Abstract and Applied Analysis

Positive Solutions for Impulsive Differential Equations with Mixed Monotonicity and Optimal Control

Lingling Zhang, Noriaki Yamazaki, and Rui Guo

Full-text: Open access

Abstract

We consider positive solutions and optimal control problem for a second order impulsive differential equation with mixed monotone terms. Firstly, by using a fixed point theorem of mixed monotone operator, we study positive solutions of the boundary value problem for impulsive differential equations with mixed monotone terms, and sufficient conditions for existence and uniqueness of positive solutions will be established. Also, we study positive solutions of the initial value problem for our system. Moreover, we investigate the control problem of positive solutions to our equations, and then, we prove the existence of an optimal control and its stability. In addition, related examples will be given for illustrations.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 974968, 11 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412279716

Digital Object Identifier
doi:10.1155/2014/974968

Mathematical Reviews number (MathSciNet)
MR3232871

Zentralblatt MATH identifier
07023435

Citation

Zhang, Lingling; Yamazaki, Noriaki; Guo, Rui. Positive Solutions for Impulsive Differential Equations with Mixed Monotonicity and Optimal Control. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 974968, 11 pages. doi:10.1155/2014/974968. https://projecteuclid.org/euclid.aaa/1412279716


Export citation

References

  • D. Guo and V. Lakshmikantham, “Coupled fixed points of nonlinear operators with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 5, pp. 623–632, 1987.
  • D. J. Guo, “Fixed points of mixed monotone operators with applications,” Applicable Analysis: An International Journal, vol. 31, no. 3, pp. 215–224, 1988.
  • A. Reinfelds and L. Sermone, “Stability of impulsive differential systems,” Abstract and Applied Analysis, vol. 2013, Article ID 253647, 11 pages, 2013.
  • S. Chang and Y. H. Ma, “Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of functional equations arising in dynamic programming,” Journal of Mathematical Analysis and Applications, vol. 160, no. 2, pp. 468–479, 1991.
  • Z. Zhao, “Existence and uniqueness of fixed points for some mixed monotone operators,” Nonlinear Analysis: Theory, Methods and Applications, vol. 73, no. 6, pp. 1481–1490, 2010.
  • Y. Wu and Z. Liang, “Existence and uniqueness of fixed points for mixed monotone operators with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 65, no. 10, pp. 1913–1924, 2006.
  • C. Zhai and L. Zhang, “New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 382, no. 2, pp. 594–614, 2011.
  • Z. Zhang and K. Wang, “On fixed point theorems of mixed monotone operators and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3279–3284, 2009.
  • B. Samet, “Existence results for a coupled system of nonlinear fourth-order differential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 324848, 9 pages, 2013.
  • N. U. Ahmed, “Impulsive evolution equations in infinite dimensional spaces,” Dynamics of Continuous, Discrete and Impulsive Systems A: Mathematical Analysis, vol. 10, no. 1–3, pp. 11–24, 2003.
  • D. J. Guo and X. Liu, “Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 177, no. 2, pp. 538–552, 1993.
  • Y. Li and Z. Liu, “Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 1, pp. 83–92, 2007.
  • J. Liang, J. H. Liu, and T. Xiao, “Nonlocal impulsive problems for nonlinear differential equations in Banach spaces,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 798–804, 2009.
  • J. Liu, “Nonlinear impulsive evolution equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 6, no. 1, pp. 77–85, 1999.
  • S. Jinli and M. Yihai, “Initial value problems for the second order mixed monotone type of impulsive differential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 247, no. 2, pp. 506–516, 2000.
  • Y. Peng, X. Xiang, and W. Wei, “Necessary conditions of optimality for second-order nonlinear impulsive differential equations,” Advances in Difference Equations, vol. 2007, Article ID 040160, pp. e1–e17, 2007.
  • P. Sattayatham, “Strongly nonlinear impulsive evolution equations and optimal control,” Nonlinear Analysis: Theory, Methods & Applications, vol. 57, no. 7-8, pp. 1005–1020, 2004.
  • X. Xiang, Y. Peng, and W. Wei, “A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces,” Discrete and Continuous Dynamical Systems A, pp. 911–919, 2005.
  • L. Zhang, N. Yamazaki, and C. Zhai, “Optimal control problem of positive solutions to second order impulsive differential equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 31, no. 2, pp. 237–250, 2012.
  • D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.
  • N. Kenmochi, “Solvability of nonlinear evolution equations with time-dependent constraints and applications,” Bulletin of the Faculty of Education, Chiba University, vol. 30, pp. 1–87, 1981. \endinput