JOURNAL Series A - Pure and Applied Mathematics

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62 At. Georgieva, St. KostadinovIn case N > 1 we setQ n η =N −12η (η ∈ X)(M n + 1)Then for x, y ∈ D(A(s)) <strong>and</strong> t k−1 < s ≤ t k ≤ t n < t ≤ t n+1 we have‖V (t, s)x − V (t, s)y‖ ≤≤ M(t,tn)M n+1 N −1 M(t n,t n−1 )(M n+1)(M n−1 +1) N −1 . . .≤ N −(n−k+1) ‖x − y‖ ≤ N‖x − y‖Let f(t, x) ≠ 0. We denote∫ tmax Nt n−1 ≤s≤t≤t n+1In case 0 < N ≤ 1 we setsQ n η =M(t k−1 ,t k )N −1 M(t k ,s)(M k−1 +1)(M k +1) M k +1L(ξ)dξ = p n > 0, (n = 0, ±1, ±2, . . .)N2(M n + 1) 2 e−pn η, (η ∈ X).‖x − y‖ ≤Let h n : X → D(A(t n )) are arbitrary Lipschitz continuous functions with Lipschitzconstants H n < 1 N epn , (n = 0, ±1, ±2, . . .).Then for t k−1 < s ≤ t k ≤ t n < t ≤ t n+1 we have‖V (t, s)x‖ ≤ ( N 2 ) n−k+1e −pn−...−p k‖x‖ < Ne −pn−...−p k‖x‖ < N‖x‖<strong>and</strong> hence for x, y ∈ D(A(s)) we obtain‖X(t, s)x − X(t, s)y‖ ≤ ‖V (t, s)(x − y)‖+∫ t+ ‖V (t, ξ)(f(ξ, x(ξ)) − f(ξ, y(ξ)))‖dξ+s+ ∑ ‖V (t, t + n )(h n (x(t n )) − h n (y(t n )))‖ ≤s
Nonlinear Semigroup for Nonlinear Abstract Impulse Differential Equations 63Applying Lemma 3 we obtain‖X(t, s)x − X(t, s)y‖ ≤≤ ( N 2 )n−k+1 e −pn−...−p k ‖x − y‖≤ ( N 2 )n−k+1 ‖x − y‖×e∫ t−p n+N L(ξ)dξt n−1∏s
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JOURNAL Series A - Pure and Applied Mathematics

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