Báo cáo hóa học: " Research Article On Some Generalized B m-Difference Riesz Sequence Spaces and Uniform Opial Property"

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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 485730, 17 pages doi:10.1155/2011/485730 Research Article On Some Generalized B m-Difference Riesz Sequence Spaces and Uniform Opial Property ¨¨ Metin Basarır and Mahpeyker Ozturk ¸ Department of Mathematics, Sakarya University, 54187 Sakarya, Turkey Correspondence should be addressed to Metin Basarır, basarir@sakarya.edu.tr ¸ Received 29 November 2010; Accepted 18 January 2011 Academic Editor: Radu Precup Copyright q 2011 M. Basarır and M. Ozturk. 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. q q q We deﬁne the new generalized diﬀerence Riesz sequence spaces r∞ p, B m , rc p, B m , and r0 p, B m q which consist of all the sequences whose B m -transforms are in the Riesz sequence spaces r∞ p , q q rc p , and r0 p , respectively, introduced by Altay and Basar 2006 . We examine some topological ¸ q q q properties and compute the α-, β-, and γ -duals of the spaces r∞ p, B m , rc p, B m , and r0 p, B m . Finally, we determine the necessary and suﬃcient conditions on the matrix transformation q q q from the spaces r∞ p, B m , rc p, B m , and r0 p, B m to the spaces l∞ and c and prove that q q have the uniform Opial property for pk ≥ 1 for all m m sequence spaces r0 p, B and rc p, B k ∈ Æ. 1. Introduction Let w be the space of real sequences. We write l∞ , c, c0 for the sequence spaces of all bounded, convergent and null sequences, respectively. Also, by bs, cs, and l1 , we denote the sequence spaces of all bounded, convergent and absolutely convergent series, respectively. A linear topological space X over the real ﬁeld R is said to be a paranormed space if there is a subadditive function g : X → R such that g θ g −x and scalar 0, g x multiplication is continuous; that is, |αn − α| → 0 and g xn − x → 0 imply g αn xn − αx → 0 for all α’s in R and all x’s in X , where θ is the zero vector in the linear space X . Assume here and after that p pk is a bounded sequence of strictly positive real numbers with max{1, H }. Then, the linear space l∞ p , c p , c0 p , and l p were sup pk H and M deﬁned by Maddox 1, 2 , Nakano 3 and Simons 4 as follows:
2 Journal of Inequalities and Applications xk ∈ w : sup|xk |pk < ∞ , l∞ p x k ∈Æ 0 for some l ∈ Ê , 1.1 xk ∈ w : lim |xk − l|pk cp x k→∞ xk ∈ w : lim |xk |pk c0 p x 0, k→∞ which are the complete spaces paranormed by sup|xk |pk /M , g1 x k ∈Æ 1.2 pk xk ∈ w : n .
Journal of Inequalities and Applications 3 The Riesz sequence spaces introduced by Altay and Basar in 5, 6 are ¸ ⎧ ⎫ pk ⎨ ⎬ k 1 xk ∈ w : n . The results related to the matrix domain of the matrix B are more general and more comprehensive than the corresponding consequences of matrix domain of Δ and include them 6, 8–13 . Basarır and Kayikci 14 deﬁned the matrix Bm bnk which reduced the diﬀerence m ¸ ¸ m−1 matrix Δ ΔΔ in case r 1, s −1 by m ⎧⎛ ⎞ ⎪ ⎪⎝ m ⎠ m−n k n−k ⎨ max{0, n − m} ≤ k ≤ n , r s; m bnk 1.10 n−k ⎪ ⎪ ⎩ 0 ≤ k < max{0, n − m} or k > n , 0;
4 Journal of Inequalities and Applications and introduced the generalized Bm -diﬀerence Riesz sequence space which is the sequence space of the sequences x whose Rq Bm -transforms are in l p . The main purpose of this paper is to introduce the Bm -diﬀerence Riesz sequence spaces q q q r∞ p, Bm , rc p, Bm , and r0 p, Bm of the sequences whose Rq Bm -transform are in l∞ p , c p , and c0 p , respectively, and to investigate some topological and geometric properties of them. For simplicity, we take the matrix Rq Bm T . 2. Bm -Difference Riesz Sequence Spaces {yn q }, which is used, as the Rq Bm Let us deﬁne the sequence y T -transform of a sequence x xk , that is, 1 n−1 n m rm n∈Æ . r m−i k si−k qi xk yn q Tx qn xn , 2.1 n i−k Qn k 0 Qn ik q q q We deﬁne the Bm -diﬀerence Riesz sequence spaces r∞ p, Bm , rc p, Bm , and r0 p, Bm by q xj ∈ w : ∈ l∞ p r∞ p , B m x Tx , n q xj ∈ w : ∈c p rc p , B m 2.2 x Tx , n q xj ∈ w : ∈ c0 p r0 p , B m x Tx . n q q q If m 1 then they are reduced the spaces r∞ p, B , rc p, B , and r0 p, B deﬁned by Basarır in ¸ q q q 15 . If we take B Δ then we have r∞ p, Δm , rc p, Δm , and r0 p, Δm . If we take B Δ and q q q m 1 then we have r∞ p, Δ , rc p, Δ , and r0 p, Δ . If we take pk p for all k then we have q q q r∞ Bm , rc Bm , and r0 Bm . We have the following. q Theorem 2.1. r0 p, Bm is a complete linear metric space paranormed by gB , deﬁned by sup| T x k |pk /M , gB x 2.3 k ∈Æ q q gB is a paranorm for the spaces r∞ p, Bm and rc p, Bm only in the trivial case with inf pk > 0 when q q q q r∞ p , B m r∞ Bm and rc p, Bm rc B m .
Journal of Inequalities and Applications 5 q q Proof. We prove the theorem for the space r0 p, Bm . The linearity of r0 p, Bm with respect to the coordinatewise addition and scalar multiplication that follow from the inequalities which q are satisﬁed for u, v ∈ r0 p, Bm 16 ⎡⎡ ⎤ ⎤ pk /M 1 ⎣ k−1⎣ k m r m qk vj ⎦ vk ⎦ m−i j i−j sup r s qi uj uk i−j k ∈Æ Qk j 0 i j Qk ⎡ ⎡ ⎤ ⎤ pk /M k −1 k m r m qk 1⎣ ⎣ s qi uj ⎦ uk ⎦ m−i j i−j ≤ sup 2.4 r i−j Qk j 0 i Qk k ∈Æ j ⎡⎡ ⎤ ⎤ pk /M 1 ⎣ k−1 ⎣ k m r m qk r m−i j si−j qi vj ⎦ vk ⎦ sup , i−j k ∈Æ Qk j 0 i j Qk and for any α ∈ Ê 1 , |α|pk ≤ max 1, |α|M . 2.5 q 0 and gB −x gB x for all x ∈ r0 p, Bm . Again, the inequalities It is clear that gB θ 2.4 and 2.5 yield the subadditivity of gB and gB αu ≤ max{1, |α|}gB u . 2.6 q Let {xn } be any sequence of the elements of the space r0 p, Bm such that gB xn − x −→ 0, 2.7 and λn also be any sequence of scalars such that λn → λ, as n → ∞. Then, since the inequality gB x n ≤ gB x gB x n − x 2.8
6 Journal of Inequalities and Applications holds by subadditivity of gB , {gB xn } is bounded, and thus we have gB λn xn − λx ⎡⎡ ⎤ ⎤ pk /M 1 ⎣ k−1 ⎣ k m r m qk − λxj ⎦ λn xk − λxk ⎦ m−i j i−j n n sup r s qi λn xj i−j k ∈Æ Qk j 0 i j Qk ⎡⎡ ⎤ ⎤ pk /M 1 ⎣ k−1 ⎣ k m r m qk n r m−i j si−j qi xj ⎦ x⎦ 1/M |λn − λ| n 2.9 sup Qk k i−j k ∈Æ Qk j 0 i j ⎡⎡ ⎤ ⎤ pk /M 1 ⎣k−1 ⎣ k m r m qk n − xj ⎦ xk − xk ⎦ m−i j i−j |λ|1/M sup n r s qi xj i−j k ∈Æ Qk j 0 i j Qk ≤ |λn − λ|1/M gB xn |λ|1/M gB xn − x , which tends to zero as n → ∞. Hence, the scalar multiplication is continuous. Finally, it q is clear to say that gB is a paranorm on the space r0 p, Bm . Moreover, we will prove the q q completeness of the space r0 p, Bm . Let xi be a Cauchy sequence in the space r0 p, Bm , where q i xi {xk } {x0 , x1 , x2 , . . .} ∈ r0 p, Bm . Then, for a given ε > 0, there exists a positive integer i i i n0 ε such that gB xi − xj < ε, 2.10 for all i, j ≥ n0 ε . If we use the deﬁnition of gB , we obtain for each ﬁxed k ∈ Æ that pk /M − T xj ≤ sup T xi − T xj T xi < ε, 2.11 k ∈Æ k k k k for i, j ≥ n0 ε which leads us to the fact that T x0 , T x1 , T x2 ,... 2.12 k k k is a Cauchy sequence of real numbers for every ﬁxed k ∈ Æ . Since Ê is complete, it converges, so we write T xi k → T x k as i → ∞. Hence, by using these inﬁnitely many limits T x 0 , T x 1 , T x 2 , . . ., we deﬁne the sequence { T x 0 , T x 1 , T x 2 , . . .}. From 2.11 with j → ∞, we have − Tx T xi < ε, 2.13 k k i ≥ n0 ε for every ﬁxed k ∈ Æ . Since xi q i {xk } ∈ r0 p, Bm , pk /M T xi 2.14 < ε, k
Journal of Inequalities and Applications 7 for all k ∈ Æ . Therefore, by 2.13 , we obtain that pk /M pk /M | T x k |pk /M ≤ − T xi T xi 2.15 Tx < ε, k k k for all i ≥ n0 ε . This shows that the sequence Tx belongs to the space c0 p . Since {xi } was q an arbitrary Cauchy sequence, the space r0 p, Bm is complete. Theorem 2.2. Let k j i−j r m−i j si−j qi / 0 for all k, m and 0 ≤ j ≤ k − 1. Then the Bm -diﬀerence m i q q q Riesz sequence spaces r∞ p, Bm , rc p, Bm , and r0 p, Bm are linearly isomorphic to the spaces l∞ p , c p , and c0 p , respectively, where 0 < pk ≤ H < ∞. q Proof. We establish this for the space r∞ p, Bm . For the proof of the theorem, we should show q the existence of a linear bijection between the space r∞ p, Bm and l∞ p for 0 < pk ≤ H < ∞. q With the notation of 2.1 , deﬁne the transformation S from r∞ p, Bm to l∞ p by x → y Sx.S is a linear transformation, morever; it is obviuos that x θ whenever Sx θ and hence S is injective. yk ∈ l∞ p and deﬁne the sequence x Let y xk by k−i−1 k −1 n 1 sk−i m 1 Qk ∀k ∈ Æ . k −n −1 xk Qn yn yk , 2.16 k −i r m qk k−i rm qi n0 in Then, ⎡ ⎤ pk /M 1 k−1 ⎣ k m r m qk s qi xj ⎦ m−i j i−j gB x sup r xk i−j k ∈Æ Qk j 0 i j Qk 2.17 pk /M k pk /M g1 y < ∞, sup δkj yj sup yk k ∈Æ k ∈Æ j0 where ⎧ ⎨1, k j, δkj 2.18 ⎩0, k / j. q Thus, we have that x ∈ r∞ p, Bm . Consequently, S is surjective and is paranorm preserving. q Hence, S is linear bijection, and this explains that the spaces r∞ p, Bm and l∞ p are linearly isomorphic. q q 3. The Basis for the Spaces rc p, Bm and r0 p, Bm q q In this section, we give two sequences of the points of the spaces r0 p, Bm and rc p, Bm which form the basis for those spaces.
8 Journal of Inequalities and Applications If a sequence space λ paranormed by h contains a sequence bn with the property that for every x ∈ λ, there is a unique sequence of scalars αn such that n lim h x − αk βk 0, 3.1 n→∞ k0 then bn is called a Schauder basis or brieﬂy basis for λ. The series αk βk which has the sum x is then called the expansion of x with respect to bn and written as x αk βk . Because of the isomorphism S is onto, deﬁned in the proof of Theorem 2.2, the inverse q image of the basis of the spaces, c0 p and c p are the basis of the new spaces r0 p, Bm and q m rc p, B , respectively. We have the following. T x k for all k ∈ Æ and 0 < pk ≤ H < ∞. Deﬁne the sequence b k q Theorem 3.1. Let μk q {bn q }n∈Æ of the elements of the space r0 p, Bm for every ﬁxed k ∈ Æ by q k ⎛ ⎞ ⎧ n−i−1 1 ⎪k n−i m 1 ⎪ s ⎪ ⎝ ⎠ Qk , n−k ⎪ −1 n>k , ⎪ ⎪ n−i ⎨i rm qi n−i k k bn q 3.2 ⎪ Qn , ⎪m ⎪r q k n, ⎪ ⎪ ⎪ n ⎩ 0, k>n . Then, one has the following. q q a The sequence {b k q }k∈Æ is a basis for the space r0 p, Bm , and any x ∈ r0 p, Bm has a unique representation of the form μk q b k q . x 3.3 k T −1 e, b k q } is a basis for the space rc p, Bm and any x ∈ rc p, Bm has q q b The set {z a unique representation of the form μk q − l b k q , x le 3.4 k where l lim T x k . 3.5 k→∞ q Proof. It is clear that {b k q } ⊂ r0 p, Bm , since for k ∈ Æ , e k ∈ c0 p , Tb k q 3.6
Journal of Inequalities and Applications 9 for 0 < pk ≤ H < ∞, where e k is the sequence whose only non-zero term is 1 in kth place for each k ∈ Æ . Let x ∈ r0 p, Bm be given. For every nonnegative integer m, we put q m xm μk q b k q . 3.7 k0 Then, we obtain by applying T to 3.7 with 3.6 that m m Tx m μk q T b k q T ke k , k0 k0 ⎧ 3.8 ⎨0, 0≤i≤m , x−x q M R ⎩ Tx , i i>m . i Given ε > 0, then there exists an integer m0 such that ε sup| T x i |pk /M < , 3.9 2 i≥m for all m ≥ m0 . Hence, ε sup| T x i |pk /M ≤ sup| T x i |pk /M < gB x − x m < ε, 3.10 2 i≥m i≥m0 q for all m ≥ m0 , which proves that x ∈ r0 p, Bm is represented as in 3.3 . To show the uniqueness of this representation, we suppose that there exists a representation λk q b k q . x 3.11 k q Since the linear transformation S from r0 p, Bm to c0 p , used in Theorem 2.2, is continuous we have n ∈ Æ, k Tb k q Tx λk q λ k q en λn q ; 3.12 n n k k which contradicts the fact that T x n μk q for all n ∈ Æ . Hence, the representation 3.3 of q x ∈ r0 p, Bm is unique. Thus, the proof of the part a of theorem is completed. q q b Since {b k q } ⊂ r0 p, Bm and e ∈ c p , the inclusion {e, b k q } ⊂ rc p, Bm q trivially holds. Let us take x ∈ rc p, Bm . Then, there uniquely exists an l satisfying 3.5 . We q thus have the fact that u ∈ r0 p, Bm whenever we set u x − le. Therefore, we deduce by part a of the present theorem that the representation of x given by 3.4 is unique and this step concludes the proof of the part b of theorem.
10 Journal of Inequalities and Applications q q q 4. The α-, β-, and γ -Duals of the Spaces rc p, Bm , r0 p, Bm , and r∞ p, Bm In this section, we prove the theorems determining the α-, β-, and γ -duals of the sequence q q spaces rc p, Bm and r0 p, Bm . For the sequence spaces λ and μ, deﬁne the set S λ, μ by zk ∈ w : xz xk zk ∈ μ ∀x ∈ λ . S λ, μ z 4.1 With the notation 4.1 , the α-, β-, γ -duals of a sequence space λ, which are, respectively, denoted by λα , λβ , and λγ are deﬁned by λα λβ λγ 4.2 S λ, l1 , S λ, cs , S λ, bs . Now, we give some lemmas which we need to prove our theorems Lemma 4.1 see 17 . A ∈ l∞ p : l1 if and only if ank K 1/pk < ∞, ∀ integers K > 1. sup 4.3 K ∈F n k ∈K Lemma 4.2 see 18 . Let pk > 0 for every k ∈ Æ . Then, A ∈ l∞ p : l∞ if and only if |ank |K 1/pk < ∞, ∀ integers K > 1. sup 4.4 n∈Æ k Lemma 4.3 see 18 . Let pk > 0 for every k ∈ Æ . Then, A ∈ l∞ p : c if and only if ∀ integers K > 1, |ank |K 1/pk converges uniformly in n, 4.5 k ∀k ∈ Æ . lim αnk , 4.6 n→∞
Journal of Inequalities and Applications 11 Æ, Theorem 4.4. For each m ∈ deﬁne the sets R1 p , R2 p , R3 p , R4 p , R5 p , and R6 p as follows: k1 Qn an ak ∈ w : sup ∇ i, n, k Qk an K 1/pk < ∞, ∀K > 1 , R1 p a r m qn N ∈F n k ∈N K >1 ik ⎧ ⎛ ⎞ ⎨ n ⎝ ak aj ⎠Qk K 1/pk converges ak ∈ w : ∇ i, n, k R2 p a ⎩ r m qk K >1 k jk1 ⎫ ⎬ ak Qk ∈ c0 , ∀K > 1 , uniformly in n and m K 1/pk ⎭ r qk ⎧ ⎛ ⎞ ⎨ n ⎝ ak aj ⎠Qk K 1/pk < ∞, ∀K > 1, ak ∈ w : ∇ i, n, k R3 p a ⎩ r m qk K >1 k jk1 ⎫ ⎬ n ak ∇ i, n, k ∈ l∞ ai Qk , ⎭ mq rk ik1 k1 Qn an K −1/pk < ∞, ∀K > 1 , ak ∈ w : sup ∇ i, n, k Qk ai R4 p a r m qn N ∈F n k ∈N K >1 ik k1 Qn an ak ∈ w : ∇ i, n, k Qk ai 1 , R6 p a ⎩ ⎭ r m qk K >1 k jk1 4.7 where n−i−1 sn−i m 1 n−k ∇ i, n, k −1 . 4.8 r m n−i n−i qi Then, α β γ q q q r∞ p , B m r∞ p , B m r∞ p , B m R1 p , R2 p , R3 p , α β q q R4 p ∩ R5 p , R6 p ∩ cs, rc p , B m rc p , B m 4.9 γ q R6 p ∩ bs, rc p , B m α β γ q q q r0 p , B m r0 p , B m r0 p , B m R4 p , R6 p .
12 Journal of Inequalities and Applications q an ∈ w. We easily Proof. We give the proof for the space r∞ p, Bm . Let us take any a derive with the notation ⎡ ⎤ 1 k−1 ⎣ k m rm r m−i j si−j qi xj ⎦ yk qk xk , 4.10 i−j Qk j 0 i j Qk that n−1 k 1 an Qn yn ∇ i, n, k an Qk yk an xn r m qn k0 ik 4.11 n unk yk Uy n , k0 n ∈ Æ , where U unk is deﬁned by ⎧k 1 ⎪ ⎪ ∇ i, n, k an Qi , ⎪ 0≤ k ≤n−1 , ⎪ ⎪ ⎨i k aQ unk 4.12 ⎪ n n, k n, ⎪ rmq ⎪ ⎪ ⎪ n ⎩ 0, k>n , for all k, n ∈ Æ . Thus we deduce from 4.6 that ax an xn ∈ l1 whenever x xk ∈ q r∞ p, Bm if and only if Uy ∈ l1 whenever y yk ∈ l∞ p . From Lemma 4.1, we obtain the desired result that α r q p, B m R1 p . 4.13 Consider the equation ⎛ ⎞ n−1 n k1 n ak Qk yk ⎝ ak aj ⎠Qk yk ∇ i, n, k ak xk r m qk r m qk 4.14 k0 k 0 ik jk1 n∈Æ , V y n, where V vnk deﬁned by ⎧⎛ ⎞ ⎪ k1 n ⎪⎝ ak ⎪ aj ⎠Qk , ⎪ ∇ i, n, k 0 ≤k ≤ n−1 , ⎪ r mq ⎪ ⎨ k ik jk1 vnk 4.15 ak Qk ⎪ ⎪m , k n, ⎪r q ⎪ ⎪ ⎪ k ⎩ 0, k>n ,
Journal of Inequalities and Applications 13 for all k, n ∈ Æ . Thus, we deduce by with 4.11 that ax ak xk ∈ cs whenever x q xk ∈ r∞ p, Bm if and only if V y ∈ c whenever y yk ∈ l∞ p . Therefor, we derive from Lemma 4.3 that ⎛ ⎞ n ⎝ ak aj ⎠Qk K 1/pk converges uniformly in n ∀K > 1, ∇ i, n, k r m qk k jk1 4.16 ak Qk K 1/pk lim 0, k → ∞ r m qk which shows that r q p, Bm β R2 p . q ak xk ∈ bs whenever x xk ∈ r∞ p, Bm As this, we deduce by 4.11 that ax if and only if V y ∈ l∞ whenever y yk ∈ l∞ p . Therefore, we obtain by Lemma 4.2 that r q p, Bm γ R3 p and this completes proof. q q Now we characterize the matrix mappings from the spaces r∞ p, Bm , rc p, Bm , and q m r0p, B to the spaces l∞ and c. Since the following theorems can be proved by using standart methods, we omit the detail. q Theorem 4.5. (i) A ∈ r∞ p, Bm : l∞ if and only if ank ∀n, M ∈ Æ , Qk M1/pk lim 0, 4.17 qk k→∞ n ank ∀M ∈ Æ ∇ i, n, k anj Qk M1/pk < ∞, sup 4.18 r m qk n∈Æ k jk1 hold. q (ii) A ∈ rc p, Bm : l∞ if and only if 4.14 , ⎛ ⎞ n ⎝ ank ∃M ∈ Æ , anj ⎠Qk M1/pk ∇ i, n, k sup 0, r m qk n∈Æ k jk1 4.19 ⎛ ⎞ n ⎝ ank anj ⎠Qk < ∞ ∇ i, n, k sup r m qk n∈Æ k jk1 hold. q (iii) A ∈ r0 p, Bm : l∞ if and only if 4.14 and 4.18 hold.
14 Journal of Inequalities and Applications q Theorem 4.6. (i) A ∈ r∞ p, Bm : c if and only if 4.14 , ⎛ ⎞ n ⎝ ank ∀M ∈ Æ , anj ⎠Qk M1/pk < ∞, ∇ i, n, k sup r m qk n∈Æ k jk1 4.20 ⎛ ⎞ ⎡ ⎤ n ⎝ ank ∃ αk ⊂ Ê anj ⎠Qk − αk M1/pk ⎦ such that lim ⎣ ∇ i, n, k 0, r m qk n→∞ k jk1 ∀M ∈ Æ hold. q (ii) A ∈ rc p, Bm : c if and only if 4.14 , 4.18 , ⎛ ⎞ n ank ∃α ⊂ Ê such that lim ⎝ m anj ⎠Qk − α ∇ i, n, k 0, 4.21 n→∞ r qk jk1 ⎛ ⎞ n ank ∃ αk ⊂ Ê ∀k ∈ Æ , such that lim ⎝ m anj ⎠Qk − αk ∇ i, n, k 0, 4.22 n→∞ r qk jk1 ⎛ ⎞ n ank ∃ αk ⊂ Ê such that sup ⎝ m anj ⎠Qk − αk M−1/pk < ∞, ∇ i, n, k 4.23 n∈Æ r qk jk1 ∃M ∈ Æ hold. q (iii) A ∈ r0 p, Bm : c if and only if 4.14 , 4.18 , 4.21 , and 4.22 hold. 5. Uniform Opial Property of Bm -Difference Riesz Sequence Spaces q In this section, we investigate the uniform Opial property of the sequence spaces r0 p, Bm q and rc p, Bm . The Opial property plays an important role in the study of weak convergence of iterates of mapping of Banach spaces and of the asymptotic behavior of nonlinear semigroup. The Opial property is important because Banach spaces with this property have the weak ﬁxed point property 19 see 20, 21 . We give the deﬁnition of uniform Opial property in a linear metric space and use the q q method in 22 , and obtain that r0 p, Bm and rc p, Bm have uniform Opial property for pk ≥ 1. xn ∈ rc p, Bm and for i ∈ Æ , we use the q q xn ∈ r0 p, Bm or x For a sequence x x 1 , x 2 , . . . , x i , 0, 0, . . . and x|Æ−i notation x|i 0, 0, . . . , 0, x i 1 , x i 2 , . . . . We know that every total paranormed space becomes a linear metric space with the q q q g x − y . It is clear that r∞ p, Bm , r0 p, Bm , and rc p, Bm are total metric given by d x, y gB x − y . paranormed spaces with d x, y Now, we can give the deﬁnition of uniform Opial property in a linear metric space.
Journal of Inequalities and Applications 15 A linear metric space X, d has the uniform Opial property if for each ε > 0 there exists τ > 0 such that for any weakly null sequence {xn } in S 0, r and x ∈ X with d x, 0 ≥ ε the following inequality holds: τ ≤ lim inf d xn r x, 0 . 5.1 n→∞ Now, we give following lemma which we need it to prove our main theorem. It can be proved by using same method in 14 we omit the detail. Lemma 5.1. If lim infk → ∞ pk > 0 then for any L > 0 and ε > 0, there exists δ δ ε, L > 0 for u, v ∈ X such that dM u v , 0 < d M u, 0 5.2 ε q q whenever dM u, 0 ≤ L and dM v, 0 ≤ δ, where X r 0 p , B m or r c p , B m . q q Theorem 5.2. If pk ≥ 1, then r0 p, Bm and rc p, Bm have uniform Opial property. q q Proof. We prove the theorem for r0 p, Bm . rc p, Bm can be proved by similiar way. For any ε > 0, we can ﬁnd a positive number ε0 ∈ 0, ε such that εM M rM 5.3 >r ε0 . 4 q Take any x ∈ r0 p, Bm with dM x, 0 ≥ εM and xn to be weakly null sequence in S 0, r . By this, we write dM xn , 0 rM. 5.4 There exists q0 ∈ Æ such that ∞ εM ε0 M |Tx k |pk < dM x|Æ−q0 , 0 < . 5.5 4 4 k q0 1 Furthermore, we have ∞ q0 |Tx k |pk |Tx k |pk , ε M ≤ d M x, 0 k0 k q0 1 q0 εM |Tx k |pk εM ≤ 5.6 , 4 k0 q0 3ε M |Tx k |pk . ≤ 4 k0
16 Journal of Inequalities and Applications Æ such By xn → 0, weakly, this implies that xn → 0, coordinatewise, hence there exists n0 ∈ that with 5.6 q0 3ε M x k |pk , ≤ |T xn k 5.7 4 k0 for all n ≥ n0 . Lemma 5.1 asserts that εM z, 0 ≤ dM y, 0 dM y , 5.8 4 whenever dM y, 0 ≤ r M and dM z, 0 ≤ ε0 . Again by xn → 0, weakly, there exists n1 > n0 such that dM xn|q0 , 0 < ε0 for all n > n1 , so by 5.8 , we obtain that εM dM xn|Æ−q0 xn|q0 , 0 < dM xn|Æ−q0 , 0 5.9 , 4 hence, ∞ εM |Txn k |pk , dM xn , 0 − < dM xn|Æ−q0 , 0 4 k q0 1 5.10 ∞ εM pk − |Txn k | , M r < 4 k q0 1 for all n > n1 . This, together with 5.5 , 5.6 , implies that for any n > n1 , q0 x k |pk |T xn k dM xn x, 0 k0 ∞ x k |pk |T xn k k q0 1 ∞ q0 x k |pk |Txn k |pk ≥ |T xn k k0 k q0 1 5.11 ∞ |Tx k |pk − k q0 1 3ε M εM εM εM − rM rM > 4 4 4 4 M >r ε0 .
Journal of Inequalities and Applications 17 q This means that dM xn x, 0 > r ε0 , so we get that the sequence space r0 p, Bm has uniform Opial property for pk ≥ 1. References 1 I. J. Maddox, “Paranormed sequence spaces generated by inﬁnite matrices,” Proceedings of the Cambridge Philosophical Society, vol. 64, pp. 335–340, 1968. 2 I. J. Maddox, “Spaces of strongly summable sequences,” Quarterly Journal of Mathematics, vol. 18, no. 2, pp. 345–355, 1967. 3 H. Nakano, “Modulared sequence spaces,” Proceedings of the Japan Academy, vol. 27, pp. 508–512, 1951. 4 S. Simons, “The sequence spaces l pv and m pv ,” Proceedings of the London Mathematical Society. Third Series, vol. 15, no. 3, pp. 422–436, 1965. 5 B. Altay and F. Basar, “On the paranormed Riesz sequence spaces of non-absolute type,” Southeast ¸ Asian Bulletin of Mathematics, vol. 26, no. 5, pp. 701–715, 2003. 6 B. Altay and F. Basar, “Some paranormed Riesz sequence spaces of non-absolute type,” Southeast ¸ Asian Bulletin of Mathematics, vol. 30, no. 4, pp. 591–608, 2006. 7 B. Altay and F. Basar, “On the ﬁne spectrum of the generalized diﬀerence operator B r, s over the ¸ sequence spaces c0 and c,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 18, pp. 3005–3013, 2005. 8 B. Altay and H. Polat, “On some new Euler diﬀerence sequence spaces,” Southeast Asian Bulletin of Mathematics, vol. 30, no. 2, pp. 209–220, 2006. 9 H. Kızmaz, “On certain sequence spaces,” Canadian Mathematical Bulletin, vol. 24, no. 2, pp. 169–176, 1981. 10 M. Basarır and M. Ozturk, “On the Riesz diﬀerence sequence space,” Rendiconti del Circolo Matematico ¨¨ ¸ di Palermo. Second Series, vol. 57, no. 3, pp. 377–389, 2008. 11 C. Aydin and F. Basar, “Some generalizations of the sequence space αr ,” Iranian Journal of Science and p Technology. Transaction A. Science, vol. 30, no. 2, pp. 175–190, 2006. 12 H. Polat and F. Basar, “Some Euler spaces of diﬀerence sequences of order m,” Acta Mathematica ¸ Scientia B, vol. 27, no. 2, pp. 254–266, 2007. 13 F. Basar, B. Altay, and M. Mursaleen, “Some generalizations of the space bvp of p-bounded variation ¸ sequences,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 2, pp. 273–287, 2008. 14 M. Basarır and M. Kayikci, “On the generalized B m -Riesz diﬀerence sequence space and β-property,” ¸ ¸ Journal of Inequalities and Applications, vol. 2009, Article ID 385029, 18 pages, 2009. 15 M. Basarır, “On the generalized Riesz B-diﬀerence sequence spaces,” Filomat, vol. 24, no. 4, pp. 35–52, ¸ 2010. 16 I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London, UK, 1970. 17 K.-G. Grosse-Erdmann, “Matrix transformations between the sequence spaces of Maddox,” Journal of Mathematical Analysis and Applications, vol. 180, no. 1, pp. 223–238, 1993. 18 C. G. Lascarides and I. J. Maddox, “Matrix transformations between some classes of sequences,” Proceedings of the Cambridge Philosophical Society, vol. 68, pp. 99–104, 1970. 19 J.-P. Gossez and E. Lami Dozo, “Some geometric properties related to the ﬁxed point theory for nonexpansive mappings,” Paciﬁc Journal of Mathematics, vol. 40, pp. 565–573, 1972. 20 J. Diestel, Geometry of Banach Spaces—Selected Topics, vol. 485 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975. 21 L. Maligranda, Orlicz Spaces and Interpolation, vol. 5 of Seminars in Mathematics, Universidade Estadual de Campinas, Departamento de Matem´ tica, Campinas, Brazil, 1989. a 22 K. Khompurngson, Geometric properties of some paranormed sequence spaces, M.S. thesis, Chiang Mai University, 2004.