Czechoslovak Mathematical Journal

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1 Czechoslovak Mathematical Journal H. K. Pathak; Shin Min Kang; Yeol Je Cho Coincidence and fixed point theorems for nonlinear hybrid generalized contractions Czechoslovak Mathematical Journal, Vol. 48 (998), No. 2, Persistent URL: Terms of use: Institute of Mathematics AS CR, 998 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Czechoslovak Mathematical Journal, 48 (23) (998), COINCIDENCE AND FIXED POINT THEOREMS FOR NONLINEAR HYBRID GENERALIZED CONTRACTIONS H. K. Pathak, Bhilai Nagar, S. M. Kang and Y. J. Cho, Chinju (Received November 3, 995) Abstract. In this paper we first prove some coincidence and fixed point theorems for nonlinear hybrid generalized contractions on metric spaces. Secondly, using the concept of an asymptotically regular sequence, we give some fixed point theorems for Kannan type multi-valued mappings on metric spaces. Our main results improve and extend several known results proved by other authors. MSC 2000 : 54H25, 47H0 Keywords: Weakly commuting, compatible and weakly compatible mappings, asymptotically regular sequence, coincidence point and fixed point, Kannan mapping. Introduction and preliminaries Let (X, d) be a metric space and let f and g be mappings from X into itself. In [9], Sessa defined f and g to be weakly commuting if d(gfx, fgx) d(gx, fx) for all x in X. It can be seen that commuting mappings are weakly commuting, but the converse is false as shown by Example in [2]. Recently, Jungck [6] extended the concept of weak commutativity in the following way: Definition.. Let f and g be mappings from a metric space (X, d) intoitself. The mappings f and g are said to be compatible if lim d(fgx n,gfx n )=0 n whenever {x n } is a sequence in X such that lim fx n = lim gx n = z for some z n n in X. 34

3 It is obvious that weakly commuting mappings are compatible, but the converse is not true. Some examples of this fact can be found in [6]. Recently, Kaneko [9] and Singh et al. [24] extended the concepts of weak commutativity and compatibility for single-valued mappings to the setting of single-valued and multi-valued mappings, respectively. Let (X, d) be a metric space and let CB(X) denote the family of all nonempty closed and bounded subsets of X. LetH be the Hausdorff metric on CB(X) induced by the metric d, i.e., { H(A, B) =max sup x A } d(x, B), sup d(y, A) y B for all A, B CB(X), where d(x, A) = inf d(x, y). y A It is well-known that (CB(X),H) is a metric space, and if a metric space (X, d) is complete, then (CB(X),H) is complete. Lemma.. [9] Let A, B CB(X) and k>. Then for each a A, there exists a point b B such that d(a, b) kh(a, B). Let (X, d) be a metric space and let f : X X and T : X CB(X) besinglevalued and multi-valued mappings, respectively. Definition.2. The mappings f and T are said to be weakly commuting if, for all x X, ftx CB(X) andh(tfx,ftx) d(fx,tx), where H is the Hausdorff metric defined on CB(X). Definition.3. The mappings f and T are said to be compatible if and only if ftx CB(X) for all x X and H(Tfx n,ftx n ) 0asn whenever {x n } X such that Tx n M CB(X) andfx n t M as n. Remark.. In [0], Kaneko and Sessa gave an example that weak commutativity implies compatibility, but the converse is not true. Recently, Pathak [6] introduced the concept of weak compatible mappings for single-valued and multi-valued mappings on a metric space as follows: Definition.4. The mappings f and T are said to be f-weak compatible if ftx CB(X) for all x X and the following limits exist and satisfy the relevant inequalities: (i) (ii) lim H(fTx n,tfx n ) lim H(Tfx n,tx n ), n n lim d(ftx n,fx n ) lim H(Tfx n,tx n ), n n 342

4 whenever {x n } is a sequence in X such that Tx n M CB(X) andfx n t M as n. It can be seen that compatible mappings f and T are weak compatible, but the converse is not true. Example in [6] and Example 2. of this paper support this fact. Note that if T is a single-valued mapping in Definitions.2,.3 and.4, we obtain the concepts of weak commutativity [2], compatibility [6] and weak compatibility [5], [7], [8] for single-valued mappings. Remark.2. We have only f-weak compatibility for single-valued and multivalued mappings in contrast to single-valued mappings for which we can define f- weak as well as T -weak compatibility. On the other hand, in [20], Rhoades et al. introduced the concept of asymptotically regular sequences in metric spaces and proved a fixed point theorem using this concept. Let T : X CB(X) be a multi-valued mapping and {x n } a sequence in X. Let f : X X be a mapping such that T (X) f(x). Then {x n } is said to be asymtotically T -regular with respect to f if d(fx n,tx n ) 0asn. In the case when f is the identity mapping on X, we simply say that the sequence {x n } is asymptotically T -reqular [8]. A point x is said to be a fixed point of a single-valued mapping f (a multi-valued mapping T )ifx = fx (x Tx). The point x is called a coincidence point of f and T if fx Tx. In this paper we give some concidence and fixed point theorems for nonlinear hybrid generalized contractions, i.e., the generalized contractive conditions including single-valued and multi-valued mappings on metric spaces. Our main results improve and generalize many results proved by many authors, Kaneko [8], [9], Kaneko and Sessa [0], Kubiak [3], Pathak [6], Nadler [9] and Singh et al. [24]. Finally, we prove some fixed point theorems for Kannan type multi-valued mappings by using the concept of asymptotically regular sequences in metric spaces. 2. Coincidence theorems and fixed point theorems In this section we give some coincidence and fixed point theorems for nonlinear hybrid generalized contractions. Theorem 2.. Let (X, d) be a complete metric space, let f : X X and T : X CB(X) be f-weak compatible continuous mappings such that T (X) 343

5 f(x) and (2.) H(Tx,Ty) h [a L(x, y)+( a) N(x, y)] for all x, y in X, where0 h<, 0 a, L(x, y) =max{d(fx,fy),d(fx,tx),d(fy,ty), 2 [d(fx,ty)+d(fy,tx)]} and N(x, y) =[max{d 2 (fx,fy),d(fx,tx) d(fy,ty),d(fx,ty) d(fy,tx), 2 d(fx,tx) d(fy,tx), 2 d(fx,ty) d(fy,ty)}]/2. Then there exists a point t X such that ft Tt, i.e., the point t is a coincidence point of f and T. Proof. Pick x 0 in X and choose x X such that fx Tx 0. This is possible since Tx 0 f(x). If h =0,weobtaind(fx,Tx ) k H(Tx 0,Tx ) = 0, i.e., fx Tx since Tx is closed. Assume that 0 <h<andsetk =/ h. By the definition of H, thereexistsapointy Tx such that d(y,fx ) k Hd(Tx,Tx 0 ). Observe that this inequality may be in the reversed direction if k by Lemma.. Since Tx f(x), let x 2 X be such that y = fx 2. In general, having chosen x n X, wemaychoosex n+ X such that y n = fx n+ Tx n and d(y n,fx n ) k H(Tx n,tx n ) for each n. Using (2.), we have where 344 d(fx n,fx n+ ) k H(Tx n,tx n ) h [a L(x n,x n )+( a) N(x n,x n )], L(x n,x n ) max{d(fx n,fx n ),d(fx n,fx n ),d(fx n,fx n+ ), 2 [d(fx n,fx n+ )+0]} max{d(fx n,fx n ),d(fx n,fx n+ ), 2 [d(fx n,fx n ) + d(fx n,fx n+ )]} max{d(fx n,fx n ),d(fx n,fx n+ )}

6 and N(x n,x n ) [max{d 2 (fx n,fx n ),d(fx n,fx n ) d(fx n,fx n+ ), 0, 0, 2 d(fx n,fx n+ ) d(fx n,fx n+ )}] /2 [max{d 2 (fx n,fx n ),d(fx n,fx n ) d(fx n,fx n+ ), 2 [d(fx n,fx n )+d(fx n,fx n+ )] d(fx n,fx n+ )}] /2 [max{d 2 (fx n,fx n ),d(fx n,fx n ) d(fx n,fx n+ ), d 2 (fx n,fx n+ )}] /2. Suppose that d(fx n,fx n+ ) > h d(fx n,fx n )forsomen N. Then we obtain d(fx n,fx n+ ) <d(fx n,fx n+ ), which is a contradiction, and so (2.2) d(fx n,fx n+ ) h d(fx n,fx n ) for all n N. Since h<andx is complete, it follows from (2.2) that {fx n } is a Cauchy sequence converging to a point t X. Also, the fact that H(Tx n,tx n ) h d(fx n,fx n )and{fx n } is a Cauchy sequence in X implies that {Tx n } is a Cauchy sequence in the complete metric space (CB(X),H). So, letting Tx n M CB(X), we have d(t, M) d(t, fx n )+d(fx n,m) d(t, fx n )+H(Tx n,m) 0 as n. Thus, since M is closed, we have t M. Also,thef-weak compatibility of f and T implies that and lim n H(fTx n,tfx n ) lim n H(Tfx n,tx n ) lim n d(ftx n,fx n ) lim n H(Tfx n,tx n ). Using the above second inequality, we obtain lim d(ffx n,fx n ) lim d(ffx n,ftx n ) + lim d(ftx n,fx n ) n n n lim d(ffx n,ftx n ) + lim H(Tfx n,tx n ). n n Since f and T are continuous, we have (2.3) H(f(M),Tt) H(Tt,M) and d(ft,t) H(Tt,M). 345

7 On the other hand, we have (2.4) d(ft,tt) d(ft,f 2 x n+ )+d(f 2 x n+,tt) d(ft,f 2 x n+ )+H(fTx n,tt) d(ft,f 2 x n+ )+H(fTx n,tfx n )+H(Tfx n,tt) d(ft,f 2 x n+ )+H(Tfx n,tx n )+ε n, where ε n 0asn. Thus, letting n in (2.4), we have d(ft,tt) H(Tt,M). Now using (2.), we have (2.5) H(Tx n,tt) h [a L(x n,t)+( a) N(x n,t)], where (2.6) and (2.7) L(x n,t)=max{d(fx n,ft),d(fx n,tx n ),d(ft,tt), 2 [d(fx n,tt)+d(ft,tx n ]} max{d(fx n,ft),d(fx n,tx n ),d(ft,tt), 2 [d(fx n,tt)+d(ft,fx n )+d(fx n,tx n )]} N(x n,t) [max{d 2 (fx n,ft),d(fx n,tx n ) d(ft,tt), d(fx n,tt) [d(ft,fx n )+d(fx n,tx n )], 2 d(fx n,tx n ) [d(ft,fx n )+d(fx n,tx n )], 2 d(fx n,tt) d(ft,tt)}] /2. Passing to the limits in (2.6) and (2.7) as n,wehave (2.8) and (2.9) 346 lim n L(x n,t) max{d(t, ft),d(t, M),d(ft,Tt), 2 [d(t, T t)+d(ft,t)+d(t, M)]} max{h(tt,m), 0,H(Tt,M), 2 [H(M,Tt)+H(Tt,M)]} = H(M,Tt) lim N(x n,t) [max{d 2 (t, ft),d(t, M) d(ft,tt), n d(t, T t) [d(ft,t)+d(t, M)], 2 d(t, M) [d(ft,t)+d(t, M)], 2d(t, T t) d(ft,tt)}]/2 [max{h 2 (Tt,M), 0,H(Tt,M)[H(Tt,M)+0], 0, 2 H2 (Tt,M)}] /2 = H(M,Tt),

8 respectively. Thus, we have, from (2.5), (2.8) and (2.9), H(M,Tt) h [a H(M,Tt)+( a) H(M,Tt)] = h H(M,Tt), which implies that H(M,Tt) =0. Therefore, d(ft,tt)=0andsoft Tt since Tt is closed. This completes the proof. Remark 2.. We recall that a non-empty subset S of X is proximinal if for each x X, there exists a point y S such that d(x, y) =d(x, S). Let PB(X) be the family of all bounded proximinal subsets of X. IfT : X PB(X) and we have chosen x n X, let x n+ X be such that y n = fx n+ Tx n and d(fx n,y)=d(fx n,tx n ). We include here an iteration scheme of Smithson [23], where T (X) is compact and hence proximinal. Since a proximinal set is closed, we have PB(X) CB(X), and it can be observed that the results of [8] and [9] follow as corollaries. Corollary 2.2. Let (X, d) be a complete metric space, let f : X X and T : X PB(X) be continuous mappings such that ftx PB(X) and H(Tfx,fTx) h d(fx,tx) for all x, y in X. IfT (X) f(x) and (2.) is satisfied for all x, y in X, where 0 h<, then there exists t X such that ft Tt. Corollary 2.3. Let (X, d) be a complete metric space, T : X CB(X) and let f be a continuous self-mapping of X such that H(Tx,Ty) h d(fx,fy) for all x, y in X, where0 h< and Tfx = ftx. If T (X) f(x), then there exists t X such that ft Tt. Remark 2.2. In Corollary 2.3, the continuity of f implies the continuity of T. Corollary 2.4. [6] Let (X, d) be a complete metric space, let f : X X and T : X CB(X) be f-weak compatible continuous mappings such that T (X) f(x) and (2.0) H(Tx,Ty) h L(x, y) for all x, y in X, where0 h<. Then there exists a point t X such that ft Tt. Corollary 2.5. Let (X, d) be a complete metric space, let f : X X and T : X CB(X) be f-weak compatible continuous mappings such that T (X) f(x) and (2.) H(Tx,Ty) h N(x, y) 347

9 for all x, y in X, where0 h<. Then there exists a point t X such that ft Tt. The following example shows that Theorem 2. is indeed a proper generalization of Theorem 2 in [0], Corollaries 2.2 and 2.3. Example 2.. Let X =[0, ) be endowed with the Euclidean metric d(x, y) = x y. Let f(x) = 3 2 (x2 + x) and Tx = [0,x 2 +2] for each x 0. Then T and f are clearly continuous and T (X) = f(x) = X. Since Tx n [0, 3] and fx n = 3 2 (x2 n + x n ) 3 [0, 3] if x n, we easily conclude that d(ftx n,fx n ) 0, H(fTx n,tfx n ) 7, H(Tfx n,tx n ) 8 if x n and therefore f and T are f-weak compatible, but they are not compatible. Thus, Theorem 2 in [0] is not applicable. Corollaries 2.2 and 2.3 are not applicable either since f and T are not weakly commuting (for x = 3) and hence they are not commuting. Again, since H(Tx,Ty)= x 2 y 2 ( 2(x + y) 3 = 3(x + y +) 2 = 2(x + y) 3(x + y +) 2 3 d(fx,fy) ) x y (x + y +) ) ( 3 2 x2 y 2 + x y h [a L(x, y)+( a) N(x, y)] for all x, y X, where h [ 2 3, ) and 0 a, all conditions of Theorem 2. are satisfied and for each t [0, ], we have ft Tt. In the sequel, we use the following lemma for our main theorem, which is a generalization of lemmas from [0] and [6]. Lemma 2.6. Let T : X CB(X) and f : X X be f-weak compatible mappings. If fw Tw for some w X and (2.) holds for all x, y in X, then ftw = Tfw. Proof. Let x n = w for all n N. Then fx n = fw fw and Tx n M = Tw as n. Hence, if fw Tw, then, by the f-weak compatibility of f and T, (2.2) H(fTw,Tfw) H(Tfw,Tw), d(f 2 w, fw) d(f 2 w, ft w)+d(ftw,fw) H(Tfw,Tw). 348

10 By (2.) we obtain (2.3) H(Tfw,Tw) h [a L(fw,w)+( a) N(fw,w)], where and L(fw,w)=max{d(f 2 w, fw),d(fw,tfw),d(fw,tw), 2 [d(f 2 w, T w)+d(fw,tfw)]} max{h(tfw,tw),h(tfw,tw), 0, 2 [d(f 2 w, fw)+d(fw,tw)+h(tw,tfw)]} max{h(tfw,tw),h(tfw,tw), 0,H(Tfw,Tw)} = H(Tfw,Tw) N(fw,w)=[max{d 2 (f 2 w, fw),d(f 2 w, T fw) d(fw,tw), d(f 2 w, T w) d(fw,tfw), 2 d(f 2 w, T fw) d(fw,tfw), 2 d(f 2 w, T w) d(fw,tw)}] /2 [max{h 2 (Tfw,Tw), 0,H 2 (Tfw,Tw), 2 H2 (Tfw,Tw), 0}] /2 = H(Tfw,Tw). Hence, (2.3) implies that H(Tfw,Tw) h [a H(Tfw,Tw)+( a) H(Tfw,Tw)] = h H(Tfw,Tw), which is a contradiction. Therefore, we have Tfw = Tw and hence (2.2) implies Tfw = ftw. This completes the proof. To obtain a fixed point theorem, we need additional assumptions as those given in [6], [9] and [0]. In the sequel, by applying the proof technique of [0] and using the above lemma, we have the following theorem: Theorem 2.7. Let f and T have the same meanings as in Theorem 2.. Assume also that for each x X either (i) fx f 2 x implies fx Tx or (ii) fx Tx implies f n x z for some z X. Then f and T have a common fixed point in X. Remark 2.3. It is not yet known whether the continuity of both f and T is necessary or not in Corollary 2.4. However, simple examples prove that the conditions f(x) T (X)andthef-weak compatibility of f and T are necessary in Corollary 2.4. Nevertheless, by weakening the inequality (2.0) for single-valued mappings f and T using the continuity of at least one of them, we can extend Theorem 2. of [3]. 349

11 Theorem 2.8. Let (X, d) be a complete metric space and let f,t : X X be f-weak compatible mappings such that T (X) f(x) and (2.4) d(tx,ty) h [a L(x, y)+( a) N(x, y)] for all x, y in X, where0 h<, 0 a, L(x, y) =max{d(fx,fy),d(fx,tx),d(fy,ty),d(fx,ty),d(fy,tx)} and N(x, y) =[max{d 2 (fx,fy),d(fx,tx) d(fy,ty),d(fx,ty) d(fy,tx), d(fx,tx) d(fy,tx),d(fy,ty) d(fx,ty)}] /2. If one of f or T is continuous, then there exists a unique common fixed point of f and T. Proof. Following the technique of Das and Naik [3], it can easily be seen that the sequence {Tx n },wheretx n = fx n+ for all n N, is a Cauchy sequence in X and, since X is complete, it follows that {Tx n } converges to some point z X. Assume that T is continuous. Then T 2 x n Tz and Tfx n Tz as n.bythe f-weak compatibility of f and T,wehave (2.5) lim d(ftx n,tfx n ) lim d(tfx n,tx n ), n n lim d(ftx n,fx n ) lim d(tfx n,tx n ). n n Now using (2.4), (2,5) and the continuity of T,wehave (2.6) d(t 2 x n,tx n ) h [a L(Tx n,x n )+( a) N(Tx n,x n )], where L(Tx n,x n )=max{d(ftx n,fx n ),d(ftx n,t 2 x n ),d(fx n,tx n ), d(ftx n,tx n ),d(fx n,t 2 x n )} max{d(ftx n,fx n ),d(ftx n,tfx n )+d(tfx n,t 2 x n ), d(fx n,tx n ),d(ftx n,fx n )+d(fx n,tx n ),d(fx n,t 2 x n )} 350

12 and N(Tx n,x n ) [max{d 2 (ftx n,fx n ),d(ftx n,t 2 x n ) d(fx n,tx n ), d(ftx n,tx n ) d(fx n,t 2 x n ),d(ftx n,t 2 x n ) d(fx n,t 2 x n ), d(ftx n,tx n ) d(fx n,tx n )}] /2, [max{d 2 (ftx n,fx n ), [d(ftx n,tfx n )+d(tfx n,t 2 x n )] d(fx n,tx n ),d(ftx n,tx n ) d(fx n,t 2 x n ), [d(ftx n,tfx n )+d(tfx n,t 2 x n )] d(fx n,t 2 x n ), [d(ftx n,tfx n )+d(tfx n,tx n )] d(fx n,tx n )}] /2. Thus, we have (2.7) lim L(Tx n,x n ) max{d(tz,z),d(tz,z), 0,d(Tz,z),d(z,Tz)} n = d(tz,z) and (2.8) lim N(Tx n,x n ) [max{d 2 (Tz,z), 0,d 2 (Tz,z),d 2 (Tz,z), 0}] /2 n = d(tz,z). Hence from (2.6) (2.8) we obtain d(tz,z) h [a d(tz,z)+( a) d(tz,z)], i.e., d(tz,z) h d(tz,z), which implies that Tz = z. Since T (X) f(x), there exists a point z such that z = Tz = Tz and, using (2.) again, we obtain d(t 2 x n,tz ) h [a max{d(ftx n,z),d(ftx n,t 2 x n ), d(z,tz ),d(ftx n,tz ),d(z,t 2 x n )} (2.9) +( a) [max{d 2 (ftx n,z),d(ftx n,t 2 x n ) d(z,tz ), d(ftx n,tz ) d(z,t 2 x n ),d(ftx n,t 2 x n ) d(z,t 2 x n ), d(z,tz ) d(ftx n,tz )}] /2]. Passing to the limit in (2.9) as n, we deduce that d(z,tz ) h d(z,tz ), i.e., z = Tz = fz and, by Lemma 2.6, fz = ftz = Tfz = Tz = z. Now, assume that f is continuous. Then f 2 x n fz and ftx n fz. By the f-weak compatibility of f and T and the continuity of f we have (2.20) lim n d(fz,tfx n) lim n d(tfx n,z), d(fz,z) lim n d(tfx n,z). 35

13 Using (2.4), (2.20) and the continuity of f, wehave d(tfx n,tx n ) h [a max{d(f 2 x n,fx n ),d(f 2 x n,tfx n ),d(fx n,tx n ), d(f 2 x n,tx n ),d(fx n,tfx n )} +( a) max[{d 2 (f 2 x n,fx n ), d(f 2 x n,tfx n ) d(fx n,tx n ),d(f 2 x n,tx n ) d(fx n,tfx n ), d(f 2 x n,tfx n ) d(fx n,tfx n ),d(fx n,tx n ) d(f 2 x n,tx n )}] /2 ], i.e., as n, i.e., as n, d(fz,z) d(tfx n,z) h [a max{d(fz,z),d(fz,tfx n ), 0,d(fz,z),d(z,Tfx n )} +( a) [max{d 2 (fz,z), 0,d(fz,z) d(z,tfx n ), d(fz,tfx n ) d(z,tfx n ), 0}] /2 ], d(fz,z) d(tfx n,z) h [a max{d(fz,z),d(tfx n,z), 0,d(Tfx n,z), d(z,tfx n )} +( a) [max{d 2 (fz,z), 0,d 2 (Tf xn,z), d 2 (Tfx n,z), 0}] /2 ]. Thus, Tfx n z as n and so fz = z. Again using (2.4) and (2.20), we obtain, as n, d(tz,tfx n ) h [a max{d(fz,f 2 x n ),d(fz,tz),d(f 2 x n,tfx n ), d(fz,tfx n ),d(f 2 x n,tz)} +( a) [max{d 2 (fz,f 2 x n ), d(fz,fz) d(f 2 x n,tfx n ),d(fz,tfx n ) d(f 2 x n,tz), d(fz,fz)d(f 2 x n,tz),d(f 2 x n,tfx n ) d(fz,tfx n )}] /2 ], i.e., Thus, we have d(tz,z) h [a max{0,d(z,tz), 0, 0,d(z,Tz)} +( a) [max{0, 0, 0,d 2 (z,tz), 0}] /2 ]. d(tz,z) h d(z,tz), which is a contradiction. Therefore, z is a common fixed point of f and T. From (2.4), the uniqueness of z follows easily. This completes the proof. 352 Next, we give an example to discuss the validity of Theorem 2.8.

14 Example 2.2. Let X =[0, ) with the usual metric. Let f and T : X X be mappings defined by fx = 2 (x2 + x) andtx= 3 (x2 + x) forx in X, respectively. Then T and f are continuous and T (X) =f(x) =X. Since fx = Tx if and only if x, we can and do choose a sequence {x n } in X such that x n. Then we have lim d(tfx n,ftx n ) = lim n n 36 x4 n + 6 x3 n 36 x2 n + 9 =0. Therefore, f and T are compatible and so f and T are f-weak compatible. Since d(tx,ty)= 3 x2 y 2 { } 2(x + y) = 3(x + y +) 2 x2 y 2 + x y 2 3 d(fx,fy), i.e., i.e., d(tx,ty) h [a d(fx,fy)+( a) [d 2 (fx,fy)] /2 ], d(tx,ty) h [a L(x, y)+( a) N(x, y)] for all x, y X, all conditions of Theorem 2.8 are fulfilled with h [ 2 3, ),0 a and the point is the unique common fixed point of f and T. 3. Fixed point theorems for Kannan type multi-valued mappings Kannan [] established a fixed point theorem for a single-valued mapping T defined on a complete metric space (X, d) satisfying d(tx,ty) α[d(x, T x)+d(y, Ty)] for all x, y X, where0<α< 2. The range of α is crucial even to the existence part of this result in the setting of a complete metric space. However, in a more restrictive yet quite natural setting, elaborate fixed point theorems exist for the case α = 2. This wider class of mappings were studied by Kannan in [2]. Recently, Beg and Azam [], [2], Shiu, Tan and Wong [22] and Wong [26] have also studied such mappings. In this section, we consider a mapping T : X CB(X) satisfying the following condition: (3.) H r (Tx,Ty) α (d(x, T x))d r (x, T x)+α 2 (d(y, Ty))d r (x, T x) 353

15 for all x, y X, whereα i : R [0, ) (i =, 2) and r is a fixed positive real number. If there exists a sequence {x n } in X such that lim d(x n,tx n )=0,then{x n } is n said to be asymptotically T -regular. In fact, we establish the following: Theorem 3.. Let (X, d) be a complete metric space and let T : X CB(X) be a multi-valued mapping satisfying (3.) for all x, y X, whereα i : R [0, ) (i =, 2) andr is a fixed positive real number. If there exists an asymptotically T - regular sequence {x n } in X, thent has a fixed point x in X. Moreover,Tx n Tx as n. Proof. By hypothesis, we have H r (Tx n,tx m ) α (d(x n,tx n ))d r (x n,tx n )+α 2 (d(x m,tx m ))d r (x m,tx m ) 0 as n, m. This shows that {Tx n } is a Cauchy sequence in (CB(X),H). Since (CB(X),H)is complete, there exists K CB(X) such that H(Tx n,k ) 0asn. Suppose x K. Then by (3.) we have d r (x,tx ) H r (K,Tx ) lim n Hr (Tx n,tx ) lim [α (d(x n,tx n ))d r (x n,tx n )+α 2 (d(x,tx ))d r (x,tx )] n α 2 (d(x,tx ))d r (x,tx ), which implies that ( α 2 (d(x,tx )))d r (x,tx ) 0, i.e., d(x,tx ) = 0, and so x Tx.Now H r (K,Tx ) = lim n Hr (Tx n,tx ) α 2 (d(x,tx ))d r (x,tx ) d r (x,tx )=0. Therefore, we obtain Tx = K = lim n Tx n. This completes the proof. Theorem 3.2. Let (X, d) be a complete metric space and let T : X CB(X) be a multi-valued mapping satisfying (3.). If there exists an asymptotically T -regular sequence {x n } in X and Tx n is compact for all n N, then each cluster point of {x n } is a fixed point of T. 354

16 Proof. Let y n Tx n be such that d(x n,y n )=d(x n,tx n ). It is obvious that a cluster point of {x n } is a cluster point of {y n }. Suppose that y is such a cluster point of {x n } and {y n }. Then as in Theorem 3., we have d r (x,tx ) H r (Tx n,tx ) α (d(x n,tx n ))d r (x n,tx n )+α 2 (d(x,tx ))d r (x,tx ) α (d(x n,tx n ))d r (x n,tx n ), which implies that y Tx. By (3.) again, d r (y,ty ) H r (Tx,Ty ) α (d(x,tx ))d r (x,tx )+α 2 (d(y,ty ))d r (y,ty ), i.e., ( α 2 (d(y,ty )))d r (y,ty ) 0. Therefore, we have y Ty. This completes the proof. Theorem 3.3. Let (X, d) be a complete metric space and let T : X CB(X) be a multi-valued mapping satisfying (3.) with α (d(x, T x)) + α 2 (d(y, Ty)). If inf{d(x, T x): x X} =0,thenT has a fixed point in X. Proof. In view of Theorem 3.2 it suffices to show that there exists an asymptotically T -regular sequence {x n } in X. Pick x 0 in X and consider a sequence {x n } in X such that x n Tx n for all n N. Then the inequality (3.) implies (3.2) d r (x n,tx n ) H r (Tx n,tx n ) α (d(x n,tx n ))d r (x n,tx n ) + α 2 (d(x n,tx n ))d r (x n,tx n ) α (d(x n,tx n )) α 2 (d(x n,tx n )) dr (x n,tx n ) d r (x n,tx n ). It follows from (3.2) that the sequence {d(x n,tx n )} is decreasing. Therefore, we have d(x n,tx n ) inf{d(x n,tx n ): n N} and so d(x n,tx n ) 0. Therefore, {x n } is asymptotically T -regular. This completes the proof. 355

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