Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Int Jr. of Mathematical Sciences & Applications Vol.4, No.1, January-June 2014 Copyright Mind Reader Publications ISSN No: 2230-9888 www.journalshub.com ON SOME FIXED POINT RESULTS IN dp-COMPLETE HAUSDORFF SPACES V. Naga Raju Department of Mathematics, College of Engineering Osmania University,Hyderabad - 500007, Andhra Pradesh,India Email : [email protected] Abstract In this paper we define d p - complete topological spaces for p 2 . A d 2 - complete topological space is a generalized d-complete topological space introduced by Troy L. Hicks. Some fixed point theorems for self-maps of d p - complete Hausdorff topological spaces are established which generalize the results of Troy L. Hicks. Keywords: d p - complete topological spaces, d-complete topological spaces, orbitally lower semi continuous and orbitally continuous maps. AMS (2000) Mathematics classification: 47H10, 54H25 1 . Introduction Troy L. Hicks [5] has introduced d-complete topological spaces, attributing the basic ideas of these spaces to Kasahara ([8], [9]) and Iseki [7] as follows: 1.1 Definition: A topological space (X, t) is said to be d- complete if there is a mapping d : X X [0, ) such that (i) d ( x, y) 0 x y and (ii) xn is a sequence in X such that d (x n 1 n , x n 1 ) is convergent implies that xn converges in (X, t). In this paper we introduce d 2 - complete topological spaces as a generalization of d-complete topological spaces. In fact , we define non-empty set X, let 1.2 X p be its p-fold cartesian product. Definition: A topological space (X, t) is said to be that (i) d p - complete topological spaces for any integer p 2 . For a d p - complete if there is a mapping d p : X p [0, ) such d p ( x1 , x 2 , . . . , x p ) 0 x1 x 2 . . . x p and (ii) xn is a sequence in X with lim d p ( xn , xn1 , xn2 , . . . , xn p 1 ) 0 implies that xn converges to some point in (X, t). A n d p - complete topological space is denoted by (X, t, d p ) 1.3 Remark: The function d in the Definition 1.1 and the function d 2 (the case p = 2) in Definition 1.2 are both defined on XX and satisfy condition (i) of the definitions which are identical. Since the convergence of an infinite series n 1 n of real numbers implies that lim n 0 , but not conversely; it follows that every dn 235 complete topological space is d 2 - complete, but not conversely. Therefore the class of d 2 - complete topological spaces is wider than the class of d-complete spaces and hence a separate study of fixed point theorems of selfmaps on d 2 - complete topological spaces is meaningful. . 2 . Preliminaries Let X be a non-empty set. A mapping p non-negative on X provided 2.1 d p : X p [0, ) is called a d p ( x1 , x 2 , . . . , x p ) 0 x1 x 2 . . . x p . Definition: Suppose (X, t) is a topological space and d p is a p non- negative on X. A sequence xn in X is d p - Cauchy sequence if d p ( x n , x n 1 , . . . , x n p 1 ) 0 as n . said to be a In view of Definition 2.1, a topological space (X, t) is p-non- negative d p - complete if there is a d p on X such that every d p - Cauchy sequence in X converges to some point in (X, t). If T is a self map of a non-empty set X and x X , then the orbit of x, OT (x) is given by OT ( x) x, Tx , T 2 x, . . . . If T is a self map of a topological space X, then a mapping G : X [0, ) is said to be T-orbitally lower semi-continuous (resp. T-orbitally continuous) at some x X with x n x* X if xn is a sequence in OT (x) for x * as n then G( x*) lim inf G( xn ) ( resp. G( x*) lim G( xn ) ). A self map T n of topological space X is said to be w-continuous at x X if n x n x as n implies Tx n Tx as n . d p is a p non-negative on a non-empty set X, and T : X X then we write, for simplicity of If notation, that (2.2) G p ( x) : d p ( x, Tx, T 2 x, . . . , T p1 x) for x X Clearly we have (2.3) G p ( x ) 0 if and only if x is a fixed point of T . (2.4) H p ( x, y) d p ( x, y, Ty, T 2 y, . . . , T p2 y) for x, y X (2.5) E p ( x, y ) d p ( x, y, y, . . . , y ) for x, y X Clearly (2.6) 3.1 H p ( x, Tx ) G p ( x) and E p ( x, x ) 0 . 3 . Main results Theorem: Suppose (X, t, dp) is a dp- complete Hausdorff topological space and T: X X is a w-continuous map such that Gp (Tx) k (Gp(x)) for all x X, where k : [0,) [0,) is a non-decreasing function with k(0) = 0. Then T has a fixed point iff there is an x X such that lim kn(Gp(x))=0. In this case lim Tnx=z and Tz=z. n n Proof : Suppose T has a fixed point, say z, so that z = Tz =…..= T p-1 z and hence Gp(z) = 0 which gives kn(Gp( z )) = 0 for all n 1. lim kn(Gp(z))=0. n Conversely, suppose that lim kn(Gp(x))=0 for some x X. n Let xn= Tnx for n 0 with x0 = x. Then 236 d p ( xn , xn1 ,....,xn p 1 ) d p (T n x, T n1 x,....,T n p 1 x) G p (T n x) k (G p (T n1 x)) k 2 G p T n2 x ........ k n G p x 0 as n which gives (xn) is a dp- Cauchy sequence in X and hence converges, say to z. That is, lim Tnx=z exists. Since n x n z as n and since T is w- continuous, we get that Tx n Tz as n . That is, x n 1 Tz as n which gives x n Tz as n . Since X is Hausdorff, we get Tz =z. 3.2 Remark: It may be noted that in view of Remark 1.3, the result proved by Troy L. Hicks ([5], Theorem 2,pp.437) is a particular case of Theorem 3.1 . 3.3 Theorem : Suppose (X,t,dp) is a dp-complete Hausdorff topological space and T : X X such that there is an x X with Gp(Ty) k (Gp(y)) for all y OT(x), where k : [0,) [0,) is a nondecreasing function with k (0) = 0. Suppose kn(Gp(x)) 0 as n .Then a) lim Tnx = z exists and n b) Tz = z iff Gp(x) is T - orbitally lower semi-continuous at z. Proof : (a) Consider dp (Tnx, Tn+1x,……, Tn+p-1x) = Gp (Tn x) k (Gp(Tn-1x)) k2 (Gp(Tn-2 x))…… kn (Gp(x)) 0 as n by hypothesis, which n gives that (T x) is a dp-Cauchy sequence in X and hence converges, say, to z. (b) Suppose Tz = z and (Tn x) is a sequence in OT(x) with lim Tnx=z. n Then Gp(z) = 0 ≤ lim inf n Gp(Tnx) which gives the T-orbitally lower semi-continuity of Gp(x) at z. Conversely, suppose that Gp(x) is T-orbitally lower semi-continuous at z. By proof of part (a), (Tnx) is a dp-Cauchy sequence and hence 0 = lim Gp(Tnx). n This together with the T-orbitally lower semi-continuity of Gp(x) at z imply Tz=z by Theorem 3.1. 3.4 Remark:The result of Troy L. Hicks ([5], Theorem 3, pp.437, 438) is a particular case of Theorem 3.3. We now present some consequences of Theorems 3.1 and 3.3. 3.5 Corollary: Suppose (X, t, dp) is a dp-complete Hausdorff topological space, T : X X is w-continuous and satisfying Hp(Tx, Ty) Hp (x, y) for all x, yX, where 0 < < 1. Then T has a unique fixed point z and z = lim Tnx for any x X. n Proof : Taking k(t) = t, where 0 < < 1. Then kn (Gp(x)) = n Gp(x) and therefore, by Theorem 3.1, we have xn z as n , where Tz = z. Suppose z and z*are two fixed points and taking x = z, y = z* in the inequality, we get Hp(Tz, Tz*) Hp(z, z*) Hp(z, z*) Hp(z, z*), which gives Hp(z, z*) = 0. That is, z = z*, proving the uniqueness. 3.6 Remark: The result proved by Troy L. Hicks ([5], Corollary 2,pp.438) is a particular case of Corolloary 3.5. 3.7 Corollary: Suppose (X, t, dp) is a dp-complete Hausdorff topological space, T : XX and Hp(Tx, Ty) (Hp(x,y)) Hp(x,y) for all x, y X, where : [0, ) [0,1) is non-decreasing. If xn = Tny, then a) lim xn= z exists and n b) Tz = z iff Gp(x) is T-orbitally lower semi-continuous at z. 237 Proof :Taking k(t) = t (t). We find by induction that kn(t) t ((t))n and the inequality of the theorem gives Hp(Tx, Ty) k(Hp(x,y)) for all x, y X. Now, taking y = Tx in the above inequality, we get Hp(Tx, T2x) k(Hp(x, Tx) for all x X Gp (Tx) k(Gp(x)) for all x X. Since (t) < 1, kn(Gp(x)) 0 as n and therefore the theorem follows from Theorem 3.3. 3.8 Remark : Note that, the result of Troy L. Hicks ([5], pp.438) is a particular case of Corollary 3.7. References: [1] J. Achari, Results on fixed point theorems, Maths. Vesnik 2 (15) (30) (1978), 219-221. [2] Ciric, B. Ljubomir, A certain class of maps and fixed point theorems, Publ. L’Inst. Math. (Beograd) 20 (1976), 73-77. [3] B. Fisher, Fixed point and constant mappings on metric spaces, Rend. Accad. Lincei 61(1976), 329 – 332. [4] K.M. Ghosh, An extension of contractive mappings, JASSY 23(1977), 39-42. [5] Troy. L. Hicks, Fixed point theorems for d-complete topological spaces I, Internet. J. Math & Math. Sci. 15 (1992), 435440. [6] Troy. L. Hicks and B.E. Rhoades, Fixed point theorems for d-complete topological spaces II, Math. Japonica 37, No. 5(1992), 847-853. [7] K. Iseki, An approach to fixed point theorems, Math. Seminar Notes, 3(1975), 193-202. [8] S. Kasahara, On some generalizations of the Banach contraction theorem, Math. Seminar Notes, 3(1975), 161-169. [9] S. Kasahara, Some fixed point and coincidence theorems in L-Spaces, Math. Seminar Notes, 3(1975), 181-187. [10] M.S. Khan, Some fixed point theorems IV, Bull. Math. Roumanie 24 (1980), 43-47. 238