1 Introduction and preliminaries
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1 Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Filomat 5: (0), 5 3 DOI: 0.98/FIL05D MULTIVALUED GENERALIZATIONS OF THE KANNAN FIXED POINT THEOREM Boško Damjanović and Dragan -Dorić Abstract In this paper we obtain multi-valed mapping generalizations of two recent theorems of Kikkawa and Suzuki [M. Kikkawa and T. Suzuki, Some similarity between contractions and Kannan mappings, Fixed Point Theory Appl., (008), Article ID , 8] and the main theorem of Enjouji et all. [Y. Enjouji, M. Nakanishi and T. Suzuki, A Generalization of Kannan s Fixed Point Theorem, Fixed Point Theory and Applications, Volume 009, Article ID 987, 0 pages]. Introduction and preliminaries Let (X, d) be a metric space and let T be a self-mapping on X. Then T is called a Kannan mapping if there exists a [0, /) such that for all x, y X. If T is a such that d(t x, T y) ad(x, T x) + ad(y, T y) () d(t x, T y) r max {d(x, T x), d(y, T y)} for some r [0, ) and all x, y X, then T is called a generalized Kannan mapping. If X is complete, then every (generalized) Kannan mapping have a unique fixed point [6]. Subrahmanyam [] proved that Kannan theorem characterizes the metric completeness of underlying spaces. It is known that the Banach theorem [] cannot characterize the metric completeness []. In [7] authors generalized Kannan mappings. 00 Mathematics Subject Classifications. 47H0. Key words and Phrases. Complete metric space, Kannan fixed point, multivalued mapping. Received: August 30, 00 Communicated by Vladimir Rakočević The authors thank to the Ministry of Science and Technological Development of Republic Serbia
2 6 Boško Damjanović and Dragan -Dorić Theorem.. (Kikkawa and Suzuki [7]) Let T be a mapping on complete metric space (X, d) and let ϕ be a non-increasing function from [0, ) onto (/, ] defined by if 0 r, ϕ(r) = + r if r <. Let α [0, /) and put r = α/( α) [0, ). Suppose that ϕ(r)d(x, T x) d(x, y) implies d(t x, T y) αd(x, T x) + αd(y, T y) () for all x, y X. Then T has a unique fixed point z and lim n T n x = z holds for every x X. Theorem.. (Kikkawa and Suzuki [7]) Let T be a mapping on complete metric space (X, d) and let θ be a non-increasing function from [0, ) onto (/, ] defined by if 0 r ( 5 ), r θ(r) = r + r Suppose that there exists r [0, ) such that if if ( 5 ) r, r <. θ(r)d(x, T x) d(x, y) implies d(t x, T y) r max {d(x, T x), d(y, T y)} (3) for all x, y X. Then T has a unique fixed point z and lim n T n x = z holds for every x X. On the other side, Nadler [0] proved multi-valued extension of the Banach contraction theorem. Theorem.3. (Nadler [0]) Let (X, d) be a complete metric space and let T be a mapping from X into CB(X). Assume that there exists r [0, ) such that H(T x, T y) rd(x, y) for all x, y X. Then there exists z X such that z T z. Many fixed point theorems have been proved by various authors as generalizations of the Nadler s theorem (see [9], [4], [], [3]). The following recent result [8] is a generalization of Nadler [0]. Theorem.4. (Kikkawa and Suzuki [8]) Let (X, d) be a complete metric space and let T be a mapping from X into CB(X). Define a strictly decreasing function η from [0, ) onto (/, ] by η(r) = + r
3 Multivalued generalizations of the Kannan fixed point theorem 7 and assume that there exists r [0, ) such that η(r)d(x, T x) d(x, y) implies H(T x, T y) rd(x, y) for all x, y X. Then there exists z X such that z T z. In this paper we obtain multi-valued version of Theorem. and then as a corollary we obtain multi-valued version of Theorem.. Main results Let (X, d) be a metric space. We denote by CB(X) the family of all non-empty closed bounded subsets of X. Let H(, ) be the Hausdorff metric, i.e., H(A, B) = max{sup d(a, B), sup d(a, b)} a A b B for A, B CB(X), where d(x, B) = inf d(x, y). y B Now we prove our main result. Theorem.. Define a non-increasing function ϕ from [0, ) into (0, ] by 5, if 0 r <, ϕ (r) = 5 r, if r <. Let (X, d) be a complete metric space and let T be a mapping from X into CB(X). Assume that ϕ(r)d(x, T x) d(x, y) implies H(T x, T y) r max {d(x, T x), d(y, T y)} (4) for all x, y X. Then there exists z X such that z T z. Proof. Let r be a real number such that 0 r < r <. Let u = X and u T u be arbitrary. Since u T u, then d(u, T u ) H(T u, T u ) and ϕ = (r)d(u, T u ) d(u, T u ) d(u, u ). Thus from the assumption (4), d(u, T u ) H(T u, T u ) r max {d(u, T u ), d(u, T u )}. Hence, as r <, we have d(u, T u ) rd(u, u ). So, there exists u 3 T u such that d(u, u 3 ) r d(u, u ). Thus, we can construct a sequence {u n } in X such that u n+ T u n and d(u n+, u n+ ) r d(u n, u n+ ). Hence, by induction, d(u n, u n+ ) r n d(u, u ).
4 8 Boško Damjanović and Dragan -Dorić Then by the triangle inequality, we have d(u n, u n+ ) r n d(u, u ) <. n= n= Hence we conclude that {u n } is a Cauchy sequence. Since X is complete, there is some point z X such that lim n u n = z. Now we shall show that d(z, T x) rd(x, T x) for all x X\{z}. (5) Since u n z, there exists n 0 N such that d(z, u n ) (/3)d(z, x) for all n n 0. Then we have ϕ (r) d(u n, T u n ) d(u n, T u n ) d(u n, u n+ ) d(u n, z) + d(u n+, z) d(x, z). 3 Since 3 d(x, z) = d(x, z) d(x, z) 3 d(x, z) d(u n, z) d(u n, x), we get ϕ (r) d(u n, T u n ) d(u n, x). Then from (4), H(T u n, T x) r max {d(u n, T u n ), d(x, T x)}. Since u n+ T u n, then d(u n+, T x) H(T u n, T x). So, it follows that d(u n+, T x) r max{d(u n, u n+ ), d(x, T x)} for all n N with n n 0. Letting n tends to, we obtain d(z, T x) rd(x, T x). Thus we proved (5). Now we show that z T z. Suppose, to the contrary, that z / T z. Consider at first the case 0 r < 5. Let a T z. Then a z and so by (5), we have d(z, T a) rd(a, T a). On the other hand, since from (4) we have ϕ (r) d(z, T z) = d(z, T z) d(z, a), H(T z, T a) r max {d(z, T z), d(a, T a)}. (6)
5 Multivalued generalizations of the Kannan fixed point theorem 9 Hence we have d(a, T a) H(T z, T a) r max {d(z, T z), d(a, T a)}. Hence we get d(a, T a) rd(z, T z). Therefore, by (5) and (6), we obtain d(z, T z) d(z, T a) + H(T a, T z) rd(a, T a) + r max {d(z, T z), d(a, T a)} rd(a, T a) + rd(z, T z) r d(z, T z) + rd(z, T z) = ( r + r ) d(z, T z). Hence, as r < 5 implies r + r <, we have d(z, T z) < d(z, T z), a contradiction. So we obtain z T z. Consider now the case 5 r <. We first prove H(T x, T z) r max {d(x, T x), d(z, T z)} for all x X. (7) If x = z, then (7) obviously holds. So we assume x z. Then for every n N, there exists y n T x such that d(z, y n ) d(z, T x) + (/n)d(x, z). We have d(x, T x) d(x, y n ) d(x, z) + d(z, y n ) d(x, z) + d(z, T x) + d(x, z) n d(x, z) + rd(x, T x) + d(x, z) n for n N. Hence ( r)d(x, T x) ( + /n)d(x, z) holds. Letting n tend to, we have ( r)d(x, T x) d(x, z). Thus ϕ(r)d(x, T x) d(x, z). From the assumption (4), we obtain (7). Therefore, as u n+ T u n, from (7) with x = u n we have d(z, T z) = lim n d(u n+, T z) lim H(T u n, T z) n lim r max {d(u n, T u n ), d(z, T z)} n lim r max {d(u n, u n+ ), d(z, T z)} n = rd(z, T z). Hence( r)d(z, T z) 0, which implies d(z, T z) = 0. Since T z is closed, we obtain z T z. This completes the proof.
6 30 Boško Damjanović and Dragan -Dorić Theorem. is a multi-valued mapping generalization of the theorem.3 of Kikkawa and Suzuki [7] and therefore the Kannan fixed point theorem [6] for generalized Kannan mappings. Corollary.. Let (X, d) be a complete metric space and let T be a mapping from X into CB(X). Let α [0, /) and put r = α. Suppose that ϕ(r)d(x, T x) d(x, y) implies H(T x, T y) αd(x, T x) + αd(y, T y) (8) for all x, y X, where the function ϕ is defined as in Theorem.. Then there exists z X such that z T z. Corollary. is a multi-valued mapping generalization of the theorem. of Kikkawa and Suzuki [7] and therefore the well known Kannan fixed point theorem [6]. Corollary. also is a multi-valued mapping generalization of the theorem 3. of Y. Enjouji et all. [5], since by symmetry the inequality (3.3) in [5] implies the inequality () in Theorem.. References [] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (9), [] E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc., 0 (959), [3] Lj. Ćirić, Multi-valued nonlinear contraction mappings, Nonlinear Analysis 7 (009) [4] P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 9 (995) [5] Y. Enjouji, M. Nakanishi and T. Suzuki, A Generalization of Kannan s Fixed Point Theorem, Fixed Point Theory and Applications, Volume 009, Article ID 987, 0 pages. [6] R. Kannan, Some results on fixed points II, Amer. Math. Monthly, 76 (969), [7] M. Kikkawa and T. Suzuki, Some similarity between contractions and Kannan mappings, Fixed Point Theory Appl., (008), Article ID , 8. [8] M. Kikkawa and T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal. 69 (008), [9] N. Mizoguchi, W. Takahashi, Fixed point theorems for multi-valued mappings on complete metric spaces, J. Math. Anal. Appl. 4 (989)
7 Multivalued generalizations of the Kannan fixed point theorem 3 [0] S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math., 30 (969), [] P.V. Semenov, Fixed points of multi-valued contractions, Funct. Anal. Appl. 36 () (00) [] P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math., 80 (975), [3] T. Suzuki, Contractive mappings are Kannan mappings, and Kannan mappings are contractive mappings in some-sense, Commentationes Mathematicae. Prace Matematyczne, vol. 45, no., pp , 005. [4] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 36 (008), Boško Damjanović Department of Mathematics, Faculty of Agriculture, University of Belgrade, 000 Beograd, Nemanjina 6, Serbia dambo@agrif.bg.ac.rs Dragan -Dorić Department of Mathematics, Faculty of Organizational Sciences, University of Belgrade, 000 Beograd, Jove Ilića 54, Serbia djoricd@fon.rs
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