Article
MSC:
34K06,
34K10,
34K11,
34K12,
34K38,
34K45 | MR 3368488 | Zbl 06486928 | DOI: 10.21136/MB.2015.144320
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
impulsive equation; nonlocal boundary value problem; Green's function; positivity of Green's function; negativity of Green's function; estimates of solutions
impulsive equation; nonlocal boundary value problem; Green's function; positivity of Green's function; negativity of Green's function; estimates of solutions
Summary:
We propose results about sign-constancy of Green's functions to impulsive nonlocal boundary value problems in a form of theorems about differential inequalities. One of the ideas of our approach is to construct Green's functions of boundary value problems for simple auxiliary differential equations with impulses. Careful analysis of these Green's functions allows us to get conclusions about the sign-constancy of Green's functions to given functional differential boundary value problems, using the technique of theorems about differential and integral inequalities and estimates of spectral radii of the corresponding compact operators in the space of essential bounded functions.
References:
[1] Agarwal, R. P., Berezansky, L., Braverman, E., Domoshnitsky, A.: Nonoscillation Theory of Functional Differential Equations with Applications. Springer Berlin (2012). MR 2908263 | Zbl 1253.34002
[2] Ashordia, M.: Criteria of correctness of linear boundary value problems for systems of generalized ordinary differential equations. Czech. Math. J. 46 (1996), 385-404. MR 1408294 | Zbl 0879.34037
[3] Azbelev, N. V., Maksimov, V. P., Rakhmatullina, L. F.: Introduction to the Theory of Linear Functional-Differential Equations. Methods and Applications. Advanced Series in Mathematical Science and Engineering 3 World Federation Publishers, Atlanta (1995). MR 1422013
[4] Benchohra, M., Henderson, J., Ntouyas, S. K.: Impulsive Differential Equations and Inclusions. Contemporary Mathematics and Its Applications 2 Hindawi Publishing, New York (2006). MR 2322133 | Zbl 1130.34003
[5] Benchohra, M., Henderson, J., Ntouyas, S. K.: An existence result for first-order impulsive functional differential equations in Banach spaces. Comput. Math. Appl. 42 (2001), 1303-1310. DOI 10.1016/S0898-1221(01)00241-3 | MR 1861529 | Zbl 1005.34069
[6] Domoshnitsky, A.: On periodic boundary value problem for first order impulsive functional-differential nonlinear equation. Funct. Differ. Equ. 4 (1997), 39-46. MR 1491788
[7] Domoshnitsky, A., Drakhlin, M.: Nonoscillation of first order impulse differential equations with delay. J. Math. Anal. Appl. 206 (1997), Article No. ay975231, 254-269. DOI 10.1006/jmaa.1997.5231 | MR 1429290 | Zbl 0870.34010
[8] Domoshnitsky, A., Drakhlin, M.: On boundary value problems for first order impulse functional-differential equations. Boundary Value Problems for Functional Differential Equations World Scientific Singapore (1995), 107-117 J. Henderson. MR 1375468 | Zbl 0842.34063
[9] Domoshnitsky, A., Drakhlin, M., Litsyn, E.: On boundary value problems for $N$-th order functional-differential equations with impulses. International Symposium on Differential Equations and Mathematical Physics, Tbilisi, 1997. Advances in Mathematical Sciences and Applications 8, 1998 987-996. MR 1657212 | Zbl 0901.34065
[10] Domoshnitsky, A., Drakhlin, M., Litsyn, E.: On boundary value problems for $N$-th order functional-differential equations with impulses. Mem. Differ. Equ. Math. Phys. 12 (1997), 50-56. MR 1636857 | Zbl 0901.34065
[11] Domoshnitsky, A., Volinsky, I.: About positivity of Green's functions for nonlocal boundary value problems with impulsive delay equations. TSWJ: Mathematical Analysis 2014 Article ID 978519, 13 pages (2014). MR 3368488
[12] Domoshnitsky, A., Volinsky, I., Shklyar, R.: About Green's functions for impulsive differential equations. Funct. Differ. Equ. 20 (2013), 55-81. MR 3328886 | Zbl 1318.34090
[13] Fan, Z., Li, G.: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal. 258 (2010), 1709-1727. DOI 10.1016/j.jfa.2009.10.023 | MR 2566317 | Zbl 1193.35099
[14] Federson, M., Schwabik, Š.: Generalized {ODE} approach to impulsive retarded functional differential equations. Differential Integral Equations 19 (2006), 1201-1234. MR 2278005 | Zbl 1212.34251
[15] Krasnosel'skii, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitskii, J. B., Stezenko, V. J.: Approximate Methods for Solving Operator Equations. Nauka Moskva (1969), Russian.
[16] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics 6 World Scientific, Singapore (1989). MR 1082551 | Zbl 0719.34002
[17] Li, J., Nieto, J. J., Shen, J.: Impulsive periodic boundary value problems of first-order differential equations. J. Math. Anal. Appl. 325 (2007), 226-236. DOI 10.1016/j.jmaa.2005.04.005 | MR 2273040 | Zbl 1110.34019
[18] Nieto, J. J., Rodríguez-López, R.: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J. Math. Anal. Appl. 318 (2006), 593-610. DOI 10.1016/j.jmaa.2005.06.014 | MR 2215172 | Zbl 1101.34051
[19] Pandit, S. G., Deo, S. G.: Differential Systems Involving Impulses. Lecture Notes in Mathematics 954 Springer, Berlin (1982). DOI 10.1007/BFb0067476 | MR 0674119 | Zbl 0539.34001
Search
Partner of
