ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS

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1 STUDIA UNIV BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Numbe 4, Decembe 2003 ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS VATAN KARAKAYA AND NECIP SIMSEK Abstact The pupose of this pape is to intoduce and study some sequence spaces which ae defined by combining the concepts of lacunay convegence, invaiant mean and the sequence of modulus functions We also examine some topological popeties of these spaces Intoduction Let l and cdenote the Banach spaces of eal bounded and convegent sequences x = (x ) nomed by x = sup x, espectively Let σ be a mapping of the set of positive integes into itself A continuous linea functional φ on l, the space of eal bounded sequences, is said to be an invaiant mean o σ-mean if and only if i φ (x) 0 when the sequence x = (x n ) has x n 0 fo all n, ii φ (e) 0, whee e= (,,,) and, iii φ ( ) x σ(n) = φ (x) fo all x l Let V σ denote the set of bounded sequence all of whose invaiant means ae equal In paticula, if σ is the tanslation n n +, then a σ-mean educe to a Banach limit (see, Banach []) and set V σ educe to ĉ, the spaces of all almost convegent sequences (see, Loentz [7] ) If x = (x n ), wite T x = T x n = ( ) x σ(n) It can be shown (Schaefe [6]) that V σ = x l : lim t n (x) = l, unifomly in n, l = σ lim x, whee t n (x) = x n + x σ (n) + + x σ (n) + Hee σ (n) denote the th iteate of the mapping σ at n The mapping σ is one to one and such that σ (n) n fo all positive integes n and Thus a σ- mean φ extends the limit functional on c, the spaces of convegent sequence, in the sense that φ (x) = lim x fo all x c (see, Musaleen []) We call V σ as the space of σ-convegent sequences A sequence x = (x ) is said to be stongly σ-convegent (Musallen [2]) if thee exists a numbe l such that lim j= xσ j (n) l = 0 unifomly in n Received by the editos: Mathematics Subject Classification Pimay 43A0, 43A20; Seconday 46H20 Key wods and phases Modulus functions 43

2 VATAN KARAKAYA AND NECIP SIMSEK We denote [V σ ] as the set of all stongly σ-convegent sequences In case σ (n) = n +, [V σ ] educe to [ĉ], the space of all stong almost convegent sequence (Maddox [8]) Also the stongly almost convegent sequences was studied by Feedman et all [4], independently By a lacunay θ = ( ); = 0,, 2, whee 0 = 0, we shall mean an inceasing sequence of non-negative integes with as The intevals detemined by θ will be denoted by I = (, ] and = The atio will be denoted by q The space of lacunay stongly convegent sequence N θ was defined by Feedman et al [4] as: N θ = x = (x ) : lim x l = 0, fo some l Recently, the concept of lacunay stong σ-convegence was intoduced by Savas [4] which is a genealization of the idea of lacunay stong almost convegence due to Das and Misha [2] A modulus function f is a function fom [0, ) to [0, ) such that i f(x) = 0 if and only if x = 0 ii f(x + y) f(x) + f(y), fo all x, y > 0 iii f is inceasing, iv f is continuous fom the ight at zeo Since f (x) f (y) f ( x y ), it follows fom conditions (ii) and (iv) that f is continuous eveywhee on [0, ) A modulus function may be bounded o unbounded Fo example, f (t) = t t+ is bounded but f (t) = t p (0 < p ) is unbounded Rucle [3] and Maddox [9], Savas [5] and othe authos used modulus function to constuct new sequence spaces Recently, Kol ( [6], [7]) gave an extension of X (f) by consideing a sequence of moduli F = (f ) ie, X (f ) = x = (x ) : (f ( x )) X In this pape by combining lacunay sequence, invaiant mean and a sequence of modulus functions, we define the following new sequence spaces: [ w 0 σ, F ] x = : lim f θ ( t n (x) ) = 0, unifomly in n = x : lim f ( t n (x l) ) = 0, unifomly in n, fo some l [wσ, F ] θ = x : sup I f ( t n (x) ) <,n 44 [w σ, F ] = x : lim m m f ( t n (x l) ) = 0, unifomly in n, fo some l = Some sequence spaces ae obtained by specializing F, θ, σ Fo example, if

3 ON LACUNARY INVARIANT SEQUENCE SPACES θ = (2 ), σ (n) = n + and f (x) = x fo all, then = ŵ (see, Das and Sahoo [3]) If σ (n) = n + and f (x) = f fo all, then = [ŵ (f)] θ and [w σ, F ] = [ŵ (f)] (see, Musaleen and Chishti [2]) When σ (n) = n +, the spaces [ wσ, 0 F ] θ, [w σ, F ] θ and [wσ, F ] θ educe to the spaces [ŵ 0, F ] θ, [ŵ, F ] θ and [ŵ, F ] θ espectively, whee [ŵ, F ] θ = x = (x ) : lim f ( d n (x l) ) = 0, and unifomly in n, fo some l d n (x) = x n + x n+ + + x n+ + If θ = (2 ), then [ w 0 σ, F ] θ = [ w 0 σ, F ], = [w σ, F ] and [w σ, F ] θ = [w σ, F ] 2 Main Results We have Theoem 2 Fo any a sequence of modulus functions F = (f ), [ w 0 σ, F ] θ, [w σ, F ] θ, [w σ, F ] θ and [w σ, F ] ae linea spaces ove the set of complex numbes Poof We shall pove the esult only fo [ wσ, 0 F ] The othes can be teated similaly θ Let x, y [ wσ, 0 F ] and α, β C Then thee exist integes H θ α and K β such that α < H α and β < K β We have h f ( t n (αx βy) ) H α f ( t n (x) ) This implies αx + βy [ w 0 σ, F ] θ We will now give a lemma +K β f ( t n (y) ) Lemma 22 Let f be a modulus and let 0 < δ < Then fo each t n (x) > δ fo all and n we have f ( t n (x) ) 2f () δ t n (x) Poof f ( t n (x) ) f ( + ( f () + t n (x) δ [ ]) ([ ]) tn (x) tn (x) f () + f δ δ ) 2f () δ t n (x) Theoem 23 Fo a sequence of modulus functions F = (f ) and any lacunay sequence θ = ( ), [w σ, F ] θ 45

4 VATAN KARAKAYA AND NECIP SIMSEK Poof Let F = (f ) be a sequence of modulus functions and x sup f () = M We can wite f ( t n (x) ) f ( t n (x l) ) + f ( l ) f ( t n (x l) ) + T l M whee T l is intege numbe such that l < T l Hence x [wσ, F ] θ Now fo any lacunay sequence θ = ( ), we give connection between and [w σ, F ] Theoem 24 Let θ = ( ) be a lacunay sequence with lim inf q > Then fo sequence of modulus functions F = (f ), [w σ, F ] [w, σf ] θ Poof Suppose that lim inf q >, then thee exists δ > 0 such that q > + δ fo all Then fo x [w σ, F ], we wite = f ( t n (x l) ) = f ( t n (x l) ) + = f ( t n (x l) ) = f ( t n (x l) ) δ f ( t n (x l) ) + δ By taing limit as unifomly in, hence we obtain x This completes the poof Theoem 25 Let θ = ( ) be a lacunay sequence with lim sup q < Then fo any sequence of modulus functions F = (f ), [w σ, F ] Poof If lim sup q <, thee exists H > 0 such that q < H fo all Let x and ε > 0 Thee exists R > 0 such that fo evey j R and all n A j = h j I j f ( t n (x l) ) < ε We can also find M > 0 such that A j < K fo all j =, 2, Now let m be any intege with < m, whee > R We have 46 m m = f ( t n (x l) ) = j= = I j f ( t n (x l) ) f ( t n (x l) ) Put

5 = ON LACUNARY INVARIANT SEQUENCE SPACES R f ( t n (x l) ) + j= I j ( ) sup A j R + ε j R j=r+ f ( t n (x l) ) I j j=r+ M R + ε (h R+ + h R ) M R + εh Since as, it follows that m m f ( t n (x l) ) 0 = unifomly in n and consequently x [w σ, F ] Hence the poof completes Theoem 26 Let θ = ( ) be a lacunay sequence < lim inf q lim sup q < Then fo any sequence of modulus functions F = (f ), = [w σ, F ] Poof Theoem 26 follows the theoems 25 and 24 Refeences [] S Banach, Theoie des opeations lineaies, Waszawa, 932 [2] G Das, S Misha, Lacunay distibution of sequences, Indian J Pue Apll Math, 20()(989), [3] G Das, A K Sahoo, On some sequence spaces, J Math Anal Appl, 64(992), [4] A R Feedman, J J Sembe, M Raphael, Some Cesao-type summability, Poc London Math Soc, 37(3)(978), [5] E Kol, On stong boundedness and summability witespect to a sequence of moduli, Acta et Comment Univ Tatu, 960(993), 4-50 [6] E Kol, Inclusion theoems fo some sequence spaces defined by a sequence of moduli, Acta et Comment Univ Tatu, 970(994), [7] G G Loentz, A contibution to the theoy of divegent seies, Acts Math 80(948), [8] I J Maddox, On stong almost convegence, Math Poc Camb Phil Soc 85(979), [9], Sequence spaces defined by a modulus, Math Poc Camb Phil Soc, 00(986), 6-66 [0] Musaleen, On some new invaiant matix methods of summability, Quat J Math Oxfod, 34(983), [] -, Matix tansfomation between some new sequence spaces, Houston J Math, 9(983), [2] Musaleen, T A Chishti, Some spaces of lacunay sequences defined by the modulus, J Analysis, 4(996), [3] W H Rucle, FK spaces in which the sequence of coodinate vectos is bounded, Can J Math 25(973), [4] E Savas, On lacunay stong σ-convegence, Indian J Pue Appl Math, 2(4) (990), [5] E Savas, On some genealized sequence spaces defined by a modulus, Indian J Pue and Apl Math, 38()(999), h j 47

6 VATAN KARAKAYA AND NECIP SIMSEK [6] P Schaefe, Infinite matices and invaiant means, Poc Ame Math Soc, 36(972), 04-0 Yüzüncü Yil Univesity, Faculty of Ats and Sciences, Depatment of Mathematics, VAN, TURKEY addess: vaya@yahoocom, nsimse@yyuedut 48

Asymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function

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