Boundary value problems on the half-line with impulses and infinite delay. (English) Zbl 1009.34059

The following problem is considered \[ \begin{cases} (Lx)(t)+ f(t, x_t)= 0,\;t\neq t_k,\;\Delta x|_{t=t_k}= I_k(x_{t_k}),\;k= 1,2,\dots, m,\\ \lambda x(0)- \beta\lim_{t\to 0} p(t) x'(t)= a,\;\gamma x(\infty)+ \delta\lim_{t\to\infty} p(t) x'(t)= b,\\ x(t)\text{ is bounded on }[0,+\infty),\end{cases}\tag{1} \] where \(x_t\) is defined by \(x_t(s)= \begin{cases} x(t+ s),\;t\geq t+ s\geq 0;\\ \phi(t+ s),\;-\infty< t+ s< 0,\end{cases}\) \[ (Lx)(t)= {1\over p(t)} (p(t) x'(t))',\;p\in C([0, +\infty), R)\cap C^1(0,+\infty),\;p(t)> 0\quad\text{for }t\in (0,\infty), \] \(\Delta x|_{t_k}= \lim_{\varepsilon\to 0^+} [x(t_k+ \varepsilon)- x(t_k- \varepsilon)]\) and \(\lambda\), \(\beta\), \(a\), \(\gamma\), \(\delta\), \(b\), \(\phi(t)\) are given. The existence and uniqueness of a solution to problem (1) are proved.
Reviewer: A.Kh.Shamilov


34K10 Boundary value problems for functional-differential equations
34K45 Functional-differential equations with impulses
Full Text: DOI


[1] Chen, S.Z.; Zhang, Y., Singular boundary value problems on a half-line, J. math. anal. appl., 195, 449-468, (1995) · Zbl 0852.34019
[2] Kawano; Yanagida; Yotsutani, Structure theorems for positive radial solutions to δu+K(|x|)up=0 in rn, Funkcial. ekvac., 36, 557-579, (1993) · Zbl 0793.34024
[3] Tan, J., The radial solutions of 2-order semilinear elliptic equations, Acta math. appl. sinica, 19, 57-64, (1996) · Zbl 0864.35040
[4] O’Regan, D., Theory of singular boundary value problems, (1994), World Scientific Singapore · Zbl 0808.34022
[5] Baxley, J.V., Existence and uniqueness for nonlinear boundary value problems on infinite intervals, J. math. anal. appl., 147, 127-133, (1990) · Zbl 0719.34037
[6] Guo, D.; Liu, X.Z., Impulsive integro-differential equations on unbounded domain in a Banach space, Nonlinear stud., 3, 49-57, (1996) · Zbl 0864.45009
[7] Guo, D.; Liu, X.Z., Multiple positive solutions of boundary value problems for impulsive differential equations, Nonlinear anal., 25, 327-337, (1995) · Zbl 0840.34015
[8] Liu, X.Z., Some existence and nonexistence principles for a class of singular boundary value problems, Nonlinear anal., 27, 1147-1164, (1996) · Zbl 0860.34010
[9] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045
[10] Corduneanu, C., Integral equations and stability of feedback systems, (1973), Academic Press New York · Zbl 0268.34070
[11] Meehan, M.; O’Regan, D., Existence theory for nonlinear Fredholm and Volterra integral equations on half-open intervals, Nonlinear anal., 35, 355-387, (1999) · Zbl 0920.45006
[12] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048
[13] Diekmann, O.; Van. Gils, S.A.; Verduyn Lunel, S.M.; Walther, H.-O., Delay equations, (1994), Springer-Verlag New York
[14] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations: asymptotic properties of the solutions, (1989), World Scientific Singapore · Zbl 0719.34002
[15] Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: asymptotic properties of the solutions, (1995), World Scientific Singapore · Zbl 0828.34002
[16] Ballinger, G.; Liu, X.Z., Existence, uniqueness results for impulsive delay differential equations, Dynam. continuous discr. impulsive systems, 5, 579-591, (1999) · Zbl 0955.34068
[17] Wen, L.; Weng, P., Weakly exponentially asymptotic stability of functional differential equation with impulse, Dynam. continuous discr. impulsive systems, 5, 251-271, (1999)
[18] Dong, Y., Perodic boundary value problems for functional differential equations with impulses, J. math. anal. appl., 210, 170-182, (1997)
[19] Fu, X.; Yan, B., The global solutions of impulsive retarded functional differential equations, Internat. J. appl. math., 2, 389-396, (2000) · Zbl 1171.34333
[20] Anokhin, A.; Berezansky, L.; Braverman, E., Exponential stability of linear delay impulsive differential equations, J. math. anal. appl., 193, 923-941, (1995) · Zbl 0837.34076
[21] Krishna, S.V.; Anokhin, A.V., Delay differential systems with discontinuous initial data and existence and uniqueness theorems for systems with impulses and delay, J. appl. math. stochastic anal., 7, 49-67, (1994) · Zbl 0802.34080
[22] Wang, K.; Huang, K., C_{h}-spaces and the boundedness of solutions for functional differential equations and periodic solutions, Acta sci. China ser. A, 3, 242-253, (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.