Abstract
This paper is concerned with the initial-value problem for nonlinear first-order impulsive integro-differential equations on time scales \(\mathbb{T}\) . We establish certain existence criteria by using a fixed-point theorem for operator on cones, under which such problems have aminimal and amaximal solution lying in a corresponding region bounded by their lower and upper solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Doctoral Dissertation (UniversitätWürzburg, 1988).
S. Hilger, “Analysis on measure chains–A unified approach to continuous and discrete calculus,” Results Math. 18 (1–2), 18–56 (1990).
R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” ResultsMath. 35 (1–2), 3–22 (1999).
R. P. Agarwal, M. Bohner, and P. Rehak, “Half–linear dynamic equations,” Kluwer Acad. Publ. 1 (5) 1–57 (2003).
M. Bohner and A. Peterson, Dynamic Equations on Time Scales (An Introduction with Applications, Birkhäser, Boston,MA, 2001).
M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales (Birkhäser, Boston, 2003)
V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, “Dynamic Systems on Measure Chains,” Kluwer Acad. Publ. 370 (1–2), 125–135 (1996).
L. Zhang, H. X. Li, and X. B. Zhang, “Periodic solutions of competition Lotka–Volterra dynamic system on time scales,” Comput. Math. Appl. 57 (7), 1204–211 (2009).
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of ImpulsiveDifferential Equations(World Scientific, Singapore, 1989).
J. H. Liang, S. Y. Tang, and Robert A. Cheke, “Beverton–Holt discrete pestmanagement models with pulsed chemical control and evolution of pesticide resistance,” Cnmmun. Nonlinear Sci. 36 327–341 (2016).
T. Xu and F. Gao, “Global Synchronous Pulse Width Modulation of Distributed Inverters,” IEEE T. Power Electr.31 (9) 6237–6253 (2016).
Y. P. Yang, Y. N. Xiao, and J. H. Wu, “Pulse HIV Vaccination: Feasibility for Virus Eradication and Optimal Vaccination Schedule,” B. Math. Biol. 75 (5) 725–751 (2013).
C. Wu, K. Teo, Y. Zhou, and W. Yan, “Solving an identification problem as an impulsive optimal parameter selection problem,” Comput. Math. Appl. 50 (1) 217–229 (2005).
C. Wu, K. Teo, Y. Zhou, and W. Yan, “An optimal control problem involving impulsive integro–differential systems,” Optim. Meth. Software, 22 (3) 531–549 (2007).
B. Ahmad and J. Nieto, “Existence of extremal solutions for non–linear impulsive functional integro–differential equations with anti–periodic boundary conditions,” Arab. J. Sci. Eng, 1 (1) 1–12 (2009).
D. J. Guo, “Extremal solutions for nth–order impulsive integro–differential equations on the half–line in Banach spaces,” Nonlinear Anal. 65 (3) 677–696 (2006).
G. X. Song and X. L. Zhu, “Extremal solutions of periodic boundary value problems for first order integro–differential equations of mixed type,” J. Math. Anal. Appl. 300 (1) 1–11 (2004).
H. K. Xu and J. Nieto, “Extremal solutions of a class of nonlinear integro–differential equations in Banach space,” Proc. Amer. Math. Soc. 125 (9) 2605–2614 (1997).
S. Heikkilä and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (Dekker, New York, 1994. )
Y. Gong and X. Xiang, “A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales,” J. Ind. Manag. Optim. 5 (1) 1–10 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
The article was submitted by the authors for the English version of the journal.
Rights and permissions
About this article
Cite this article
Zhang, L., Xing, Y.F. Extremal Solutions for Nonlinear First-Order Impulsive Integro-Differential Dynamic Equations. Math Notes 105, 123–131 (2019). https://doi.org/10.1134/S0001434619010139
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434619010139