On Some I-Convergent Double Sequence Spaces Defined by a Modulus Function

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1 Engineering, 03, 5, Published Online Ma 03 ( On Some -Convergent Double Sequence Spaces Deined b a Modulus Function Vakeel A Khan, Nazneen Khan Department o Mathematics, Aligarh Muslim Universit, Aligarh, ndia vakhanmaths@gmailcom, nazneen4maths@gmailcom Received Februar 5, 03; revised March 7, 03; accepted March 6, 03 Copright 03 Vakeel A Khan, Nazneen Khan This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited ABSTRACT n 000, Kostrko, Salat, and Wilcznski introduced and studied the concept o -convergence o sequences in metric spaces where is an ideal The concept o -convergence has a wide application in the ield o Number Theor, trigonometric series, summabilit theor, probabilit theor, optimization and approximation theor n this article we in- troduce the double sequence spaces c0, c and l or a modulus unction and stud some o the properties o these spaces Kewords: deal; Filter; Modulus Function; Lipschitz Function; -Convergence Field; -Convergent; Monotone and Solid Double Sequence Spaces ntroduction The notion o -Convergence is a generalization o the concept statistical convergence which was irst introduced b H Fast [] and later on studied b J A Frid [,3] rom the sequence space point o view and linked it with the summabilit theor At the initial stage -Convergence was studied b Kostrko, Salat and Wileznski [4] Further it was studied b Salat, Tripath, Ziman [5] and Demirci [6] Throughout a double sequence is denoted b x x Also a double sequence is a double ininite arra o elements x kl or all kl, The inital works on double sequences is ound in Bromwich [7], Basarir and Solancan [8] and man others Deinitions and Preliminaries Throughout the article N, R, and denotes the set o natural, real, complex numbers and the class o all sequences respectivel Let X be a non empt set A set X X ( denoting the power set o X) is said to be an ideal i is additive ie A, B A B and hereditar ie A, B A B A non-empt amil o sets X is said to be ilter on X i and onl i, or A, B we have A B and or each A and A B implies B An deal X is called non-trivial i X A non-trivial ideal X is called admissible i x : x X A non-trivial ideal is maximal i there cannot exist an non-trivial ideal J containing as a subset For each ideal, there is a ilter corresponding to c c ie K N : K, where K N K The idea o modulus was structured in 953 b Nakano (See [9]) A unction :0, 0, is called a modulus i () t 0 i and onl i t 0, () tu t u or all tu, 0, (3) is nondecreasing, and (4) is continuous rom the right at zero Ruckle [0] used the idea o a modulus unction to construct the sequence space X xxk: xk k This space is an FK space, and Ruckle[0] proved that the intersection o all such X spaces is, the space o all inite sequences The space X() is closel related to the space l which is an X() space with x x or all real x 0 Thus Ruckle [] proved that, or an modulus Copright 03 SciRes

2 36 V A KHAN, N KHAN where X l and X l X k : kxk k The space X is a Banach space with respect to the norm x x (See [0]) k k Spaces o the tpe X are a special case o the spaces structured b B Gramsch in [] From the point o view o local convexit, spaces o the tpe X are quite pathological Thereore smmetric sequence spaces, which are locall convex have been requentl studied b D J H Garling [3,4], G Kothe [5] and W H Ruckle [0,6] Deinition A sequence space E is said to be solid or normal i E implies E or all sequence o scalars with or all i, j N (see [7]) Deinition Let K ni, kj: i, j N; n n n3 and k k k3 N N and E be a double sequence space A E is a sequence space E x : x E K K -step space o Deinition 3 A cannonical preimage o a sequence x E is a sequence b E deined as ollows ni, kj b nk, nk, ank,,or n, k K, (see [8]) 0, otherwise Deinition 4 A sequence space E is said to be monotone i it contains the cannonical preimages o all its stepspaces (see [9]) Deinition 5 A sequence space E is said to be convergence ree i E, whenever E and 0 implies 0 Deinition 6 A sequence space E is said to be a sequence algebra i E whenever E E Deinition 7 A sequence space E is said to be smmetric i x E x E where i i j whenever and j is a permutation on N Deinition 8 A sequence is said to be -convergent to a number L i or ever 0 i, j N N : L n this case we write -lim L The space c o all -convergent sequences to L is given b c x : i, jn N : L, or some L C Deinition 9 A sequence x is said to be -null i L 0 n this case we write -lim 0 Deinition 0 A sequence x is said to be -cauch i or ever 0 there exists a number m m and n n such that i, j NN : xmn Deinition A sequence x is said to be -bounded i there exists M 0 such that i, j N N : M Deinition A modulus unction is said to satis condition i or all values o u there exists a constant K 0 such that Lu KL u or all values o L Deinition 3 Take or the class o all inite subsets o N Then is a non-trivial admissible ideal and convergence coincides with the usual convergence with respect to the metric in X (see [4]) Deinition 4 For and A N with A 0 respectivel is a non-trivial admissible ideal, -convergence is said to be logarithmic statistical convergence (see [4]) Deinition 5 A map deined on a domain D X ie : D X R is said to satis Lipschitz condition i x K x where K is known as the Lipschitz constant The class o K-Lipschitz unctions deined on D is denoted b DK, (see [0]) Deinition 6 A convergence ield o -convergence is a set F x xk l :thereexists limx R F is a closed linear sub- The convergence ield space o l with respect to the supremum norm, F l c (See [5]) Deine a unction : F R such that x lim x, or all x F, then the unction : F R is a Lipschitz unction (see [0]) (c [8,0-30]) Throughout the article l, c, c0, m and m 0 represent the bounded, -convergent, -null, bounded -convergent and bounded -null sequence spaces respectivel n this article we introduce the ollowing classes o sequence spaces Copright 03 SciRes

3 V A KHAN, N KHAN 37 0 and c x : lim x Lorsome L c x : lim x 0 l x x :sup We also denote b m c l m c l 0 0 The ollowing Lemmas will be used or establishing some results o this article Lemma () Let E be a sequence space E is solid then E is monotone Lemma () Let K and M N M, then M N Lemma (3) M N 3 Main Results N and M N M, then Theorem 3 For an modulus unction, the classes o sequences c, c0, m and m0 are linear spaces Proo: We shall prove the result or the space c The proo or the other spaces will ollow similarl Let, c and let, be scalars Then lim x L 0, or some L c; lim L 0, or some L c; That is or a given 0, we have A i, jn N : x L, A i, j NN : L Since is a modulus unction, we have x L L x L L x L L Now, b () and (), i, jn : x L L A A () () Thereore x c Hence c is a linear space Theorem 3 A sequence x m is -convergent i and onl i or ever 0 there exists, J N such that i, j N N : x x m (3), J Proo: Suppose that L lim x Then B i, j NN : L m For all 0 Fix an, J B Then we have x J x J L L which holds or all i, j B Hence i, jnn : x J m Conversel, suppose that i, jnn : x J m That is i, jnn : x J m or all 0 Then the set C i, j N N : x xj, x J m or all 0 Let N xj, x J we ix an 0 then we have C m as well as C m Hence C C m that is that is This implies that N N i, j NN : N m diam N diam N where the diam o N denotes the length o interval N n this wa, b induction we get the sequence o closed intervals N 0 with the propert that diam diam i j or i, j,3,4, and i, j N N : x m or i, j,,3,4, Copright 03 SciRes

4 38 V A KHAN, N KHAN Then there ex ists a where i, j N such that lim x So that lim x, that is L lim x Theorem 33 Let and g be modulus unctions that satis the -condition X is an o the spaces c, c0, m and m 0 etc, then the ollowing assertions hold (i) X g X g, X X g X g (ii) Proo: (i) Let c0 g Then 0 lim g (4) Let 0 and choose with 0 such that t or 0 t Write g x and consider lim lim lim We have lim lim (5) For, we have Since is non-decreasing,it ollows that Since satisies the -condition, we have K K K Hence lim max, K lim (6) From (4), (5) and (6), we have x c g 0 Thus c0 g c0 g The other cases can be proved similarl (ii) Let c0 c0 g Then lim x 0 and lim g 0 lim Thereore g lim g g lim lim 0 g lim 0 which implies x X g, that is X X g X g Corollar 34 X X or X c, c0, m and m 0 Proo: The result can be easil proved using x x or x X Theorem 35 The spaces c0 and m0 are solid and monotone Proo: We shall prove the result or c0 Let x c Then 0 lim x 0 (7) Let be a sequence o scalars with or all i, j N Then we have lim lim 0 lim x x x lim x 0 or all i, jn which implies that c0 Thereore the space c0 is solid The space c is monotone ollows rom Lemma () For 0 the result can be proved similarl Theorem 36 The spaces c and m m 0 are neither solid nor monotone in general Proo: Here we give a counter example Let and x x or all x 0, Consider the K-step space X K o X deined as ollows, Let x X X K be such that and let x,i i, jis even, 0, otherwise Consider the sequence x deined b or all i, j N Then c but its K-stepspace preimage does not belong to c is not mo- c Thus notone Hence c is not solid Theorem 37 The spaces c and c0 are sequence algebras Proo: We prove that c0 is a sequence algebra x, c Then and Let Then we have 0 lim x 0 lim 0 lim x 0 x c0 Thus is a sequence algebra For the space c, the result can be proved similarl Theorem 38 The spaces c and c0 are not convergence ree in general Proo: Here we give a counter example Copright 03 SciRes

5 V A KHAN, N KHAN 39 3 Let and x x 0, Consider the sequence x and deined b and i j or all i, jn i j Then x c and c or all x and, but c c 0 Hence the spaces c and c0 are not convergence ree Theorem 39 is not maximal and, then the spaces c and c0 are not smmetric Proo: Let A be ininite and x x or all x 0,, or i, j A, 0, otherwise Then b Lemma (3) we have c0 c Let K N be such that K and N K Let : K A and : N K N A be bections, then the map π : N N deined b,or i, j K, π,otherwise is a permutation on N, but x c and x c 0 Hence c 0 and c are not smmetric Theore m 30 Let be a modulus unction Then c0 c l and the inclusions are proper Proo: The inclusion c0 c is obvious Let x c Then there exists L C such that 0 lim x L 0 We have L L Taking the supremum over iand j on bot h sides we get l Next we show that the inclusion is pr oper c c (i) 0 Let x c then lim x L or so me L 0 C, which implies x c0 Hence the inclusion is proper c l Let x x l then (ii) lim x lim lim x LL lim x L L lim x L lim x L 0 Thereore x c, and hence the inclusion is proper Theorem 3 The unction : m R is the Lipschitz unction, where m c l, and hence uniorml continuous x, m, x Then the sets Proo: Let x j N N A i, : x x x, A i, j NN : x Thus the sets, B i, j NN : x x x m x,, j : x m B B m, so that B B i N N Hence also x Now taking i, j in B, x x 3 x Thus is a Lipschitz unction For m 0 the result can be proved similarl Theorem 3 x, m, then x m and x x Proo: For 0 B i, j NN : x x m, x B (, i j) NN : m Now, x x x x M and M As m l, there exists an M R such that Using Equation (8) we get x x MM M For all i, jbx B m Hence x m and x x For m the result can be proved si 0 milarl 4 Acknowledgements The authors would like to record their gratitude to the reviewer or his careul reading and making some useul correction s which improved the presentation o the paper (8) Copright 03 SciRes

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