A STUDY ON TOPOLOGICAL AND GEOMETRICAL CHARACTERISTICS OF NEW BANACH SEQUENCE SPACES

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1 Gulf Journal of Mathematics Vol 3, Issue 4 (2015) A STUDY ON TOPOLOGICAL AND GEOMETRICAL CHARACTERISTICS OF NEW BANACH SEQUENCE SPACES MURAT CANDAN 1 AND EMRAH EVREN KARA 2 Abstract. A short extract of this study is to submit a new sequence space l p ( F (r, s)) under the domain of the matrix F (r, s) constituted by using Fibonacci sequence non-zero real number r s, of l p previously defined. Additionally, we construct some inclusion relations concerning with this space determine its the α-, β-, γ-duals. Moreover, we characterize some matrix classes on the space l p ( F (r, s)) examine some geometric properties of this space. 1. Introduction preliminaries As usual, the symbol ω denotes the space of all real valued sequences. Also, we use the symbols l, c, c 0 l p (1 p < ), to represent the sets of all bounded, convergent, null sequences p-absolutely convergent series, respectively. Since any vector subspace of ω is called as a sequence space, each of these sets is a sequence space. Additionally, it will be used the conventions that e = (1, 1,...) e (n) is the sequence whose only non-zero term is 1 in the n th place for each n N, where N = 0, 1, 2,...}. We also write x = (x ) A = (a n ) instead of x = (x ) A = (a n) n,, respectively. Let η ϑ be two sequence spaces A = (a n ) be an infinite matrix of real numbers a n, where n, N. Then, we say that A defines a matrix mapping from η into ϑ we denote it by writing A : η ϑ, if for every sequence x = (x ) η the sequence Ax = A n (x)}, the A-transform of x, is in ϑ; where A n (x) = a n x ; (n N). (1.1) For simplicity in notation, here in what follows, the summation without limits runs from 0 to. Now, let us give the definition of matrix mapping which will be used especially in section 4. By (η, ϑ), we denote the class of all matrices A such that A : η ϑ. Thus, A (η, ϑ) if only if the series on the right side of (1.1) converges for each n N every x η we have Ax ϑ for all x η. Date: Received: Feb 16, 2015; Accepted: Jun 2, Corresponding author Mathematics Subject Classification. 11B39, 46A45, 46B45, 46B20. Key words phrases. Sequence spaces, Fibonacci numbers, difference matrix, α-, β-, γ- duals, matrix transformations, fixed point property, Banach-Sas type p. 67

2 68 M. CANDAN, E.E. KARA The matrix domain has fundamental importance for this study. So, the concept is stated in this paragraph. The matrix domain ψ A of an infinite matrix A in a sequence space ψ is defined by ψ A = x = (x ) ω : Ax ψ} (1.2) which is a sequence space. In recent years, some researchers have addressed the approach to constructing a new sequence space by means of the matrix domain of a particular limitation method, see [2, 5, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 42, 46, 49, 50, 51, 52, 53, 54], the references therein. The difference matrix = (δ n ) can be written in two different ways as follows ( 1) n (n 1 n) δ n = 0 (0 < n 1 or > n) or δ n = ( 1) n (n n + 1) 0 (0 < n or > n + 1). As already nown, the matrix domain λ is said the difference sequence space whenever λ is a normed or paranormed sequence space. In 1981, Kızmaz [31] firstly defined difference operator ( x ) = (x x +1 ) examined difference sequence spaces established difference operator as follows: φ( ) = x = (x ) ω : (x x +1 ) φ} for φ = l,c,c 0 }. Now, let us give some fundamental study on this concept. The difference space bv p as the set of all sequences whose transforms is the space l p, i.e., bv p = x = (x ) ω : (x x +1 ) l p }. It is obvious that bv p = (l p ), see [3, 8, 20]. The paranormed difference sequence space λ(p) = x = (x ) ω : (x x +1 ) λ(p)} was examined by Ahmad Mursaleen [1] Malowsy [37], where λ(p) is any of the paranormed spaces l (p), c(p) c 0 (p) defined by Simons [55] Maddox [36]. Recently, Başar et al. [4] have defined the sequence spaces bv(u, p) bv (u, p) by bv(u, p) = x = (x ) ω : u (x x 1 ) p < } bv (u, p) = x = (x ) ω : u (x x 1 ) p < }, N where u = (u ) is an arbitrary fixed sequence 0 < p H < for all N. These spaces are generalization of the space bv p for 1 p. Quite recently, Kirişçi Başar [32] have introduced studied the generalized difference sequence spaces X = x = (x ) ω : B(r, s)x X}

3 A STUDY ON TOPOLOGICAL AND GEOMETRICAL CHARACTERISTICS OF NEW for X = l, l p, c c 0, where 1 p < B(r, s)x = (sx 1 + rx ) (r, s 0). Following Kirişçi Başar [32], Can [10] has examined the sequence space X( B) as the set of all sequences whose B( r, s)-transforms are in the space X l, l p, c, c 0 }, where B( r, s) denotes the double sequential b matrix B( r, s) = b n (r n, s n )} defined by b n ( r, s) = r n, ( = n), s n, ( = n 1), 0, (0 < n 1 or > n), for all, n N, where r = (r n ) n=0 s = (s n) n=0 be given convergent sequences of positive real numbers 1 p <. Also in [9, 19, 23, 24, 25, 28, 38, 39, 40, 45, 48], the authors studied some difference sequence spaces. In this study, we have introduce the generalized Fibonacci difference matrix F (r, s) constituted by using Fibonacci sequence f n } non-zero real numbers r s, earlier defined by Can [15]. Also, we have defined new sequence spaces l p ( F (r, s)) l ( F (r, s)) related to matrix domain of F (r, s) in the sequence spaces l p l, respectively; where 1 p <. The rest of this study is organized as follows: At the beginning of the next section, it is given some historical developments about Fibonacci numbers served some notations basic concepts including BK-space which are going to be needed in later sections. After then, we consider a b matrix, constituted by using Fibonacci sequences non-zero real number r s. Furthermore, we introduce the new br sequence spaces l p ( F (r, s)) l ( F (r, s)). Additionally, we find out some inclusion relations concerning with these spaces establish the basis of the space l p ( F (r, s)) for 1 p <. In Section 3, we compute the α-, β-, γ-duals of the spaces l p ( F (r, s)) l ( F (r, s)). In Section 4, we characterize the classes (l p ( F (r, s)), X) (l ( F (r, s)), X), where 1 p < X is any of the spaces l, l 1, c c 0. In the final section of the study, we devote to investigate some geometric properties of the space l p ( F (r, s)) for 1 < p <. 2. The Fibonacci Difference Sequence Spaces l p ( F (r, s)) l ( F (r, s)) Before we go to the main topic of our study, let us give a history about the Fibonacci numbers. In 1202, Fibonacci numbers were first noted by the Italian mathematician Leonardo of Pisa whose nicname was Fibonacci in his Liber Abaci which famous in the world. For information on the important properties uses of the Fibonacci numbers see [35]. Define the sequence f n } n=0 of Fibonacci numbers given by the linear recurrence relations f 0 = f 1 = 1 f n = f n 1 + f n 2 ; n 2.

4 70 M. CANDAN, E.E. KARA Fibonacci numbers have many interesting properties applications in arts, sciences architecture. For instance, the ratio sequences of Fibonacci numbers converges to the golden ratio which is important in sciences arts. Additionally, some well-nown basic properties of Fibonacci numbers are given as follows: f n+1 lim = = α (Golden Ratio), n f n 2 f = f n+2 1; (n N), 1 f converges, f n 1 f n+1 fn 2 = ( 1) n+1 ; (n 1) (Cassini Formula). Substituting for f n+1 in Cassini s formula yields fn f n f n 1 fn 2 = ( 1) n+1. Now, we are going to define the generalized Fibonacci b matrix F (r, s) = ( f n (r, s)) constituted by using Fibonacci sequence non-zero real numbers r s. Additionally, we firstly introduce the sequence spaces l p ( F (r, s)) l ( F (r, s)), where 1 p <. Also, we present some inclusion theorems construct Schauder basis of the space l p ( F (r, s)) for 1 p <. Let f n be the nth Fibonacci number for every n N. Then, we define the infinite matrix F (r, s) = ( f n (r, s)) by s f n+1 f n, ( = n 1), f n (r, s) = r fn f n+1, ( = n), (n, N). 0, (0 < n 1 or > n), Now, we introduce the Fibonacci difference sequence spaces l p ( F (r, s)) l ( F (r, s)) as the set of all sequences such that their F (r, s)-transforms are in the space l p l, respectively, i.e., l p ( F (r, s)) = x = (x n ) ω : r f n x n + s f } p n+1 x n 1 f n+1 < ; 1 p < l ( F (r, s)) = x = (x n ) ω : n n N f n r f n x n + s f n+1 f n+1 f n x n 1 } <. It is natural that the spaces l p ( F (r, s)) l ( F (r, s)) may be rewritten by with the notation of (1.2) that l p ( F (r, s)) = (l p ) F (r,s) (1 p < ) l ( F (r, s)) = (l ) F (r,s). (2.1) It is remarable that the sequence y = (y n ) which will be frequently used is the F -transform of a sequence x = (x n ) everywhere in study, in other words, y n = F n (r, s)(x) = r f 0 f 1 x 0 = rx 0 (n = 0) r fn f n+1 x n + s f n+1 f n x n 1 (n 1) ; (n N). (2.2)

5 A STUDY ON TOPOLOGICAL AND GEOMETRICAL CHARACTERISTICS OF NEW Now, we should state that the matrix F (r, s) can be reduced to the matrix F in case r = 1 s = 1. Therefore, the results related to the spaces l p ( F (r, s)) l ( F (r, s)) are more general more comprehensive than the corresponding consequences of the spaces l p ( F ) l ( F ) more recently defined by Kara in [30]. For additional details may be refer to the following references [15, 17, 18]. Before presenting the next theorem, let us consider the following concepts. A sequence space X is called F K space if it is a complete linear metric space with continuous coordinates p n : X R (n N), where R denotes the real field p n (x) = x n for all x = (x ) X every n N. A BK space is a normed F K space, that is, a BK space is a Banach space with continuous coordinates. The space l p (1 p < ) is BK space with x p = ( x p ) 1/p c 0, c l are BK spaces with x = x. Now, we may begin with the following theorem which is essential in the text. Theorem 2.1. When p satisfied the condition 1 p. The newly defined sequence space l p ( F (r, s)) is a BK-space with the norm x lp( F (r,s)) = F (r, s)x p, in other words ( ) x lp( F (r,s)) = F 1/p n (r, s)(x) p ; (1 p < ) n x l ( F (r,s)) = n N F n (r, s)(x). Proof. The proof can be directly reached when we assess as right both the hypothesis well-now Theorem of Wilansy [57]. Indeed, we now that (2.1) is valid the spaces l p l recalled earlier section are BK-spaces with respect to their natural norms the matrix F (r, s) is a triangle; Theorem of Wilansy results in the fact that the spaces l p ( F (r, s)) l ( F (r, s)) are BK-space with the given norms, where 1 p <. This mars the end of the proof. Remar 2.2. One can easily chec that the absolute property does not hold on the spaces l p ( F (r, s)) l ( F (r, s)), that is x lp( F (r,s)) x l p( F (r,s)) x l ( F (r,s)) x l ( F (r,s)) for at least one sequence in the spaces l p( F (r, s)) l ( F (r, s)), this shows that l p ( F (r, s)) l ( F (r, s)) are the sequence spaces of non-absolute type, where x = ( x ) 1 p <. Theorem 2.3. The generalized Fibonacci difference sequence space l p ( F (r, s)) of non-absolute type are linearly isomorphic to the space l p, that is l p ( F (r, s)) = l p for 1 p. Proof. To verify the fact that l p ( F (r, s)) = l p, we need to show the existence of a linear bijection between the spaces l p ( F (r, s)) l p when p satisfied the condition 1 p, from the definition of linear isomorphism. To do this, we consider the transformation T defined above, with the aid of notation of (2.2),

6 72 M. CANDAN, E.E. KARA from l p ( F (r, s)) to l p by x y = T x. In that case T x = y = F (r, s)x l p for every x l p ( F (r, s)). Since the linearity of the map T is not difficult to prove, we omit the detail. Further, it is trivial that x = 0 whenever T x = 0 hence T is injective. Furthermore, let y = (y ) l p for 1 p define the sequence x = (x ) by x = 1 ( ) j s f+1 2 y j ; ( N). r r f j f j+1 Thus, in the cases 1 p < p =, we have ( x lp( F (r,s)) = r f x + s f +1 ) p 1/p x 1 f +1 f ( f ( ) j s f+1 2 = y j + sf 1 ( ) j 1 +1 s f 2 y j f +1 r f j f j+1 rf r f j f j+1 ( ) 1/p = y p = y p < x l ( F (r,s)) = N F (r, s)(x) = y <, respectively. Then, we get x l p ( F (r, s)) (1 p ). Therefore, T is surjective norm preserving. Consequently, T is a linear bijection which tells us the result that the spaces l p ( F (r, s)) l p are linearly isomorphic for 1 p. This ends the proof. Now, we give some inclusion relations concerning with the space l p ( F (r, s)). Theorem 2.4. The inclusion l p l p ( F (r, s)) strictly holds for 1 p. Proof. The proof will be done in two steps. In the first step, we will show that the inclusion l p l p ( F (r, s)) is valid for 1 p. If every x lies in l p, then it is sufficient to prove the existence of a number M > 0 such that x lp( F (r,s)) M x p. To do this, let us assume that x l p 1 < p. Since the f inequalities f +1 1 f +1 f 2 always hold for every N, we can deduce with the notation of (2.2), F (r, s)(x) p 2 p 1 ( rx p + 2sx 1 p ) ( 2 2p 1 max r, s } x p + ) x 1 p N F (r, s)(x) 3max r, s } x, N p) 1/p

7 A STUDY ON TOPOLOGICAL AND GEOMETRICAL CHARACTERISTICS OF NEW which together result in, as expected, x lp( F (r,s)) 4max r, s } x p (2.3) for 1 < p. In the last step, we must find an element which belong to l p ( F (r, s)) but which does not belong to l p. Since the sequence x = (x ) = (1/r( s/r) f+1 2 ) is in l p ( F (r, s)) l p, the inclusion l p l p ( F (r, s)) is strictly valid for 1 < p. Similarly, one can easily prove that inequality (2.3) also holds in the case p = 1 so we omit the details. This step finishes the proof. Theorem 2.5. If 1 p < s, then l p ( F (r, s)) l s ( F (r, s)). Proof. The proof of the theorem is quite stard easy. If two real numbers p s satisfy the condition 1 p < s x l p ( F (r, s)). Then we get from Theorem 2.1 that y l p, where y is the sequence given by (2.2). Therefore, the well-nown inclusion l p l s results in y l s. This shows that x l s ( F (r, s)) then, the inclusion l p ( F (r, s)) l s ( F (r, s)) is valid. In fact, this is exactly what we want to prove. After, we remember the concept of Schauder basis, we are going to give a sequence of the points of the space l p ( F (r, s)) which forms a basis for the space l p ( F (r, s)) (1 p < ). A sequence (b n ) in a normed space X is called a Schauder basis for X if for every x X there is a unique sequence (α n ) of scalars such that x = n α nb n, i.e., lim m x m n=0 α nb n = 0. Theorem 2.6. Let 1 p < define the sequence c () l p ( F (r, s)) for every fixed N by 0 (n < ) (c () ) n = ( 1 s ) n f 2 n+1 ; (n N). (2.4) r r f f +1 (n ) Then, the sequence (c () ) is a basis for the space l p( F (r, s)) every x l p ( F (r, s)) has a unique representation of the form x = F (r, s)(x)c (). (2.5) Proof. Let us assume that 1 p <. Then, it is not hard to verify of the relation F (r, s)(c () ) = e () l p ( N) by using (2.4), hence c () l p ( F (r, s)) for all N. Also, let us tae any given sequence x l p ( F (r, s)). For all non-negative integer m, we put m x (m) = F (r, s)(x)c (). Then, we have that F (r, s)(x (m) ) = m F (r, s)(x) F (r, s)(c () ) = m F (r, s)(x)e ()

8 74 M. CANDAN, E.E. KARA hence F n (r, s)(x x (m) ) = 0, (0 n m) F n (r, s)(x), (n > m) ; (n, m N). Now, for any given ε > 0 there is a non-negative integer m 0 such that F n (r, s)(x) p ( ε ) p. 2 n=m 0 +1 Therefore, we have for every m m 0 that ( x x (m) lp( F = (r,s)) n=m+1 ( n=m 0 +1 ε 2 < ε ) F 1/p n (r, s)(x) p ) F 1/p n (r, s)(x) p which shows that lim m x x (m) lp( F = 0 hence x is represented as (r,s)) in (2.5). Eventually, we must show the uniqueness of the representation (2.5) of x l p ( F (r, s)). To do this, let us assume that x = µ (x)c (). Since the linear transformation T defined from l p ( F (r, s)) to l p, in the proof of Theorem 3.2, is continuous, we observe that F n (r, s)(x) = µ (x) F n (r, s)(c () ) = µ (x)δ n = µ n (x); (n N). Therefore, the newly calculated equalities indicates that the representation (2.5) of x l p ( F (r, s)) is unique. This last step concludes the proof. 3. The α-, β- γ-duals of the space l p ( F (r, s)) In this section, after we recall the α-, β- γ-duals of the an arbitrary sequence space, we compute the α-, β- γ-duals of the newly defined sequence space l p ( F (r, s)) of nonabsolute type. The α-, β- γ-duals of a sequence space X are respectively defined by } X α = a = (a ) ω : a x < for all x = (x ) X, X β = X γ = a = (a ) ω : a = (a ) ω : } a x convergent for all x = (x ) X, } a x bounded for all x = (x ) X,

9 A STUDY ON TOPOLOGICAL AND GEOMETRICAL CHARACTERISTICS OF NEW [7]. We assume throughout that p, q 1 with p 1 + q 1 = 1 denote the collection of all finite subsets of N by F. The following nown results [56] are fundamental for our investigation. Lemma 3.1. A = (a n ) (l p, l 1 ) if only if a n < ; 1 < p. K F n K Lemma 3.2. A = (a n ) (l p, c) if only if n N lim a n exists for all N, (3.1) n a n q < ; 1 < p <. (3.2) Lemma 3.3. A = (a n ) (l, c) if only if (3.1) holds lim a n = lim a n. n n Lemma 3.4. A = (a n ) (l p, l ) if only if (3.2) holds with 1 < p. Since the case p = 1 can be proved by analogy, we omit the proof of that case consider only the case 1 < p in the proof of Theorems , respectively. Theorem 3.5. The α-dual of the space l p ( F (r, s)) is the set ( ) n 1 s f d 2 n+1 1 (r, s) = a = (a ) ω : a K F n r r f f +1 where 1 < p. n K q < Proof. Let us assume that 1 < p. Consider any sequence a = (a n ) ω, define the matrix B = (b n ) by ( 1 s ) n f 2 n+1 b n = r r f f +1 a n (0 n) 0 ( > n) for all n, N. Additionally, for every x = (x n ) ω we put y = F x. Thus, it is obtained by (2.2) that ( ) n 1 s fn+1 2 a n x n = a n y = B n (y); (n N). (3.3) r r f f +1 Hence, we observe by (3.3) that ax = (a n x n ) l 1 whenever x l p ( F (r, s)) iff By l 1 whenever y l p. Thus, we derive by using Lemma 3.1 that ( ) n 1 s f 2 q n+1 a n < r r f f +1 K F n K which results in that (l p ( F (r, s))) α = d 1 (r, s). },

10 76 M. CANDAN, E.E. KARA Theorem 3.6. Define the sets d 2 (r, s), d 3 (r, s) d 4 (r, s) by ( ) } j 1 s fj+1 d 2 2 (r, s) = a = (a ) ω : a j exists for all N, r r f f +1 d 3 (r, s) = j= a = (a ) ω : n N ( ) j 1 s f 2 j+1 a j r r f f +1 j= q < ( ) } j 1 s f d 2 j+1 4 (r, s) = a = (a ) ω : lim a j n r r f f +1 j= = a = (a ) ω : ( ) } j 1 s f 2 j+1 a j r r f f +1 <. j= Then (l p ( F (r, s))) β = d 2 (r, s) d 3 (r, s) (l ( F (r, s))) β = d 2 (r, s) d 4 (r, s), where 1 < p <. Proof. The proof is reached by considering the definition of β dual. For this, let us assume that a = (a ) ω after then we compute ( ( ) ) j 1 s f+1 2 a x = a y j r r f j f j+1 ( ( ) ) j 1 s fj+1 2 = a j y (3.4) r r f f +1 = D n (y), where D = (d n ) is defined by ( 1 s ) j f 2 j+1 r r f d n = f +1 a j (0 n) j= ; n, N. 0 ( > n) j= In that case, we have the right to say, from Lemma 3.2 with (3.4), that ax = (a x ) cs whenever x = (x ) l p ( F (r, s)) if only if Dy c whenever y = (y ) l p. Then, (a ) (l p ( F (r, s))) β if only if (a ) d 2 (r, s) (a ) d 3 (r, s) by (3.1) (3.2), respectively. Hence, (l p ( F (r, s))) β = d 2 d 3. It is clear that one can also prove the case p = by the same technique used in the proof of the case 1 < p < with Lemma 3.3 instead of Lemma 3.2. So, we leave the detailed proof to the reader. Theorem 3.7. (l p ( F (r, s))) γ = d 3 (r, s), where 1 < p. Proof. According to the definition of γ dual, this proof can be directly reached by using both (3.4) Lemma 3.4. },

11 A STUDY ON TOPOLOGICAL AND GEOMETRICAL CHARACTERISTICS OF NEW Some matrix transformations related to the sequence space l p ( F (r, s)) In this section, we focus on the characterize the classes (l p ( F (r, s)), X), in which 1 p X l, l 1, c, c 0 }. For simplicity in notation, we write ã n = ( ) j 1 s fj+1 2 a nj r r f f +1 j= for all, n N. The following lemma is essential for our results. Lemma 4.1. (see [32], Theorem 4.1]) Let λ be an F K-space, U be a triangle, V be its inverse µ be arbitrary subset of ω. Then we have A = (a n ) (λ U, µ) if only if ( C (n) = c (n) m ) (λ, c) for all n N C = (c n ) (λ, µ), m where c (n) m = j= a njv j (0 m) 0 ( > m), m, n N. c n = j= a njv j for all Now, we list the following conditions. m m ( ) j 1 s f 2 q j+1 a m N nj < (4.1) r r f f +1 j= m ( ) j 1 s fj+1 2 lim a nj = ã n ; n, N (4.2) m r r f f +1 j= m m ( ) j 1 s f 2 j+1 lim a m nj r r f f j+1 = ã n for each n N (4.3) j= ã n q < (4.4) n N N F q ã n < (4.5) n N lim n ãn = α ; N (4.6) ã n = α (4.7) ã n = 0 (4.8) lim n lim n n, N ã n < (4.9)

12 78 M. CANDAN, E.E. KARA m ( ) j 1 s f 2 j+1 a,m N nj r r f f +1 < (4.10) j= ã n < (4.11) N N,K F n ã n <. (4.12) n N K Then, by combining Lemma 4.1 with the results in [56], we immediately derive the following results. Theorem 4.2. (a) A = (a n ) (l 1 ( F (r, s)), l ) if only (4.2), (4.9) (4.10) hold. (b) A = (a n ) (l 1 ( F (r, s)), c) if only if (4.2), (4.6), (4.9) (4.10) hold. (c) A = (a n ) (l 1 ( F (r, s)), c 0 ) if only if (4.2), (4.6) with α = 0, (4.9) (4.10) hold. (d) A = (a n ) (l 1 ( F (r, s)), l 1 ) if only (4.2), (4.10) (4.11) hold. Theorem 4.3. Let 1 < p <. Then, we have (a) A = (a n ) (l p ( F (r, s)), l ) if only if (4.1), (4.2) (4.4) hold. (b) A = (a n ) (l p ( F (r, s)), c) if only if (4.1), (4.2), (4.4) (4.6) hold. (c) A = (a n ) (l p ( F (r, s)), c 0 ) if only if (4.1), (4.2), (4.4) (4.6) with α = 0 hold. (d) A = (a n ) (l p ( F (r, s)), l 1 ) if only if (4.1), (4.2) (4.5) hold. Theorem 4.4. (a) A = (a n ) (l ( F (r, s)), l ) if only (4.2), (4.3) (4.4) with q = 1 hold. (b) A = (a n ) (l ( F (r, s)), c) if only (4.2), (4.3), (4.6) (4.7) hold. (c) A = (a n ) (l ( F (r, s)), c 0 ) if only (4.2), (4.3) (4.8) hold. (d) A = (a n ) (l ( F (r, s)), l 1 ) if only (4.2), (4.3) (4.12) hold. 5. Some geometric properties of the space l p ( F (r, s)) (1 < p < ) In this section, we study some geometric properties of the space l p ( F (r, s)) for 1 < p <. For these properties, one can see [21, 22, 26, 27, 33, 34, 41, 43, 44, 53]. A Banach space X is said to have the Banach-Sas property if every bounded sequence (x n ) in X admits subsequence (z n ) such that the sequence t (z)} is convergent in the norm in X [43], where t (z) = (z 0 + z z ); ( N). (5.1) A Banach space X is said to have the wea Banach-Sas property whenever given any wealy null sequence (x n ) X there exists a subsequence (z n ) of (x n ) such that the sequence t (z)} strongly convergent to zero.

13 A STUDY ON TOPOLOGICAL AND GEOMETRICAL CHARACTERISTICS OF NEW In [43], García-Falset introduce the following coefficient: } R(X) = lim inf x n x : (x n ) B(X), x w n 0, x B(X), n where B(X) denotes the unit ball of X. Remar 5.1. A Banach space X with R(X) < 2 has the wea fixed point property [27]. Let 1 < p <. A Banach space is said to have the Banach-Sas type p or property (BS) p, if every wealy null sequence (x ) has a subsequence (x j ) such that for some C > 0, x j < C(n + 1) 1/p, (5.3) for all n N ( see [34]). For a normed linear space E, Gurarii s modulus of convexity is defined by } β (E) (ε) = inf 1 inf αx + (1 α)y : x, y S(E), x y = ε, 0 α 1 where S(E) denotes the unit sphere of E 0 ε 2 [29, 47]. Now, we give some geometric properties of the space l p ( F (r, s)), where 1 < p <. Theorem 5.2. The space l p ( F (r, s)) has the Banach-Sas type p, where 1 < p <. Proof. Let (ε n ) be a sequence of positive numbers such that ε n 1/2, (x n ) be a wealy null sequence in B(l p ( F (r, s))). Set z 0 = x 0 = 0 z 1 = x n1 = x 1. Then, there exists 1 N such that z 1 (i)e (i) < ε 1. i= 1 +1 l p( F (r,s)) Since (x n ) is wealy null sequence implies x n 0 coordinatewise, there is an n 2 N such that 1 x n (i)e (i) < ε 1 lp( F (r,s)) i=0 for n n 2. Set z 2 = x n2. Then, there exists an 2 > 1 such that z 2 (i)e (i) < ε 2. i= 2 +1 l p( F (r,s)) Again using the fact that x n 0 coordinatewise, there exists an n 3 n 2 such that 2 x n (i)e lp( (i) < ε 2 F (r,s)) for n n 3. i=0

14 80 M. CANDAN, E.E. KARA If we continue this process, we can find two increasing subsequences ( i ) (n i ) such that j x n (i)e (i) < ε j for each n n j+1 i=0 i= j +1 z j (i)e (i) where z j = x nj. Hence, j 1 z j = z j (i)e (i) + lp( F (r,s)) i=0 j z j (i)e (i) i= j 1 +1 lp( F (r,s)) lp( F (r,s)) j i= j 1 +1 lp( F (r,s)) < ε j, z j (i)e (i) ε j. i= j +1 z j (i)e (i) lp( F (r,s)) On the other h, it can be seen that x lp( F (r,s)) < 1. Therefore, we have p j j z j (i)e (i) = r f i z j (i) + s f i+1 z j (i 1) f i+1 f i i= j 1 +1 Hence, we obtain l p( F (r,s)) j i= j 1 +1 i= j 1 +1 r f i z j (i) + s f i+1 z j (i 1) f i+1 f i i=0 (n + 1). z j (i)e (i) (n + 1)1/p. By using the fact that 1 (n + 1) 1/p for all n N 1 < p <, we have z j (n + 1) 1/p + 1 2(n + 1) 1/p lp( F (r,s)) which means that l p ( F (r, s)) has the Banach-Sas type p. Remar 5.3. Since l p ( F (r, s)) is linearly isomorphic to l p, we have R(l p ( F (r, s))) = R(l p ) = 2 1/p. From Remars (5.1) (5.3), we have: Theorem 5.4. The space l p ( F (r, s)) has the wea fixed point property, where 1 < p <. p p

15 A STUDY ON TOPOLOGICAL AND GEOMETRICAL CHARACTERISTICS OF NEW Theorem 5.5. Let 1 p <. The modulus of convexity for the space l p ( F (r, s)) is ( ε ) p ) 1/p β lp( F (r,s)) (1 (ε) 1, 2 where 0 ε 2. Proof. For x l p ( F (r, s)), we have ( ) 1/p x lp( F (r,s)) = F (r, s)x p = F n (r, s) p. Let 0 ε 2 consider the following sequences: ( u = F 1 (r, s)(1 (ε/2) p ) 1/p, F ) 1 (r, s)(ε/2), 0, 0,... n ( v = F 1 (r, s)(1 (ε/2) p ) 1/p, F ) 1 (r, s)( ε/2), 0, 0,..., where F 1 (r, s) is the inverse of the matrix F (r, s). Clearly, the F (r, s)-transforms of u v are as follows: F (r, s)u = ( (1 (ε/2) p ) 1/p, ε/2, 0, 0,... ) F (r, s)v = ( (1 (ε/2) p ) 1/p, ε/2, 0, 0,... ). Then, F (r, s)u p = u lp( F (r,s)) = 1 F (r, s)v p = v lp( F (r,s)) = 1, that is, u, v S(l p ( F (r, s))). Also, we have F (r, s)u F (r, s)v p = u v lp( F (r,s)) = ε. For 0 α 1, we have Hence, we obtain αu + (1 α)v p l p( F = α F (r, s)u + (1 α) F (r, s)v p (r,s)) p ( ε ) p ( = 1 + 2α 1 p ε ) p. 2 2 ( ( ε ) p ) 1/p inf αu + (1 α)v 0 α 1 l p( F (r,s)) = 1. 2 Consequently, for u, v S(l p ( F (r, s))) with F (r, s)u F (r, s)v p = u v lp( F (r,s)) = ε, we have ( ε ) p ) 1/p β lp( F (r,s)) (1 (ε) 1, 2 where 1 p <. Thus the proof is completed.

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18 84 M. CANDAN, E.E. KARA 54. E. Savaş M. Mursaleen, Matrix transformations in some sequence spaces, Istanbul univ. Fen Fa. Mat. Derg (1993). 55. S. Simons, The sequence spaces l(p v ) m(p v ), Proc. Lond. Math. Soc. 3(15), (1965). 56. M. Stieglitz H. Tietz, Matrix transformationen von folgenräumen eine ergebnisübersicht, Math. Z. 154, 1 16 (1977). 57. A. Wilansy, Summability through Functional Analysis. North-Holl Mathematics Studies 85, Elsevier Science Publishers, Amsterdam: New Yor: Oxford, FACULTY OF ARTS SCIENCES, DEPARTMENT OF MATHEMATICS, İNÖNÜ UNIVERSITY, MALATYA-44280, TURKEY address: murat.can@inonu.edu.tr 2 DEPARTMENT OF MATHEMATICS, TURKEY address: eevrenara@duzce.edu.tr DÜZCE UNIVERSITY, DÜZCE-81620,