id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ma-134	Zhang, Dongwen; Rassias, John Michael; Liu, Qi;  Li, Yongjin	A Note on the Stability of Functional Equations via a Celebrated Direct Method	2022-11-10	19	.pdf	application/pdf	6903	505	81	−vf (−e(x))‖6 δ(x), x ∈ X, (2.1) where u, v ∈ (−∞, +∞), and the mappings e : X → X, δ : X → [0,∞) satisfy that e is even ( i.e., e(−x) = e(x) for every x ∈ X). We will prove that ‖f (x) −unf (en(x)) −vnf (−en(x))‖6 γn(x), x ∈ X, (2.3) where γn(x) := n−1∑ i=0 [ |ui|δ ( ei (x) ) + |vi|δ ( −ei (x) )] , x ∈ X,n ∈N. First of all, consider with every m,n ∈N and it is easy to observe that un+1 = uun + uvn, vn+1 = vvn + vun, and un+m = umun + vmun, vn+m = umvn + vmvn.	cache/ma-134.pdf	txt/ma-134.txt
