A Locally Uniform Rotund and Property (β) of Generalized Cesàro (ces(p),d) Metric Space

Transcription

1 Int. Journal of Math. Analysis, Vol. 5,, no., A Locally Uniform Rotund and Property β of Generalized Cesàro cesp,d Metric Space W. Sanhan and C. Mongoleha Department of Mathematics, Faculty of Liberal Arts and Science Kasetsart University, Kamphaeng-Saen Campus Nahonpathom 734, Thailand Abstract In this paper we study some geometric properties of Cesàro metric linear space cesp,d where p =p is a bounded sequence of positive real number. The main result of this study is to show that the space cesp,dislur space and has property H and β. Mathematics Subject Classification: 46B, 46B45, 46B99 Keywords: Cesàro cesp,d metric linear space, locally uniforml rotund, property H and property β Introduction A lot of mathematicians are interested in Cesàro sequence space. Y. cui and H. Hudzi [] indicated that Cesàro sequence space is NUC and has uniform Opail property where p>. Y. Cui and C. Meng [] pointed that Cesàro sequence space has the Banach-Sas of type p if p>, and it was shown that has property β. S. Suantai [8] revealed that generalize Cesàro sequence space has property H. In 3, S. Suantai [9] introduced a new modular space which is a generalization of Cesàro sequence space, and then, some geometric properties on such new modular space were considered eqiuipped with the Luxemburg norm. The notion of rotundity or strictly convexity in metric linear spaces was introduced by G. C. Ahuja and others in 977, and the concept of uniform rotundity in metric linear spaces was introduced in 979 by K. P. R. Sastry. W. Junde and C. Lianchang [3] showed that in a complete metric linear space uniform rotundity implies reflexivity. In [4], W. Junde and T. D. Narang showed that if a metric linear spaces X, d isur then X, d Corresponding author; C.Mongoleha addresses: winate s@yahoo.com, cm.mongol@hotmail.com

2 55 W. Sanhan and C. Mongoleha has property H. In the same way N. Petrot [7] mentioned that linear metric space X, d islur then X, d has property H. According to the notion above, I am interested in studying geometric properties of generalize Cesàro sequence metric linear speaces. In this paper we defined metrics on cesp, and we show that a metric linear space cesp islur space and has property H and β. Preliminary Notes Definition.. For a real vector space X, a function ρ : X [, ] is called a modular if it satisfies the following conditions : i ρx =if and only if x =; ii ραx =ρx for all scalar α with α =; iii ραxβy ρxρy, for all x, y X and all α, β with αβ =. The modular ρ is called convex if iv ραx βy αρx βρy, for all x, y X and all α, β with α β =. and If ρ is modular in X, we define X ρ = {x X : ρλx asλ }, X ρ = {x X : ρλx < for some λ>}. It clear that X ρ X ρ.ifρ is a convex modular, for x X ρ we define x = inf{λ >:ρ x }. λ Orlicz [6] proved that if ρ is convex modular on X, then X ρ = Xρ and. is a norm on X ρ for which it is a Banach space. The norm. defined as in is called the Luxemburg norm. Let l be the space of all real sequences and p =p a bounded sequence of positive real numbers with p for all N, the Cesàro sequence space ces p, for short is defined by ces p = {x l : n= n p n xi < } i=

3 Locally uniform rotund and property β 55 equipped with the norm x = n= n p p n xi. The Generalized Cesàro Sequence Space cesp is defined by where ρx = n= i= cesp ={x l : ϱλx < for someλ >}, n pn xi is a convex modular on cesp. We consider i= cesp equipped with the Luxemburg norm x = inf{ε >:ρ x ε }. Definition.. A metric space is an order pair X, d, where X is a nonempty set and d a metric on X, that is d : X X R is a function satisfies the following conditions: i dx, y for all x, y X; ii dx, y =if and only if x = y; iii dx, y =dy, x for all x, y X; iv dx, z dx, ydy, z for all x, y, z X. From now on, let X be a vector space over the scalar field of real numbers R and d an invariant linear metric on X. We denoted B d X ={x X dx, r} and S d X ={x X dx, = r}. We begin by introduce some concepts about geometric properties in metric linear space. Definition.3. Let X, d be a metric space. A sequence x n of member of X converges to x X, if lim dx n,x=. When x n converges to x, we n write lim x n = x or x n x. n Definition.4. A metric linear space X, d is called locally uniformly rotund written LUR for shortif, for each r >, ε > and for each x S d,r there exists δ = δ r,x,ε > such that y B d,r and dx, y ε imply d xy, <r δ. Definition.5. A metric linear space X, d has the property β if and only if for each r> and ε> there exists δ> such that for each element x B d,r and each sequence x n in B d,r with sepx n ε there is an index for which d xx, δ where sepx n = inf{dx n,x m :n m} > ε.

4 55 W. Sanhan and C. Mongoleha Definition.6. A metric linear space X, d is said to have property H if for each x S d,r and x n S d,r such that x n w x, implies x n x. Theorem.7. If X, d is LUR metric linear space, then X, d possesses property H. Proof. See [7]. 3 Main Results Let l be the space of all real sequences and p =p be a bounded sequnces of positive real number with p > for all N and let M = supp and m = inf p. By the convexity of the function t p where as p> we have the following mapping dx, y = xi yi p M is a metric on the = i= space cesp. First, we will show that the space cesp under the metric given is LUR. Theorem 3.. The space cesp,d is LUR. Proof. Let r> be given. Let x S d,r and x n B d,r for all n N such that d xnx, r as n, we want to show that dx, x n as n. First, we show that x n i xi asn for all i N. Suppose that there exits i N, which x n i xi asn. Without loss of generality we may assume that i = and there exists η>such that x n x p >ηfor all n N, this implies that p x n p x p >η for all n N. Since t t p is uniformly convex function, so there exists δ> such that x nx p < δ xn p x p for all n N.

5 Locally uniform rotund and property β 553 Thus, d x M n x, = x p nixi = i= = x n x p x p nixi = i= xn p x p < δ x nixi = i= xn p x p < δ p x n i = i= p xi = i= = p x n i p xi = i= = i= xn p x p δ Hence, rm rm = r M δη p. δη p d x n x, < r M δη M p δη <r a where a,r r M p M. This implies that d xnx, r as n, a contradiction. Next, let ε> be given. Since x n i xi asn there exists ε,ε M and,n N such that = i= = xi p < ε 3 M, i= = p x n i > = i= i= x n i xi p < ε 3, and xi p ε 3 M p

6 554 W. Sanhan and C. Mongoleha for all n>n. Thus for all n>n, we have p dx n x, M = x n i xi = i= p p = x n i xi x n i xi = i= = i= < ε p 3 x n i xi = i= ε p p 3 M x n i xi = i= = i= ε p p 3 M r M x n i xi = i= = i= ε p 3 M r M xi ε 3 M = i= p M xi = i= ε p 3 M xi ε 3 M = i= p M xi = i= ε p 3 M xi ε 3 = i= ε ε 3 M ε 3 M 3 = ε 3 ε 3 ε 3 = ε <ε M. This means that, dx n x, <ε. Therefore, dx n x, asn. By Theorem.7, we have the following results Corollary 3.. The space cesp,d has property H.

7 Locally uniform rotund and property β 555 Next, we will show that The space cesp,d has property β. Lemma 3.3. Lety,z cesp,d. Ifβ,, then dy z, M dy, M M β dy, M M β M dz, M. Proof. Let y, z cesp,d and <β<. Then = p M dy z, = yizi = i= β yi β = i= p β yi β = M = = M β M This completes the proof. i= p yi M β i= i= = i= zi p β yi p M β zi i= = = p yi zi β i= = i= yi i= yi i= p yi zi β p p = dy, M M β dy, M M β M dz, M. Lemma 3.4. Let y, z cesp,d. Then for any ε> and L> there exists δ> such that dy z, M dy, M <ε, p whenever dy, M L and dz, M δ. Proof. Given ε> and L>. We choose δ = εβm M when β = ε M Lε.

8 556 W. Sanhan and C. Mongoleha By Lemma 3.3, we have and we have dy z, M dy, M M β dy, M M β M dz, M dy, M M β L M β δ M = dy, M M ε L M L ε M εβ M β M M < dy, M ε ε = dy, M ε. dy, M dy z, M M β dy z, M M β M d z, M < dy z, M M β dy, M ε M β M δ dy z, M M βl ε M εβ M β M M = dy z, M M ε M L ε L ε ε = dy z, M ε ε = dy z, M ε. Hence, we obtain dy z, M dy, M <ε. Theorem 3.5. The space cesp,d has property β. Proof. Let ε> and x n Bcesp with sepx n ε and x Bcesp. N {}}{ we define y N =,,...,, N yi,yn,yn,... Since for each i= i N, x n i n= is bounded, by using the diagonal method, we have that for each N N we can find subsequence x nj ofx n such that x nj i converge for each i N with i N. Therefore, for each N N there exists r N N such that sepx N n j>r N ε. Hence there is a sequence of positive integers r N N= with r <r <r 3 <... such that dx N r N, ε for all N N. Then

9 Locally uniform rotund and property β 557 there exists η> such that =N p x rn η i= for all N N. From Lemma 3.4 there exits δ such that dy z, M M η dy, < 3 m whenever dy, M r M and dz, M δ. Since x Bcesp and dx, M r M then there exits N N such that dx N, M δ. Put y = x N r N =N and z = x N. Then i= xix p r N i =N i= x p r N i η. 4 m By using, 3, 4 and convexity of function ft = t p, for all N, we have d y z M, = xix p r N i = i= N = xix p r N i xix r N i = i= =N i= N xix p r N i x p r N i η = i= =N i= N p xi N p x rn i = i= = i= p x m rn i η m =N i= N p xi p x rn i = i= = i= p m x m rn i η m =N i= < rm rm m m η η m = r M η. p m

10 558 W. Sanhan and C. Mongoleha Thus d yz, < rm η M <r δ, where δ,r r M η M. Hence, the space cesp,d has property β. Acnowledgements. The authors would lie to than the Faculty of Liberal Arts and Science, Kasetsart University Kamphaeng-Saen Campus for financial supported. References [] Y. Cui and H. Hudzi. Some geometric properties related to fixed point theory in Cesàro space, Collect. Math., 5 999, [] Y. Cui, C. Meng and R. Ucienni. Banach - Sa property and property β in Cesàro sequence space. Southeast Asian Bull. Math, 4,. [3] W. Jude and C. Lianchang. Reflexivity of Uniform convexity in metric linear space and application. Advances in Math. China, 3 994, [4] W. Jude and T. D. Narang. H - property, Normal strucre and fixed points of non expensive mapping in metric linear space. Acta Methemitica Vietnamica., 5, 3 8. [5] D. N. Kutzarova. A nearly Uniformly convex space which is not a β space. Acta Univ. Cololin. Math., 3 989, [6] W. Orlicz. A note on modular spaces I. Bull. Acad. Polon. Sci.Sér. Sci. Math. Astronom. Phys., 9 96, [7] N. Petrot,S. Suantai.Locally Uniform Rotunbdity in Metric Linear Spaces Italian Journal of Pure and Applied mathematics, 5. [8] S. Suantai. On H- Property of Some Banach sequence space. Arch. Math.Brno., 39 3, [9] S. Suantai. On some convexity properties of genneralized Cesàro sequence spaces. Georgian Math. J., 3, 93. Received: September,