Applied Mathematics, 2013, 4, 1296-1300 http://dx.doi.org/10.4236/am.2013.49175 Published Online September 2013 (http://www.scirp.org/journal/am)
Modular Spaces Topology Ahmed Hajji Laboratory of Mathematics, Computing and Application, Department of Mathematics, Faculty of Sciences, Mohammed V-Agdal University, Rabat, Morocco Email:
[email protected] Received April 29, 2013; revised May 29, 2013; accepted June 7, 2013 Copyright © 2013 Ahmed Hajji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT In this paper, we present and discuss the topology of modular spaces using the filter base and we then characterize closed subsets as well as its regularity. Keywords: Topology of Modular Spaces; ∆2-Condition; Filter Base
1. Introduction In the theory of the modular spaces X , the notion of ∆2-condition depends on the convergence of the sequences in modular space X . More precisely, it reads: for any sequence xn n in X , if lim 2 xn 0 , n
we have lim 2 xn 0 . This condition has been used n
to study the topology of modular spaces, see J. Musielak [1], and to establish some fixed point theorems in modular spaces, see [2-7]. Some fixed point theorems without ∆2-condition can be found in [8,9]. In this paper, we present a new equivalent form for the ∆2-condition in the modular spaces X which is used to show that the corresponding topology is separate and to establish some associated topological properties, including the characterization of the -closed subsets as well as its regularity. The present work is an improved English version of a pervious preprint in French [10].
2. Preliminaries We begin by recalling some definitions. Definition 2.1 Let X be an arbitrary vector space over K or . 1) A functional : X 0, is called modular if x 0 implies x 0 . a) x x for any x X when K , and
b) eit x x for any real t when K . c) x y x y 1.
for
2) If we replace c) by the following x y x y for Copyright © 2013 SciRes.
, 0
and
, 0
and
1 , then the modular is called convex. 3) For given modular in X, the X x X x 0 as 0 is called a modular space. 4) a) If is a modular in X, then x x inf u 0, u u
is a F-norm. b) Let be a convex modular, then x
x inf u 0, 1 u
is called the Luxemburg norm.
3. Topology τ in Modular Spaces In this section, we introduce the property 0 for a modular , which will be used to show that the corresponding topology, noted by , on modular space X is separate, and to characterize their closed subsets. We begin with the following Proposition 3.1 Consider the family B 0, r r 0 , where B 0, r x X x r .
Then 1) The family is a filter base. 2) Any element of is balanced and absorbing. Furthermore, if is convex, then any element of is convex. Proof. 1) is a filter base. Indeed, we have AM
A. HAJJI
a) because any B 0, r . b) Let B 0, r1 and B 0, r2 be in and set r inf r1 , r2 . Then, for any z B 0, r we have z r r1 z r r2
and therefore z B 0, r1 B 0, r2 . That is B 0, r B 0, r1 B 0, r2 .
Hence is a filter base for the existence of B 0, r such that B 0, r B 0, r1 B 0, r2 .
2) Let B 0, r . a) B 0, r is balanced. Indeed, for given ei with and 1 , and given x B 0, r , we have
x e x x x < r . i
This means that x B 0, r . b) B 0, r is absorbing. Indeed, for given x X we have lim x 0 . Whence, for all r > 0 there 0
exists > 0, such that 0 < < and x < r . Hence, there exists > 0 such that x B 0, r . This shows that B 0, r is absorbing. Now, assume that is in addition convex and let B 0, r . For given x, y B 0, r and 0,1 , we have
1297
we see that for z x y B 0, 0 B 0, 0 with x 0 y 0 .
We obtain y z x B 0, 0 . This implies z x 0 and x 0 L . Thence
z x 0 r r. This infers that z B 0, r , and so B 0, 0 B 0, 0 B 0, r .
Hence the family is a fundamental system of neighborhoods of zero, then the unique topology defined by in X is given by G , G X if x G ,
then V such that x V G ,
so that X is a topological vector space. To show that X , is separate, let x, y in X such that x y and assume that for any Vx neighborhood of x and Vy neighborhood of y we have Vx Vy . So that one can consider 1 1 z x B 0, y B 0, n n
for certain n * . Then
x 1 y x 1 y < r ,
1 x z n y z 1 . n
x 1 y B 0, r .
Since satisfies the property 0 , then there exist for any 0 , two reals L 0 and 0 , such that
then Thence B 0, r is convex. Definition 3.1 We say that satisfies the property 0 if for all > 0 , there exist L > 0 and > 0 such that y x for every x, y satisfying x L and x y < . Theorem 3.1 Assume that the modular satisfies the property 0 . Then X is a separate topological vector space. Proof. In Proposition 3.1, we have seen that the family is a filter base, and furthermore any element of is balanced and absorbing. On the other hand, for any B 0, r , there exists 0 > 0 such that B 0, 0 B 0, 0 B 0, r .
In fact, let ; r > > 0 . Since satisfies the property 0 , there are L > 0 and > 0 , such that for x < L and x y < we have y x < . Thus, if we set
0 inf r , L, , Copyright © 2013 SciRes.
y x
0 there exists N 0 such that xn x B 0, whenever n > N 0 . Note that the property 0 is a necessary condition to show the uniqueness of the limit when exists. Thus, the -convergence need the property 0 and it is easy to see that -convergence and -convergence are equivalent. Definition 3.4 Let be a modular satisfying the property 0 . A subset B of X is said to be -closed if and only if the complimentary of B in X , noted by C XB , is an element of . The following lemma shows that the property 0 makes sense in the theory of modular spaces. Lemma 3.1 Let be a modular and X be a modular space. Then satisfies the 2 -condition if and only if satisfies the property 0 . Proof. To prove “if”, let xn n be a sequence in X such that xn 0 as n . This implies that for all > 0 , there exists n0 such that for any n > n0 we have
xn inf L, , . 2 Copyright © 2013 SciRes.
X n xn Yn X n inf L, , . 2
This yields Yn 2 xn
xn when2 ever n n0 . Whence, the sequence 2 xn n tends to zero as n goes to , and therefore satisfies the 2 -condition. For “only if”, let be a modular satisfying the 2 -condition, and suppose that there exists > 0 such that for any L > 0 and for any > 0 , there exist x, y X satisfying x < L, x y < and 1 y x . In particular, for L there n exist xn , yn X such that
1 1 and n n yn xn ,
xn , yn xn
which implies xn 0 and yn xn 0 n . However, we have
as
yn yn xn xn 2 xn yn 2 xn . Now, since satisfies the ∆2-condition, then yn 0 as n . It follows that
yn xn 0 as n , which contradicts the fact that yn xn > 0 for any n . Finally, for all > 0 , there are L > 0 and > 0 such that if x < and y x < , we have y x < . This completes the proof of Lemma 3.1. In the following theorem, we show that the -topology and the 1 -topology are the same. Theorem 3.2 Let be a modular satisfying the ∆2condition and F X , then F is -closed if and only if F is -closed. The following result is needed to show Theorem 3.2. Proposition 3.2 Let be a modular satisfying the ∆2-condition and F a -closed subset of X . Then x F 0, B x, F .
Proof. For x X , we have F F x F x C X , C X is an open set of the -topology
B 0, x B 0, B x, C XF > 0, such that B x, F . AM
A. HAJJI
Finally, x F > 0, B x, F .
and Proof of Theorem 3.2. Let F be -closed xn n be a sequence in F such that xn x . Then, for any > 0 , there exists n0 such that for every n > n0 , we have xn B x, . This implies that > 0, B x, F .
Whence, making use of Proposition 3.1, we get that xF . Conversely, assume that F is not -closed, then C XF is not an open set for the -topology. There exists then x C XF satisfying B x, C XF and so 1 B x, F for any > 0 . Therefore, for k 1 there exists xk B x, F . Thence, the obtained k sequence
xn n F
satisfies xn x . This implies
x F , which is in contradiction with the fact that x C XF . In conclusion, F is -closed. Remark 3.1 Observe that
1299
x yn
0 . Next, since satisfies r the 2 -condition then by Lemma 3.1, for > 0 , 3 there exist L 0 , and 0 such that if x L and y x we have y x . Morer over, there exists m0 * such that inf L, m whenever m m0 . Now, let m1 max 3, m0 and we consider the open neighborhood of x0
satisfies the 2 -condition As consequence, we see that under the assumption that
such that yn x . Whence x A .
Inversely, let x A , then by Theorem 3.2, there ex-
r Vx0 x0 B 0, . m1
satisfies the property 0 .
satisfies the 0 property, we have
1 this implies that there exists a sequence n
Suppose next that Vx0 A and let y Vx0 A . Since A is closed we make use of Proposition 3.1 to exhibit a sequence
yn n A
such that yn y . So
that one considers X n y yn and Yn x0 yn . Since yn A and x0 A , then Yn r . On the other hand, note that
X n y yn
r inf L, , m1
whenever n n0 and
X n Yn x0 y
r inf L, . m1
Therefore r Yn y yn
r r 2r m1 3 3
whenever n n0 , a contradiction. Thus Vx0 A . Remark 3.2 If satisfies Fatou property, then B 0, r B 0, r x X x r
is a closed ball of the topology . We note by B f x, r all closed ball centered at x with the radius r 0 (see [7]). Corollary 3.1 Under the same hypotheses of Theorem
AM
A. HAJJI
1300
3.3, and if the modular satisfies Fatou property, then
Vx0 A .
Proof. Making appeal of Theorem 3.3, there exists r Vx0 x0 B 0, such that Vx0 A . Then, we m1 r x0 B f 0, . Indeed, let y Vx0 and m1 note that from Proposition 3.1, there exists a sequence r yn n B f 0, such that m1
have Vx0
By Proposition 3.1, there exists
yn n B 0,
r m1
such that yn y . Moreover, the sequence
x0 yn n Vx0
satisfying x0 yn x0 y . Hence
x0 y Vx0 .
Finally, we take the same arguments as in the proof of Theorem 3.3, we have
Vx0 A .
x0 yn y,
REFERENCES
which implies that yn y x0 . Indeed, it is easy to see
[1]
J. Musielak, “Orlicz Spaces and Modular Spaces,” Lecture Notes in Mathematics, Vol. 1034, 1983.
[2]
A. Ait Taleb and E. Hanebaly, “A Fixed Point Theorem and Its Application to Integral Equations in Modular Function Spaces,” Proceedings of the American Mathematical Society, Vol. 128, 2000, pp. 419-426. doi:10.1090/S0002-9939-99-05546-X
[3]
A. Razani and R. Moradi, “Common Fixed Point Theorems of Integral Type in Modular Spaces,” Bulletin of the Iranian Mathematical Society, Vol. 35, No. 2, 2009, pp. 11-24.
[4]
A. Razani, E. Nabizadeh, M. B. Mohammadi and S. H. Pour, “Fixed Point of Nonlinear and Asymptotic Contractions in the Modular Space,” Abstract and Applied Analysis, Vol. 2007, 2007, Article ID: 40575.
[5]
A. P. Farajzadeh, M. B. Mohammadi and M. A. Noor, “Fixed Point Theorems in Modular Spaces,” Mathematical Communications, Vol. 16, 2011, pp. 13-20.
[6]
M. A. Khamsi, “Nonlinear Semigroups in Modular Function Spaces,” Thèse d'état, Département de Mathématiques, Rabat, 1994.
[7]
M. A. Khamsi, W. Kozlowski and S. M.-Reich, “Fixed Point Theory in Modular Function Spaces,” Nonlinear Analysis, Theory, Methods and Applications, Vol. 14, No. 11, 1990, pp. 935-953.
[8]
F. Lael and K. Nourouzi, “On the Fixed Points of Correspondences in Modular Spaces,” ISRN Geometry, Vol. 2011, 2011, Article ID: 530254. doi:10.5402/2011/530254
[9]
M. A. Khamsi, “Quasicontraction Mappings in Modular Spaces without ∆2-Condition,” Fixed Point Theory and Applications, Vol. 2008, 2008, Article ID: 916187.
that Yn yn y x0 0 and since satisfies the
2 -condition we have also X n 2 yn y x0 0 .
Thence, for 0 , there are L 0 and 0 such that
X n inf L, , , 2 and
Yn X n Yn inf L, , , 2
whenever n n0 , then
Yn yn y x0 inf L, , , 2 2 2 2 whenever n n0 . Therefore
r yn y x0 B 0, B f m1
r 0, . m1
It follows r y x0 y x0 x0 B f 0, , m1
and hence r Vx0 x0 B f 0, . m1
[10] A. Hajji, “Forme Equivalente à la Condition ∆2 et Certains Résultats de Séparations dans les Espaces Modulaires,” 2005. http://arXiv.org/abs/math.FA/0509482
Inversely, let r x0 y x0 B f 0, . m1
Copyright © 2013 SciRes.
AM
