id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-4247	Rafeeq, Ava Shafeeq	Periodic Solution of Caputo-Fabrizio Fractional Integro–differential Equation with Periodic and Integral Boundary Conditions	2022	14	.pdf	application/pdf	4916	275	71	g ( s, u ( s, uk0 )) ds ) − 2(1−α) (2−α)N(α) 1 T ∫ T 0 h ( s, u ( s, uk0 ) , ∫ a(s) 0 g ( τ, u ( τ,uk0 )) dτ ) ds+ 2α (2−α)N(α) ∫ t 0 (h(s, u ( s, uk0 ) , ∫ a(s) 0 g ( τ, u ( τ,uk0 )) dτ)− 1 T ∫ T 0 h(s, u ( s, uk0 ) , ∫ a(s) 0 g ( τ, u ( τ,uk0 )) dτ)ds)ds (4.24) where k = 1, 2, from (4.24), we get∣∣u (t, u10)− u ( t, u20 )∣∣ ≤| u10 − u20 ] + 4(1−α) (2−α)N(a) (k1 + aTL1k2) ∣∣u (t,u10)− u ( t, u20 )∣∣+ αT (2− α)N(α) (k1 + aTL1k2) ∣∣u (t, u10)− u ( t, u20 )∣∣ (4.25) Existence and uniqueness of periodic Solution of (1.1) with integral boundary condition In this section, we investigate the periodic solution of the problem (1.1) with integral boundary conditions: u(0)− u(T ) = ∫ T 0 H(u(s))ds (4.28) Where the function H(u(s)) defined and continuous on are compact subset of R and periodic on t of periodic T. A. S. Rafeeq / Eur.	cache/ejpam-4247.pdf	txt/ejpam-4247.txt