바로가기메뉴

본문 바로가기 주메뉴 바로가기

logo

  • P-ISSN1226-0657
  • E-ISSN2287-6081
  • KCI

ISSN : 1226-0657

Article Contents

Next
e-Submission

Vol.21 No.3

Citation Share

EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2014, v.21 no.3, pp.147-163
https://doi.org/10.7468/jksmeb.2014.21.3.147
Miao, Chunmei
Ge, Weigao
Zhang, Zhaojun
Miao,, C. , Ge,, W. , & Zhang,, Z. (2014). EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 21(3), 147-163, https://doi.org/10.7468/jksmeb.2014.21.3.147

Abstract

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions <TEX>$$\{u^{{\prime}{\prime}}(t)+q(t)f(t,u(t),u^{\prime}(t))=0,\;t{\in}\mathbb{J}^{\prime},\\{\Delta}u(t_k)=I_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\{\Delta}u^{\prime}(t_k)=-L_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\u=(0)={\int}_{0}^{1}g(t)u(t)dt,\;u^{\prime}=0,$$</TEX>) where the nonlinearity f(t, u, v) may be singular at v = 0. The proof is based on the theory of Leray-Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved.

keywords
singular impulsive diffrential equations, integral boundary conditions, positive solutions, Leray-Schauder degree

Reference

1.

R.P. Agarwal & D. O'Regan. (2000). Multiple nonnegative solutions for second-order impulsive differential equations. Appl. Math. Comput., 114, 51-59. 10.1016/S0096-3003(99)00074-0.

2.

R.P. Agarwal, D. Franco & D. O'Regan. (2005). Singular boundary value problems for first and second order impulsive di'erential equations. Aequationes Math., 69, 83-96. 10.1007/s00010-004-2735-9.

3.

J. Chu & J.J. Nieto. (2008). Impulsive periodic solutions of first-order singular differential equations. Bull. London Math. Soc., 40(1), 143-150. 10.1112/blms/bdm110.

4.

M. Choisy & J.F. Guegan & P. Rohani. (2006). Dynamics of infectious deseases and pulse vaccination: teasing apart the embedded resonance effects. Phys. D., 223, 26-35. 10.1016/j.physd.2006.08.006.

5.

A. D'Onofrio. (2005). On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Appl. Math. Lett., 18, 729-732. 10.1016/j.aml.2004.05.012.

6.

M.Q. Feng & D.X. Xie. (2009). Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J. Comput. Appl. Math., 223, 438-448. 10.1016/j.cam.2008.01.024.

7.

M.Q. Feng, B. Du & W.G. Ge. (2009). Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian. Nonlinear Anal., 70, 3119-3126. 10.1016/j.na.2008.04.015.

8.

T. Jankowski. (2008). Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments. Appl. Math. Comput., 197, 179-189. 10.1016/j.amc.2007.07.081.

9.

R.K. George & A.K. Nandakumaran & A. Arapostathis. (2000). A note on contrability of impulsive systems. J. Math. Anal. Appl., 241, 276-283. 10.1006/jmaa.1999.6632.

10.

G. Jiang & Q. Lu. (2007). Impulsive state feedback control of a predator-prey model. J. Comput. Appl. Math., 200, 193-207. 10.1016/j.cam.2005.12.013.

11.

D.Q. Jiang, J.F. Chu & Y. He. (2007). Multiple positive solutions of Sturm-Liouville problems for second order impulsive differential equations. Dynam. Systems Appl., 16, 611-624.

12.

T. Jankowski. (2008). Positive solutions to second order four-point boundary value problems for impulsive differential equations. Appl. Math. Comput., 202, 550-561. 10.1016/j.amc.2008.02.040.

13.

X. Lin & D. Jiang. (2006). Multiple positive solutions of Dirichlet boundary value problems for second-order impulsive differential equations. J. Math. Anal. Appl., 321, 501-514. 10.1016/j.jmaa.2005.07.076.

14.

V. Lakshmikantham, D.D. Bainov & P. S. Simeonov. Theory of Impulsive Differential Equations.

15.

E.K. Lee & Y.H. Lee. (2004). Multiple positive solutions of singular two point boundary value problems for second-order impulsive differential equations. Appl. Math. Comput., 158, 745-759. 10.1016/j.amc.2003.10.013.

16.

Y.H. Lee & X.Z. Liu. (2007). Study of singular boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl., 331, 159-176. 10.1016/j.jmaa.2006.07.106.

17.

S.H. Liang & J.H. Zhang. (2009). The existence of countably many positive solutions for some nonlinear singular three-point impulsive boundary value problems. Nonlinear Anal., 71, 4588-4597. 10.1016/j.na.2009.03.016.

18.

C.M. Miao, W.G Ge & J. Zhang. (2009). A singular boundary value problem with integral boundary condition for a first-order impulsive differential equation. (Chinese) Mathematics in Practice and Theory, 39(15), 182-186.

19.

J.J. Nieto. (1997). Basic theory for nonresonance impulsive periodic problems of first order. J. Math. Anal. Appl., 205(2), 423-433. 10.1006/jmaa.1997.5207.

20.

S. Nenov. (1999). Impulsive controllability and optimization problems in population dynamics. Nonlinear Anal., 36, 881-890. 10.1016/S0362-546X(97)00627-5.

21.

J.J. Nieto & D. O'Regan. (2009). Variational approach to impulsive differential equations. Nonlinear Anal. RWA, 10, 680-690. 10.1016/j.nonrwa.2007.10.022.

22.

J.J. Nieto. (2010). Variational formulation of a damped Dirichlet impulsive problem. Appl. Math. Lett., 23, 940-942. 10.1016/j.aml.2010.04.015.

23.

I. Rachunkova & J. Tomecek. (2006). Singular Dirichlet problem for ordinary differential equation with impulses. Nonlinear Anal., 65, 210-229. 10.1016/j.na.2005.09.016.

24.

Y. Tian & W.G. Ge. (2010). Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations. Nonlinear Anal., 72, 277-287. 10.1016/j.na.2009.06.051.

25.

Y. Tian & W.G. Ge. (2012). Multiple solutions of impulsive Sturm-Liouville boundary value problem via lower and upper solutions and variational methods. J. Math. Anal. Appl., 387, 475-489. 10.1016/j.jmaa.2011.08.042.

26.

J.F. Xu, D. O'Regan & Z.L. Yang. (2014). Positive Solutions for a nth-Order Impulsive Differential Equation with Integral Boundary Conditions. Differ Equ Dyn Syst., 22, 427-439. 10.1007/s12591-013-0176-4.

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

e-Submission OA Policy