id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-109	Burton, Theodore A.; Purnara, Ioannis K.	Open mappings: The case for a new direction in fixed point theory	2022	30	.pdf	application/pdf	12917	676	84	Moreover, writing the second summand in the right-hand-side of (9.10) as β t2 + 1 sin ( t2 + x2(t) t2 + 1 )∫ t 0 s (t2 + 1)te−t + |x(s)|p 1 + s2 + t2 ds t ≥ 0, it is not difficult to see that for any bounded function x it holds 0 ≤ β t2 + 1 ∣∣∣ sin( t2 + x2(t) t2 + 1 )∣∣∣ ∫ t 0 s (t2 + 1)te−t + |x(s)|p 1 + s2 + t2 ds ≤ βte−t t2 + 1 ∫ t 0 sds+ β‖x‖p t2 + 1 ∫ t 0 s 1 + s2 + t2 ds ≤ βt3e−t 2(t2 + 1) + β‖x‖p t2 + 1 ∫ t 0 s 1 + t2 ds ≤ βte−t + β‖x‖pt2 2(t2 + 1)2 , and so lim t→∞ β t2 + 1 sin ( t2 + x2(t) t2 + 1 )∫ t 0 s (t2 + 1)te−t + |x(s)|p 1 + s2 + t2 ds = 0. + ∫ t 0 K(t, s)v(t, s, x(s))ds, for their importance in applied mathematics and the fact that the integral term un- der a wide set of conditions will define a compact map	cache/ejde-109.pdf	txt/ejde-109.txt