id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
iajs-448	Nemah, Eman M.	Modified Iterative Solution of Nonlinear Uniformly Continuous Mappings Equation in Arbitrary Real Banach Space	2017	8	.pdf	application/pdf	3556	189	83	2𝑐𝑛〈𝑢𝑛 − 𝑞, 𝑗(𝑥𝑛+1 − 𝑞)〉 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛〈𝑇𝑛𝑥𝑛+1 − 𝑞, 𝑗(𝑥𝑛+1 − 𝑞)〉 + 2𝑏𝑛〈𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1, 𝑗(𝑥𝑛+1 − 𝑞)〉 + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛〈𝑇𝑛𝑥𝑛+1 − 𝑞, 𝑗(𝑥𝑛+1 − 𝑞)〉 + 2𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖‖𝑥𝑛+1 − 𝑞‖ + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛(1 − 𝑘)‖𝑥𝑛+1 − 𝑞‖2 + 2𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖‖𝑥𝑛+1 − 𝑞‖ + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛(1 − 𝑘)‖𝑥𝑛+1 − 𝑞‖2 + 2𝑏𝑛𝑀‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖ + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛(1 − 𝑘)‖𝑥𝑛+1 − 𝑞‖2 + 2𝑏𝑛𝑑𝑛 + 2𝑀2𝑐𝑛 …(2.3) Where 𝑑𝑛 = 𝑀‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖ … (2.4) From (1.10) we have ‖𝑦𝑛 − 𝑥𝑛+1‖ = �(�̀�𝑛−𝑎𝑛)𝑥𝑛 + �̀�𝑛𝑇𝑛𝑥𝑛 + 𝑐�̀��̀�𝑛 + 𝑏𝑛𝑇𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛)� …(2.5) By conditions (ii-iii) and (2.5) then; lim𝑛→∞‖𝑦𝑛 − 𝑥𝑛+1‖ = 0 ⇒ lim𝑛→∞‖𝑇𝑦𝑛 − 𝑇𝑥𝑛+1‖ = 0 ⇒ lim𝑛→∞‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖ = 0 lim 𝑛→∞ 𝑑𝑛 = 0. …(2.6) Substi ‖𝑥𝑛+1 − 𝑞‖2 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛(1 − 𝑘)‖𝑥𝑛+1 − 𝑞‖2 + 2𝑏𝑛𝑑𝑛 + 2𝑀2𝑐𝑛 ≤ �1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛((1 − 𝑏𝑛) ‖𝑥𝑛 − 𝑞‖2 + 𝑀2𝑏𝑛 + 𝑀2𝑐𝑛� +2𝑏𝑛𝑑𝑛 + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2 ‖𝑥𝑛 − 𝑞‖2 +	cache/iajs-448.pdf	txt/iajs-448.txt