JOURNAL Series A - Pure and Applied Mathematics

c○Journal of Technical University at PlovdivFundamental Sciences and Applications, Vol. 11, 2005-2006Series A-Pure and Applied MathematicsBulgaria, ISSN 1310-8271L p - Equivalence between a Linear andNonlinear perturbed impulsivedifferential equations with a Generalizeddichotomous linear partAlbena Kosseva, Stepan KostadinovAbstractBy help of Schauder-Tychonoff’s fixed point theorem sufficient conditionsfor L p - equivalence between a linear and nonlinear perturbed impulsivedifferential equations with the some linear part are obtained.1 IntroductionWe consider linear and nonlinear perturbed impulsive differential equations withthe some linear part. The linear equatin is supposed to be generalized dichotomous.That means that there exist two nonnegative functions h(t) and k(t) such thatfollowing relations‖U(t)P 1 U −1 (s)‖ ≤ Mh(t)h −1 (s),(0 ≤ s < t);‖U(t)P 2 U −1 (s)‖ ≤ Mk −1 (t)k(s), (0 ≤ t ≤ s)are satisfied (P i : X → X, i = 1, 2 are projectors with P 1 + P 2 = I , g −1 (t) = 1g(t) .Itmay be noted that for ordinary differential equations the notion of (h, k)-dichotomywas introduced in [3].0 2000 Mathematics Subject Classification: 34A37, 34G10, 34D09 .0 Key words and phrases: Impulse differential equations; L p - equivalence; (h, k) - dichotomous.0 Received September 15, 200629