Filomat 6:5 (01), 1045 1053 DOI 10.98/FIL105045A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Common fixed point of multivalued mappings in dered generalized metric spaces Mujaid Abbas a, Abdul Rahim Khan b, Talat Nazir a a Department of Mathematics, Lahe University of Management Sciences, Lahe 5479, Pakistan b Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 3161, Saudi Arabia Abstract. In this paper, study of necessary conditions f the existence of common fixed point of multivalued mappings satisfying generalized contractive conditions in the setting of dered generalized metric spaces is initiated. Examples to suppt our results are presented. These results establish most general common fixed point theems f multivalued maps in dered generalized metric spaces. 1. Introduction and preliminaries Over the past two decades the development of fixed point they in metric spaces has attracted considerable attention due to numerous applications in areas such as variational and linear inequalities, optimization, and approximation they. Mustafa and Sims [13] generalized the concept of a metric space. Based on the notion of generalized metric spaces, Mustafa et al. ([1, 14, 15]) obtained some fixed point theems f mappings satisfying different contractive conditions. Abbas and Rhoades [5] initiated the study of common fixed point they in generalized metric spaces. Saadati et al. [19] proved some fixed point results f contractive mappings in partially dered G-metric spaces. Choudhury and Maity [8] established coupled fixed point theems in a partially dered G-metric space. Recently, Abbas et al. [3] obtained fixed points of multivalued mapping satisfying Ćirić type contractive conditions in G-metric spaces. Existence of fixed points in partially dered metric spaces was first investigated in 004 by Ran and Reurings [18], and then by Nieto and Lopez [16]. Further results in this direction under different contraction conditions were proved in [1,, 6, 7]. The aim of this paper is to prove various common fixed points results f multivalued mappings taking closed values in generalized metric spaces. It is wth mentioning that our results do not rely on the notion of continuity of the mappings involved therein. Our results extend and unify various comparable results ([3, 4, 7, 10, 11, 17]). Consistent with Mustafa and Sims [13], the following definitions and results will be needed in the sequel. Definition 1.1. Let X be a nonempty set. Suppose that a mapping G : X X X R + satisfies: 010 Mathematics Subject Classification. Primary 47H10; Secondary 54H5 Keywds. Multivaled mapping, common fixed point, non symmetric, generalized metric spaces, partially dered set Received: 9 January 01; Accepted: 10 March 01 Communicated by Qamrul Hasan Ansari and Ljubiša D.R. Kočinac The auths A. R. Khan and M. Abbas are grateful to King Fahd University of Petroleum and Minerals f suppting research project IN 101037 Email addresses: mujahid@lums.edu.pk (Mujaid Abbas), arahim@kfupm.edu.sa (Abdul Rahim Khan), talat@lums.edu.pk (Talat Nazir)
M. Abbas et al. / Filomat 6:5 (01), 1045 1053 1046 (a) G(x, y, z) = 0 if x = y = z; (b) 0 < G(x, y, z) f all x, y X, with x y; (c) G(x, x, y) G(x, y, z) f all x, y, z X, with y z; (d) G(x, y, z) = G(p{x, y, z), where p is a permutation of x, y, z (symmetry); (e) G(x, y, z) G(x, a, a) + G(a, y, z) f all x, y, z, a X. Then G is called a G-metric on X and (X, G) is called a G-metric space. Definition 1.. A sequence {x n in a G-metric space X is: (i) a G-Cauchy sequence if, f any ε > 0, there is an n 0 N ( the set of natural numbers ) such that f all n, m, l n 0, G(x n, x m, x l ) < ε, (ii) a G-Convergent sequence if, f any ε > 0, there is an x X and an n 0 N, such that f all n, m n 0, G(x, x n, x m ) < ε. A G-metric space is said to be G-complete if every G-Cauchy sequence in X is G-convergent in X. It is known that {x n G-converges to x X if and only if G(x m, x n, x) 0 as n, m [13]. Proposition 1.3. ([13]) Let X be a G-metric space. Then the following are equivalent: 1. {x n is G-convergent to x.. G(x n, x n, x) 0 as n. 3. G(x n, x, x) 0 as n. 4. G(x n, x m, x) 0 as n, m. Definition 1.4. A G-metric on X is said to be symmetric if G(x, y, y) = G(y, x, x) f all x, y X. Proposition 1.5. Every G-metric on X will define a metric d G on X by d G (x, y) = G(x, y, y) + G(y, x, x), x, y X. F a symmetric G-metric space d G (x, y) = G(x, y, y), x, y X. However, if G is not symmetric, then the following inequality holds: 3 G(x, y, y) d G(x, y) 3G(x, y, y), x, y X. Definition 1.6. Let X be a nonempty set. Then (X,, G) is called an dered generalized metric space if and only if: (i) G is a generalized metric on X and (ii) is a partial der on X. Definition 1.7. Let (X, ) be a partially dered set. Then x, y X are called comparable if x y y x holds. Definition 1.8. ([7]) Let (X, ) be a partially dered set and A and B be two nonempty subsets of (X, ). If f every a A, there exist b B such that a b, then we say A 1 B.
M. Abbas et al. / Filomat 6:5 (01), 1045 1053 1047 Kannan [10] established existence of a fixed point f a selfmap T on a complete metric space X which satisfies d(tx, Ty) h[d(x, Tx) + d(y, Ty)] f all x, y in X and f a fixed h [0, 1 ). A mapping T which satisfies the above contractive condition is called Kannan mapping. Ćirić [9] considered a mapping T : X X satisfying the following contractive condition: d(tx, Ty) q max{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx) where q [0, 1). He proved the existence of a fixed point when X is a T-bitally complete metric space. We denote by P(X), the family of all nonempty subsets of X, and by P cl (X), the family of all nonempty closed subsets of X. A point x in X is called a fixed point of a multivalued mapping T : X P cl (X) provided x Tx. The collection of all fixed point of T is denoted by F(T). Latif and Beg [11] introduced the notion of a K-multivalued mapping, which is an extension of Kannan mapping, to multivalued mappings. Rus [17] coined the term R-multivalued mapping, which is a generalization of a K-multivalued mapping. Abbas and Rhoades [4] studied common fixed point problems f multivalued mappings and introduced the notion of generalized R-multivalued mappings which in turn generalizes R-multivalued mappings. They of multivalued maps has applications in control they, convex optimization, differential equations and economics. The aim of this paper is to prove some common fixed point results f multivalued mappings taking closed values in dered generalized metric spaces. Let (X, ) be a partially dered set. We define 1 = {(x, y) X X x y Definition 1.9. Let (X, ) be a partially dered set and T 1, T : X P cl (X) be multivalued mappings. A pair (T 1, T ) is said to satisfy the property (P 1 ) if f every (x, y) 1 and u x T i (x), there exists u y T j (y) such that (u x, u y ) 1 and G(u x, u y, u y ) G(x,u y,u y )+G(y,u x,u x ), G(u x, u x, u y ) h max{g(x, x, y), G(x, x, u x ), G(y, y, u y ), G(x,x,u y )+G(y,y,u x ), holds where h [0, 1), i, j {1, with i j. (P ) if f every (x, y) 1 and u x T i (x), there exists u y T j (y) such that (u x, u y ) 1 and (1) () G(u x, u y, u y ) a 1 G(x, y, y) + a G(x, x, y) + a 3 G(x, u x, u x ) + a 4 G(x, x, u x ) +a 5 G(y, u y, u y ) + a 6 G(y, y, u y ), G(u x, u x, u y ) a 1 G(x, x, y) + a G(x, y, y) + a 3 G(x, x, u x ) + a 4 G(x, x, u x ) +a 5 G(y, y, u y ) + a 6 G(y, u y, u y ), holds where a i 0 f i = 1,,..., 6 and a 1 + a 3 + a 5 + (a + a 4 + a 6 ) < 1, i, j {1, with i j. (P 3 ) if f every (x, y) 1 and u x T i (x), there exists u y T j (y) such that (u x, u y ) 1 and G(u x, u y, u y ) αg(x, y, y) + βg(x, u x, u x ) + γg(y, u y, u y ), (5) G(u x, u x, u y ) αg(x, x, y) + βg(x, x, u x ) + γg(y, y, u y ), (6) holds where α, β, γ 0, α + β + γ < 1 and i, j {1, with i j. (3) (4)
M. Abbas et al. / Filomat 6:5 (01), 1045 1053 1048 (P 4 ) if f every (x, y) 1 and u x T i (x), there exists u y T j (y) such that (u x, u y ) 1 and G(u x, u y, u y ) h[g(x, u x, u x ) + G(y, u y, u y )], (7) G(u x, u x, u y ) h[g(x, u x, u x ) + G(y, y, u y )], (8) where 0 h < 1/ and i, j {1, with i j. (P 5 ) if f every (x, y) 1 and u x T i (x), there exists u y T j (y) such that (u x, u y ) 1 and G(u x, u y, u y ) λg(x, y, y), (9) G(u x, u x, u y ) λg(x, x, y), (10) where 0 λ < 1 and i, j {1, with i j.. Fixed point theems In this section, we obtain common fixed point theems f a multivalued mapping satisfying certain generalized contractive conditions in the frame wk of an dered generalized metric space. Theem.1. Let (X,, G) be an dered complete G-metric space and T 1, T : X P cl (X) be multivalued mappings. If the pair (T 1, T ) satisfies the property (P 1 ), then F(T 1 ) = F(T ) provided that the following conditions hold: (i) F each x 0 X, {x 0 1 T 1 (x 0 ) {x 0 1 T (x 0 ). (ii) If x n x in X with (x n, x n+1 ) 1 f all n N, we have (x n, x) 1. Proof. Let x T 1 (x ). Then there exists an x T (x ) such that G(x, x, x) h max{g(x, x, x ), G(x, x, x ), G(x, x, x), G(x, x, x) + G(x, x, x ) = h max{g(x, x, x), G(x, x, x) = hg(x, x, x), which implies that G(x, x, x) = 0 and so we obtain x = x. Thus F(T 1 ) F(T ). Similarly, F(T ) F(T 1 ). So F(T 1 ) = F(T ). Suppose that x 0 X and the assumption (1) holds. Then there exists x 1 T 1 (x 0 ) such that (x 0, x 1 ) 1. Now, f x 1 T 1 (x 0 ), there exists x T (x 1 ) with (x 1, x ) 1 such that G(x 1, x, x ) h max{g(x 0, x 1, x 1 ), G(x 0, x 1, x 1 ), G(x 1, x, x ), G(x 0, x, x ) + G(x 1, x 1, x 1 ) h max{g(x 0, x 1, x 1 ), G(x 1, x, x ), G(x 0, x 1, x 1 ) + G(x 1, x, x ) = h max{g(x 0, x 1, x 1 ), G(x 1, x, x ) Now, max{g(x 0, x 1, x 1 ), G(x 1, x, x ) = G(x 0, x 1, x 1 ) implies that G(x 1, x, x ) hg(x 0, x 1, x 1 ).
If max{g(x 0, x 1, x 1 ), G(x 1, x, x ) = G(x 1, x, x ), then G(x 1, x, x ) hg(x 1, x, x ), M. Abbas et al. / Filomat 6:5 (01), 1045 1053 1049 which implies that x 1 = x. Next f this x T (x 1 ), there exists x 3 T 1 (x ) with (x, x 3 ) 1 such that G(x, x 3, x 3 ) h max{g(x 1, x, x ), G(x 1, x, x ), G(x, x 3, x 3 ), G(x 1, x 3, x 3 ) + G(x, x, x ) h max{g(x 1, x, x ), G(x, x 3, x 3 ), G(x 1, x, x ) + G(x, x 3, x 3 ) = h max{g(x 1, x, x ), G(x, x 3, x 3 ) Now, max{g(x 1, x, x ), G(x, x 3, x 3 ) = G(x 1, x, x ) implies that G(x, x 3, x 3 ) hg(x 1, x, x ). If max{g(x 1, x, x ), G(x, x 3, x 3 ) = G(x, x 3, x 3 ), then G(x, x 3, x 3 ) hg(x, x 3, x 3 ), which implies that x = x 3. Continuing this process, f x n T (x n 1 ), there exists x n+1 T 1 (x n ) with (x n, x n+1 ) 1 such that G(x n, x n+1, x n+1 ) h max{g(x n 1, x n, x n ), G(x n 1, x n, x n ), G(x n, x n+1, x n+1 ), G(x n 1, x n+1, x n+1 ) + G(x n, x n, x n ) h max{g(x n 1, x n, x ), G(x n, x n+1, x n+1 ), G(x n 1, x n, x n ) + G(x n, x n+1, x n+1 ) = h max{g(x n 1, x n, x ), G(x n, x n+1, x n+1 ) If max{g(x n 1, x n, x ), G(x n, x n+1, x n+1 ) = G(x n 1, x n, x n ), then G(x n, x n+1, x n+1 ) hg(x n 1, x n, x n ). When max{g(x n 1, x n, x ), G(x n, x n+1, x n+1 ) = G(x n, x n+1, x n+1 ), we obtain G(x n, x n+1, x n+1 ) hg(x n, x n+1, x n+1 ), and x n = x n+1. In a similar manner, f x n+1 T 1 (x n ), there exists x n+ T (x n+1 ) with (x n+1, x n+ ) 1 such that G(x n+1, x n+, x n+ ) hg(x n, x n+1, x n+1 ). Therefe, we obtain a sequence {x n in X such that f x n T 1 (x n 1 ), there exists x n+1 T (x n ) with (x n, x n+1 ) 1 such that Hence G(x n, x n+1, x n+1 ) hg(x n 1, x n, x n ). G(x n, x n+1, x n+1 ) hg(x n 1, x n, x n ) h G(x n, x n 1, x n 1 )... h n G(x 0, x 1, x 1 )
f all n N, and so f m > n, we have G(x n, x m, x m ) hn 1 h G(x 0, x 1, x 1 ). M. Abbas et al. / Filomat 6:5 (01), 1045 1053 1050 It follows that {x n is a Cauchy sequence in X. Since X is complete, there exists an element x X such that x n x as n. If the assumption (ii) holds, then (x n, x ) 1 f all n. F x n T j (x n 1 ), there exists u n T i (x ) with (x n, u n ) 1 such that Note that G(x n, u n, u n ) h max{g(x n 1, x, x ), G(x n 1, x n, x n ), G(x, u n, u n ), G(x n 1, u n, u n ) + G(x, x n, x n ) G(u n, u n, x ) G(u n, u n, x n ) + G(x n, x n, x ) which further implies h max{g(x n 1, x, x ), G(x n 1, x n, x n ), G(x, u n, u n ), G(x n 1, u n, u n ) + G(x, x n, x n ) + G(x n, x n, x ) h max{g(x n 1, x, x ), G(x n 1, x n, x n ), G(x, u n, u n ), G(x n 1, x, x ) + G(x, u n, u n ) + G(x, x n, x n ) + G(x n, x n, x ) h[g(x n 1, x, x ) + G(x n 1, x n, x n ) + G(x, u n, u n )] +(1 + h/)g(x n, x n, x ), G(u n, u n, x ) 1 1 h [hg(x n 1, x, x ) + hg(x n 1, x n, x n )] (1 + h/) + 1 h G(x n, x n, x ). This shows that {u n is a bounded sequence, so lim supu n = u and lim inf u n = u both exist. Since, f n n x n T (x n 1 ), there exists u n T 1 (x ) with (x n, u n ) 1 such that G(x n, u n, u n ) h max{g(x n 1, x, x ), G(x n 1, x n, x n ), G(x, u n, u n ), G(x n 1, u n, u n ) + G(x, x n, x n ) Which on taking lim sup as n implies G(x, u, u ) h max{g(x, x, x ), G(x, x, x ), G(x, u, u ), G(x, u, u ) + G(x, x, x ) = hg(x, u, u ) and so x = u. Similarly, taking lim inf as n implies x = u. Thus u n x as n. Since T 1 (x ) is closed, x F(T j ) = F(T i ). The proof f the inequality () is similar. The above theem extends, Theem 1.9 in [4] to dered generalized metric spaces.
M. Abbas et al. / Filomat 6:5 (01), 1045 1053 1051 Example.. Let X = [0, ) be endowed with usual der and G(x, y, z) = max{ x y, y z, z x be a G-metric on X. Define T 1, T : X P cl (X) as T 1 x = [0, x 8 ] and T x = [0, x 6 ]. Note that f x = y = 0, (1) is satisfied as u x = u y = 0. F, x = y 0, u x T 1 x. Take u y = 0. Then G(u x, u y, u y ) = u x x 8 < 3 4 (7 8 x) hg(x, u x, u x ) Thus (1) is satisfied with h = 3 4. G(x, u y, u y ) + G(y, u x, u x ) Now if x = 0, y 0, then (1) is satisfied f u y = 0. If x 0, y = 0, then f any u x T 1 x, we have u y = 0, and G(u x, u y, u y ) = u x x 8 < 3 4 (7 8 x) hg(x, u x, u x ) G(x, u y, u y ) + G(y, u x, u x ) Now if x < y, u x T 1 x, take u y = 0. Then G(u x, u y, u y ) = u x x 8 < 3 4 y hg(y, u y, u y ) G(x, u y, u y ) + G(y, u x, u x ) Finally, f, y < x, u x T 1 x, take u y = y 6. Then G(u x, u y, u y ) = ux u y ux + u y Thus (1) is satisfied with h = 3 4. x 8 + y 6 < 7 4 x < 3 4 (7 8 x) hg(x, x, u x) G(x, u y, u y ) + G(y, u x, u x ) Now we show that f x, y X, u x T x, there exist u y T 1 y such that (1) is satisfied. F x = y = 0, (1) is satisfied as u x = u y = 0. F x = y 0, u x T x. Take u y = 0, then we have G(u x, u y, u y ) = u x x 6 < 3 4 x = hg(y, u y, u y ) Now if x = 0, y 0, then (1) is satisfied f u y = 0. If x 0, y = 0, then f any u x T x, we have u y = 0, and G(u x, u y, u y ) = u x < 3 4 (x u x) = hg(x, u x, u x )
Now f 0 < x < y, u x T x, take u y = 0. Then we have G(u x, u y, u y ) 1 6 x < 3 4 y = hg(y, u y, u y ) M. Abbas et al. / Filomat 6:5 (01), 1045 1053 105 F, 0 < y < x, u x T x, we have three possibilities: f y = u x, take u y = 0 and G(u x, u y, u y ) = u x < 3 4 (x u x) = hg(x, y, y) F u x > y, take u y = 0. Then we have G(u x, u y, u y ) = u x < 3 4 (x u x) = hg(x, u x, u x ) Finally, f u x < y, take u y = y. Then we have 8 G(u x, u y, u y ) = u x y < 3 8 4 (x u x) = hg(x, u x, u x ) with h = 3. Thus f all x, y in X, (1) is satisfied. Hence all the conditions of Theem.1 are satisfied. 4 Meover, 0 is the common fixed point of T 1 and T with F(T 1 ) = F(T ). Collary.3. Let (X,, G) be an dered complete G-metric space and T 1, T : X P cl (X) be multivalued mappings. If the pair (T 1, T ) satisfies the property (P ), then F(T 1 ) = F(T ) provided that the following conditions hold: (i) F each x 0 X, {x 0 1 T 1 (x 0 ) {x 0 1 T (x 0 ). (ii) If x n x in X with (x n, x n+1 ) 1 f all n N, we have (x n, x) 1. Proof. Note that (3) implies that G(u x, u y, u y ) G(x, x, y), G(x, x, u x) where h = a 1 + a 3 + a 5 + (a + a 4 + a 6 ) < 1. This further implies that, G(y, y, u y), G(u x, u y, u y ) and the result follows from Theem.1.
M. Abbas et al. / Filomat 6:5 (01), 1045 1053 1053 The following collary generalizes, Theem 3.3 of Rus et al. [17] to dered G-metric spaces. Collary.4. Let (X,, G) be an dered complete G-metric space and T 1, T : X P cl (X) be multivalued mappings. Suppose that the pair (T 1, T ) satisfies the property (P 3 ). Then F(T 1 ) = F(T ) provided that the following conditions hold: (i) F each x 0 X, {x 0 1 T 1 (x 0 ) {x 0 1 T (x 0 ). (ii) If x n x in X such that (x n, x n+1 ) 1 f all n N, implies (x n, x) 1. The following collary generalizes, Theem 4.1 of Latif and Beg [11] to G-metric spaces. Collary.5. Let (X,, G) be an dered complete G-metric space and T 1, T : X P cl (X) be multivalued mappings. If the pair (T 1, T ) satisfies the property (P 4 ), then F(T 1 ) = F(T ) provided that the following conditions hold: (i) F each x 0 X, {x 0 1 T 1 (x 0 ) {x 0 1 T (x 0 ). (ii) If x n x in X with (x n, x n+1 ) 1 f all n N, we have (x n, x) 1. Collary.6. Let (X,, G) be an dered complete G-metric space and T 1, T : X P cl (X) be multivalued mappings. If the pair (T 1, T ) satisfies the property (P 5 ), then F(T 1 ) = F(T ) provided that the following conditions hold: (i) F each x 0 X, {x 0 1 T 1 (x 0 ) {x 0 1 T (x 0 ). (ii) If x n x in X with (x n, x n+1 ) 1 f all n N, we have (x n, x) 1. Proof. Take α = λ and β = γ = 0 in Collary.4. References [1] M. Abbas, M.A. Khamsi, A.R. Khan, Common fixed point and invariant approximation in hyperbolic dered metric spaces, Fixed Point They Appl. 011, 011:5. [] M. Abbas, T. Nazir, S. Radenović, Common fixed points of four maps in partially dered metric spaces. Appl. Math. Lett. 4 (011) 150 156. [3] M. Abbas, T. Nazir, B.E. Rhoades, Fixed points of multivalued mapping satisfying Ćirić type contractive conditions in G-metric spaces, Hacettepe J. Math. Stat., to appear. [4] M. Abbas, B.E. Rhoades, Fixed point theems f two new classes of multivalued mappings, Appl. Math. Lett. (009) 1364 1368. [5] M. Abbas, B.E. Rhoades, Common fixed point results f non-commuting mappings without continuity in generalized metric spaces, Appl. Math. Comput. 15 (009) 6 69. [6] I. Altun, H. Simsek, Some fixed point theems on dered metric spaces and application, Fixed Point They Appl. Vol. 010, Article ID 6149, 17 pages. [7] I. Beg, A.R. Butt, Common fixed point f generalized set valued contractions satisfying an implicit relation in partially dered metric spaces, Math. Commun. 15 (010) 65 76. [8] B.S. Coudhury, P. Maity, Coupled fixed point results in generalized metric spaces, Math. Comput. Model. 54 (011) 73 79. [9] Lj. Ćirić, Generalization of Banach s contraction principle, Proc. Amer. Math. Soc. (45) (1974) 67 73. [10] R. Kannan, Some results on fixed points, Bull. Calcutta. Math. Soc. 60 (1968) 71 76. [11] A. Latif, I. Beg, Geometric fixed points f single and multivalued mappings, Demonstratio Math. 30 (1997) 791 800. [1] Z. Mustafa, B. Sims, Some remarks concerning D-metric spaces, Proc. Int. Conf. Fixed Point They Appl., Valencia (Spain), July, 003, pp. 189 198. [13] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (006) 89 97. [14] Z. Mustafa, H. Obiedat, F. Awawdehand, Some fixed point theem f mapping on complete G-metric spaces, Fixed Point They Appl., Vol. 008, Article ID 189870, 1 pages. [15] Z. Mustafa, B. Sims, Fixed point theems f contractive mapping in complete G-metric spaces, Fixed Point They Appl., Vol. 009, Article ID 917175, 10 pages. [16] J.J. Nieto, R.R. Lopez, Contractive mapping theems in partially dered sets and applications to dinary differential equations, Order (005) 3 39. [17] I.A. Rus, A. Petrusel, A. Sintamarian, Data dependence of fixed point set of some multivalued weakly Picard operats, Nonlinear Anal. 5 (003) 1944 1959. [18] A.C.M. Ran, M.C.B. Reurings, A fixed point theem in partially dered sets and some application to matrix equations, Proc. Amer. Math. Soc. 13 (004) 1435 1443. [19] R. Saadati, S.M. Vaezpour, P. Vetro, B.E. Rhoades, Fixed point theems in generalized partially dered G-metric spaces, Math. Comput. Model. 5 (010) 797 801.