Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems

A.  Domoshnitsky; Iu.  Mizgireva; V.  Raichik

doi:10.37863/umzh.v73i7.473

A. Domoshnitsky Ariel Univ., Israel
Iu. Mizgireva Ariel Univ., Israel
V. Raichik Ariel Univ., Israel

DOI: https://doi.org/10.37863/umzh.v73i7.473

Keywords: second order impulsive differential equations, semi-nonoscillation intervals

Abstract

UDC 517.9

We consider the second order impulsive differential equation with delays

$$(Lx)(t)\equiv x''(t) + \sum\limits _{j = 1}^{p} a_j(t) x'(t-\tau_j(t)) + \sum\limits _{j = 1}^{p} b_j(t) x(t-\theta_j(t)) = f(t), \quad t \in [0, \omega], $$ $$x(t_k) = \gamma_k x(t_k-0), \quad x'(t_k) = \delta_k x'(t_k-0),\quad k = 1, 2, \ldots , r,$$

where $\gamma_k > 0,$ $\delta_k >0$ for $k = 1, 2, \ldots , r.$ In this paper, we obtain the conditions of semi-nonoscillation for the corresponding homogeneous equation on the interval $[0, \omega].$ Using these results, we formulate theorems on sign-constancy of Green's functions for two-point impulsive boundary-value problems in terms of differential inequalities.

Author Biography

V. Raichik, Ariel Univ., Israel

References

R. P. Agarwal, L. Berezansky, E. Braverman, A. Domoshnitsky, Nonoscillation theory of functional differential equations with applications, Springer (2012), https://doi.org/10.1007/978-1-4614-3455-9 DOI: https://doi.org/10.1007/978-1-4614-3455-9

R. P. Agarwal, D. O’Regan, Multiple nonnegative solutions for second order impulsive differential equations, Appl. Math. and Comput., 114, № 1, 51 – 59 (2000), https://doi.org/10.1016/S0096-3003(99)00074-0 DOI: https://doi.org/10.1016/S0096-3003(99)00074-0

N. V. Azbelev, About zeros of solutions of linear differential equations of the second order with delayed argument, Differents. Uravn., 7, № 7, 1147 – 1157 (1971).

N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, Introduction to the theory of functional differential equations: methods and applications, Contemp. Math. and Its Appl., vol. 3, Hindawi, New York (2007), https://doi.org/10.1155/9789775945495 DOI: https://doi.org/10.1155/9789775945495

A. A. Boichuk, A. M. Samoilenko, Generalized inverse operators and Fredholm boundary-value problems, VSP, Utrecht; Boston (2004), https://doi.org/10.1515/9783110944679 DOI: https://doi.org/10.1515/9783110944679

A. Domoshnitsky, Wronskian of fundamental system of delay differential equations, Funct. Different. Equat., 7, 445 – 467 (2002).

A. Domoshnitsky, M. Drakhlin, Nonoscillation of first order impulsive differential equations with delay, J. Math. Anal. and Appl., 206, № 1, 254 – 269 (1997), https://doi.org/10.1006/jmaa.1997.5231 DOI: https://doi.org/10.1006/jmaa.1997.5231

A. Domoshnitsky, G. Landsman, Semi-nonoscillation intervals in the analysis of sign constancy of Green’s functions of Dirichlet, Neumann and focal impulsive problems, Adv. Differerence Equat., 81,№1 (2017), https://doi.org/10.1186/s13662-017-1134-1 DOI: https://doi.org/10.1186/s13662-017-1134-1

A. Domoshnitsky, G. Landsman, S. Yanetz, About sign-constancy of Green’s functions of one-point problem for impulsive second order delay equations, Funct. Different. Equat., 21, № 1-2, 3 – 15 (2014). DOI: https://doi.org/10.7494/OpMath.2014.34.2.339

A. Domoshnitsky, V. Raichik, The Sturm separation theorem for impulsive delay differential equations, Tatra MT Math. Publ., 71, № 1, 65 – 70 (2018), https://doi.org/10.2478/tmmp-2018-0006 DOI: https://doi.org/10.2478/tmmp-2018-0006

A. Domoshnitsky, I. Volinsky, About differential inequalities for nonlocal boundary-value problems with impulsive delay equations, Math. Bohem., 140>, № 2, 121 – 128 (2015). DOI: https://doi.org/10.21136/MB.2015.144320

A. Domoshnitsky, I. Volinsky, About positivity of Green’s functions for nonlocal boudary value problems with impulsive delay equations, Sci. World J. (TSWJ): Math. Anal., Article ID 978519, https://doi.org/10.1155/2014/978519 (2014). DOI: https://doi.org/10.1155/2014/978519

A. Domoshnitsky, I. Volinsky, R. Shklyar, About Green’s functions for impulsive differential equations, Funct. Different. Equat., 20, № 1-2, 55 – 81 (2013).

M. Feng, D. Xie, Multiple positive solutions of multi-point boundary-value problem for second-order impulsive differential equations, J. Math. Anal. and Appl., 223, № 1, 438 – 448 (2009), https://doi.org/10.1016/j.cam.2008.01.024 DOI: https://doi.org/10.1016/j.cam.2008.01.024

L. Hu, l. Liu, Y. Wu, Positive solutions of nonlinear singular two-point boundary-value problems for second-order impulsive differential equations, Appl. Math. and Comput., 196, № 2, 550 – 562 (2008), https://doi.org/10.1016/j.amc.2007.06.014 DOI: https://doi.org/10.1016/j.amc.2007.06.014

T. Jankowsky, Positive solutions to second order four-point boundary-value problems for impulsive differential equations, Appl. Math. and Comput., 202, № 2, 550 – 561 (2008), https://doi.org/10.1016/j.amc.2008.02.040 DOI: https://doi.org/10.1016/j.amc.2008.02.040

D. Jiang, X. Lin, Multiple positive solutions of Dirichlet boundary-value problems for second order impulsive differential equations, J. Math. Anal. and Appl., 321, № 2, 501 – 514 (2006), https://doi.org/10.1016/j.jmaa.2005.07.076 DOI: https://doi.org/10.1016/j.jmaa.2005.07.076

M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreyko, Y. B. Ruticki, V. Y. Stet’senko, Approximate solution of

operator equations, Nauka, Moscow (1969) (in Russian).

S. M. Labovskii, Condition of nonvanishing of Wronskian of fundamental system of linear equation with delayed argument>, Differents. Uravn., 10, № 3, 426 – 430 (1974).

V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, World Sci., Singapore (1989), https://doi.org/10.1142/0906 DOI: https://doi.org/10.1142/0906

E. K. Lee, Y. H. Lee, Multiple positive solutions of singular two point boundary-value problems for second order impulsive differential equations, Appl. Math. and Comput., 158, № 3, 745 – 759 (2004), https://doi.org/10.1016/j.amc.2003.10.013 DOI: https://doi.org/10.1016/j.amc.2003.10.013

Iu. Mizgireva, On Positivity of Green’s functions of two-point impulsive problems, Funct. Different. Equat., № 3-4 (2018).

S. G. Pandit, S. G. Deo, Differential systems involving impulses, Lect. Notes Math., vol. 954, Springer-Verlag, New York (1982). DOI: https://doi.org/10.1007/BFb0067476

A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, World Sci., Singapore (1992), https://doi.org/10.1142/9789812798664 DOI: https://doi.org/10.1142/9789812798664

S. G. Zavalishchin, A. N. Sesekin, Dynamic impulse systems: theory and applications, Math. and Appl., 394, Kluwer, Dordrecht (1997), https://doi.org/10.1007/978-94-015-8893-5 DOI: https://doi.org/10.1007/978-94-015-8893-5