h-Stability for Differential Systems Relative to Initial Time Difference

and Applied Analysis 3 Moreover, using the condition (iii), we obtain z (t) ≤ h (t) h −1 (τ 0 ) ( 󵄩󵄩󵄩󵄩y0 − x0 󵄩󵄩󵄩󵄩 + 󵄨󵄨󵄨󵄨η 󵄨󵄨󵄨󵄨) . (12) Then from (12), we get 󵄩󵄩󵄩󵄩x (t, τ0, y0) − x (t − η, t0, x0) 󵄩󵄩󵄩󵄩 ≤ [ 󵄩󵄩󵄩󵄩y0 − x0 󵄩󵄩󵄩󵄩 + 󵄨󵄨󵄨󵄨τ0 − t0 󵄨󵄨󵄨󵄨] h (t) h −1 (τ 0 ) . (13) So by Definition 1 with c = 1, the solution x(t, τ 0 , y 0 ) of (2) is ITDhS with respect to the solution x(t − η, t 0 , x 0 ). This completes the proof. Remark 5. Set h(t) = e0, and then we can obtain Theorem 3.4 in [8]. Theorem 6. Let y(t, τ 0 , y 0 ) be the solution of (3) through (τ 0 , y 0 ). Assume that (i) the solution x(t, τ 0 , y 0 ) of (2) is ITDhS with respect to the solution x(t − η, t 0 , x 0 ) for t ≥ τ 0 , where x(t, t 0 , x 0 ) is any solution of (1); (ii) there exist c ≥ 1, α > 0 and a positive bounded continuous function h defined on R+ such that 󵄩󵄩󵄩󵄩Φ (t, s, y (s)) 󵄩󵄩󵄩󵄩 ≤ ch (t) h −1 (s) , 󵄩󵄩󵄩󵄩R (s, y (s)) 󵄩󵄩󵄩󵄩 ≤ r (s) 󵄩󵄩󵄩󵄩y (s) 󵄩󵄩󵄩󵄩 , (14) provided that y(s, τ 0 , y 0 ) ≤ α, r(s) ∈ C(R + , R + ) and