RATE PRESERVATION OF DOUBLE SEQUENCES UNDER l-l TYPE TRANSFORMATION

Following the concepts of divergent rate preservation for ordinary sequences, we present a notion of rates preservation of divergent double sequences under l-l type transformations. Definitions for Pringsheim limit inferior and superior are also presented. These definitions and the notion of asymptotically equivalent double sequences, are used to present necessary and sufficient conditions on the entries of a four-dimensional matrix such that, the rate of divergence is preserved for a given double sequences under l-l type mapping where l=: {xk,l : ∑∞,∞ k,l=1,1 |xk,l|<∞}.

1. Introduction.The concepts of rates preservation for P -convergence and P -divergence of asymptotically equivalent of double sequences under four-dimensional matrix transformation is presented in [2].This paper presents necessary and sufficient conditions on the entries of a four-dimensional matrix such that the asymptotic properties of the given sequences are preserved under l-l type mapping where l =: {x k,l :

Definitions, notation, and preliminary results
Definition 2.1 (see [5]).A double sequence [x] = {x k,l } has Pringsheim limit L (denoted by P -lim x = L) provided that given > 0 there exists N ∈ N such that |x k,l − L| < whenever k > N and l > N. We will describe such an [x] more briefly as "P -convergent."A double sequence [x] is bounded if and only if there exists a positive number M such that |x k,l | < M for all k and l.Note that a convergent double sequence need not be bounded.In 1900, Pringsheim gave the following definition: a double sequence [x] is called definite divergent if for every (arbitrarily large) G > 0, there exist two natural numbers n 1 and Definition 2.2 (see [4]).A number β is called a Pringsheim limit point of the double sequence [x] = {x n,k } provided that there exists a sequence [y] = {y n,k } of {x n,k } that has Pringsheim limit β : P -lim y n,k = β.Definition 2.3 (see [4]).The double sequence [y] is a double subsequence of the sequence [x] provided that there exist two increasing double index sequences {n j } and {k j } such that if {z j } = {x n j ,k j }, then [y] is formed by Remark 2.4.The definition of a Pringsheim limit point can also be stated as follows: β is a Pringsheim limit point of x provided that there exist two increasing index sequences {n i } and {k i } such that lim i x n i ,k i = β.In addition to this reformulation of subsequence definition it should be noted that a finite number of unbounded rows and/or columns does not affect the P -convergence or P -divergence of [x] and its subsequences.
We consider the following notation: exists}, P δ is the set of all real double number sequences such that x k,l ≥ δ > 0 for all k, and l}, and P 0 is the set of all nonnegative sequences which have at most a finite number of columns and/or rows with zero entries.Definition 2.6 (see [2]).Two nonnegative double sequences [x] and [y] are said to be asymptotically equivalent if (2.2) Definition 2.8 (see [3]).Let [x] = {x k,l } be a double sequence of real numbers and for each n, let α n = sup n {x k,l : k > n ≥ and l ≥ n}.The Pringsheim limit superior of [x] is defined as follows: (1) if α = +∞ for each n, then P -lim sup[x] := +∞; (2) if α < ∞ for some n, then P -lim sup[x] := inf n {α n }.Similarly, let β n = inf n {x k,l : k ≥ n and l ≥ n} then the Pringsheim limit inferior of [x] is defined as follows: (1) if (b) for some p and q Proof.We will begin by proving that (2) implies (1).Suppose (2) holds and [x] and [y] are such that x P ∼ y, [x] ∈ P 0 , and [y] ∈ P δ for some δ > 0. First we will show that [Ax] and [Ay] are not in l .Since [y] is in P δ and 2(a) grant us that we obtain the following inequality Thus [Ay] is not in l .Also since x P ∼ y, given > 0 there exist K and L such that |x k,l /y k,l − 1| < whenever k > K and l > L. This implies that As a consequence of (3.5), the following inequality holds: (3.7) As a consequence of the last inequality, we obtain the following: Observe that condition 2(b) implies the following: (3.9)    2).This completes the proof of this theorem.

Definition 2 . 7 .
For any double sequence [x], let [Sx] denote the sequence of partial double sums whose m and nth term is given by

3 .
Main results.For sequences x and y not in l = {x k :∞ k=1 |x k | < ∞}, Marouf in[1] used partial sum to characterize divergence rate preserving for such sequences under l-l transformation.The following theorem is a multiple-dimensional analog of Marouf theorem.Theorem 3.1.If A is a nonnegative real four-dimensional matrix, then the following statements are equivalent: (1) if [x] and [y] are bounded double sequences such that x P ∼ y, [x] ∈ P 0 , and [y] ∈ P δ for some δ > 0, then (a) [Ax] and [Ay] are not in l and ] ∉ l .Second we show that S m,n (Ax) P ∼ S m,n (Ay).Consider the following: l=1,1 a m,n,k,l .(3.11) Condition 2(b) implies that each of the nonconstant terms of the last inequality has a Pringsheim limit of zero this yields , S m,n (Ax) P ∼ S m,n (Ay)).This completes the sufficiency part of this theorem.To establish the necessary part of this theorem (that is, (1) implies (2)).We form two double sequences [x] and [y] such that x P ∼ y, 1(a) and 1(b) holds but [S m,n (Ax)] is not asymptotically equivalent to [S m,n (Ay)].Let y k,l = 1 for all k and l and x k,l :=    1, if k ≥ α, l ≥ β; 0, otherwise.(3.14) Observe that x P ∼ y, [x] ∈ P 0 , and [y] ∈ P 1 .Assume that condition 2(a) does not hold.Also let β and α be two fixed positive integers and consider the following inequality l=1,1 a m,n,k,l − p,q m,n=1,1 a m,n,α−1,β p,q m,n=1,1 ∞,∞ k,l=1,1 a m,n,k,l − p,q m,n=1,1 a m,n,α−1,β−1 p,q m,n=1,1 ∞,∞ k,l=1,1 a m,n,k,l.[S m,n (Ax)] is not asymptotically equivalent to [S m,n (Ay)].We need only to show that 2(a) holds.If we let y k,l = 1 for all k and l, then(Ay) m,n = ∞,∞ k,l a m,n,k,l .(3.19)Since [Ay] ∉ l then