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

Author:M. Mursaleen & S.A. Mohiuddine
Language: eng
Format: epub
Publisher: Springer India, New Delhi

(6.1)

if and only if

(6.2)

(6.3)

(6.4)

(6.5)

Proof

Since each of the functions 1, x, y, x 2+y 2 belongs to C(I 2), conditions (6.2)–(6.5) follow immediately from (6.1). By the continuity of f on I 2, we can write |f(x,y)|≤M, a≤x,y≤b, where M=∥f∥∞. Therefore,

(6.6)

Also, since f∈C(I 2), for every ϵ>0, there is δ>0 such that

(6.7)

Using (6.6), (6.7) and putting ψ 1=ψ 1(s,x)=(s−x)2 and ψ 2=ψ 2(t,y)=(t−y)2, we get

that is,

Now, operate T j,k (1;x,y) to this inequality. Since T j,k (f;x,y) is monotone and linear, we obtain

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