Convergence theorem for multi-step Noor fixed point iterative procedure with errors

CONVERGENCE THEOREM OF MULTI-STEP NOOR FIXED POINT ITERATIVE PROCEDURE

6.2 Convergence theorem for multi-step Noor fixed point iterative procedure with errors

In this section we sate and prove a convergence theorem for multi-step iterative procedure by using Zamfirescu operator, which have generate the analogous results of different fixed point iterative procedures.

Theorem 6.2.1.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 be a Zamfirescu operator. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined by multi-step Noor fixed point iterative procedurewith errors (5.1), for each 𝑘𝑘 = 1, 2, 3, … ,𝑙𝑙and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇) ≠ ∅, ∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘) = ∞, and

�𝑣𝑣_𝑛𝑛^(𝑘𝑘⁾ − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘)�,

for each 𝑘𝑘 = 1, 2, 3, … ,𝑙𝑙 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘)�converges strongly to a fixed point of 𝑇𝑇.

Proof.According to our assumption 𝑇𝑇 is a Zamfirescu operator, so by Theorem 1.6.5, we know that 𝑇𝑇 has a unique fixed point in 𝐵𝐵, say 𝑝𝑝

i.e., 𝑇𝑇𝑝𝑝 =𝑝𝑝. (6.1)

Now, we combine the Zamfirescu conditions according to the approach of V. Berinde [49-52]. Since 𝑇𝑇is a Zamfirescu operator, hence 𝑇𝑇 is satisfied at least one of the Zamfirescu conditions (𝑧𝑧₁), (𝑧𝑧₂) and (𝑧𝑧₃) defined by the Theorem 1.6.5.

If 𝑇𝑇 satisfies (𝑧𝑧₂), then for all 𝑥𝑥,𝑦𝑦 ∈ 𝐵𝐵 we have

‖𝑇𝑇𝑥𝑥 − 𝑇𝑇𝑦𝑦‖ ≤ 𝑏𝑏[‖𝑥𝑥 − 𝑇𝑇𝑥𝑥‖+‖𝑦𝑦 − 𝑇𝑇𝑦𝑦‖]

≤ 𝑏𝑏[‖𝑥𝑥 − 𝑇𝑇𝑥𝑥‖+‖𝑦𝑦 − 𝑥𝑥‖+‖𝑥𝑥 − 𝑇𝑇𝑥𝑥‖+‖𝑇𝑇𝑥𝑥 − 𝑇𝑇𝑦𝑦‖], which implies

‖𝑇𝑇𝑥𝑥 − 𝑇𝑇𝑦𝑦‖ ≤ _1−𝑏𝑏^𝑏𝑏 ‖𝑥𝑥 − 𝑦𝑦‖+_1−𝑏𝑏^2𝑏𝑏 ‖𝑥𝑥 − 𝑇𝑇𝑥𝑥‖ . (6.2) If 𝑇𝑇 satisfies (𝑧𝑧₃), then for all 𝑥𝑥,𝑦𝑦 ∈ 𝐵𝐵 similarly we obtain

‖𝑇𝑇𝑥𝑥 − 𝑇𝑇𝑦𝑦‖ ≤ ^𝑐𝑐 ‖𝑥𝑥 − 𝑦𝑦‖+ ^2𝑐𝑐 ‖𝑥𝑥 − 𝑇𝑇𝑥𝑥‖ . (6.3)

Now, if we take

𝛿𝛿 = 𝑙𝑙𝑎𝑎𝑥𝑥 �𝑎𝑎, _1−𝑏𝑏^𝑏𝑏 ,_1−𝑐𝑐^𝑐𝑐 �. (6.4)

Then we have 0 ≤ 𝛿𝛿 < 1 and in view of (𝑧𝑧₁) and (6.2) to (6.4), we obtained the following inequality.

‖𝑇𝑇𝑥𝑥 − 𝑇𝑇𝑦𝑦‖ ≤ 𝛿𝛿‖𝑥𝑥 − 𝑦𝑦‖+ 2𝛿𝛿‖𝑥𝑥 − 𝑇𝑇𝑥𝑥‖ . (6.5)

If we suppose�𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a multi-step Noor fixed point iterative procedurewith errors defined by (5.1) and 𝑢𝑢₀ ∈ 𝐵𝐵 arbitrary, then we have

‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖ =�𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)} +𝑏𝑏_𝑛𝑛^(𝑙𝑙⁾𝑢𝑢_𝑛𝑛 +𝑐𝑐_𝑛𝑛^(𝑙𝑙)𝑣𝑣_𝑛𝑛^(𝑙𝑙) − 𝑝𝑝�. (6.6) Since 𝑎𝑎_𝑛𝑛^(𝑙𝑙) +𝑏𝑏_𝑛𝑛^(𝑙𝑙⁾+𝑐𝑐_𝑛𝑛^(𝑙𝑙) = 1, hence from (2.6) we have

‖𝑢𝑢𝑛𝑛+1 − 𝑝𝑝‖ = �(1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾)(𝑢𝑢_𝑛𝑛 − 𝑝𝑝) +𝑎𝑎_𝑛𝑛^(𝑙𝑙)�𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)} − 𝑝𝑝�+𝑐𝑐_𝑛𝑛^(𝑙𝑙⁾(𝑣𝑣_𝑛𝑛^(𝑙𝑙⁾− 𝑢𝑢_𝑛𝑛)�

≤ (1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾)‖𝑢𝑢𝑛𝑛 − 𝑝𝑝‖+𝑎𝑎_𝑛𝑛^(𝑙𝑙)�𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)} − 𝑝𝑝�+𝑐𝑐_𝑛𝑛^(𝑙𝑙)�𝑣𝑣_𝑛𝑛^(𝑙𝑙) − 𝑢𝑢_𝑛𝑛�. (6.7) But according to our assumption, we have

�𝑣𝑣_𝑛𝑛^(𝑙𝑙) − 𝑢𝑢_𝑛𝑛� = 0(𝑎𝑎_𝑛𝑛^(𝑙𝑙)) Hence from (6.7), we have

‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖ ≤ (1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙))‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖+𝑎𝑎_𝑛𝑛^(𝑙𝑙)�𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)}− 𝑝𝑝�+𝑐𝑐_𝑛𝑛^(𝑙𝑙)0(𝑎𝑎_𝑛𝑛^(𝑙𝑙)). (6.8) Now, if we put 𝑥𝑥 =𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)}and 𝑦𝑦 =𝑝𝑝 in (6.5), we obtain

�𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)} − 𝑇𝑇𝑝𝑝� ≤ 𝛿𝛿 �𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)}− 𝑝𝑝�, (6.9) where𝛿𝛿 is given by (6.4).

Combining (6.8) and (6.9), we obtain

‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖ ≤ (1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙))‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖+𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝛿𝛿 �𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)} − 𝑝𝑝�+𝑐𝑐_𝑛𝑛^(𝑙𝑙)0(𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾) (6.10) Further by the definition of multi-step Noor fixed point iterative procedurewith errors (5.1), we have

�𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)}− 𝑝𝑝� =�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)}+𝑏𝑏_𝑛𝑛^{(𝑙𝑙−1)}𝑢𝑢_𝑛𝑛 +𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}𝑣𝑣_𝑛𝑛^{(𝑙𝑙−1)}− 𝑝𝑝�^(6.11) Since 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}+𝑏𝑏_𝑛𝑛^{(𝑙𝑙−1)} +𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)} = 1, hence from (6.11) we have

�𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)}− 𝑝𝑝� ≤ (1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)})‖𝑢𝑢𝑛𝑛 − 𝑝𝑝‖+𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}�𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)} − 𝑝𝑝�

+𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}�𝑣𝑣_𝑛𝑛^{(𝑙𝑙−1)}− 𝑢𝑢_𝑛𝑛�. (6.12) But according to our assumption�𝑣𝑣_𝑛𝑛^{(𝑙𝑙−1)}− 𝑢𝑢_𝑛𝑛� = 0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}), hence from (6.12) we have

�𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)}− 𝑝𝑝� ≤ (1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)})‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖

+𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}�𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)}− 𝑝𝑝�+𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}) (6.13) Now, if we put 𝑥𝑥 =𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)}and 𝑦𝑦 =𝑝𝑝 in (6.5), then we have

�𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)} − 𝑇𝑇𝑝𝑝� ≤ 𝛿𝛿 �𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)}− 𝑝𝑝�, (6.14) where𝛿𝛿 is given by (6.4).

Combining (6.13) and (6.14), we obtain

�𝑢𝑢_𝑛𝑛^{(𝑙𝑙−1)}− 𝑝𝑝� ≤ (1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)})‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖+𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝛿𝛿 �𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)} − 𝑝𝑝�

+𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}). (6.15) From (6.10) and (6.15), we have

‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖ ≤ (1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙))‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖+𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝛿𝛿 � (1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)})‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖

+𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝛿𝛿 �𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)} − 𝑝𝑝�� + 𝑐𝑐_𝑛𝑛^(𝑙𝑙)0(𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾) +𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)})

= (1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾ +𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝛿𝛿(1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}))‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖

+𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}�𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)} − 𝑝𝑝�

+ 𝑐𝑐_𝑛𝑛^(𝑙𝑙)0(𝑎𝑎_𝑛𝑛^(𝑙𝑙)) +𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}) (6.16)

Further by the definition of multi-step Noor fixed point iterative procedurewith errors (5.1), we have

�𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)}− 𝑝𝑝� =�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−3)}+𝑏𝑏_𝑛𝑛^{(𝑙𝑙−2)}𝑢𝑢_𝑛𝑛 +𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}𝑣𝑣_𝑛𝑛^{(𝑙𝑙−2)}− 𝑝𝑝�(6.17) Since 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}+𝑏𝑏_𝑛𝑛^{(𝑙𝑙−2)} +𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)} = 1, hence from (6.17) we have

�𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)}− 𝑝𝑝� ≤ (1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)})‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖+𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}�𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−3)} − 𝑝𝑝�

+𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}�𝑣𝑣_𝑛𝑛^{(𝑙𝑙−2)}− 𝑢𝑢_𝑛𝑛�. (6.18)

But according to our assumption�𝑣𝑣_𝑛𝑛^{(𝑙𝑙−2)}− 𝑢𝑢_𝑛𝑛� = 0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}), hence from (6.18) we have

+𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}) (6.19)

Now, if we put 𝑥𝑥 =𝑢𝑢_𝑛𝑛^{(𝑙𝑙−3)}and 𝑦𝑦 =𝑝𝑝 in (6.5), then we have

�𝑇𝑇𝑢𝑢_𝑛𝑛^{(𝑙𝑙−3)} − 𝑇𝑇𝑝𝑝� ≤ 𝛿𝛿 �𝑢𝑢_𝑛𝑛^{(𝑙𝑙−3)}− 𝑝𝑝�, (6.20) where𝛿𝛿 is given by (2.4).

Combining (6.19) and (6.20), we obtain

�𝑢𝑢_𝑛𝑛^{(𝑙𝑙−2)}− 𝑝𝑝� ≤ (1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)})‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖+𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}𝛿𝛿 �𝑢𝑢_𝑛𝑛^{(𝑙𝑙−3)} − 𝑝𝑝�

+𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}). (6.21)

From (6.16) and (6.21), we have

‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖

≤ (1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾)‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖

+𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝛿𝛿 �(1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)})‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖+𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝛿𝛿 � (1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)})‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖

+𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}𝛿𝛿 �𝑢𝑢_𝑛𝑛^{(𝑙𝑙−3)}− 𝑝𝑝��

+𝑐𝑐_𝑛𝑛^(𝑙𝑙)0(𝑎𝑎_𝑛𝑛^(𝑙𝑙)) +𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}) +𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)})

= (1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙) +𝛿𝛿𝑎𝑎_𝑛𝑛^(𝑙𝑙)(1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}) +𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}(1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}))‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖

+𝛿𝛿³𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}�𝑢𝑢_𝑛𝑛^{(𝑙𝑙−3)} − 𝑝𝑝�+𝑐𝑐_𝑛𝑛^(𝑙𝑙)0�𝑎𝑎_𝑛𝑛^(𝑙𝑙)�

+𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}) + 𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}0(𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}) (6.22)

Now if we continue the above process until the initial equation of multi-step Noor fixed point iterative procedurewith errors (5.1) have been used, then the inequality (6.22) can written as follows.

‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖ ≤ [1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙) +𝛿𝛿𝑎𝑎_𝑛𝑛^(𝑙𝑙)(1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}) +𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}�1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}� +⋯+𝛿𝛿^𝑙𝑙−1𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}…𝑎𝑎_𝑛𝑛⁽³⁾𝑎𝑎_𝑛𝑛⁽²⁾�1− 𝑎𝑎_𝑛𝑛⁽¹⁾�]‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖

+𝛿𝛿^𝑙𝑙𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}…𝑎𝑎_𝑛𝑛⁽³⁾𝑎𝑎_𝑛𝑛⁽²⁾𝑎𝑎_𝑛𝑛⁽¹⁾‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖

+𝑐𝑐_𝑛𝑛^(𝑙𝑙)0�𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾� 𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}� +𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}0�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}�

+⋯+𝛿𝛿^𝑙𝑙−1𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}…𝑎𝑎_𝑛𝑛⁽³⁾𝑎𝑎_𝑛𝑛⁽²⁾𝑐𝑐_𝑛𝑛⁽¹⁾0(𝑎𝑎_𝑛𝑛⁽¹⁾)

= (1− 𝑎𝑎_𝑛𝑛^(𝑙𝑙) +𝛿𝛿𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾(1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}) +𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}�1− 𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}� +⋯+𝛿𝛿^𝑙𝑙−1𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}…𝑎𝑎_𝑛𝑛⁽³⁾𝑎𝑎_𝑛𝑛⁽²⁾�1− 𝑎𝑎_𝑛𝑛⁽¹⁾�

+𝛿𝛿^𝑙𝑙𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}…𝑎𝑎_𝑛𝑛⁽³⁾𝑎𝑎_𝑛𝑛⁽²⁾𝑎𝑎_𝑛𝑛⁽¹⁾)‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖+𝑐𝑐_𝑛𝑛^(𝑙𝑙)0�𝑎𝑎_𝑛𝑛^(𝑙𝑙)� +𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}�+𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}0�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}� +⋯+𝛿𝛿^𝑙𝑙−1𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}…𝑎𝑎_𝑛𝑛⁽³⁾𝑎𝑎_𝑛𝑛⁽²⁾𝑐𝑐_𝑛𝑛⁽¹⁾0(𝑎𝑎_𝑛𝑛⁽¹⁾),

which implies,

‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖ ≤ [1−(1− 𝛿𝛿)𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾�1− 𝛿𝛿𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}� �1− 𝛿𝛿𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}� �1− 𝛿𝛿𝑎𝑎_𝑛𝑛^{(𝑙𝑙−3)}�… … (1− 𝛿𝛿𝑎𝑎_𝑛𝑛⁽¹⁾)]‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖ +𝑐𝑐_𝑛𝑛^(𝑙𝑙)0�𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾�

+𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}�+𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}0�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}� +⋯+ 𝛿𝛿^𝑙𝑙−1𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}…𝑎𝑎_𝑛𝑛⁽³⁾𝑎𝑎_𝑛𝑛⁽²⁾𝑐𝑐_𝑛𝑛⁽¹⁾0(𝑎𝑎_𝑛𝑛⁽¹⁾). (6.23)

But it is clear that,

[1−(1− 𝛿𝛿)𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾�1− 𝛿𝛿𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}� �1− 𝛿𝛿𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}� �1− 𝛿𝛿𝑎𝑎_𝑛𝑛^{(𝑙𝑙−3)}�… (1− 𝛿𝛿𝑎𝑎_𝑛𝑛⁽¹⁾)]

≤ [1−(1− 𝛿𝛿)^𝑙𝑙𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾]. Hence form (6.23), we obtain

‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖ ≤ [1−(1− 𝛿𝛿)^𝑙𝑙𝑎𝑎_𝑛𝑛^(𝑙𝑙)]‖𝑢𝑢_𝑛𝑛 − 𝑝𝑝‖ +𝑐𝑐_𝑛𝑛^(𝑙𝑙⁾0�𝑎𝑎_𝑛𝑛^(𝑙𝑙)� +𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑐𝑐_𝑛𝑛^{(𝑙𝑙−1)}0�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}�+𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}0�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}� +⋯+ 𝛿𝛿^𝑙𝑙−1𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}…𝑎𝑎_𝑛𝑛⁽³⁾𝑎𝑎_𝑛𝑛⁽²⁾𝑐𝑐_𝑛𝑛⁽¹⁾0(𝑎𝑎_𝑛𝑛⁽¹⁾),𝑛𝑛 ∈ ℕ. (6.24)

By (6.24) inductively, we obtain

‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖ ≤ ∏^𝑛𝑛_𝑟𝑟₌₀�1−(1− 𝛿𝛿)^𝑙𝑙𝑎𝑎_𝑟𝑟^(𝑙𝑙)�‖𝑢𝑢₀ − 𝑝𝑝‖

+𝛿𝛿²𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}𝑐𝑐_𝑛𝑛^{(𝑙𝑙−2)}0�𝑎𝑎_𝑛𝑛^{(𝑙𝑙−2)}� +⋯+𝛿𝛿^𝑙𝑙−1𝑎𝑎_𝑛𝑛^(𝑙𝑙)𝑎𝑎_𝑛𝑛^{(𝑙𝑙−1)}…𝑎𝑎_𝑛𝑛⁽³⁾𝑎𝑎_𝑛𝑛⁽²⁾𝑐𝑐_𝑛𝑛⁽¹⁾0(𝑎𝑎_𝑛𝑛⁽¹⁾), 𝑛𝑛 ∈ ℕ. (6.25)

Now since 0≤ 𝛿𝛿 < 1, 𝑎𝑎_𝑛𝑛^(𝑙𝑙⁾ ∈ (0, 1)and ∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑙𝑙) = ∞, hence by Lemma 6.1.2 we can write

lim_{𝑛𝑛→∞} ∏^𝑛𝑛_𝑟𝑟=0[1−(1− 𝛿𝛿)^𝑙𝑙𝑎𝑎_𝑟𝑟^(𝑙𝑙)] = 0. (6.26) Taking limit as 𝑛𝑛 → ∞ on both sides of (6.25) and using (6.26), we get

lim_{𝑛𝑛→∞}‖𝑢𝑢_𝑛𝑛+1 − 𝑝𝑝‖ = 0.

This implies that �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�, 𝑘𝑘 = 1, 2, 3, … ,𝑙𝑙converges strongly to 𝑝𝑝 ∈ 𝐹𝐹(𝑇𝑇). This completes our proof.∎

Corollary 6.2.2.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 satisfies the Kannan’s contractive conditions defined by (1.6). Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘)�be a sequence defined by multi-step Noor fixed point iterative procedurewith errors (5.1), for each 𝑘𝑘 = 1, 2, 3, … ,𝑙𝑙and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇) ≠ ∅,

∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘⁾ =∞, and �𝑣𝑣_𝑛𝑛^(𝑘𝑘) − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘)�, for each 𝑘𝑘 = 1, 2, 3, … ,𝑙𝑙 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘)�converges strongly to a fixed point of 𝑇𝑇.

Corollary 6.2.3.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 satisfies the Chatterjea’s contractive conditions defined by (1.7) respectively. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined by multi-step Noor fixed point iterative procedurewith errors (5.1), for each 𝑘𝑘 = 1, 2, 3, … ,𝑙𝑙and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇) ≠ ∅, ∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘) = ∞, and �𝑣𝑣_𝑛𝑛^(𝑘𝑘⁾ − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘)�, for each 𝑘𝑘 = 1, 2, 3, … ,𝑙𝑙 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘)�converges strongly to a fixed point of 𝑇𝑇.

Corollary 6.2.4.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 be a Zamfirescu operator. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined

by Noor iterative procedurewith errors (5.2), for each 𝑘𝑘 = 1, 2, 3and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇) ≠ ∅, ∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘) = ∞, and �𝑣𝑣_𝑛𝑛^(𝑘𝑘⁾ − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘)�, for each 𝑘𝑘 = 1, 2, 3 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�converges strongly to a fixed point of 𝑇𝑇.

Corollary 6.2.5.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 be a Zamfirescu operator. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined by Noor iterative procedure(5.3), for each 𝑘𝑘 = 1, 2, 3and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇) ≠ ∅,

∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘⁾ =∞, and �𝑣𝑣_𝑛𝑛^(𝑘𝑘) − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘)�, for each 𝑘𝑘 = 1, 2, 3 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�converges strongly to a fixed point of 𝑇𝑇.

Corollary 6.2.6.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 be a Zamfirescu operator. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined by Ishikawa iterative procedurewith errorsdefined by Y. Xu (5.4), for each 𝑘𝑘 = 1, 2and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇)≠ ∅, ∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘⁾ =∞, and �𝑣𝑣_𝑛𝑛^(𝑘𝑘) − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘⁾�, for each 𝑘𝑘 = 1, 2 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�converges strongly to a fixed point of 𝑇𝑇.

Corollary 6.2.7.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 be a Zamfirescu operator. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined by Ishikawa iterative procedure with errorsdefined by L.S.Lu(5.6), for each 𝑘𝑘 = 1, 2and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇)≠ ∅, ∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘⁾ =∞, and �𝑣𝑣_𝑛𝑛^(𝑘𝑘) − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘⁾�, for each 𝑘𝑘 = 1, 2 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�converges strongly to a fixed point of 𝑇𝑇.

Corollary 6.2.8.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 be a Zamfirescu operator. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined

by Ishikawa iterative procedure (5.5), for each 𝑘𝑘 = 1, 2and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇) ≠ ∅,

∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘⁾ =∞, and

�𝑣𝑣_𝑛𝑛^(𝑘𝑘⁾ − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘)�,

for each 𝑘𝑘 = 1, 2 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘)�converges strongly to a fixed point of 𝑇𝑇.

Corollary 6.2.9.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 be a Zamfirescu operator. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined by Mann iterative procedure with errorsdefined by Y. Xu (5.7), for each 𝑘𝑘 = 1and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇) ≠ ∅, ∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘) = ∞, and

�𝑣𝑣_𝑛𝑛^(𝑘𝑘⁾ − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘)�,

for each 𝑘𝑘 = 1 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�converges strongly to a fixed point of 𝑇𝑇.

Corollary 6.2.10.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 be a Zamfirescu operator. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined by Mann iterative procedure with errorsdefined by L.S. Lu (5.10), for each 𝑘𝑘 = 1and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇)≠ ∅, ∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘⁾ =∞, and

�𝑣𝑣_𝑛𝑛^(𝑘𝑘⁾ − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘)�,

for each 𝑘𝑘 = 1 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�converges strongly to a fixed point of 𝑇𝑇.

Corollary 6.2.11.Let 𝐵𝐵 be a nonempty closed convex subset of an arbitrary normed space𝑋𝑋. Let 𝑇𝑇:𝐵𝐵 → 𝐵𝐵 be a Zamfirescu operator. Let �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�be a sequence defined by Mann iterative procedure(5.8), for each 𝑘𝑘 = 1and 𝑛𝑛 ∈ ℕ. If 𝐹𝐹(𝑇𝑇) ≠ ∅,

∑^∞_𝑛𝑛=1𝑎𝑎_𝑛𝑛^(𝑘𝑘⁾ =∞, and

�𝑣𝑣_𝑛𝑛^(𝑘𝑘⁾ − 𝑢𝑢_𝑛𝑛� = 0�𝑎𝑎_𝑛𝑛^(𝑘𝑘)�,

for each 𝑘𝑘 = 1 and 𝑛𝑛 ∈ ℕ. Then �𝑢𝑢_𝑛𝑛^(𝑘𝑘⁾�converges strongly to a fixed point of 𝑇𝑇.

Dalam dokumen A Study on Fixed Point Iterative Procedures (Halaman 76-84)