Positive solutions of nth-order impulsive differential equations with integral boundary conditions

Tokmak Fen, Fatma; Yaslan Karaca, Ilkay

doi:10.1515/auom-2016-0014

Abstract

This paper is concerned with the existence and nonexistence of positive solutions of nth-order impulsive boundary value problem with integral boundary conditions. The fixed point theorem of cone expansion and compression is used to investigate the existence of at least one positive solution. Also, an example is given to illustrate the effective of our results.

References

[1] B. Ahmad, A. Alsaedi and B. S. Alghamdi, Analytic approximation of so- lutions of the forced Duffing equation with integral boundary conditions, Nonlinear Anal. Real World Appl. 9 (2008) 1727-1740.10.1016/j.nonrwa.2007.05.005
Search in Google Scholar Back to article
[2] B. Ahmad and A. Alsaedi, Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions, Nonlinear Anal. Real World Appl. 10 (2009) 358-367.10.1016/j.nonrwa.2007.09.004
Search in Google Scholar Back to article
[3] B. Ahmad and S. K. Ntouyas, Boundary value problems for n-th or- der differential inclusions with four-point integral boundary conditions, Opuscula Math. 32 (2012) 205-226.10.7494/OpMath.2012.32.2.205
Search in Google Scholar Back to article
[4] B. Ahmad and S. K. Ntouyas, A study of higher-order nonlinear ordinary differential equations with four-point nonlocal integral boundary conditions, J. Appl. Math. Comput. 39 (2012) 97-108.10.1007/s12190-011-0513-0
Search in Google Scholar Back to article
[5] M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer, New York, 2010.10.1007/978-1-4419-6581-3
Search in Google Scholar Back to article
[6] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, New York, USA, 2006.10.1155/9789775945501
Search in Google Scholar Back to article
[7] M. Benchohra, F. Berhoun and J. J. Nieto, Existence results for impulsive boundary value problem with integral boundary conditions, Dynam. Systems Appl. 19 (2010) 585-597.
Search in Google Scholar Back to article
[8] A. Boucherif, Second-order boundary value problems with integral boundary conditions, Nonlinear Anal. 70 (2009) 364-371.10.1016/j.na.2007.12.007
Search in Google Scholar Back to article
[9] J. R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math. 21 (1963) 155-160.10.1090/qam/160437
Search in Google Scholar Back to article
[10] R. Yu. Chegis, Numerical solution of a heat conduction problem with an integral condition, Litovsk. Mat. Sb. 24 (1984) 209-215.
Search in Google Scholar Back to article
[11] W. Ding and Y. Wang, New result for a class of impulsive differential equation with integral boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 1095-1105.10.1016/j.cnsns.2012.09.021
Open DOI Search in Google Scholar Back to article
[12] M. Feng, X. Zhang and X. Yang, Positive solutions of nth-order non- linear impulsive differential equation with nonlocal boundary conditions, Bound. Value Probl. Art. ID 456426 (2011) 19 pp.10.1155/2011/456426
Search in Google Scholar Back to article
[13] M. Feng, Multiple positive solutions of four-order impulsive differential equations with integral boundary conditions and one-dimensional p-Laplacian, Bound. Value Probl. Article ID 654871 (2011) 26 pp.10.1155/2011/654871
Search in Google Scholar Back to article
[14] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, Mass, USA, 1988.
Search in Google Scholar Back to article
[15] X. Hao, L. Liuand Y. Wu, Positive solutions for second order impulsive differential equations with integral boundary conditions, Commun. Non- linear Sci. Numer. Simul. 16 (2011) 101-111.10.1016/j.cnsns.2010.04.007
Search in Google Scholar Back to article
[16] N. I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Differencial'nye Uravnenija 13 (1977) 294-304.
Search in Google Scholar Back to article
[17] M. Jia and X. P. Liu, Multiple nonnegative solutions to boundary value problems for a class of second-order impulsive differential equations with integral boundary conditions, J. Jilin Univ. Sci. 49 (2011) 594-600.
Search in Google Scholar Back to article
[18] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.10.1142/0906
Search in Google Scholar Back to article
[19] Y. Li and F. Li, Sign-changing solutions to second-order integral boundary value problems, Nonlinear Anal. 69 (2008) 1179-1187.10.1016/j.na.2007.06.024
Search in Google Scholar Back to article
[20] Y. H. Li, H. Y. Zhang and Z. L. Zhang, Existence of three positive solutions to a singular integral boundary value problem for systems of non- linear nth-order ordinary differential equations, Appl. Math. J. Chinese Univ. Ser. A 27 (2012) 168-174.
Search in Google Scholar Back to article
[21] Z. W. Lv, J. Liang and T. J. Xiao, Multiple positive solutions for second order impulsive boundary value problems in Banach spaces, Electron. J. Qual. Theory Differ. Equ. 28 (2010) 15 pp.
Search in Google Scholar Back to article
[22] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.10.1142/2892
Search in Google Scholar Back to article
[23] Z. Yang, Existence of nontrivial solutions for a nonlinear Sturm-Lioville problem with integral boundary conditions, Nonlinear Anal. 68 (2008) 216-225.10.1016/j.na.2006.10.044
Search in Google Scholar Back to article
[24] X. Zhang, M. Feng and W. Ge, Existence results for nonlinear boundary- value problems with integral conditions in Banach spaces, Nonlinear Anal. 69 (2008) 3310-3321.10.1016/j.na.2007.09.020
Search in Google Scholar Back to article
[25] X. Zhang, X. Yang and W. Ge, Positive solutions of nth-order impulsive boundary value problems with integral boundary conditions in Banach spaces, Nonlinear Anal. 71 (2009) 5930-5945.10.1016/j.na.2009.05.016
Search in Google Scholar Back to article
[26] X. Zhang, M. Feng and W. Ge, Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces, J. Comput. Appl. Math. 233 (2010) 1915-1926.10.1016/j.cam.2009.07.060
Search in Google Scholar Back to article

Positive solutions of nth-order impulsive differential equations with integral boundary conditions

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