This paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure of noncompactness, and the linear part generates only a strongly continuous evolution system.
@article{bwmeta1.element.doi-10_2478_s11533-013-0367-9,
author = {Leszek Olszowy},
title = {Existence of mild solutions for semilinear differential equations with nonlocal and impulsive conditions},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {623-635},
zbl = {06271198},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0367-9}
}
Leszek Olszowy. Existence of mild solutions for semilinear differential equations with nonlocal and impulsive conditions. Open Mathematics, Tome 12 (2014) pp. 623-635. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0367-9/
[1] Akhmerov R.R., Kamenskii M.I., Potapov A.S., Rodkina A.E., Sadovskii B.N., Measure of Noncompactness and Condensing Operators, Oper. Theory Adv. Appl., 55, Birkhäuser, Basel, 1992 http://dx.doi.org/10.1007/978-3-0348-5727-7
[2] Akhmerov R.R., Kamenskii M.I., Potapov A.S., Sadovskii B.N., Condensing operators, J. Soviet Math., 1982, 18(4), 551–592 http://dx.doi.org/10.1007/BF01084869 | Zbl 0477.47032
[3] Banaś J., Goebel K., Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math., 60, Marcel Dekker, New York, 1980 | Zbl 0441.47056
[4] Bothe D., Multivalued perturbation of m-accretive differential inclusions, Israel. J. Math., 1998, 108, 109–138 http://dx.doi.org/10.1007/BF02783044 | Zbl 0922.47048
[5] Cardinali T., Rubbioni P., Impulsive semilinear differential inclusions: Topological structure of the solution set and solutions on non-compact domains, Nonlinear Anal., 2008, 69(1), 73–84 http://dx.doi.org/10.1016/j.na.2007.05.001 | Zbl 1147.34045
[6] Cardinali T., Rubbioni P., Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces, Nonlinear Anal., 2012, 75(2), 871–879 http://dx.doi.org/10.1016/j.na.2011.09.023 | Zbl 1252.34068
[7] Fan Z., Impulsive problems for semilinear differential equations with nonlocal conditions, Nonlinear Anal., 2010, 72(2), 1104–1109 http://dx.doi.org/10.1016/j.na.2009.07.049 | Zbl 1188.34073
[8] Fan Z., Li G., Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 2010, 258(5), 1709–1727 http://dx.doi.org/10.1016/j.jfa.2009.10.023 | Zbl 1193.35099
[9] Ji S., Li G., Existence results for impulsive differential inclusions with nonlocal conditions, Comput. Math. Appl., 2011, 62(4), 1908–1915 http://dx.doi.org/10.1016/j.camwa.2011.06.034
[10] Liang J., Liu J.H., Xiao T.-J., Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Modelling, 2009, 49(3–4), 798–804 http://dx.doi.org/10.1016/j.mcm.2008.05.046
[11] Mönch H., Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 1980, 4(5), 985–999 http://dx.doi.org/10.1016/0362-546X(80)90010-3
[12] Olszowy L., On existence of solutions of a quadratic Urysohn integral equation on an unbounded interval, Comment. Math., 2008, 48(1), 103–112 | Zbl 1179.45005
[13] Olszowy L., On some measures of noncompactness in the Fréchet spaces of continuous functions, Nonlinear Anal., 2009, 71(11), 5157–5163 http://dx.doi.org/10.1016/j.na.2009.03.083 | Zbl 1179.45007
[14] Olszowy L., Solvability of some functional integral equation, Dynam. Systems Appl., 2009, 18(3–4), 667–676
[15] Olszowy L., Fixed point theorems in the Fréchet space C(ℝ+) and functional integral equations on an unbounded interval, Appl. Math. Comput., 2012, 218(18), 9066–9074 http://dx.doi.org/10.1016/j.amc.2012.03.044 | Zbl 1245.45006
[16] Olszowy L., Existence of mild solutions for semilinear nonlocal Cauchy problems in separable Banach spaces, Z. Anal. Anwend., 2013, 32(2), 215–232 http://dx.doi.org/10.4171/ZAA/1482 | Zbl 1272.34082
[17] Olszowy L., Existence of mild solutions for the semilinear nonlocal problem in Banach spaces, Nonlinear Anal., 2013, 81, 211–223 http://dx.doi.org/10.1016/j.na.2012.11.001 | Zbl 1298.34107
[18] Pazy A., Semigroup of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44, Springer, New York, 1983 http://dx.doi.org/10.1007/978-1-4612-5561-1
[19] Sadovskii B.N., Limit-compact and condensing operators, Russian Math. Surveys, 1972, 27(1), 85–156 http://dx.doi.org/10.1070/RM1972v027n01ABEH001364
[20] Wang J., Wei W., A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces, Results Math., 2010, 58(3–4), 379–397 http://dx.doi.org/10.1007/s00025-010-0057-x