Fu Paper Maejo Internationa Journa of Science and Technoogy ISSN 1905-7873 Avaiabe onine at www.mijst.mju.ac.th A study on Lucas difference sequence spaces (, ) (, ) and Murat Karakas * and Ayse Metin Karakas Department of Statistics, Bitis Eren University, Bitis, 13000, Turkey * Corresponding author, e-mai: mkarakas@beu.edu.tr Received: 23 February 2017 Accepted: 26 March 2018 Pubished: 3 Apri 2018 Abstract: A new Banach sequence space estabished by using Lucas sequences and nonzero rea numbers and is presented. At the same time, some incusion reations are given and aso some geometrica properties such as Banach-Saks type, weak fixed-point property and the moduus of convexity for this space are anaysed. Keywords: difference sequence spaces, Lucas numbers, weak fixed-point property, Banach-Saks property, moduus of convexity INTRODUCTION Let be the space of a rea and compex vaued sequences. A inear subspace of is caed a sequence space. If is a compete inear metric space, then a K-space X is caed an FKspace. An FK-space whose topoogy is normabe is caed BK-space. The symbos,, and for 1 < represent the sequence spaces of a bounded, convergent, nu sequences and p- absoutey convergent series respectivey, normed by = and = ( ). Difference sequence spaces were introduced for the first time by Kizmaz [1] in the form of ( ) = { : } for =,,. Later, Et and Coak [2] generaised these spaces such as ( ) = { : }, =,,. An infinite matrix is a doube sequence = of rea or compex numbers defined by a function from the set N N into the compex fied C( R ) where N = {0,1,2, }. The treatment of infinite matrices is absoutey different from that of finite matrices. There are various reasons for this. In some instances the most genera inear operator among two sequence spaces is presented by an infinite matrix. Let and be any two sequence spaces. A defines a matrix mapping from to if = { } for every = where
71 =. (1) The cass of a matrices such that : is symboised by( : ). In this way, ( : ) if and ony if the series on the right hand side of (1) converges for each N and every, and we get = { } for a. The notion of matrix domain for an infinite matrix in the sequence space is given by = { : }, (2) which is a sequence space [3]. During the recent years, some sequence spaces by means of the matrix domain for a triange infinite matrix have been introduced by many mathematicians, e.g. Mursaeen and Noman [4,5], Karakas [6], Candan and Kara [7], Kara [8], Mursaeen et a. [9], Basar and Atay [10], Savas et a. [11] and Kara and Basarır [12]. In the iterature, if is a normed or paranormed sequence space, then the matrix domain is said to be a difference sequence space. For =, this space is caed the space of sequences of p-bounded variation, i.e.. Aso, it is cear that =. Now et us give the foowing two triange summabiity matrices, named as backward difference matrix and forward difference matrix for a, N : = ( 1), 1 0, > 0 < 1 and = ( 1), + 1 0, > + 1 0 <, respectivey. Kirisci and Basar [13] have atey defined and examined the difference sequence spaces = { : (, ) }, 1 <, =,,, where (, ) = ( + );, 0. At the same time, difference sequence spaces have been anaysed in some studies by Et [14,15], Mursaeen [16], Coak and Et [17], Et and Basarır [18], Bektas et a. [19], Et and Esi [20], Gaur and Mursaeen [21], Atın [22] and Poat et a. [23]. Now according to the we-known concept of Schauder basis, a sequence is said to be Schauder basis for a normed sequence space if invoves a sequence such that there is a unique sequence of scaars for every with and im ( + + + ) = 0. The aim of this note is to introduce and investigate the new sequence spaces (, ) (, ) by constructing the generaised Lucas difference matrix (, ) by the hep of Lucas sequence { } and, R {0}. Besides, we show that these spaces are BK-spaces and ineary isomorphic to space for 1. Additionay, we study a number of incusion reations and give the basis for the space (, ), 1 < and aso examine severa geometric properties of (, ), 1 < <. RESULTS AND DISCUSSION The sequence { } of Lucas numbers given by the Fibonacci recurrence reation and different initia conditions is defined as = 2, = 1 and = +, 2.
72 Lucas numbers have severa interesting properties and appications [24, 25]. Some of them are as foows: = 3 ; = 1 = 2 ; = 4 = ( 5)( 1). If the term obtained: is deducted from both sides of the ast equaity, the foowing formua is + = ( 5)( 1). In the view of the above information, we estabish the generaised Lucas band matrix (, ) = (, ) by ( = 1) (, ) = (, ) = ( = ). (3) 0 ( > 0 < 1) Aso, we define transform of a sequence = : = (, ) = +, 1. (4) Now et us introduce the foowing Lucas difference sequence spaces (, ) and (, ) by using (3) and (4) in the forms of (, ) = : + <, 1 < and (, ) = : + <. These spaces may be redefined by the hep of (2) as (, ) = (, ) and (, ) = ( ) (, ). (5) Theorem 1. The sequence space (, ) is a BK-space for 1, normed by (, ) = (, ) and (, ) = (, ). Proof. Since the matrix (, ) is a triange, (5) hods and the spaces and are BK-spaces according to their usua norms. By the theorem 4.3.12 of Wiansky [26], the proof of theorem can be straight-forwardy obtained. So our spaces are BK-spaces with the above norms. This competes the proof. Remark 1. (, ) and (, ) are the sequence spaces of non-absoute type. Indeed (, ) (, ) and (, ) (, ). This eads to the fact that the
73 absoute property does not hod for the spaces (, ) and (, ) for at east one sequence where = ( ) and 1 <. Theorem 2. The generaised Lucas difference sequence space ineary isomorphic to the space in the case 1. (, ) of non-absoute type is Proof. Let us consider the transformation : (, ) defined by = with (4). Hence for (, ), we have = = (, ). So it is easy to see that is inear and we omit it. Additionay, we can simpy show that = 0 whenever = 0 and thus is injective. Additionay, et us take = and define the sequence = in the form of =. (6) In that case we get (, ) = + = + = < and (, ) = + = = <. This gives us (, ), 1. Hence we see that is surjective and norm preserving. In concusion is a inear bijection and so the spaces (, ) and are ineary isomorphic. Theorem 3. If 1, then the incusion (, ) stricty hods. Proof. Let us assume that, 1 <. Due to the inequaities 2 and 3, we have from (4): (, ) 6 ( 2 + 3 ) and 6 {, }( + ) (, ) 5 {, }. From here, we obtain for 1 < : (, ) 36 {, } and (, ) 5 {, }. (7)
74 Lasty, the incusion (, ) is strict for 1 < because the sequence = = (1 ( ) ) is in (, ). It can be easiy seen that (7) aso hods for = 1. Thus, the proof is compete. Theorem 4. (, ) (, ) for 1 <. Proof. Under the conditions 1 < and (, ), we obtain from Theorem 1 that if we consider the sequence given by (4). In view of the fact that the incusion hods, we have. This ceary impies that (, ) and so the incusion (, ) (, ) is true. Theorem 5. The sequence defined by =, 0, > (8) is a basis for the space (, ) for 1 <. Aso, every (, ) has a unique representation of the form = (, ). (9) Proof. By use of (8), it is trivia that (, ) = and this gives us ( (, ). Moreover, et us consider (, ) and take ) = (, ) for every non-negative integer. Then it is obtained that (, ) = (, ) (, ) = (, ) and aso, (, ) = (, ), > 0, 0. So there is a non-negative integer such that (, ) < for any > 0. Hence we have that (, ) = (, ) (, ) < 2 for every, i.e. im (, ) = 0. Finay, to show the uniqueness of (9), et us assume that = for (, ). From the continuity of the inear transformation predefined in Theorem 2, we get (, ) = (, ) = =. This competes the proof.
75 Now we examine the geometrica properties of (, ). Let be a normed inear space and, be the unit sphere and unit ba of respectivey. Carkson [27] defined the moduus of convexity as foows: = 1 :,, =, [0,2]. In the iterature there are a few equivaent definitions for moduus of convexity. One of them is Gurarii s moduus of convexity, which was defined by Gurarii [28] and Sanchez and Uan [29] in the form = 1 [, ] + (1 ) :,, = 2, [0,2]. If 0 < < 1, then is uniformy convex and if 1, then is stricty convex. Theorem 6. The Gurarii s moduus of convexity for the space (, ), 1 < is (, ) 1 1, [0,2]. Proof. Let (, ). Then (, ) = (, ) = (, ). (10) Let [0,2] and consider the sequences beow where (, ) is the inverse of the matrix (, ): = = (, ) 1, (, ), 0,0,, = = (, ) 1, (, ), 0,0,. (11) Therefore, it is easy to see that the (, ) transforms of the sequences given by (11) are (, ) = 1,, 0,0, and (, ) = 1,, 0,0,. Hence (, ) = (, ) = 1 and (, ), (, ) and (, ) (, ) Now for [0,1], = (, ) = 1. This means that = (, ) =. From here, + (1 ) (, ) = (, ) + (1 ) (, ) = 1 + 2 1. (12) Therefore, for 1 <, [, ] + (1 ) (, ) = 1. (13) (, ) 1 1 2.
76 Coroary 1. i) If = 2, then (, ) 1 and thus (, ) is stricty convex. ii) If 0 < < 2, then 0 < (, ) < 1 and thus (, ) is uniformy convex. A Banach space is said to have Banach-Saks property if any bounded sequence in admits a subsequence whose arithmetic mean converges in norm. Simiary, we say that a Banach space has weak Banach-Saks property if any weaky nu sequence in admits a subsequence whose arithmetic mean strongy converges in norm. Let be a Banach space. Garcia-Faset [30] defined the coefficient as foows: = (im + ), where the supremum is taken over a weaky nu sequences of the unit ba and a points of the unit ba. He aso proved that a Banach space with < 2 has the weak fixed point property. Some studies on geometrica properties of a sequence space have been done by Karakas et a. [31], Et et a. [32], Mursaeen et a. [33] and Karakas et a. [34]. Theorem 7. The space (, ) has the Banach-Saks property of type. Proof. It can be demonstrated with a standard method. Remark 2. (, ) = = 2 by reason that the space (, ) is ineary isomorphic to. Now we point out the foowing resut due to the fact that (, ) < 2. Coroary 2. The Lucas difference sequence space 1 < <. (, ) has weak fixed-point property for CONCLUSIONS An approach to constructing a new sequence space using matrix domain of a trianguar infinite matrix defined by Lucas numbers has been presented. REFERENCES 1. H. Kizmaz, On certain sequence spaces, Canad. Math. Bu., 1981, 24, 169-176. 2. M. Et and R. Coak, On some generaized difference sequence spaces, Soochow J. Math., 1995, 21, 377-386. 3. F. Başar, Summabiity Theory and Its Appications, Bentham e-books, Istanbu, 2011. 4. M. Mursaeen and A. K. Noman, On the spaces of convergent and bounded sequences, Thai J. Math., 2010, 8, 311-329. 5. M. Mursaeen and A. K. Noman, On some new sequence spaces of non-absoute type reated to the spaces and I, Fiomat, 2011, 25, 33-51. 6. M. Karakas, A new reguar matrix defined by Fibonacci numbers and its appications, BEU J. Sci., 2015, 4, 205-210. 7. M. Candan and E. E. Kara, A study on topoogica and geometrica characteristics of new Banach sequence spaces, Guf J. Math., 2015, 3, 67-84. 8. E. E. Kara, Some topoogica and geometrica properties of new Banach sequence spaces, J. Inequa. App., 2013, 2013, 38.
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