space (X,d). Since Tx 2n k ∈ F j (x 2n k −1 )foranyeveninteger j ∈ N and Sx 2n k −1 ∈ K. Using pointwise R-weak commutativity of (F j ,S), we have d  SF j  x 2n k −1  ,F j  Sx 2n k −1  ≤ R 1 · d  F j  x 2n k −1  ,Sx 2n k −1  (3.39) for any ev en integer j ∈ N with some R 1 > 0. Also d  SF j  x 2n k −1  ,F j (z)  ≤ d  SF j  x 2n k −1  ,F j  Sx 2n k −1  + H  F j  Sx 2n k −1  ,F j (z)  . (3.40) Making k →∞in (3.39)and(3.40) and using continuity of F j as well as S,wegetd(Sz, F j (z)) ≤ 0 yielding thereby Sz ∈ F j (z)foranyevenintegerj ∈ N. Since y 2n k +1 ∈ F i (x 2n k )and{Tx 2n k }∈K, pointwise R-weak commutativity of (F i ,T) implies d  TF i  x 2n k  ,F i  Tx 2n k  ≤ R 2 · d  F i  x 2n k  ,Tx 2n k  (3.41) for any odd integer i ∈ N with some R 2 > 0, besides d  TF i  x 2n k  ,F i (z)  ≤ d  TF i  x 2n k  ,F i  Tx 2n k  + H  F i  Tx 2n k  ,F i (z)  . (3.42) Therefore, as earlier the continuity of F i as well as T implies d(Tz,F i (z)) ≤ 0giving thereby Tz ∈ F i (z)ask →∞. If we assume that there exists a subsequence {Sx 2n k +1 } contained in Q ◦ ,thenanalogous arguments establish the earlier conclusions. This concludes the proof.  290 Hybrid fixed point theorems in metrically convex spaces In the next theorem, we utilize the closedness of TK and SK to replace the continuity requirements besides minimizing the commutativity requirements to merely coincidence points. Theorem 3.5. Let (X, d) be a complete metrically convex metric space and K anonempty closed subset of X.Let{F n } ∞ n=1 : K → CB(X) and S,T : K → X satisfying (3.1), (iv) and (v). Suppose that (xii) TK and SK are clos e d subspaces of X. Then ()(F i ,T) has a point of coincidence, ()(F j ,S) has a point of coincidence. Moreover , (F i ,T) has a common fixed point if T is quasi-coincidentally commuting and coincidentally idempotent w.r.t F i whereas (F j ,S) has a common fixed point provided S is quasi-coincidentally commuting and coincidentally idempotent w.r.t F j . Proof. On the lines of Theorem 3.1, one assumes that there exists a subsequence {Tx 2n k } which is contained in P ◦ and TK as well as SK are closed subspaces of X.Since{Tx 2n k } is Cauchy in TK, it converges to a point u ∈ TK.Letv ∈ T −1 u,thenTv = u.Since{Sx 2n k +1 } is a subsequence of Cauchy sequence, {Sx 2n k +1 } converges to u as well. Using (3.1), one can write q · d  F i (v),Tx 2n k  ≤ H  F i (v),F j  x 2n k −1  ≤ a · max  1 2 d  Tv,Sx 2n k −1  ,d  Sx 2n k −1 ,F j  x 2n k −1  ,d  Tv,F i (v)   + b  d  Tv,F j  x 2n k −1  + d  Sx 2n k −1 ,F i (v)  , (3.43) which on letting k →∞,reducesto q · d  F i (v),u  ≤ a · max  0,d  u,F i (v)  ,0  + b  0+d  F i (v),u  ≤ (a + b) · d  u,F i (v)  , (3.44) yielding thereby u ∈ F i (v) which implies that u = Tv ∈ F i (v)asF i (v)isclosed. Since Cauchy sequence {Tx 2n } con verges to u ∈ K and u ∈ F i (v), u ∈ F i (K) ∩ K ⊆ SK, there exists w ∈ K such that Sw = u. Again using (3.1), one gets q · d  Sw,F j (w)  = q · d  Tv,F j (w)  ≤ H  F i (v),F j (w)  ≤ a · max  1 2 d(Tv,Sw),d  Tv,F i (v)  ,d  Sw,F j (w)   + b  d  Tv,F j (w)  + d  Sw,F i (v)  ≤ (a + b) · d  Sw,F j (w)  , (3.45) implying thereby Sw ∈ F j (w), that is w is a coincidence point of (S,F j ). [...]... [13] A Ahmad and M Imdad, On common fixed point of mappings and multivalued mappings, Rad Mat 8 (1992), no 1, 147–158 , Some common fixed point theorems for mappings and multi-valued mappings, J Math Anal Appl 218 (1998), no 2, 546–560 A Ahmad and A R Khan, Some common fixed point theorems for non-self -hybrid contractions, J Math Anal Appl 213 (1997), no 1, 275–286 N A Assad and W A Kirk, Fixed point theorems. .. mappings, Acta Math Hungar 67 (1995), no 1-2, 69–78 V Popa, Coincidence and fixed points theorems for noncontinuous hybrid contractions, Nonlinear Anal Forum 7 (2002), no 2, 153–158 M Imdad: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail address: mhimdad@yahoo.co .in Ladlay Khan: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail address:... Kaneko and S Sessa, Fixed point theorems for compatible multi-valued and single-valued mappings, Int J Math Math Sci 12 (1989), no 2, 257–262 294 [14] [15] [16] [17] [18] [19] [20] Hybrid fixed point theorems in metrically convex spaces M S Khan, Common fixed point theorems for multivalued mappings, Pacific J Math 95 (1981), no 2, 337–347 M S Khan, H K Pathak, and M D Khan, Some fixed point theorems in. .. O Hadˇ i´ , On coincidence points in convex metric spaces, Univ u Novom Sadu Zb Rad Prirod.zc Mat Fak Ser Mat 19 (1986), no 2, 233–240 O Hadˇ i´ and Lj Gaji´ , Coincidence points for set-valued mappings in convex metric spaces, zc c Univ u Novom Sadu Zb Rad Prirod.-Mat Fak Ser Mat 16 (1986), no 1, 13–25 N J Huang and Y J Cho, Common fixed point theorems for a sequence of set-valued mappings, J Korean... metrically convex spaces, Georgian Math J 7 (2000), no 3, 523–530 K Kuratowski, Topology Vol I, Academic Press, New York, 1966 S B Nadler Jr., Multi-valued contraction mappings, Pacific J Math 30 (1969), no 2, 475–488 R P Pant, Common fixed points of noncommuting mappings, J Math Anal Appl 188 (1994), no 2, 436–440 H K Pathak, Fixed point theorems for weak compatible multi-valued and single-valued mappings, ... Math Soc 4 (1997), no 1, 1–10 M Imdad and A Ahmad, On common fixed point of mappings and set-valued mappings with some weak conditions of commutativity, Publ Math Debrecen 44 (1994), no 1-2, 105–114 M Imdad, A Ahmad, and S Kumar, On nonlinear nonself hybrid contractions, Rad Mat 10 (2001), no 2, 233–244 S Itoh, Multivalued generalized contractions and fixed point theorems, Comment Math Univ Carolinae... for any odd integer i ∈ N Thus we have shown that {Fn } and T have a common point of coincidence By setting Fi = F (for every odd integer i ∈ N), F j = G (for every even integer j ∈ N) and T = IK in Theorem 3.11, we deduce the following corollary for a pair of set-valued ´ c mappings which is a partial generalization of Theorem 2.3 of Ciri´ and Ume [5] due to the reasons already stated in respect of. ..M Imdad and L Khan 291 If one assumes that there exists a subsequence {Sx2nk +1 } contained in Q◦ with TK as well as SK are closed subspaces of X, then noting that {Sx2nk +1 } is Cauchy in SK, the foregoing arguments establish that Tv ∈ Fi (v) and Sw ∈ F j (w) Since v is a coincidence point of (Fi ,T) therefore using quasi-coincidentally commuting property of (Fi ,T) and coincidentally idempotent... point theorems for set-valued mappings of contractive type, Pacific J Math 43 (1972), no 3, 553–562 ´ c Lj B Ciri´ and J S Ume, On an extension of a theorem of Rhoades, Rev Roumaine Math Pures Appl 49 (2004), no 2, 103–112 B C Dhage, U P Dolhare, and A Petrusel, Some common fixed point theorems for sequences of ¸ nonself multivalued operators in metrically convex metric spaces, Fixed Point Theory 4 (2003),... multi-valued version of a result due to Khan et al [15] Remark 3.10 By setting Fi = F (for any odd integer i ∈ N), F j = G (for any even integer j ∈ N) and S = T = IK in Theorem 3.5, one deduces a sharpened and generalized form of a result due to Khan [14] Finally, we prove a theorem when “closedness of K” is replaced by “compactness of K.” 292 Hybrid fixed point theorems in metrically convex spaces Theorem . FIXED POINT THEOREMS FOR A FAMILY OF HYBRID PAIRS OF MAPPINGS IN METRICALLY CONVEX SPACES M. IMDAD AND LADLAY KHAN Received 30 December 2004 and in revised form 24 March 2005 The present paper. F.30- 246/2004(SR)). References [1] A. AhmadandM.Imdad,On common fixed point of mappings and multivalued mappings, Rad. Mat. 8 (1992), no. 1, 147–158. [2] , Some common fixed point theorems for mappings and multi-valued mappings, J.Math. Anal fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994), no. 2, 436–440. [19] H. K. Pathak, Fixed point theorems for weak compatible multi-valued and single-valued map- pings, Acta