FIBONACCI DIFFERENCE SEQUENCE SPACES FOR MODULUS FUNCTIONS

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1 LE MATEMATICHE Vol. LXX (2015) Fasc. I, pp doi: / FIBONACCI DIFFERENCE SEQUENCE SPACES FOR MODULUS FUNCTIONS KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL In the present paper we introduce the Fibonacci difference sequence spaces l( ˆF,F, p,u) l ( ˆF,F, p,u) by using a sequence of modulus functions a new b matrix ˆF. We also mae an effort to study some inclusion relations, topological geometric properties of these spaces. Furthermore, the α,β,γ duals matrix transformation of the space l( ˆF,F, p,u) are determined. 1. Introduction Preliminaries Let w be the space of all real or complex-valued sequences. By l, c, c 0 l p (1 p < ), we denote the sets of all bounded, convergent, null sequences p-absolutely convergent series, respectively. The notion of difference sequence spaces was introduced by Kızmaz 18], who studied the difference sequence spaces l ( ), c( ) c 0 ( ). The notion was further generalized by Et Çola 11] by introducing the spaces l ( m ),c( m ) c 0 ( m ). In 1981, Kızmaz 18] defined the sequence spaces X( ) = {x = (x ) w : (x x +1 ) X} Entrato in redazione: 15 maggio 2014 AMS 2010 Subject Classification: 11B39, 46A45, 46B45, 46B20. Keywords: Fibonacci numbers, modulus function, paranorm space, α, β, γ duals, matrix transformation.

2 138 KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL for X = l,c c 0. The difference space bν p consisting of all sequences (x ) such that (x x 1 ) is in the sequence space l p, was studied in the case 0 < p < 1 by Altay Başar 4] in the case 1 p by Başar Altay 7] Çola et al. 9]. The paranormed difference sequence space λ(p) = {x = (x ) w : (x x +1 ) λ(p)} was examined by Ahmad Mursaleen 6] Malowsy 21], where λ(p) is any of the paranormed spaces l (p), c(p) c 0 (p) defined by Simons 29] Maddox 22]. Recently, Altay et al. 5] have defined the sequence spaces bν(u, p) bν (u, p) by bν(u, p) = { x = (x ) w : u (x x 1 ) p < } bν (u, p) = { x = (x ) w : sup 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 bν p for 1 p. Definition 1.1. A modulus function is a function f : 0, ) 0, ) such that 1. f (x) = 0 if only if x = 0, 2. f (x + y) f (x) + f (y), for all x,y 0, 3. f is increasing, 4. f is continuous from the right at 0. It follows that f must be continuous everywhere on 0, ). The modulus function may be bounded or unbounded. For example, if we tae f (x) = x x+1, then f (x) is bounded. If f (x) = x p,0 < p < 1 then the modulus function f (x) is unbounded. Subsequently, modulus function has been discussed in (2], 3], 23], 24], 28]) references therein. Definition 1.2. Let X be a linear metric space. A function p : X R is called paranorm, if (P1) p(x) 0 for all x X, (P2) p( x) = p(x) for all x X, (P3) p(x + y) p(x) + p(y) for all x,y X,

3 FIBONACCI DIFFERENCE SEQUENCE SPACES 139 (P4) if (λ n ) is a sequence of scalars with λ n λ as n (x n ) is a sequence of vectors with p(x n x) 0 as n, then p(λ n x n λx) 0 as n. A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm the pair (X, p) is called a total paranormed space. It is well nown that the metric of any linear metric space is given by some total paranorm (see 30, Theorem , p. 183]). Definition 1.3. Let X Y be two sequence spaces A = (a n ) be an infinite matrix of real or complex numbers a n, where n, N. Then we say that A defines a matrix mapping from X into Y if for every sequence x = (x ) =0 X, the sequence Ax = {A n (x)} n=0 the A-transform of x is in Y, where A n (x) = =0 a n x (n N). (1) By (X,Y ) we denote the class of all matrices A such that A : X Y. Thus A (X,Y ) if only if the series on the right-h side of (1) converges for each n N every x X we have Ax Y for all x X. Definition 1.4. The matrix domain X A of an infinite matrix A in a sequence space X is defined by which is a sequence space. X A = {x = (x ) w : Ax X} (2) The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors (see 1], 19], 26]). In 17] Kara introduce the Fibonacci difference sequence spaces l p ( ˆF) l ( ˆF) as { l p ( ˆF) = x = (x n ) w : n { l ( ˆF) = x = (x n ) w : sup f n f n+1 x n f n+1 f n x n 1 p < }, 1 p <, n N f n f n+1 x n f n+1 f n x n 1 < }. The sequence { f n } n=0 of Fibonacci numbers is given by the linear recurrence relations f 0 = f 1 = 1 f n = f n 1 + f n 2, n 2. Fibonacci numbers have many interesting properties applications in arts, sciences architecture. For example, the ratio sequences of Fibonacci numbers converges to the golden

4 140 KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL ratio which is important in sciences arts. Also, in 14] some basic properties of Fibonacci numbers are given as follows: f n+1 lim = = α (golden ratio), n f n 2 n =0 f = f n+2 1 (n N), 1 f converges, f n 1 f n+1 f 2 n = ( 1) n+1 (n 1) (Cassini formula). Substituting for f n+1 in Cassini s formula yields f 2 n 1 + f n f n 1 f 2 n = ( 1) n+1. Let f n be the nth Fibonacci number for every n N. Then we define the infinite matrix ˆF = ( ˆf n ) by f n+1 f n ( = n 1), f ˆf n = n f n+1 ( = n), 0 (0 < n 1 or > n) where n, N (see 17]). Define the sequence y = (y n ) by the ˆF transform of a sequence x = (x n ), i.e., y n = ˆF n (x) = { f0 f1 x 0 = x 0 (n = 0), f n f n+1 x n f n+1 f n x n 1 (n 1) (n N). (3) Definition 1.5. A sequence space X with a linear topology is called a K-space, provided each of the maps p n : X R defined by p n (x) = x n is continuous for all n N. A K-space X is called an FK-space provided X is complete linear metric space. An FK-space whose topology is normable is called a BK-space. ( The space l p (1 p < ) is a BK-space with x p = x p) 1 p c 0, c =0 l are BK-spaces with x = sup x. Let F = (F ) be a sequence of modulus functions. Let p = (p ) be any bounded sequence of positive real numbers u = (u ) be a sequence of strictly positive real numbers. ˆF = ( ˆf n ) denotes a Fibonacci b matrix f is the th Fibonacci number for every N. In this paper we define the following sequence spaces: ( { f l( ˆF,F, p,u) = x = (x ) w : u F x f p } +1 x 1 <, f +1 f

5 FIBONACCI DIFFERENCE SEQUENCE SPACES 141 ( { f l ( ˆF,F, p,u) = x = (x ) w : sup u F N f +1 x f +1 f p } x 1 <. With the notation of (2), the sequence spaces l( ˆF,F, p,u) l ( ˆF,F, p,u) may be redefined as follows l( ˆF,F, p,u) = {l(f, p,u)} ˆF (1 p < ) l ( ˆF,F, p,u) = {l (F, p,u)} ˆF. (4) The following inequality will be used throughout the paper. Let p = (p ) be a sequence of positive real numbers with 0 < p sup p = H, let D = max{1,2 H 1 }. Then, for the factorable sequences (a ) (b ) in the complex plane, we have a + b p D( a p + b p ). (5) In this paper, we define Fibonacci difference sequence spaces defined by a sequence of modulus functions Fibonacci matrix ˆF. We investigate some topological properties of these new sequence spaces establish some inclusion relations concerning these spaces. Also we determine the α,β γ duals of the space l( ˆF,F, p,u) l ( ˆF,F, p,u) in third section of this paper. In the fourth section of the paper we construct the matrix transformation of the space (l( ˆF,F, p,u),x) (l ( ˆF,F, p,u),x), where 1 p < X is any of the spaces l, l 1, c c 0. In the last section, we characterize some geometric properties of the space l( ˆF,F, p,u). 2. Some topological properties of the spaces l( ˆF,F, p,u) l ( ˆF,F, p,u) Theorem 2.1. Let F = (F ) be a sequence of modulus functions p = (p ) be a bounded sequence of positive real numbers u = (u ) be a sequence of strictly positive real numbers. Then l( ˆF,F, p,u) l ( ˆF,F, p,u) are linear spaces over the complex field C. Proof. Let x,y l( ˆF,F, p,u). Then ( f u F x f p +1 x 1 < f +1 f ( f u F f +1 y f +1 f p y 1 <.

6 142 KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL For λ,µ C, there exist integers M λ N µ such that λ M λ µ N µ. Using inequality (5) definition of modulus function, we have ( λ ( f u F x f +1 f +1 ( f u F λ x f +1 f +1 f x 1 ) ( f + u F µ y f +1 f +1 f ( DMλ H f u F x f +1 f +1 f ( + DNµ H f u F y f +1 f +1 f < f ( f + µ p x 1 p y 1 p x 1 p y 1 f +1 y f +1 f ) p y 1 so that λx + µy l( ˆF,F, p,u). This proves that l( ˆF,F, p,u) is a linear space. Similarly we can prove that l ( ˆF,F, p,u) is a linear space. Theorem 2.2. Let F = (F ) be a sequence of modulus functions p = (p ) be a bounded sequence of positive real numbers u = (u ) be a sequence of strictly positive real numbers. Then l( ˆF,F, p,u) is a paranormed space with g(x) = sup ( ( f u F f +1 x f +1 where 0 < p sup p = H < K = max(1,h). f p ) 1 K x 1 Proof. Clearly g(x) = g( x), for all x l( ˆF, F, p, u). It is trivial that f +1 x f +1 f x 1 = 0, for x = 0. Since p K 1, using Minowsy inequality, we have ( ( ( f u F x f ) ( +1 f x 1 + y f ) p ) 1 K +1 y 1 f +1 f f +1 f ( ( ) ( f u F x f +1 f x 1 + u F y f p ) 1 K +1 y 1 f +1 f f +1 f ( ( f u F x f +1 f +1 f p ) 1 K x 1 f

7 + ( ( f u F FIBONACCI DIFFERENCE SEQUENCE SPACES 143 f +1 y f +1 f p ) 1 K y 1. Hence g(x) is subadditive. For the continuity of multiplication let us tae any complex number α. By definition, we have g(αx) = sup ( C H K α g(x) ( ( f u F α x f ) p ) 1 +1 K x 1 f +1 f where C α is a positive integer such that α C α. Now, Let α 0 for any fixed x with g(x) = 0. By definition for α < 1, we have ( f u F f +1 x f +1 f p x 1 < ε for n > n 0 (ε). (6) Also for 1 n < n 0, taing α small enough. Since F is continuous, we have ( f u F f +1 x f +1 Now from equation (6) (7), we have This completes the proof. f g(αx) 0 as α 0. p x 1 < ε. (7) Theorem 2.3. Let F = (F ) be a sequence of modulus functions. If p = (p ) q = (q ) are bounded sequences of positive real numbers with 0 p q < for each, then l( ˆF,F, p,u) l( ˆF,F,q,u). Proof. Let x l( ˆF,F, p,u). Then ( f u F f +1 x f +1 f p x 1 <. This implies that ( f u F x f p +1 x 1 1, f +1 f

8 144 KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL for sufficiently large values of (say) 0, for some fixed 0 N. Since F is increasing p q we have ( f u F x f q ( +1 f x 1 f +1 u F x f p +1 x 1 f +1 0 f 0 <. Hence x l( ˆF, F, q, u). This completes the proof. Theorem 2.4. Let F = (F ) be a sequence of modulus functions β = F (t) lim > 0. Then l( ˆF,F, p,u) l( ˆF, p,u). t t Proof. In order to prove that l( ˆF,F, p,u) l( ˆF, p,u). Let β > 0. By definition of β, we have F (t) β(t), for all t > 0. Since β > 0, we have t 1 β F (t) for all t > 0. Let x = (x ) l( ˆF,F, p,u). Thus, we have ( f u x f p ( +1 x 1 1 f +1 f β f u F x f +1 f +1 f which implies that x = (x ) l( ˆF, p,u). This completes the proof. f p x 1 Theorem 2.5. Let F = (F ) F = (F ) are sequences of modulus functions, then l( ˆF,F, p,u) l( ˆF,F, p,u) l( ˆF,F + F, p,u). Proof. Let x = (x ) l( ˆF,F, p,u) l( ˆF,F, p,u). Therefore ( p u F f < Then, we have x f +1 x 1 f +1 f ( u F f x f p +1 x 1 <. f +1 f ( u (F + F f ) x f +1 x 1 f +1 f { ( f K + K { u F u F ( f x f +1 x 1 f +1 f x f +1 x 1 f +1 f p p } p }.

9 FIBONACCI DIFFERENCE SEQUENCE SPACES 145 ( Thus, u (F + F f ) x f p +1 x 1 <. f +1 f Therefore, x = (x ) l( ˆF,F + F, p,u) this completes the proof. Theorem 2.6. Let F = (F ) F = (F ) be two sequences of modulus functions, then l( ˆF,F, p,u) l( ˆF,FoF, p,u). Proof. Let ε > 0 choose δ > 0 with 0 < δ < 1 such that F (t) < ε for 0 t δ. ( Write y = u F f f +1 x f +1 f x 1 consider F (y p = F (y p +F (y p 1 where the first summation is over y δ second summation is over y > δ. Since F is continuous, we have for y > δ, we use the fact that By the definition, we have for y > δ 2 F (y p < ε H (8) 1 y < y δ 1 + y δ. F (y ) < 2F (1) y δ. Hence ( F (y p max 1,(2F (1)δ 1 ) H) y ] p. (9) 2 From equation (8) (9), we have This completes the proof. l( ˆF,F, p,u) l( ˆF,FoF, p,u). Theorem 2.7. Let 1 p H for all N. Then l( ˆF,F, p,u) is a BKspace with the norm x l( ˆF,F,p,u) = ˆFx p, i.e., x l( ˆF,F,p,u) = ( n ] p ) 1 K u F ( ˆF n (x) ) p. x l ( ˆF,F,p,u) = sup u F ( ˆF n (x) n N

10 146 KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL Proof. Since (4) holds, l p l are BK-spaces with respect to their natural norms the matrix ˆF is a triangle; Theorem of Wilansy 30, p.63] gives the fact that the spaces l( ˆF,F, p,u) l ( ˆF,F, p,u) are BK-spaces with the given norms, where 1 p H < for all N. This completes the proof. Remar 2.8. One can easily chec that the absolute property does not hold on the spaces l( ˆF,F, p,u) l ( ˆF,F, p,u), that is x l( ˆF,F,p,u) x l( ˆF,F,p,u) x l ( ˆF,F,p,u) x l ( ˆF,F,p,u) for at least one sequence in both the spaces l( ˆF,F, p,u) l ( ˆF,F, p,u); this shows that l( ˆF,F, p,u) l ( ˆF,F, p,u) are the sequence spaces of non-absolute type, where x = ( x ) 1 p H < for all N. Theorem 2.9. The sequence space l( ˆF,F, p,u) of non absolute type is linearly isomorphic to the space l p, that is, l( ˆF,F, p,u) = l p, for 1 p H < for all N. Proof. It is enough to show that the existence of a linear bijection between the spaces l( ˆF,F, p,u) l p for 1 p H for all N. Consider the transformation T defined with the notation of (3), from l( ˆF,F, p,u) to l p by x y = T x. Then T x = y = ˆFx l p, for every x l( ˆF,F, p,u). Also, the linearity of T is clear. Further it is trivial that x = 0 whenever T x = 0 hence T is injective. We assume that y = (y ) l p, for 1 p H for all N define the sequence x = (x ) by x = j=0 f 2 +1 f j f j+1 y j, N. Then in the case 1 p H < for all N p =, we get ( x l( ˆF,F,p,u) ( = f u F = ( u F ( f ( ) 1 = y p K = y p < f +1 x f +1 f +1 j=0 f p ) 1 K x 1 f+1 2 y j f 1 +1 f j f j+1 f j=0 f 2 p ) 1 K y j f j f j+1 x l ( ˆF,F,p,u) = sup p u F ( ˆF (x) = y <, N

11 FIBONACCI DIFFERENCE SEQUENCE SPACES 147 respectively. Hence T is linear bijection which shows that the spaces l p l( ˆF,F, p,u) are linearly isomorphic for 1 p H for all N. This concludes the proof. We wish to exhibit some inclusion relations concerning the space l( ˆF,F, p,u). Theorem The inclusion l p l( ˆF,F, p,u) strictly holds for 1 p H for all N. Proof. To prove the validity of the inclusion l p l( ˆF,F, p,u) for 1 p H for all N, it suffices to show the existence of a number M > 0 such that x l( ˆF,F,p,u) M x p for every x l p. Let x l p 1 < p H for all N. Since the inequalities F ( f f +1 ) 1 F ( f+1 f ) 2 hold for every N, we obtain with the notation of (3), u F ( ˆF (x) p sup N 2 p p 1 u F ( x + 2x 1 2 2p 1 which together yield, as expected, u F ( ˆF (x) p 3sup N u F x ] p + u F ( x p, ] p ) u F 2x 1 x l( ˆF,F,p,u) 4 x p for 1 < p. (10) Further, since the sequence x = (x ) = ( f 2 +1 ) = (1,22,3 2,5 2,...) belongs to l( ˆF,F, p,u) l p, the inclusion l p l( ˆF,F, p,u) is strict for 1 < p H for all N. Similarly, one can easily prove that the inequality (10) also holds in the case p = 1, so we omit the details. This completes the proof. 3. The α,β γ duals of the space l( ˆF,F, p,u) The α,β γ duals of the sequence space X are respectively defined by X α = {a = (a ) w : ax = (a x ) l 1 for all x = (x ) X}, X β = {a = (a ) w : ax = (a x ) cs for all x = (x ) X} X γ = {a = (a ) w : ax = (a x ) bs for all x = (x ) X},

12 148 KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL where cs bs are the sequence spaces of all convergent bounded series, respectively 8]. We assume throughout that p,q 1 with 1 p + 1 q = 1 denote the collection of all finite subsets of N by H. In this section, we determine α,β γ duals of the sequence space l( ˆF,F, p,u) of non-absolute type. Since the case p = 1 can be proved by same analogy, we omit the proof of that case consider only the case 1 < p H for all N in the proof of Theorem 3.5. In 27] the following nown results are fundamental for our investigation. Lemma 3.1. A = (a n ) (l p,l 1 ) if only if sup K H a n <, 1 < p. n K Lemma 3.2. A = (a n ) (l p,c) if only if lim a n exists for all N, (11) n sup n N a n q <, 1 < p <. (12) Lemma 3.3. A = (a n ) (l,c) if only if (11) holds lim a n = lim a n. (13) n n Lemma 3.4. A = (a n ) (l p,l ) if only if (12) holds with 1 < p. Theorem 3.5. The α dual of the space l( ˆF,F, p,u) is the set { ( dˆ 1 = a = (a ) w : sup f u F 2 p n+1 q a n < }, K H n K f f +1 where 1 < p H for all N. Proof. Let 1 < p H for all N. For any fixed sequence a = (a n ) w, we define the matrix B = (b n ) by ( f b n = 2 p n+1 u F a n (0 n), f f +1 0 ( > n) for all n, N. Also, for every x = (x n ) w, we put y = ˆFx. Then it follows by (3) that fn+1 2 p a n y = B n (y) (n N). (14) f f +1 ( n a n x n = u F =0

13 FIBONACCI DIFFERENCE SEQUENCE SPACES 149 Thus, we observe by (14) that ax = a n x n l 1 whenever x l( ˆF,F, p,u) if only if B(y) l 1 whenever y l p. Therefore, we drive by using Lemma 3.1 that ( sup f u F 2 p n+1 q a n <, K H n K f f +1 which implies that l( ˆF,F, p,u α = ˆ d 1. Theorem 3.6. Define the sets dˆ 2, dˆ 3 dˆ { ( 4 by f dˆ 2 = a = (a ) w : u F 2 p j+1 a j exists for all N}, j= f f +1 { ( n n f dˆ 3 = a = (a ) w : sup u F n N 2 p j+1 q a j < } =0 j= f f +1 ˆ d 4 = { a = (a ) w : lim n n =0 u F ( n j= f j+1 2 p a j = f f +1 ( u F j= f j+1 2 p a j < }. f f +1 Then l( ˆF,F, p,u β = dˆ 2 dˆ 3 l ( ˆF,F, p,u β = dˆ 2 dˆ 4 where 1 < p H < for all N. Theorem 3.7. l( ˆF,F, p,u γ = ˆ d 3, where 1 < p H for all N. Proof. The result can be obtained from Lemma Matrix transformations related to the sequence space l( ˆF,F, p,u) In this section, we characterize the classes (l( ˆF,F, p,u),x), where 1 p H for all N X is any of the spaces l, l 1, c c 0. For simplicity in notation, we write ã n = u F ( j= f 2 j+1 f f +1 a n j The following lemma is essential for our results. p for all,n N. Lemma 4.1 (15], Theorem 4.1). Let λ be an FK-space, U be a triangle, V be its inverse µ be an arbitrary subset of w. Then we have A = (a n ) (λ U, µ) if only if C (n) = ( c (n) m ) (λ,c) for all n N

14 150 KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL where c (n) m = c n = j= C = (c n ) (λ,µ), m j= a n j v j (0 m), 0 ( > m) a n j v j for all,m,n N. Now, we list the following conditions: m lim m =0 lim m m sup m N =0 u F ( m j= u F ( m j= f j+1 2 p q a n j <, (15) f f +1 f j+1 2 p a n j = ã n, n, N, (16) f f +1 ( m f j+1 u F 2 p a n j = j= f f +1 ã n for each n N, (17) sup,m N sup n N sup N H ã n q <, (18) q ã n <, (19) n N lim = ã ; N, (20) n ãn lim n lim n ã n = ã, (21) ã n = 0, (22) sup ã n <, (23) n, N ( m u F j= f j+1 2 p a n j <, f f +1 (24)

15 FIBONACCI DIFFERENCE SEQUENCE SPACES 151 sup ã n <, N (25) sup ã n <. (26) N,K H n N K Then by combining Lemma 4.1 with the results in (see 27]), we immediately derive the following results. Theorem 4.2. (a) A = (a n ) (l 1 ( ˆF,F, p,u),l ) if only if (16), (23) (24) hold. (b) A = (a n ) (l 1 ( ˆF,F, p,u),c) if only if (16), (20), (23) (24) hold. (c) A = (a n ) (l 1 ( ˆF,F, p,u),c 0 ) if only if (16), (20) with ã = 0, (23) (24) hold. (d) A = (a n ) (l 1 ( ˆF,F, p,u),l 1 ) if only if (16), (24) (25) hold. Theorem 4.3. Let 1 < p H < for all N. Then we have (a) A = (a n ) (l( ˆF,F, p,u),l ) if only if (15), (16) (18) hold. (b) A = (a n ) (l( ˆF,F, p,u),c) if only if (15), (16), (18) (20) hold. (c) A = (a n ) (l( ˆF,F, p,u),c 0 ) if only if (15), (16), (18) (20) with ã = 0 hold. (d) A = (a n ) (l( ˆF,F, p,u),l 1 ) if only if (15), (16) (19) hold. Theorem 4.4. (a) A = (a n ) (l ( ˆF,F, p,u),l ) if only if (16), (17) (18) with q = 1 hold. (b) A = (a n ) (l ( ˆF,F, p,u),c) if only if (16), (17), (20) (21) hold. (c) A = (a n ) (l ( ˆF,F, p,u),c 0 ) if only if (16), (17) (22) hold. (d) A = (a n ) (l ( ˆF,F, p,u),l 1 ) if only if (16), (17) (26) hold. 5. Some geometric properties of the space l( ˆF,F, p,u) In this section, we study some geometric properties of the space l( ˆF,F, p,u) for (1 < p H < ) for all N. For these properties, (see 10], 20], 25]). A Banach space X is said to have the Banach-Sas property if every bounded sequence (x n ) in X admits a subsequence (z n ) such that the sequence {t (z)} is convergent in the norm in X 19], where t (z) = 1 (z 0 + z z ) ( N). (27) 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

16 152 KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL (x n ) such that the sequence {t (z)} is strongly convergent to zero. In 12], García-Falset introduces the following coefficient: R(X) = sup { w lim inf x n x : (x n ) B(X),x n 0, x B(X) }, (28) 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 13]. Let 1 < p <. A Banach space is said to have the Banach-Sas type p or the property (BS) p if every wealy null sequence (x ) has a subsequence (x l ) such that for some C > 0, n x l < C(n + 1) 1 p (29) l=0 for all n N (See 16]). Now, we may give the following results related to some geometric properties of the space l( ˆF,F, p,u), where 1 < p H < for all N. Theorem 5.2. Let 1 < p H < for all N. Then the space l( ˆF,F, p,u) has the Banach-Sas type p. Proof. Let (ε n ) be a sequence of positive numbers for which ε n 1 2, also let (x n ) be a wealy null sequence in B(l( ˆF,F, p,u)). Set a 0 = x 0 = 0 a 1 = x n1 = x 1. Then there exists m 1 N such that 1 (i)e i=m 1 +1a (i)l( < ε 1. (30) ˆF,F,p,u) Since (x n ) being a wealy null sequence implies x n 0 coordinatewise, there is an n 2 N such that m 1 n (i)e i=0x (i)l( < ε 1, ˆF,F,p,u) when n n 2. Set a 2 = x n2. Then there exists an m 2 > m 1 such that 2 (i)e i=m 2 +1a (i)l( < ε 2. ˆF,F,p,u) Again using the fact that x n 0 coordinatewise, there exists an n 3 n 2 such that m 2 n (i)e i=0x (i)l( < ε 2, ˆF,F,p,u)

17 FIBONACCI DIFFERENCE SEQUENCE SPACES 153 when n n 3. If we continue this process, we can find two increasing subsequences (m i ) (n i ) such that m j n (i)e i=0x (i)l( < ε j ˆF,F,p,u) for each n n j+1 j (i)e i=m j +1a (i)l( < ε j, ˆF,F,p,u) where b j = x n j. Hence, n jl( j=0a ˆF,F,p,u) ( n m j 1 m j = a j (i)e (i) + a j (i)e (i) + j (i)e j=0 i=0 i=m j 1 +1 i=m j +1a (i))l( ˆF,F,p,u) ( ) ( n m j 1 a j (i)e (i) n m j + j (i)e j=0 i=0 l( ˆF,F,p,u) j=0 i=m j 1 +1a (i))l( ˆF,F,p,u) ( n + j (i)e j=0 i=m j +1a (i))l( ˆF,F,p,u) ( n m j j (i)e j=0 i=m j 1 +1a (i))l( n + 2 ε j. ˆF,F,p,u) j=0 On the other h, it can be seen that x l( ˆF,F,p,u) < 1. Therefore, we have that ( ) n m j p a j (i)e (i) j=0 i=m j 1 +1 l( ˆF,F,p,u) ( n m j f i = u F a j (i) f p i+1 a j (i 1) j=0 i=m j 1 +1 f i+1 f i ( n f i u F a j (i) f p i+1 a j (i 1) (n + 1). f i+1 f i j=0 i=0 Hence, we obtain n j=0 ( ) m j a j (i)e (i) p (n + 1) 1. i=m j 1 +1

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20 156 KULDIP RAJ - SURUCHI PANDOH - SEEMA JAMWAL 29] S. Simons, The sequence spaces l(p ν ) m(p ν ), Proc. Lond. Math. Soc. 3 (1965), ] A. Wilansy, Summability through functional analysis, North-Holl Math. Stud. 85, KULDIP RAJ School of Mathematics Shri Mata Vaishno Devi University Katra , J&K, India uldeepraj68@rediffmail.com SURUCHI PANDOH School of Mathematics Shri Mata Vaishno Devi University Katra , J&K, India suruchi.poh87@gmail.com SEEMA JAMWAL School of Mathematics Shri Mata Vaishno Devi University Katra , J&K, India seemajamwal8@gmail.com