Almost Convergent Sequence Space Derived by Generalized Fibonacci Matrix and Fibonacci Core

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1 Bitish Jounal of Mathematics & Compute Science 7(: 50-67, 05, Aticle no.bjmcs.05. ISSN: SCIENCEDOMAIN intenational Almost Convegent Sequence Space Deived by Genealized Fibonacci Matix and Fibonacci Coe Muat Candan and Kuddusi Kayaduman Faculty of Ats Sciences, Depatment of Mathematics, İnönü Univesity, Malatya-4480, Tuey. Faculty of Ats Sciences, Depatment of Mathematics, Gaziantep Univesity, Gaziantep-730, Tuey. Aticle Infomation DOI: /BJMCS/05/593 Edito(s: ( Rado Mesia, Depatment of Mathematics, Faculty of Civil Engineeing, Slova Univesity of Technology Batislava, Slovaia. Reviewes: ( W. Obeng-Denteh, Mathematics Depatment, Kwame Numah Univesity of Science and Technology, Kumasi, Ghana. ( Anonymous, Tuey. Complete Pee eview Histoy: Oiginal Reseach Aticle Received: 6 Decembe 04 Accepted: 9 Januay 05 Published: 0 Febuay 05 Abstact Consideable inteest in this aticle is to intoduce the sequence space ĉ f(,s deived by genealized diffeence Fibonacci matix in which, s R \ 0, also to discuss and compae with some wellnown spaces defined peviously. In addition to those, afte demonstating that the spaces ĉ f(,s and ĉ ae linealy isomophic, we have detemined the β and γ duals of space ĉ f(,s and have chaacteized some matix classes on this space. As a conclusion, we have also found out that the space has not a Schaude basis. Lastly, we have pesented the Fibonacci coe of a complex-valued sequence and deal with inclusion theoems with espect to Fibonacci coe type. Keywods: Sequence spaces, almost convegence, Fibonacci matix, β-dual, matix tansfomations, coe theoems. 00 Mathematics Subject Classification: 46A45; 40A05; 46A35. Intoduction By w, we denote the space of all eal o complex-valued sequences x = (x. Any vecto subspace of w is called a sequence space. As usual, we wite c 0, c and l denote the sets of sequences that *Coesponding autho: muat.candan@inonu.edu.t

2 ae convegent to zeo, convegent and bounded, espectively. In addition to these, the symbols bs and cs ae nown the spaces of all bounded and convegent seies, espectively. The almost convegence has fundamental impotance fo this aticle. So, the concept is stated in this paagaph. The class ĉ of almost convegent sequences was intoduced by G.G. Loentz [], using the idea of the Banach its. A Banach it L is defined on l, as a non negative linea functional, such that L(ϕx = L(x and L(e =, whee ϕ is shift opeato and e = (,,,. The existence of Banach its was poven by Banach [] in his boo. A sequence x = (x l is nown to be almost convegent to the genealized it α if all Banach its of x is α [], and denoted by ĉ x = α. Let ϕ j be the composition of ϕ with itself j times and define t mn(x fo a sequence x = (x by t mn(x := m + m ϕ j n(x fo all m, n N. Loentz [] poved that ĉ x = α iff t mn(x = α, unifomly in n. It is well nown that a convegent sequence is almost convegent such that its odinay and genealized its ae equal. As mentioned in the above, by ĉ 0 and ĉ, we denote the space of all almost null and almost convegent sequences, that is ĉ 0 := ĉ := x = (x l : m =0 x = (x l : α C x n+ = 0 unifomly in n m + m =0 It is nown that ĉ is a Banach space with the nom [3] m x n+ x ĉ = sup m +. m, x n+ = α unifomly in n m + Anothe notion we need is that of matix tansfomation. Fo this eason, in this paagaph, we shall be concened with matix tansfomation fom a sequence space X to a sequence space Y. Given any infinite matix A = (a n of eal numbes a n, whee n, N, any sequence x, we wite Ax = ( (Ax n, the A-tansfom of x, if (Axn = a nx conveges fo each n N. Fo simplicity in notation, hee and in what follows, the summation without its uns fom 0 to. If x X implies that Ax Y then we say that A defines a matix mapping fom X into Y and denote it by A : X Y. By (X : Y, we mean the class of all infinite matices such that A : X Y. When X and Y have its X and Y, espectively, and fo all x X, A (X : Y and Y n A n(x = X x is valid; we have the ight to say that A egulaly maps X into Y and also shown it as A (X : Y eg. A matix A = (a n is called a tiangle if a n = 0 fo > n and a nn 0 fo all n N. It is tivial that A(Bx = (ABx holds fo the tiangle matices A, B and a sequence x. Futhe, a tiangle U uniquely has an invese U = V that is also a tiangle matix. Then, x = U(V x = V (Ux holds fo all x ω. Fo an abitay sequence space µ, µ A is nown the domain an infinite matix A as µ A = x = x ω : Ax µ. Since µ A is a linea subspace of the space w of all eal o complex-valued sequences, it is also a sequence space. In ecent yeas, the appoach to constuct a new sequence space by means of the,. 5

3 matix domain of a paticula tiangle has been used by some of the wites in many eseach aticles [4, 5, 6, 7, 8]. Fo an oveview of the liteatue on new almost convegent sequence space, see [9, 0,, ] and the efeences theein. Since we ae motivated by the efeences, especially the spaces ĉ R t, ĉ C, ĉ B(,s, and ĉ B(, s have been studied in [9, 0,, ], espectively, whee R t is the Riesz mean, C is the Cesào matix of ode one, B(, s = b n (, s and B(, s = b n ( n, s n ae the genealized diffeence matix and double sequential band matix, espectively defined by b n (, s =, = n, s, = n, 0, othewise b n (, s = n, = n, s n, = n, 0, othewise fo all, n N, whee, s R\0 and = ( n n=0 and s = (sn n=0 be given convegent sequences of positive eal numbes. In ecent yeas, Kaa and Elmaağaç [3] defined and examined u diffeence almost sequence space ĉ u = (ĉ A u, whee A u = (a u n denote u diffeence matix. To wite in a moe clea way, a u ( n u n =, n n, 0, 0 < n o > n fo all, n N. Puely fo the development of almost convegence and some genealizations, the excellent esults [4, 5, 6, 7, 8, 9, 0,,, 3], ae ecommended. The plan of the pesent pape is oganized as follows. Afte collecting all the necessay definitions and esults, we have fistly intoduced new sequence space ĉ f(,s unde the domain of the matix F (, s, constituted by using Fibonacci sequences and non-zeo eal numbes and s, of ĉ peviously defined. Late, we give some inclusion theoems and demonstate that ĉ f(,s is linealy isomophic to the space ĉ. As a conclusion, we also show that the newly defined space has not a Schaude basis and detemine the β and γ duals of the space ĉ f(,s and chaacteize the classes of infinite matices elated to sequence space ĉ f(,s. In the last section, we have defined B F (,s coe of a sequence and chaacteized cetain class of matices fo which B F (,s coe(ax K coe(x, K coe(ax B F (,s coe(x, B F (,s coe(ax B F (,s coe(x and B F (,s coe(ax st coe(x fo all x l. The Sequence Space ĉ f(,s Deived by the Domain of the Matix F (, s In this subsection, befoe stating the new almost sequence space deived genealized diffeence matix which established both Fibonacci sequences and, s R, we pesent some histoical infomation about Fibonacci sequences. In 0, the Fibonacci numbes fist came out in the boo Libe Abaci, which means The Boo of Calculation among the fist westen boos was a histoic boo on aithmetic witten by Leonado of Pisa, commonly nown as Fibonacci. Thee ae many ways to intoduce the Fibonacci sequence, each of which is an equivalent way of defining the same thing. Hee, let us explain this concept. A numeic sequence is a set of odeed numbes geneated by well-defined algoithm. The easiest method of geneating a numbe sequence is to use one o two enel values and an suitable ecusive equation. One of the most well-nown numbe sequence is Fibonacci sequence. This sequence is obtained by the following ecusive fomula f n = f n + f n with n. That is, each tem in the sequence is equal to the sum of the pevious two tems. This sequence equies the enel values f 0 and f. Thoughout ou study, we will tae f 0 and f as. 5

4 Now, we ae taing a loo at some of the famous popeties such as Golden Ratio, and Cassini fomula of the Fibonacci sequence [4]. f n+ = + 5 = ϕ (Golden Ratio, n f n n f = f n+ fo each n N, =0 f conveges, f n f n+ f n = ( n+ fo all n (Cassini Fomula. It can easily be deived by eplacing f n+ in Cassini s fomula namely f n +f nf n f n = ( n+. Many authos used the Fibonacci numbes to establish a sequence space. In paticula, we would lie to mentioned cetain esults. Kaa [5] defined the sequence space l p( F as follows: l p( F = x ω : F x l p, ( p, whee F = ( f n is the double band matix defined by the sequence (f n of Fibonacci numbes as follows f n+ f n, = n, f f n = n f n+, = n, 0, 0 < n o > n fo all, n N. Also, Kaa et al. [6] chaacteized some classes of compact opeatos on the spaces l p( F and l ( F, whee p <. Futhemoe, the sequence spaces λ( F and µ( F, p ae studied by Başaı et al. [7], and Kaa and Demiiz [8], espectively, whee λ c 0, c and µ c 0, c, l. Recently, Candan [9] has intoduced the sequence spaces c 0( F (, s and c( F (, s afte then, Candan and Kaa [30] have examined the space l p( F (, s in which p and the matix F (, s = ( f n (, s constituted by using Fibonacci sequences and non-zeo eal numbes and s, i.e., s f n+ f n, = n, f n (, s = fn f n+, = n, 0, 0 < n o > n. Now, we define the sequence space ĉ f(,s and give an isomophism between the spaces ĉ f(,s and ĉ espectively. Late, we detemine the β dual of the space ĉ f(,s. We intoduce the sequence space ĉ f(,s as the set of all sequence whose F (, s tansfoms ae in the space ĉ, that is m ĉ f(,s y n+j = x = x l : α C = α unifomly in n, m + whee y = (y n is the F (, s-tansfom of a sequence x = (x n, i.e., y n = F x 0, n = 0, (, s n(x = fn f n+ x n + s f n+ f n x n, n. (. 53

5 It is clea that the space ĉ f(,s can be edefined as ĉ f(,s = ĉ F (,s. When α = 0, we will paticulaly denote the space ĉ f(,s by symbol ĉ f(,s 0. We should state hee that the matix F (, s can be educed to the matix F in case = and s =. Theefoe, the esults elated to the space ĉ f(,s ae moe geneal and moe compehensive than the coesponding consequences of the space ĉ f moe ecently defined by Demiiz et al. in [3]. Fo ou late use we ecall the following two lemmas hee. Lemma.. [3] An infinite matix A = (a n tansfoms each almost convegent sequence into an almost convegent sequence if and only if A = sup a n < ĉ a n = α n ĉ a n = α fo each N n q (a n+i, α + α a n+i, = 0 unifomly in n. q q + i=0 Lemma.. [33] A = (a n (ĉ : l if and only if A < holds. Theoem.3. The sequence spaces ĉ f(,s 0 and ĉ f(,s stictly include the spaces ĉ 0 and ĉ, espectively. Poof. Since the matix F (, s satisfies the conditions of Lemma., it belongs to the class (ĉ : ĉ. So, F (, sx ĉ 0 o F (, sx ĉ wheneve x ĉ 0 o x ĉ which shows that the inclusions ĉ 0 ĉ f(,s 0 and ĉ ĉ f(,s hold. ( Let us define the sequence ( n = ( s n f (n+ fo all n N. If λ ĉ 0, ĉ, then F (, s( n = n+ (, 0,, 0, = e 0 λ. Hence, ( n λ F (,s \ λ. This means that the inclusions ĉ 0 ĉ f(,s 0 and ĉ ĉ f(,s ae stict. Theoem.4. If s/ < /4 then the inclusions ĉ f(,s 0 l and ĉ f(,s l stictly hold. Poof. To veify the validity of the inclusion ĉ f(,s l, let us assume that s/ < /4 and tae an abitay x ĉ f(,s. Then, y = F (, sx ĉ l. The invese matix F (, s satisfies the condition of Lemma., indeed A = sup f n (, s f inf n f n+ sup sf n+ f n+ f inf n f n+ ( 4s <, it belongs to the class (ĉ : l by vitue of assumption. So, x = F (, sy l. Hence, the inclusion ĉ f(,s l holds. Let s/ /4. Let us conside the bounded sequence u = (u defined by u = (0,..., 0,,...,, 54

6 0,..., 0,,...,,..., whee the blocs 0 s ae inceasing by factos of 00 and the blocs of s ae inceasing by factos of 0 (cf. Mille and Ohan [34]. Then, the sequence F (, su is not almost convegent. This shows that u l \ ĉ f(,s which means that the inclusion ĉ f(,s l stictly holds. One can show by analogy that the inclusion ĉ f(,s 0 l stictly holds. So, we omit the detail. ĉ. Now, we may give following theoem concening the isomophism between the spaces ĉ f(,s and Theoem.5. The sequence space ĉ f(,s is linealy isomophic to the space ĉ, that is, ĉ f(,s = ĉ. Poof. Befoe we emba in poving the theoem, we need to be sue the tansfomation L exists between the spaces ĉ f(,s and ĉ. Fo this pupose, let us tae the tansfomation L mentioned above, with the help of the notation of (. fom the space ĉ f(,s to the space ĉ by x y = Lx = F (, sx. Since it is clea to show that both L is linea and injective, we omit the details. To pove that the tansfomation L is sujective, we fistly conside an abitay sequence y = (y ĉ and late obtain the following sequence x = (x using the invese matix F (, s as follows x = F (, sy = j f + y j (. f jf j+ fo all N. When we use the sequence (x deived only just come out, we easily get f x + s f + x = f f + f f + fo all N which esults in the fact that + s f + f j f + y j f jf j+ j f y j = y f jf j+ m f +j f ++j x +j + s f ++j m + f +j x +j m = y +j unifomly in m + = ĉ y. This biefly tells us that x = (x ĉ f(,s. Namely L is sujective. As a conclusion, L is a linea bijection, which means that the spaces ĉ f(,s and ĉ ae linealy isomophic. This mas the end of the poof. We now collect some elementay impotant facts elated to Schaude bases which will be used in the poof of the next coollay. Rema.6. [35, Rema.4] The matix domain µ A of a linea metic sequence space µ has a basis iff µ has a basis. Lemma.7. [, Coollay 3.3] The Banach space ĉ has no Schaude basis. Coollay.8. The space ĉ f(,s has no Schaude basis. 55

7 Poof. The poof can easily be obtained fom Rema.6 using the fact that not only the matix F (, s is a tiangle but also the space ĉ has not got a Schaude basis in view of Lemma.7. In this paagaph, let us fistly define S(λ, µ multiplie space of any sequence spaces λ and µ. If λ, µ w and z abitay sequence, we can wite and z λ = x = (x w : xz λ S(λ, µ = x λ x µ. We then go on define α, β and γ duals of an abitay space λ. If we choose µ = l, cs and bs, then we obtain the α, β and γ duals of the space λ, espectively as λ α = S(λ, l = a = (a w : ax = (a x l fo all x λ λ β = S(λ, cs = a = (a w : ax = (a x cs fo all x λ λ γ = S(λ, bs = a = (a w : ax = (a x bs fo all x λ. The following lemma is essential to compute β dual of the space ĉ f(,s. Lemma.9. [36] A = (a n (ĉ : c if and only if thee ae α, α C such that a n = α fo each N, (.3 n n n a n = α, (.4 (a n α = 0, (.5 sup whee (a n α = (a n α (a n,+ α + (n, N. a n <, (.6 Theoem.0. Define the sets d (, s, d (, s, d 3(, s, d 4(, s and d 5(, s by n d (, s = a = (α j f j+ ω : a j exists n f f +, d (, s = d 3(, s = d 4(, s = a = (α ω : n a = (α ω : n a = (a ω : j= n n =0 n =0 n =n+ j= i=+ j f j+ a j exists f f +, j f i+ a i f f + = 0, ϱ(, s, f, f +, f +, a = 0, 56

8 whee and Then, ϱ(, s, f, f +, f +, a = f ( a + + f f + sf + i=+ d 5(, s = a = (α ω : sup n n =0 j= ĉ f β = 5 i= d i(, s. j f j+ j f i+ a i f f + a j f f + <. Poof. Although the technical details ae somewhat involved, the idea of the poof is quite simple, howeve, the teatment of the details given hee. Conside an abitay sequence a = (a ω. In that case, we get the following equalities with the help of (. n a x = =0 n ( a =0 n n = =0 = E n(y, j= j f + j f j+ f jf j+ y j a j y (.7 f f + fo all n N, whee E = (e n is defined by n ( s j f j+ f e n = f + a j (0 n j= ; n, N. 0 ( > n Then, it is easily obseved fom the appoach we followed above i.e., fom (.7 that ax = (a x cs wheneve x = (x ĉ f(,s iff Ey c wheneve y = (y ĉ. Thus, we obtain fom Lemma.9 that ax = (a x cs wheneve x = (x ĉ f(,s iff a = (a 5 i=d i(, s. This gives that c f β = 5 i= d i(, s. In fact, this is exactly what we want to pove. Theoem.. The γ dual of the space ĉ f(,s is the set d 5(, s. Poof. The basic idea of the poof is the same as in the way of Theoem.0. The only diffeence is put the space of all bounded seies bs instead of the space of all convegent seies cs. 3 Some Matix Tansfomations Related to the Sequence Space ĉ f(,s In this section, the study will be focused on the chaacteize the matix tansfomations fom ĉ f(,s into any given sequence space X and fom a given sequence space X into ĉ f(,s. Fo the sae of simplicity, hee and in what follows, we will wite that ã n = j= j f j+ a nj, f f + 57

9 and ā n = fn f n+ a n + s fn+ f n a n,, a (n, = a (n,, m = m + n a j m a n+j, (m N fo all, n N. As ĉ f(,s = ĉ, it is obvious that the equivalence x ĉ f(,s if and only if y ĉ holds. Now, let us state the following two theoems to detemine matix classes on the space ĉ f(,s. Theoem 3.. Suppose that the enties of the infinite matices A = (a n and T = (t n ae connected with the elation t n = ã n (3. fo all, n N and X be any given sequence space. Then, A β ĉ f(,s fo all n N and T (ĉ : X. ( ĉ f(,s : X if and only if a n N Poof. Fist of all, we now that the spaces ĉ f(,s and ĉ ae linealy isomophic fom Theoem.5. In ode to pove the theoem, we will follow the same analysis employed befoe Başa and Kiişçi []. To do this, let us suppose that both X be a sequence space and condition (3. is valid fo the matices A = (a n and T = (t n. ( In poving necessity, we assume that A ĉ f(,s : X and tae any sequence y = (y ĉ. Unde these assumptions, it is clea that T F (, s exist and a n N 5 i=d i So, t n N l fo each n N. In that case, T y exist and we easily get t n y = a n x fo all n N when on account of condition (3.. Newly obtained fomula says us T y = Ax, which clealy indicates that T (ĉ : Y. The aguments we use in poving sufficiency ae a n N ĉ f(,s β fo all n N and T (ĉ : X and a taen sequence x = (x ĉ f(,s. By ou assumption, clealy Ax exists. Using a simple calculus, we can deive the following equality m m m a n x = j f j+ a nj y fo all n N. f f + =0 =0 j= By passing to it as m it is seen that T y = Ax and this illustates that A In fact, this is exactly what we want to pove. ( ĉ f(,s : X. Theoem 3.. Suppose that the enties of the infinite matices A = (a n and R = ( n ae connected with the elation n = ā n fo all, n N and X be given sequence space. Then, A (X : ĉ f(,s if and only if R (X : ĉ. 58

10 Poof. Let us tae any sequence x = (x X and we deal with the following equalities F (Ax n = fn (Ax f n + s fn+ (Ax n+ f n n = fn a n x + s fn+ f n+ f n = ( fn f n+ a n + s fn+ f n a n, a n, x x = (Rx n fo all n N, fom elementay calculus. By passing to genealized it in newly obtained fomula. It is not had to say that Ax ĉ f(,s if and only if Rx ĉ. This completes the poof. Now, we give the following conditions: sup a n <, (3. a n = 0 fo each fixed n N, (3.3 ĉ a n = α exists fo each fixed N, (3.4 a (n,, m α = 0 unifomly in n, (3.5 ĉ a n = α, (3.6 [a (n,, m α ] = 0 unifomly in n, (3.7 q [a (n + i, α ] = 0 unifomly in n, (3.8 q q + i=0 a (n, <, (3.9 sup a n = α fo each fixed n N, (3.0 n n a n = α, (3. n [a (n, α ] = 0. (3. Since it will help vey much in the implementation pocess, let us state the peviously obtained esults elated to almost convegence as a Lemma. Lemma 3.3. [] Let A = (a n be an infinite matix. Then, the following statements hold: (i A = (a n (ĉ : l if and only if (.6 holds. (ii A = (a n (l : ĉ if and only if (.6, (3.4 and (3.5 hold. (iii A = (a n (ĉ : ĉ if and only if (.6, (3.4, (3.6 and (3.7 hold. (iv A = (a n (c : ĉ if and only if (.6, (3.4 and (3.6 hold. (v A = (a n (bs : ĉ if and only if (3., (3.3, (3.4 and (3.8 hold. (vi A = (a n (cs : ĉ if and only if (3. and (3.4 hold. (vii A = (a n (ĉ : cs if and only if (3.9 (3. hold. 59

11 Late, using Theoems 3. and 3. with togethe with Lemmas.4 and 3.3 will esults in the following coollaies. Coollay 3.4. The following statements hold: (i A = (a n (ĉ f(,s : l if and only if a n N with ã n instead of a n. ( (ii A = (a n ĉ f(,s : c if and only if a n N hold with ã n instead of a n. ( (iii A = (a n ĉ f(,s : ĉ if and only if a n N ĉ f(,s β fo all n N and (.6 hold ĉ f(,s β fo all n N and (.3 (.6 ĉ f(,s β fo all n N and (.6, (3.4, (3.6 and (3.7 hold with ã n instead of a n. ( β (iv A = (a n ĉ f(,s : bs if and only if a n N ĉ f(,s fo all n N and (3.9 holds. ( β (v A = (a n ĉ f(,s : cs if and only if a n N ĉ f(,s fo all n N and (3.9 (3. hold with ã n instead of a n. Coollay 3.5. The following statements hold: (i A = (a n (l : ĉ f(,s if and only if (.6, (3.4 and (3.5 hold with ā n instead of a n. (ii A = (a n (c : ĉ f(,s if and only if (.6, (3.4 and (3.6 hold with ā n instead of a n. (iii A = (a n (ĉ : ĉ f(,s if and only if (.6, (3.4, (3.6 and (3.7 hold with ā n instead of a n. (iv A = (a n (bs : ĉ f(,s if and only if (3., (3.3, (3.4 and (3.8 hold with ā n instead of a n. (v A = (a n (cs : ĉ f(,s if and only if (3. and (3.4 hold with ā n instead of a n. 4 Coe Theoems Let x = (x be a sequence in C, the set of all complex numbes, and R be the least convex closed egion of complex plane containing x, x +, x +,.... The Knopp Coe (o K coe of x is defined by the intesection of all R (=,,..., (see [37], pp.37. In [38], it is shown that K coe(x = z C B x(z fo any bounded sequence x, whee B x(z = w C : w z sup x z. Let E be a subset of N. The natual density δ of E is defined by δ(e = n : E n n whee n : E denotes the numbe of elements of E not exceeding n. A sequence x = (x is said to be statistically convegent to a numbe l, if δ( : x l ε = 0 fo evey ε. In this case we wite st x = l, [39]. By st we denote the space of all statistically convegent sequences. In [40], the notion of the statistical coe (o st coe of a complex valued sequence has been intoduced by Fidy and Ohan and it is shown fo a statistically bounded sequence x that st coe(x = z C C x(z, 60

12 whee C x(z = w C : w z st sup x z. The coe theoems have been studied by many authos. Fo instance see [4, 4, 43, 44, 45, 46] and the othes. Using the convegence domain of the matix F (, s = (f n (, s, the new sequence spaces c 0( F (, s and c( F (, s have been constucted and thei some popeties have been investigated by Candan [9]. In this section we will conside the sequences with complex enties and by l (C denote the space of all bounded complex valued sequences. Following Knopp, a coe theoem is chaacteized a class of matices fo which the coe of the tansfomed sequence is included by the coe of the oiginal sequence. Fo example Knopp Coe Theoem [37, p. 38] states that K coe(ax K coe(x fo all eal valued sequences x wheneve A is a positive matix in the class (c : c eg. Hee, we will define B F (,s -coe of a complex valued sequence and chaacteize the class of matices to yield B F (,s coe(ax K coe(x, K coe(ax B F (,s coe(x, B F (,s coe(ax B F (,s coe(x and B F (,s coe(ax st coe(x fo all x l. Now, let us wite t mn(x = m ( fn+j x n+j + s fn++j x n +j. m + f n++j f n+j Then, we can define B F (,s coe of a complex sequence as follows: Definition 4.. Let H n be the least closed convex hull containing t mn(x, t m+,n(x, t m+,n(x,... Then, B F (,s coe of x is the intesection of all H n, i.e., B F (,s coe(x = H n. Note that, actually, we define B F (,s coe of x by the K coe of the sequence (t mn(x. Hence, we can constuct the following theoem which is an analogue of K coe, [38]. Theoem 4.. Fo any z C, let G x(z = ω C : ω z sup sup t mn(x z. Then, fo any x l, n= B F (,s coe(x = z C G x(z. Now, we pove some lemmas which will be useful to the main esults of this section. To do these, we need to chaacteize the classes (c : ĉ f(,s eg and (st l : ĉ f(,s eg. Fo bevity, in what follows we wite ã(m, n, in place of fo all m, n, N. m ( fn+j a n+j, + s fn++j m + f n++j Lemma 4.3. A (l : ĉ f(,s if and only if f n+j a n +j, A = sup m,n ã(m, n, <, (4. ã(m, n, = α fo each, (4. 6

13 ã(m, n, α = 0, unifomly in n. (4.3 Lemma 4.4. A (c : ĉ f(,s eg if and only if (4. and (4. of the Lemma 4.3 hold with α = 0 fo all N and ã(m, n, = unifomly in n. (4.5 Lemma 4.5. A (st l : ĉ f(,s eg if and only if A (c : ĉ f(,s eg and ã(m, n, = 0 unifomly in n (4.6 E fo evey E N with natual density zeo. Poof. Let A (st l : ĉ f(,s eg. Then A (c : ĉ f(,s eg immediately follows fom the fact that x, E c st l. Now, define a sequence t = (t fo x l as t = whee E any 0, / E. subset of N with δ(e = 0. Then, st t n = 0 and t st 0, so we have At ĉ f(,s 0. On the othe hand, since (At n = E a an, E nt, the matix B = (b n defined by b n = fo all n, 0, / E. must belong to the class (l : ĉ f(,s 0. Hence, the necessity of (4.6 follows fom Lemma 4.3. Convesely, let x st l with st x = l. Then, the set E defined by E = : x l ε has density zeo and x l ε if / E. Now, we can wite Since ã(m, n, x = ã(m, n, (x l + l ã(m, n,. (4.7 ã(m, n, (x l x ã(m, n, + ε A, E letting m in (4.7 and using (4.5 with (4.6, we have ã(m, n, x = l. This implies that A (st l : ĉ f(,s eg and the poof is completed. Now, we may give some inclusion theoems. Fistly, we need a lemma. Lemma 4.6. [47, Coollay ] Let A = a m (n defined by a m (n = ã(m, n, fo all m, n, N be a matix satisfying A = a m (n < and m sup a m (n = 0. Then, thee exists an y l with y such that sup sup ã(m, n, y = sup sup ã(m, n,. Theoem 4.7. B F (,s coe(ax K coe(x fo all x l if and only if A (c : ĉ f(,s eg and sup sup ã(m, n, =. (4.8 6

14 Poof. Let the B F (,s coe(ax K coe(x and tae x c with x = l. Then, since K coe(x l, B F (,s coe(ax l. So, ĉ f(,s Ax = l which means that A (c : ĉ f(,s eg. Since A (c : ĉ f(,s eg, the matix A = ã(m, n, is satisfy the conditions of Lemma 4.6. So, thee exists a y l with y such that ω C : ω sup sup ã(m, n, y = ω C : ω sup sup ã(m, n,. On the othe hand, since K coe(y A (0, by the hypothesis ω C : ω sup sup ã(m, n, A (0 = ω C : ω which implies (4.8. Convesely, let ω B F (,s coe(ax. Then, fo any given z C, we can wite ω z sup = sup sup + sup = sup sup t mn(ax z sup z ã(m, n, x sup ã(m, n, (z x sup z ã(m, n, sup ã(m, n, (z x. Now, let L(x = sup x z. Then, fo any ε > 0, x z L(x + ε wheneve 0. Hence, one can wite that ã(m, n, (z x = ã(m, n, (z x + ã(m, n, (z x < 0 0 sup z x ã(m, n, < 0 + [L(x + ε] ã(m, n, (4.0 0 sup z x ã(m, n, < 0 + [L(x + ε] ã(m, n,. 0 Theefoe, applying sup sup unde the light of the hypothesis and combining (4.9 with (4.0, we have ω z sup sup ã(m, n, (z x L(x + ε (4.9 63

15 which means that ω K coe(x. This completes the poof. The poof of the following two theoems ae entiely analogous to the Theoem 4.7. So, we omit the detail. Theoem 4.8. K coe(ax B F (,s coe(x fo all x l if and only if A (ĉ f(,s : c eg and (4.8 holds. Theoem 4.9. B F (,s coe(ax B F (,s coe(x fo all x l if and only if A (ĉ f(,s : ĉ f(,s eg and (4.8 holds. Theoem 4.0. B F (,s coe(ax st coe(x fo all x l if and only if A (st l : ĉ f(,s eg and (4.8 holds. Poof. Fistly, we assume that B F (,s coe(ax st coe(x fo all x l. By taing x st l, one can see that A (st l : ĉ f(,s eg. Also, since st coe(x K coe(x [4] fo any x, the necessity of the condition (4.8 follows fom Theoem 4.7. Convesely, let A (st l : ĉ f(,s eg and (4.8 holds and tae ω B F (,s coe(ax. Then we can wite again equality (4.9. Now, let β = st sup z x. If we wite E = : x z β +ε, then δ(e = 0 and z x β + ε wheneve / E. Hence we have ã(m, n, (z x = ã(m, n, (z x + ã(m, n, (z x E / E z x ã(m, n, + ã(m, n, z x E / E z x ã(m, n, + [β + ε] ã(m, n,. E / E Thus, applying the opeato sup sup and using the hypothesis (4.8 with (4.6, we obtain that sup sup ã(m, n, (z x β + ε. (4. Thus, (4.9 and (4. implies that ω z β + ε. Since ε is abitay, this means ω st coe(x, which completes the poof. Conclusion In the cuent study, the sequence space ĉ f(,s deived by genealized diffeence Fibonacci matix in which, s R \ 0 has been intoduced and compaed with some well-nown spaces defined peviously. Then, it has been found out that the space has not a Schaude basis. In conclusion, the Fibonacci coe of a complex-valued sequence has been pesented and inclusion theoems with espect to Fibonacci coe type ae shown. Acnowledgment The authos would lie to expess thei sincee gatitude to the efeees fo thei valuable comments and suggestions about the pape, which led to a numbe of impovements in this pape. 64

16 Competing Inteests The authos declae that no competing inteests exist. Refeences [] Loentz GG, A contibution to the theoy of divegent sequences, Acta Math. 948;80: [] Banach S. Théoie des opéations linéaies. Chelsea Publishing company. New Yo; 978. [3] Boos J. Classical and moden methods in summability, Oxfod Univesity Pess Inc, New Yo; 000. [4] Candan M. Domain of the double sequential band matix in the classical sequence spaces, J. Inequal. Appl. 0;8. [5] Candan M. A new sequence space isomophic to the space l(p and compact opeatos, J. Math. Comput. Sci. 04; 4(: [6] Candan M. Domain of the double sequential band matix in the spaces of convegent and null sequences, Adv. Diffeence Equ. 04;04:63. [7] Candan M, Güneş A. Paanomed sequence space of non-absolute type founded using genealized diffeence matix, Poc. Nat. Acad. Sci. India Sect. A, in pess. [8] Candan M. Some new sequence spaces deived fom the spaces of bounded, convegent and null sequences, Int. J. Mod. Math. Sci. 04;(: [9] Şengönül M, Kayaduman K. On the Riesz almost convegent sequence space, Abst. Appl. Anal. 0;0:8. Aticle ID DOI:0.55/0/ [0] Kayaduman K, Şengönül M. The spaces of Cesào almost convegent sequences and coe theoems, Acta Math. Sci. 0;3B(6: [] Başa F, Kiişçi M. Almost convegence and genealized diffeence matix, Comput. Math. Appl. 0;6:60-6. [] Candan M. Almost convegence and double sequential band matix, Acta. Math. Sci. 04;34B(: [3] Kaa EE, Elmaağaç K. On the u-diffeence almost sequence space and matix tansfomations, Inte. J. Moden Math. Sci. in pess. [4] Musaleen, Invaiant means and some matix tansfomations, Indian J. Pue Appl. Math. 994;5(3: [5] Musaleen M, Savaş E, Aiyub M, Mohuiddine SA. Matix tansfomations between the spaces of Cesào sequences and invaiant means, Int. J. Math. Math. Sci. 006;006(8.Aticle ID DOI:0.55/IJMMS/006/7439. [6] Başa F, Solaİ. Almost-coecive matix tansfomations, Rend. Mat. Appl. 99;(7 ( [7] Başa F, Çola R. Almost-consevative matix tansfomations, Tuish J. Math. 989;3(3:9-00. [8] Başa F. f consevative matix sequences, Tamang J. Math. 99;(:05-. [9] King JP, Almost summable sequences, Poc. Ame. Math. Soc. 966;7:9-5. [0] Qamauddin, Mohuiddine SA. Almost convegence and some matix tansfomations, Filomat. 007;(:

17 [] Nanda S. Matix tansfomations and almost boundedness, Glasni Mat. 979;34(4: [] Gupai SA. Some new sequence spaces and almost convegence, Filomat. 008;(: [3] Das G, Kuttne B, Nanda S. On absolute almost convegence, J. Math. Anal. Appl. 99;6(: [4] Koshy T. Fibonacci and Lucas Numbes with Applications. Wiley; 00. [5] Kaa EE. Some topological and geometical popeties of new Banach sequence spaces, J. Inequal. Appl.03;03(38:5. [6] Kaa EE, Başaı M, Musaleen M. Compact opeatos on the Fibonacci diffeence sequence spaces l p( F and l ( F, st Intenational Euasian Conf. on Math. Sci. and Appl, Pishtine- Kosovo, Septembe 3-7, 0. [7] Başaı M, Başa F, Kaa EE. On the spaces of Fibonacci diffeence null and convegent sequences, axiv: [8] Kaa EE, Demiiz S. Some new paanomed Fibonacci diffeence sequence spaces, st Intenational Euasian Conf. on Math. Sci. and Appl, Saajevo-Bosnia and Hezegouina, August 6-9, 03. [9] Candan M. A new appoach on the spaces of genealized Fibonacci diffeence null and convegent sequences, unde communication. [30] Candan M, Kaa EE. A study on topological and geometical chaacteistics of new Banach sequence spaces, unde communication. [3] Demiiz S, Kaa EE, Başaı M. On the Fibonacci almost sequence space and Fibonacci coe, Kyungpoo Math. J. in pess. [3] Duan JP. Infinite matices and almost convegence, Math. Z. 97;8: [33] Başa F. Summability theoy and its applications, Bentham Science Publishes, e-boos, Monogaphs, İstanbul; 0. [34] Mille HI, Ohan C, On almost convegent and statistically convegent subsequences, Acta Math. Hung. 00;93:35-5. [35] Jaah AM, Malowsy E. BK spaces, bases and linea opeatos, Ren. Cic. Mat. Palemo II. 990;5:77-9. [36] Sıddıqi JA. Infinite matices summing evey almost peiodic sequences, Pacific. J. Math. 97;39 (:35-5. [37] Cooe RG. Infinite matices and sequence spaces, Mcmillan, New Yo; 950. [38] Shchebaov AA. Kenels of sequences of complex numbes and thei egula tansfomations, Math. Notes. 977;: [39] Steinhaus H. Quality contol by sampling, Collog. Math. 95;: [40] Fidy JA, Ohan C. Statistical coe theoems, J. Math. Anal. Appl. 997;08: [4] Allen HS. T -tansfomations which leave the coe of evey bounded sequence invaiant, J. London Math. Soc. 944;9:4-46. [4] Conno J, Fidy JA, Ohan C. Coe equality esults fo sequences, J. Math. Anal. Appl. 006;3: [43] Çaan C, Çoşun H. Some new inequalities elated to the invaiant means and unifomly bounded function sequences, Appl. Math. Lett. 007;0(6: [44] Çoşun H, Çaan C. A class of statistical and σ-consevative matices, Czechoslova Math. J. 005;55(3:

18 [45] Çoşun H, Çaan C, Musaleen. On the statistical and σ-coes, Studia Math. 003;54(:9-35. [46] Kayaduman K, Fuan H. Infinite matices and σ (A -coe, Demonstatio Math. 006;39: [47] Simons S. Banach its, infinite matices and sublinea functionals, J. Math. Anal. Appl. 969;6: c 05 Candan & Kayaduman; This is an Open Access aticle distibuted unde the tems of the Ceative Commons Attibution License which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wo is popely cited. Pee-eview histoy: The pee eview histoy fo this pape can be accessed hee (Please copy paste the total lin in you bowse addess ba 67