id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
cana-1536	V. Rajitha	Tripled Fixed Point Results in G_b-Metric Spaces	2024	8	.pdf	application/pdf	3376	161	79	Since ℱ commutes with ℋ and by equation (3.4.1) ₢𝑏(ℱℱℐ𝑛+1, ℋ(𝜌, 𝜎, 𝔡), ℋ(𝜌, 𝜎, 𝔡)) + ₢𝑏(ℱℱή𝑛, ℋ(𝜎, 𝔡, 𝜌), ℋ(𝜎, 𝔡, 𝜌)) + ₢𝑏(ℱℱϱ𝑛 , ℋ(𝔡, 𝜌, 𝜎), ℋ(𝔡, 𝜌, 𝜎)) = ₢𝑏 (ℱ(ℋ(ℐ𝑛, ή𝑛, ϱ𝑛)), ℋ(𝜌, 𝜎, 𝔡), ℋ(𝜌, 𝜎, 𝔡)) + ₢𝑏 (ℱ(ℋ(ή𝑛, ϱ𝑛, ℐ𝑛)), ℋ(𝜎, 𝔡, 𝜌), ℋ(𝜎, 𝔡, 𝜌)) + ₢𝑏 (ℱ(ℋ(ϱ𝑛, ℐ𝑛, ή𝑛)), ℋ(𝔡, 𝜌, 𝜎), ℋ(𝔡, 𝜌, 𝜎)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 437 https://internationalpubls.com = ₢𝑏 (ℋ((ℱℐ𝑛, ℱή𝑛, ℱϱ𝑛)), ℋ(𝜌, 𝜎, 𝔡), ℋ(𝜌, 𝜎, 𝔡)) + ₢𝑏 (ℋ((ℱή𝑛, ℱϱ𝑛, ℱℐ𝑛)), ℋ(𝜎, 𝔡, 𝜌), ℋ(𝜎, 𝔡, 𝜌)) + ₢𝑏 (ℋ((ℱϱ𝑛, ℱℐ𝑛, ℱή𝑛)), ℋ(𝔡, 𝜌, 𝜎), ℋ(𝔡, 𝜌, 𝜎)) ≤ 𝜃[₢𝑏(ℱℱℐ𝑛, ℱ𝜌, ℱ𝜌) + ₢𝑏(ℱℱή𝑛, ℱ𝜎, ℱ𝜎) + ₢𝑏(ℱℱϱ𝑛, ℱ𝔡, ℱ𝔡)]. From (3.4.4)we get ₢𝑏(ℱℱℐ𝑛+1, ℋ(𝜌, 𝜎, 𝔡), ℋ(𝜌, 𝜎, 𝔡)) + ₢𝑏(ℱℱή𝑛, ℋ(𝜎, 𝔡, 𝜌), ℋ(𝜎, 𝔡, 𝜌)) + ₢𝑏(ℱℱϱ𝑛 , ℋ(𝔡, 𝜌, 𝜎), ℋ(𝔡, 𝜌, 𝜎)) → 0 𝑎𝑠 𝑛 → ∞. On the other hand, ℱ is continuous, ₢𝑏(ℱℱℐ𝑛+1, ℋ(𝜌, 𝜎, 𝔡), ℋ(𝜌, 𝜎, 𝔡)) + ₢𝑏(ℱℱή𝑛, ℋ(𝜎, 𝔡, 𝜌), ℋ(𝜎, 𝔡, 𝜌)) + ₢𝑏(ℱℱϱ𝑛, ℋ(𝔡, 𝜌, 𝜎), ℋ(𝔡, 𝜌, 𝜎)) → ₢𝑏(ℱ𝜌, ℋ(𝜌, 𝜎, 𝔡), ℋ(𝜌, 𝜎, 𝔡)) + ₢𝑏(ℱ𝜎, ℋ(𝜎, 𝔡, 𝜌), ℋ(𝜎, 𝔡, 𝜌)) + ₢𝑏(ℱ𝔡, ℋ(𝔡, 𝜌, 𝜎), ℋ(𝔡, 𝜌, 𝜎)) 𝑛 → ∞. It gives ₢𝑏(ℱ𝜌, ℋ(𝜌, 𝜎, 𝔡), ℋ(𝜌, 𝜎, 𝔡)) + ₢𝑏(ℱ𝜎, ℋ(𝜎, 𝔡, 𝜌), ℋ(𝜎, 𝔡, 𝜌)) + ₢𝑏(ℱ𝔡, ℋ(𝔡, 𝜌, 𝜎), ℋ(𝔡, 𝜌, 𝜎))	cache/cana-1536.pdf	txt/cana-1536.txt