Convergence Methods for Double Sequences and Applications by M. Mursaleen, S.A. Mohiuddine

By M. Mursaleen, S.A. Mohiuddine

This book's committed specialize in the 'almost' convergence and statistical convergence of double sequences demystifies the idea that utilizing a number of unveiling examples, highlighting the appliance of double sequences in key parts of natural and utilized arithmetic.

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Extra resources for Convergence Methods for Double Sequences and Applications

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J,k The existence of the limit BP- lim p,q α(j, k, p, q, s, t)xj k j,k = aj k zj k + BP- lim p,q j,k α(j, k, p, q, s, t) x k − ξ + BP- lim + BP- lim p,q α(j, k, p, q, s, t)(xj − ξ ) j,k p,q j,k α(j, k, p, q, s, t)ξ j,k then would follow if the limits on the right side exist. The third limit α(j, k, p, q, s, t)ξ = ξ v BP- lim p,q j,k exists by (iii). We will show that the first limit equals N ∈ N be such that ε |xj − ξ | ≤ 4M j vj (xj − ξ ). To this end, let ε > 0 and for j ≥ N. By (v) we can choose S ∈ N such that for p, q ≥ S and any s, t ∈ N, N −1 α(j, k, p, q, s, t) − vj ≤ j =1 k ε 4 x ∞ Then α(j, k, p, q, s, t) − vj (xj − ξ ) j k N −1 ≤2 α(j, k, p, q, s, t) − vj x j =1 k ∞ .

16) For the expression in the second bracket, we have m n−1 1 (m + 1)n = = = 1 n 1 n 1 n p=0 q=0 n−1 n−1 m−1 n−1 τpqst p=0 q=0 m m−1 τpqst − (m + 1) p=0 n−1 m−1 τpqst p=0 1 (m + 1)τmqst − m(m + 1) q=0 n−1 τmqst q=0 1 − m(m + 1)n n−1 τmqst − q=0 τpqst p=0 1 mτmqst − m(m + 1) q=0 1 mn 1 mn 1 m m(m + 1) q=0 1 = mn = τpqst − m τpqst p=0 m n−1 τpqst p=0 q=0 1 dm,n−1,s,t . 15), we get dmnst − dm−1,n,s,t − dm,n−1,s,t + dm−1,n−1,s,t = = 1 m(n + 1) n τmqst − q=0 1 n mn(n + 1) 1 mn n−1 τmqst − q=0 n n−1 τmqst − (n + 1) q=0 1 1 dmnst + dm,n−1,s,t m m τmqst − q=0 1 (dmnst − dm,n−1,s,t ) m 32 2 = = = 1 nτmnst − mn(n + 1) n−1 1 (dmnst − dm,n−1,s,t ) m τmqst − q=0 1 (n + 1)τmnst − mn(n + 1) 1 1 τmnst − mn (n + 1) Almost Convergence of Double Sequences n τmqst − q=0 n τmqst − q=0 1 (dmnst − dm,n−1,s,t ) m 1 (dmnst − dm,n−1,s,t ).

17) with the last two inequalities provides yMN − s ≤ 7 + |s| + C if min(M, N ) ≥ max(M1 , M2 , M3 , M4 , M5 ). Since > 0 is arbitrary, this proves the convergence of the A-means ymn to s as M and N tend to ∞, that is, the A-summability of x to the same limit. 2(b) and the boundedness of x. ) Let A = (amnj k ) be a strongly regular matrix. 2(b). 4) is not satisfied. Then there exists > 0 such that, for infinitely many pairs of M and N both tending to ∞, we have ∞ ∞ | 10 aMN mn | ≥ 12 . 18) j =0 k=0 or ∞ ∞ | 10 aM,N,2j +1,k | ≥ 6 .

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