id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-1812	Liang, Jin; Mu, Yunyi; Xiao, Ti-Jun	Evolution psi-Hilfer fractional differential equations in Banach spaces	2025	17	.pdf	application/pdf	7228	512	78	respectively:( Dα,β;ψx ) (t) = Ax(t) + f ( t, x(t), ∫ t 0 ρ(t, s)x(s)ds ) , 0 < α < 1, 0 ≤ β ≤ 1, t ∈ (0, b], I1−γ;ψx(0) = x0, α ≤ γ = α+ β − αβ < 1, (4.4) and( Dα′,β′;ψy ) (t) = Ay(t) + f ( t, y(t), ∫ t 0 ρ(t, s)y(s)ds ) , 0 < α′ < 1, 0 ≤ β′ As a matter of fact, for each x1, x2 ∈ C1−α−β(1−α);ψ(J,X) and t ∈ J , we can obtain ψ1−α−β(1−α)(t)∥(Tx1)(t)− (Tx2)(t)∥ ≤ ψ1−α−β(1−α)(t) ∫ t 0 ∥Kα(ψ(t)− ψ(s))[f(s, x1(s))− f(s, x2(s))]∥ψ′(s)ds ≤ Mℓ2 Γ(α) ψ1−α−β(1−α)(t) ∫ t 0 (ψ(t)− ψ(s))α−1ψα+β(1−α)−1(s) 6 J. LIANG, Y. MU, T.-J. XIAO EJDE-2025/109 × ψ1−α−β(1−α)(s)∥x1(s)− x2(s)∥ψ′(s)ds ≤ Γ(α+ β(1− α))Mℓ2ψ α(t) Γ(2α+ β(1− α)) ∥x1	cache/ejde-1812.pdf	txt/ejde-1812.txt