AN EXTENSION ABOUT WIJSMAN I ASYMPTOTICALLY λ STATISTICAL EQUIVALENCE

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1 Electronic Journal of Mathematical Analysis and Applications Vol. 4() July 06, pp IN: (online) AN EXTENION ABOUT WIJMAN I AYMPTOTICALLY λ TATITICAL EQUIVALENCE HAFIZE GUMU Abstract. In this study we extend the notions Wijsman I asymptotically λ statistical equivalent sequences, Wijsman strongly I asymptotically λ equivalent sequences and Wijsman strongly Cesáro I asymptotically equivalent sequences by using the sequence p = (p k ) which is the sequence of positive real numbers and λ = ( ) is a non-decreasing sequence of positive numbers tending to such that + + and λ =.. Introduction Convergence is one of the most important notions in Mathematics. We know that statistical convergence extends the notion. We can easily show that any convergent sequence is statistically convergent, but not conversely. Let E be a subset n of N, the set of all natural numbers. d(e) := lim n n j= χ E(j) is said to be natural density of E whenever the limit exists, where χ E is the characteristic function of E. A number sequence (x k ) is statistically convergent to x provided that for every ε > 0, d k N : x k x ε = 0 or equivalently there exists a subset K N with d(e) = and n 0 (ε) such that k > n 0 (ε) and k K imply that x k x < ε. In this case we write st lim x k = x. tatistical convergence was given by Zygmund in the first edition of his monograph published in Warsaw in 935. It was formally introduced by Fast ([5]) and teinhaus ([8])and later was reintroduced by choenberg ([7]). It has become an active area of research in recent years. This concept has applications in different fields of mathematics such as number theory by Erdös and Tenenbaum ([4]), measure theory by Miller ([]), trigonometric series by Zygmund ([]), summability theory by Freedman and ember ([6]), etc. The idea of I convergence for single sequences was introduced by Kostyrko, alat and Wilczynski ([0]). We can say that this concept is a generalization of statistical convergence which is based on the structure of the ideal I of subsets of the set of natural numbers. Recently, it has become one of the most active areas of research in classical analysis. I convergence of real sequences coincides with 000 Mathematics ubject Classification. 40G5, 40A35. Key words and phrases. et sequences, ideal convergence, asymtotically statistical equivalent sequences, Wijsman convergence. ubmitted Aug. 9, 05. 9

2 EJMAA-06/4() WIJMAN I AYMPTOTICALLY λ TATITICAL EQUIVALENCE 93 the ordinary convergence if I is the ideal of all finite subsets of N and with the statistical convergence if I is the ideal of subsets of N of natural density zero. In recent years, the concept of convergence of sequences of numbers has been extended by several authors to convergence of sequences of sets. The one of these such extensions considered in this paper is the concept of Wijsman convergence. ([]-[]-[9]-[4]-[0]) Before, Mursaleen defined λ statistical convergence and he denoted this new method by λ and found its relation to statistical convergence, [C, ] summability and [V, λ] summability ([3]) where λ = ( ) is a non-decreasing sequence of positive numbers tending to such that + + and λ =. In this situation Λ will be the set of all non-decreasing sequences and I n = [n +, n]. Gumus and avas gave the definition of λ L(I) asymptotically statistical equivalence and interested in some relations with Vλ L(I) and CL (I) spaces ([7]) In 0, Nuray and Rhodes extended Mursaleen s study and gave the definition of statistical convergence for sequences of sets ([4]). Kisi and Nuray used I convergence for the similar concepts and they introduced new convergence notions for sequences of sets, which are called by Wijsman I convergence and Wijsman λ statistical convergence ([8]-[9]). ome authors like avas and Gumus, generalized these concepts by using p = (p k ) positive real number sequence ([6]). Another basic topic for us is asymptotically equivalent sequences. Marouf presented definitions for asymptotically equivalent sequences and asymptotic reguler matrices ([]). In 003, Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices ([5]).Two nonnegative sequences x = (x k ) and y = (y k ) are said to be asymptotically equivalent if x k lim = k y k and it is denoted by x y. In this work, we investigate that which results we will have if we apply the p = (p k ) positive real number sequence to previous results. We also investigate what will happen if p = (p).. Definitons and notations A family of sets I N is called an ideal if and only if (i) I (ii) For each A, B I we have A B I (iii) For each A I and each B A we have B I An ideal is called non-trivial if N / I and a non-trivial ideal is called admissible if n I for each n N. A family of sets F N is a filter in N if and only if (i) / F (ii) For each A, B F we have A B F (iii) For each A F and each B A we have B F I is a non-trivial ideal in N if and only if F = F (I) = M = N\A : A I

3 94 HAFIZE GUMU EJMAA-06/4() is a filter in N. A real sequence x = (x k ) is said to be I convergent to L R if and only if for each ε > 0 the set A ε = k N : x k L ε belongs to I. The number L is called the I limit of the sequence x. Take for I class the I f of all finite subsets of N. Then I f is a non-trivial admissible ideal and I f convergence coincides with the usual convergence. Let (X, d) is a metric space. For any point x X and any non-empty subset A of X, we define the distance from x to A by d(x, A) = inf d(x, A). a A ([]) Let (X, d) is a metric space. For any non-empty closed subsets A, A k X we say that the sequence A k is Wijsman convergent to A if lim d(x, A k) = d(x, A) k for each x X. In this case we write W lim k A k = A. Lets define the following sequence of circles in the (x, y) plane. A k = (x, y) : x + y + kx = 0. As k the sequence is Wijsman convergent to the y axis A = (x, y) : x = 0. ([4]) Let (X, d) is a metric space. For any non-empty closed subsets A, A k X; one says that the sequence A k is Wijsman statistically convergent to A if for ε > 0 and for each x X, lim k n : d(x, A k) d(x, A) ε = 0. n In this case we write st lim W A k = A or A k A(W ) where W denotes the set of Wijsman statistical convergent sequences. ([]) Let (X, d) is a metric space. For any non-empty closed subset A k of X, we say that the sequence A k is bounded if sup < k for each x X. In this case we write A k L. ([]) Let (X, d) is a metric space. For any non-empty closed subsets A, A k X, we say that the sequence A k is Wijsman Cesáro summable to A if lim n n n = d(x, A) k= for each x X and we say that A k is Wijsman Cesáro summable to A if n lim d(x, A) = 0 n n k= for each x X. ([9]) Let (X, d) is a metric space. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for each x X, we say that the sequences

4 EJMAA-06/4() WIJMAN I AYMPTOTICALLY λ TATITICAL EQUIVALENCE 95 A k and B k are asymptotically equivalent (Wijsman sense) if for each for each x X, lim k d(x, B k ) = and this is denoted by A k B k Choose the following sequences of circles in the (x, y) plane. A k = (x, y) R : x + y + ky = 0, B k = (x, y) R : x + y ky = 0. ince, lim k d(x, B k ) = then A k and B k are asymptotically equivalent (Wijsman sense). ([9]) Let (X, d) is a metric space. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for each x X, we say that the sequences A k and B k are asymptotically statistically equivalent (Wijsman sense) provided that for each for each x X, lim n n k n : d(x, B k ) L ε = 0 W and this is denoted by A L k Bk. ([9]) Let (X, d) is a metric space, I N is an admissible ideal. For any non-empty closed subsets A, A k X we say that the sequence A k is Wijsman I convergent to A if for each ε > 0 and for each x X the set A(x; ε) = k N : d(x, A) ε belongs to I. This is denoted by I W lim A k = A or A k A(I W ) where I W is the set of Wijsman I convergent sequences. Let X = R and let A k is the following sequence: (x, y) : x A k = + y ky = 0, if k n (x, y) : y =, if k = n and A = (x, y) : y = 0. The sequence A k is not Wijsman convergent to A but if we take I = I d, then A k is Wijsman I convergent to A where I d is the ideal of sets that have zero density. ([8]) Let (X, d) is a metric space. For any non-empty closed subsets A, A k X, we say that the sequence A k is Wijsman λ statistically convergent or W λ convergent to A provided that for every ε > 0 and for each x X, lim n k I n : d(x, A) ε = 0. This is denoted by λ lim W A k = A or A k A(W λ ) where W λ = A k : A X, λ lim A k = A. W If = n, Wijsman λ statistical convergence is the same as Wijsman statistical convergence for the sequences of sets. ([9])Let (X, d) is a metric space and I N is an admissible ideal. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for

5 96 HAFIZE GUMU EJMAA-06/4() each x X we say that the sequences A k and B k are asymptotically Wijsman I equivalent of multiple L if for every ε > 0 and for each x X, k N : d(x, B k ) L ε I I and it is denoted by A W k Let I N is an non-trivial ideal, (X, d) is a metric space, A k, B k X are non-empty closed subsets. Let X = R, A k, B k are following sequences. (x, y) R A k = : 0 x n, 0 y n x, if k n (0, 0), otherwise and (x, y) R B k = : 0 x n, 0 y n x, if k n (0, 0), otherwise If we take I = I d, the sequences A k and B k are asymptotically Wijsman I equ-ivalent where I d is the ideal of sets which have zero density. ([9])Let (X, d) is a metric space. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for each x X we say that the sequences A k and B k are strong Cesáro I asymptotically equivalent (Wijsman sense) of multiple L provided that for every ε > 0 and for each x X, n n d(x, B k ) L ε I k= C and this is denoted by A L (I W ) k ([8]) Let (X, d) is a metric space. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for each x X we say that the sequences A k and B k are Wijsman I asymptotically statistical equivalent of multiple L provided that for every δ, ε > 0 and for each x X, n k n : d(x, B k ) L ε δ I and this is denoted by A L (I W ) k Let I is an admissible ideal in N.and X = R. For any non-empty closed subsets A k, B k X, A k and B k are following sequences: (x, y) R A k = : x + (y ) = k, if k n (0, 0), otherwise (x, y) R B k = : x + (y + ) = k, if k n (0, 0), otherwise If we take I = I d we have, k N : d(x, B k ) ε I thus A k I W ([8]) Let (X, d) is a metric space. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for each x X, we say that the

6 EJMAA-06/4() WIJMAN I AYMPTOTICALLY λ TATITICAL EQUIVALENCE 97 sequences A k and B k are strongly λ I asymptotically equivalent (Wijsman sense) of multiple L provided that for every ε > 0 and for each x X, d(x, B k ) L ε I V and this is denoted by A L k ([9]) Let (X, d) is a metric space. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for each x X, we say that the sequences A k and B k are I asymptotically λ statistically equivalent (Wijsman sense) of multiple L provided that for every ε, δ > 0 and for each x X, k I n : d(x, B k ) L ε δ I and in this case we write A k L 3. Main Results Let (X, d) is a metric space, I N is a non-trivial ideal, p = (p k ) is a sequence of positive real numbers and λ Λ. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for each x X, the sequences A k and B k are said to be Wijsman strongly λ I asymptotically p equivalent or λ (I W ) asymptotically equivalent of multiple L provided that for every ε > 0 and every x X, d(x, B k ) L ε I. pk In this situation we write A k V Lp λ If we take p k = p for all k N we write A (I W ) k B k instead of A k B k. Let (X, d) is a metric space, I N is a non-trivial ideal, p = (p k ) is a sequence of positive real numbers and λ Λ. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for each x X, the sequences A k and B k are said to be Wijsman strongly Cesáro I asymptotically p equivalent of multiple L if every ε > 0 and every x X, n n d(x, B k ) L ε I. pk k= σ In this case we write A L(p) (I W ) k Let (X, d) is a metric space and I N is an admissible ideal and λ Λ. For any non-empty closed subsets A k, B k X such that > 0 and d(x, B k ) > 0 for each x X we say that the sequences A k and B k are Wijsman I asymptotically λ statistical equivalent of multiple L if every ε, δ > 0 and for each x X, k I n : d(x, B k ) L ε δ I

7 98 HAFIZE GUMU EJMAA-06/4() and it is denoted by A k L Let (X, d) is a metric space, I N A k V Lp B k then A k L V Lp λ Proof. (i) Let A (I W ) k B k and ε > 0. For each x X, d(x,b k ) L p and so ε p is an admissible ideal and λ Λ. If d(x,a k) & d(x,a k ) ε d(x,b k ) L p ε p k I n : d(x,a k) d(x,b k ) L ε d(x,b k ) L k I n : d(x,a k) d(x,b k ) L ε. Then for any δ > 0, k I n : k I r d(x,a k) d(x,b k ) L ε δ d(x,b k ) L ε p δ I. Therefore A k L Let (X, d) is a metric space, I N is an admissible ideal and λ Λ. If A k, B k L and A L V Lp λ k B k then A (I W ) k Proof. A k, B k L and A L k Then there is an M such that d(x,a k) d(x,b k ) L M for each x X and all k. For each ε > 0 Then for any δ > 0, d(x,b k ) L p = + d(x,b k ) L p d(x,a k) & d(x,a k ) ε d(x,b k ) L p d(x,a k) & d(x,a k ) <ε M p k I n : + ε p k I n : M p k I n : d(x,b k ) L ε d(x,b k ) L < ε d(x,b k ) L ε + ε p. d(x,b k ) L ε k I n k I n : d(x,a k) d(x,b k ) L ε ε p M I p d(x,a k)

8 EJMAA-06/4() WIJMAN I AYMPTOTICALLY λ TATITICAL EQUIVALENCE 99 and we have the proof. A L k B k L = A k V Lp B k L. Proof. This follows from Theorem 3.. and Theorem 3.. Let λ Λ and inf p k = h and sup p k = H. A k B k implies A L k B k. Proof. Assume that A k B k and ε > 0. Then, = + d(x,a k) & d(x,a k ) ε d(x,a k) & d(x,a k ) <ε d(x,a k) & d(x,a k ) ε (ε) pk & d(x,a k ) ε min & d(x,a k ) ε min (ε) h, (ε) H k I n : (ε) h, (ε) H d(x,b k ) L ε and k I n : d(x,a k) d(x,b k ) L ε δ d(x,a k) δ min (ε) h, (ε) H I. Thus we have x L y. Let A k and B k are bounded sequences, inf k p k = h and sup k p k = H. Then, A L k B k implies A k

9 00 HAFIZE GUMU EJMAA-06/4() Proof. uppose that A k and B k are bounded and ε > 0. Then there is an integer K such that d(x,a k) d(x,b k ) L K for all k. = + & d(x,a k ) ε & d(x,a k ) k I n : + k I n : d(x,a k) <ε d(x,b k ) L ε max K h, K H d(x,b k ) L < ε max(ε) p k max K h, K H k I n : d(x,b k ) L ε + maxε h,ε H and d(x,a k) k I n k I n : d(x,a k) ε d(x,b k ) L ε ε maxε h,ε H maxk h,k H belongs to I so we have A k λ ( I w) σ Let λ Λ and I is an admissible ideal in N. If A k B k then A (p) (I W ) k B k. Proof. Assume that A k B k and ε > 0. Then, n n = n k= + n + n λn k= n d(x,a k) k= and so, n n d(x,a k) ε k= ε and it completes the proof.

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