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Sufficient conditions for the lp-equivalence between two nonlinear impulse differential equations

Sufficient conditions for the Lp-equivalence between two nonlinear impulse differential equations with unbounded linear parts and possibly unbounded nonlinearity parts are given. An example of two nonlinear impulse differential parabolic equations is considered. | Turk J Math 32 (2008) , 451 – 466. ¨ ITAK ˙ c TUB Sufficient Conditions for the Lp-Equivalence Between two Nonlinear Impulse Differential Equations A. Georgieva, S. Kostadinov Abstract Sufficient conditions for the Lp-equivalence between two nonlinear impulse differential equations with unbounded linear parts and possibly unbounded nonlinearity parts are given. An example of two nonlinear impulse differential parabolic equations is considered. Key words and phrases: Impulse differential equations, Lp-equivalence, Partial impulse differential equations of parabolic type. 1. Introduction We consider the Lp-equivalence between two nonlinear impulse differential equations with unbounded linear parts and possibly unbounded nonlinearity parts. This means that every solution of the one equation which lies in a closed, convex set induces a solution of another equation which lies in a possibly another closed and convex set and vice versa. Moreover the difference of the two solutions lies in the space Lp (1 0 (i = 1, 2) such that for t ∈ R+ (λI + Ai (t)) −1 ≤ ci 1 + |λ| (4) for each λ which Reλ ≤ 0 hold. From (4) it follows that A−α i (t) ≤ ci (5) Vi (t, s) = Ui (t, tn )Qin Ui (tn , tn−1 )Qin−1 .Qik Ui (tk , s) (6) for α ∈ (0, 1) and t ∈ R+ (i = 1, 2). It is not hard to check that 452 GEORGIEVA, KOSTADINOV (0 ≤ s ≤ tk < tn < t; i = 1, 2) are the Cauchy operators of the linear part of the impulse equations (1), (2). Lemma 1 Let condition (H1) holds and integral equations wi (t) t α = Vi (t, 0)Aα (0)ξ + Ai (0)Vi (t, s)fi (s, A−α i i i (0)wi (s))ds+ 0 −α + i + Aα i (0)Vi (t, tn )hn (Ai (0)wi (tn )) (7) 0