id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
iajs-2285	Mohamed Hasan, Zahra Mahmood; Abed, Salwa Salman	Strongly Convergence of Two Iterations For a Common Fixed Point with an Application	2019	12	.pdf	application/pdf	4873	231	87	‖𝑎𝑛+1 − 𝑤𝑛+1‖ = ‖𝑆𝑏𝑛 − (1 − 𝛼𝑛)𝑤𝑛 − 𝛼𝑛𝑆𝑢𝑛‖ ≤ ‖𝑆𝑏𝑛 − 𝑤𝑛‖ + 𝛼𝑛‖𝑆𝑢𝑛 − 𝑤𝑛‖ ≤ (𝛿 + 2𝜇 + 2𝜔)‖𝑏𝑛 − 𝑎 ∗‖ + 𝛼𝑛(𝛿 + 2𝜇 + 2𝜔)‖𝑢𝑛 − 𝑎 ∗‖ + (1 + 𝛼𝑛) (𝛿 + 2𝜇 + 2𝜔)‖𝑤𝑛 − 𝑎 ∗‖ ≤ ‖𝑏𝑛 − 𝑎 ∗‖ + 𝛼𝑛‖𝑢𝑛 − 𝑎 ∗‖ + (1 + 𝛼𝑛)‖𝑤𝑛 − 𝑎 ∗‖ ‖𝑏𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛼𝑛)‖𝑎𝑛 − 𝑎 ∗‖ + 𝛼𝑛(𝑀 + ‖𝑎 ∗‖) ‖𝑣𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛾𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛾𝑛‖𝑇𝑤𝑛 − 𝑎 ∗‖ Since T is lipschitzain and firmly nonexpansive, setting 𝜉 = 𝑘 − 𝑘𝑡 + 𝑡 ≤ (1 − 𝛾𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛾𝑛𝜉‖𝑤𝑛 − 𝑎 ∗‖ ≤ ‖𝑤𝑛 − 𝑎 ∗‖ ‖𝑢𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛽𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛽𝑛‖𝑇𝑣𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛽𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛽𝑛𝜉‖𝑣𝑛 − 𝑎 ∗‖ ≤ ‖𝑤𝑛 − 𝑎 ∗‖ ≤ 𝑀 + ‖𝑎∗‖ Proof: 1-‖𝑏𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛼𝑛)‖𝑎𝑛 − 𝑎 ∗‖ + 𝛼𝑛‖𝑇𝑎𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛼𝑛)‖𝑎𝑛 − 𝑎 ∗‖ + 𝛼𝑛[(1 − 𝑡)𝑘 + 𝑡]‖𝑎𝑛 − 𝑎 ∗‖ Suppose 𝜉 = (1 − 𝑡)𝑘 + 𝑡 ≤ (1 − (1 − 𝜉)𝛼𝑛)‖𝑎𝑛 − 𝑎 ∗‖ ‖𝑎𝑛+1 − 𝑎 ∗‖ = ‖𝑆𝑏𝑛 − 𝑎 ∗‖ ≤ 𝛿‖𝑏𝑛 − 𝑎 ∗‖ + 𝜇{‖𝑎∗ − 𝑎∗‖ + ‖𝑏𝑛 − 𝑆𝑏𝑛‖} + 𝜔{‖𝑎 ∗ − 𝑆𝑏𝑛‖ + ‖𝑏𝑛 − 𝑎 ∗‖} ≤ (𝛿 + 2𝜇 + 2𝜔)‖𝑏𝑛 − 𝑎 ∗‖ ≤ ‖𝑏𝑛 − 𝑎 ∗‖ ≤ (1 − (1 − 𝜉)𝛼𝑛)‖𝑎𝑛 − 𝑎 ∗‖ By induction: ‖𝑎𝑛+1 − 𝑎 ∗‖ ≤ ∏ (1 − (1 − 𝜉)𝛼𝑖)‖𝑎0 − 𝑎 ∗‖𝑛 𝑖=0 ≤ ‖𝑎0 − 𝑎 ∗‖𝑒−(1−𝜉)∑ 𝛼𝑖 𝑛 𝑖=0 Since ∑ 𝛼𝑖 ∞ 𝑖=0 = ∞, 𝑒−(1−𝜉)∑ 𝛼𝑖 𝑛 𝑖=0 → 0 𝑎𝑠 𝑛 → ∞. Thus, lim𝑛→∞‖𝑎𝑛 − 𝑎 ∗‖ = 0. 2-‖𝑣𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛾𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛾𝑛‖𝑇𝑤𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛾𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛾𝑛[(1 − 𝑡)𝑘 + 𝑡]‖𝑤𝑛 − 𝑎 ∗‖ Setting 𝜉 = (1 − 𝑡)𝑘 + 𝑡 ≤ (1 − 𝛾𝑛 + 𝛾𝑛𝜉)‖𝑤𝑛 − 𝑎 ∗‖ ‖𝑢𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛽𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛽𝑛‖𝑇𝑣𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛽𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛽𝑛𝜉‖𝑣𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛽𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛽𝑛𝜉(1 − 𝛾𝑛 + 𝛾𝑛𝜉)‖𝑤𝑛 − 𝑎 ∗‖ ≤ (1 − 𝛽𝑛)‖𝑤𝑛 − 𝑎 ∗‖ + 𝛽𝑛(1 − 𝛾𝑛 + 𝛾𝑛𝜉)‖𝑤𝑛 − 𝑎 ∗‖ 88 Ibn Al-Haitham Jour.	cache/iajs-2285.pdf	txt/iajs-2285.txt