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Theorem 1. Let (X, d ) be a complete cone metric space, P be a normal cone with normal constant K . Suppose the mapping T : X → X satisfies the contractive condition d(Tx,Ty) kd(x,y), for all x,y ∈X, where k ∈ [0,1) is a constant. Then T has a unique fixed point in X. And for any x ∈ X, iterative sequence {T nx} converges to the fixed point. Proof. Choosex0 ∈X.Setx1 =Tx0,x2 =Tx1 =T2x0,...,xn+1 =Txn =Tn+1x0,.... We have d(xn+1, xn) = d(T xn, T xn−1) kd(xn, xn−1) k2d(xn−1,xn−2) ··· knd(x1,x0). d(xn,xm) d(xn,xn−1)+d(xn−1,xn−2)+···+d(xm+1,xm) n−1n−2 m km (k +k +···+k )d(x1,x0) 1−kd(x1,x0). I can not understand how we move from this line to another ihope someone explain this to me Thanks,

R.M.B
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  • Theorem 1. Let $(X, d )$ be a complete cone metric space, $P$ be a normal cone with normal constant $K$ . Suppose the mapping T : X → X satisfies the contractive condition $d(Tx,Ty)\le kd(x,y)$, for all $x,y ∈X$, where $k ∈ [0,1)$ is a constant. Then $T$ has a unique fixed point in $X$. And for any $x ∈ X$, iterative sequence ${T_nx)}$ converges to the fixed point.

    Proof. Choose $x_0 ∈X$. Set $x_1 =T(x_0)$, $x_2 =T(x_1)$ .........(edit )

     – Piquito Nov 07 '17 at 22:40
  • With more details https://math.stackexchange.com/questions/2225451/contraction-mapping-in-the-context-of-fx-n-x-n1/ – rtybase Nov 07 '17 at 22:46

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