In the paper, the four-functionals fixed-point theorem is used to study the existence of positive solutions for nonlinear third-order m-point impulsive boundary-value problems on time scales. As an application, we propose an example demonstrating our results.
Similar content being viewed by others
References
M. Arhmet, Principles of Discontinuous Dynamical Systems, Springer, New York (2010).
M. Arhmet and M. O. Fen, “Chaotification of impulsive systems by perturbations,” Int. J. Bifurcat. Chaos, 24 (2014).
M. Arhmet and M. O. Fen, “Li–Yorke chaos in hybrid systems on a time scale,” Int. J. Bifurcat. Chaos, 25 (2015).
M. Arhmet and M. O. Fen, Replication of Chaos in Neural Networks, Economics and Physics, Springer, Berlin–Heidelberg (2016).
R. Avery, J. Henderson, and D. O’Regan, “Four functionals fixed point theorem,” Math. Comput. Modelling, 48, 1081–1089 (2008).
D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect, Ellis Horwood, Chichester (1989).
M. Benchohra, S. K. Ntouyas, and A. Ouahab, “Extremal solutions of second order impulsive dynamic equations on time scales,” J. Math. Anal. Appl., 324, 425–434 (2006).
M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston (2001).
M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston (2003).
H. Chen and H. Wang, “Triple positive solutions of boundary-value problems for p-Laplacian impulsive dynamic equations on time scales,” Math. Comput. Modelling, 47, 917–924 (2008).
M. Feng, B. Du, and W. Ge, “Impulsive boundary-value problems with integral boundary conditions and one-dimensional p-Laplacian,” Nonlin. Anal., 70, 3119–3126 (2009).
D. Guo, “Existence of solutions of boundary-value problems for nonlinear second order impulsive differential equations in Banach spaces,” J. Math. Anal. Appl., 181, 407–421 (1994).
S. Hilger, Ein Masskettenkalkül mit Anwendug auf Zentrumsmanningfaltigkeiten: Ph. D. Thesis, Univ. Würzburg (1988).
L. Hu, L. Liu, and Y. Wu, “Positive solutions of nonlinear singular two-point boundary-value problems for second-order impulsive differential equations,” Appl. Math. Comput., 196, 550–562 (2008).
I. Y. Karaca, O. B. Ozen, and F. Tormak, “Multiple positive solutions of boundary value problems for p-Laplacian impulsive dynamic equations on time scales,” Fixed Point Theory, 15, 475–486 (2014).
I. Y. Karaca and F. Tormak Fen, “Positive solutions of nth-order m-point impulsive boundary value problems,” Georg. Math. J., 22, 373–384 (2015).
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Sci., Singapore (1989).
V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer, Dordrecht (1996).
J. Li and J. Shen, “Existence results for second-order impulsive boundary-value problems on time scales,” Nonlin. Anal., 70, 1648–1655 (2009).
Y. Li and Y. Li, “Existence of solutions of boundary-value problems for a nonlinear third-order impulsive dynamic system on time scales,” Different. Equat. Appl., 3, 309–322 (2011).
S. Liang and J. Zhang, “The existence of countably many positive solutions for some nonlinear singular three-point impulsive boundary-value problems,” Nonlin. Anal., 71, 4588–4597 (2009).
S. Liang and J. Zhang, “Existence of three positive solutions of three-order with m-point impulsive boundary-value problems,” Acta Appl. Math., 110, 353–365 (2010).
R. Ma, “Multiple positive solutions for nonlinear m-point boundary-value problems,” Appl. Math. Comput., 148, 249–262 (2004).
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Sci., Singapore (1995).
F. Tormak and I. Y. Karaca, “Positive solutions of an impulsive second-order boundary value problem on time scales,” Dynam. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal., 20, 695–708 (2013).
F. Tormak Fen and I. Y. Karaca, “On positive solutions of m-point boundary value problems for p-Laplacian impulsive dynamic equations on time scales,” Indian J. Pure Appl. Math., 46, 723–738 (2015).
X. Zhang and W. Ge, “Impulsive boundary-value problems involving the one-dimensional p-Laplacian,” Nonlin. Anal., 70, 1692–1701 (2009).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 3, pp. 409–423, March, 2016.
Rights and permissions
About this article
Cite this article
Karaca, I.Y., Fen, F.T. Existence of Positive Solutions for Nonlinear Third-Order m-Point Impulsive Boundary-Value Problems on Time Scales. Ukr Math J 68, 458–474 (2016). https://doi.org/10.1007/s11253-016-1234-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-016-1234-1