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Impulsive retarded differential equations in Banach spaces via Bochner-Lebesgue and Henstock integrals. (English) Zbl 1011.34070
By means of the integral of Henstock, the authors provide conditions for the existence and uniqueness of solutions as well as continuous dependence on the initial conditions of the system $x'(t)=f(t,x_{t}), \quad t\not=t_{k}, \quad k=1,\ldots,m,\quad \Delta x(t_{k})=I_{k}(x(t_{k})), \quad k=1,\dots,m,\;x_{0}=\phi,$ where $$\phi$$ is a given continuous function defined on $$[-r,t_{0}], \;r\geq 0, \;f$$ is a continuous map from an open set $$\Omega\subset \mathbb{R}\times C([-r,t_{0}],X),$$ $$t_{k}, \;k=1,\dots,m$$, are pre-assigned moments of impulses from the interval $$[t_{0},t_{0}+a], \;a\geq 0, \;x\to I_{k}(x)$$ maps the Banach space $$X$$ into itself and $$\Delta x(t_{k}):=x(t_{k}^{+})-x(t_{k}^{-}), \;k=1,\dots, m.$$

##### MSC:
 34K45 Functional-differential equations with impulses 34K30 Functional-differential equations in abstract spaces
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##### References:
 [1] Artstein, Z., Topological dynamics of ordinary differential equations and kurzweil equations, J. differential equations, 23, 224-243, (1977) · Zbl 0353.34044 [2] Bongiorno, B., Relatively weakly compact sets in the Denjoy space, J. math. stud., 27, 37-43, (1994) · Zbl 1045.26502 [3] Bullen, P.S.; Sarkhel, D.N., On the solution of (dy/dx)ap=f(x,y), Math. anal. appl., 127, 365-376, (1987) · Zbl 0644.34005 [4] Chew, T.S.; Flordeliza, F., On $$ẋ=f(t,x)$$ and henstock – kurzweil integrals, Differential integral equations, 4, 4, 861-868, (1991) · Zbl 0733.34004 [5] Chew, T.S.; van-Brunt, B.; Wake, G.C., On retarded functional differential equations and henstock – kurzweil integrals, Differential integral equations, 9, 3, 569-580, (1996) · Zbl 0873.34054 [6] Federson, M., The monotone convergence theorem for multi-dimensional kurzweil vector integrals, Seminár. brasil. anál., 45, 827-833, (1997) [7] Federson, M., The fundamental theorem of calculus for multidimensional Banach space-valued Henstock vector integral, Real anal. exchange, 25, 1, (1999 2000) [8] Federson, M.; Bianconi, R., Linear integral equations of Volterra concerning the integral of Henstock, Real anal. exchange, 25, 1, (1999 2000) [9] Gengian, L., On necessary conditions for Henstock integrability, Real anal. exchange, 18, 522-531, (1992 93) [10] Guo, D., Initial value problem for nonlinear second order impulsive integro-differential equations in Banach spaces, J. math. anal. appl., 200, 1-13, (1996) · Zbl 0851.45012 [11] Hale, J., Theory of functional differential equations, (1977), Springer Berlin [12] Hale, J., Ordinary differential equations, Pure and appl. math., Vol. 21, (1980), Wiley-Interscience New York [13] Henstock, R., The general theory of integration, Oxford math. monographs, (1991), Clarendon Press Oxford [14] Hönig, C.S., Volterra-Stieltjes integral equations, Math. studies, Vol. 16, (1975), North-Holland Amsterdam [15] Hönig, C.S., There is no natural Banach space norm on the space of kurzweil – henstock – denjoy – perron integrable functions, Seminár. brasil. anál., 30, 387-397, (1989) [16] Hönig, C.S., A Riemanian characterization of the bochner – lebesgue integral, Seminár. brasil. anál., 35, 351-358, (1992) [17] Kurzweil, J., Nichtabsolut konvergente integrale, (1980), Leipzig [18] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 [19] Lee, P.Y., Lanzhou lectures on Henstock integration, (1989), World Scientific Singapore · Zbl 0699.26004 [20] Mawhin, J., Introdunction a l’analyse, (1983), Cabay Louvain-la-Neuve [21] Mc Leod, R.M., The generalized Riemann integral, Carus math. monog., 20, (1980), The Math. Assoc. of America [22] McShane, E.J., A unified theory of integration, Amer. math. monthly, 80, 349-359, (1973) · Zbl 0266.26008 [23] Pfeffer, W.F., The Riemann approach to integration, (1993), Cambridge · Zbl 0554.26008 [24] L. Schwartz, Méthodes mathématiques pour les sciences physiques, Hermann, Paris, 1961 (English translation: Mathematics for physical sciences, Addison-Wesley, Reading, MA, 1966). [25] Schwabik, S., The Perron integral in ordinary differential equations, Differential integral equations, 6, 4, 863-882, (1993) · Zbl 0784.34006
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