Hello, everyone Iwant to ask about Cauchy seq.

How to prove that d(x2m+1,x2m)and d(x2m-1,x2m) are cauchy sequences

if T:X->X is a convex contraction mapping of order 2 , X is complete metric space ,xn+1=Txn and k=max{d(x1,x0),d(x1,x2)}

(Hi)

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- April 29th 2010, 12:13 PMmath3000Cauchy sequence
Hello, everyone Iwant to ask about Cauchy seq.

How to prove that d(x2m+1,x2m)and d(x2m-1,x2m) are cauchy sequences

if T:X->X is a convex contraction mapping of order 2 , X is complete metric space ,xn+1=Txn and k=max{d(x1,x0),d(x1,x2)}

(Hi) - April 29th 2010, 12:15 PMDrexel28
- April 29th 2010, 12:43 PMmath3000
T is a convex contraction of order 2 means

1)d(T²x,T²y)≤ad(Tx,Ty)+bd(x,y)

2)a,b and a+b∈(0,1) - April 29th 2010, 09:03 PMDrexel28
- April 30th 2010, 01:38 AMmath3000
From the inequality

d(T^{2m+1}x0,T^{2m}x0)≤ad(T^{2m}x0,T^{2m-1}x0)+bd(T^{2m-1}x0,T^{2m-2}x0) and for m≥1 .I get

d(T³x0,T²x0)≤ad(T²x0,Tx0)+bd(Tx0,x0) ≤(a+b)k

d(T5x0,T⁴x0)≤ad(T⁴x0,T³x0)+bd(T³x0,T²x0) ≤(a+b)²k and so on.

and From the inequality

d(T^{2m-1}x0,T^{2m}x0)≤ad(T^{2m-2}x0,T^{2m-1}x0)+bd(T^{2m-3}x0,T^{2m-2}x0) and for m≥2 .I get

d(T³x0,T⁴x0)≤ad(T³x0,T²x0)+bd(T²x0,Tx0) ≤(a+b)k

d(T5x0,T6x)≤ad(T⁴x0,T5x0)+bd(T⁴x0,T³x0) ≤(a+b)²k and so on.

Then I arrived to

d(T^{2m+1}x0,T^{2m}x0)≤k(a+b)^{m} and also

d(T^{2m-1}x0,T^{2m}x0)≤k(a+b)^{m}

I want to show that d(T^{m+1}x0,T^{m}x0)≤ A value

then I use it to sow that

∀m<n d(T^{m}x0,Tⁿx0)≤d(T^{m}x0,T^{m+1}x0)+d(T^{m+1}x0,T ^{m+2}x0)+...+d(Tⁿ-¹x0,Tⁿx0)

I want a hint

because I am trying every thing and there is no clearly answer