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Vol. 51 No. 2 (2025)
Serdica Mathematical Journal

Articles

On boundary value problems for second order ordinary differential equation with constant delay

https://doi.org/10.55630/serdica.2025.51.133-152

Published 20-10-2025

  • Heramb B. Aiya
    • ,
  • Yeshwant Shivrai Valaulikar

Heramb B. Aiya
Goa University

Yeshwant Shivrai Valaulikar
Goa University

Abstract

The paper deals with establishing existence of solution theorems for boundary value problem of second order ordinary differential equation with constant delay of the type y′′(t) = f(t, y(t), y(t − r), y′(t)). We use Banach fixed point theorem to establish existence and uniqueness theorem for boundary value problem by assuming f to be Lipschitz in second, third and fourth argument. We also use Schauder’s fixed point theorem to prove existence ofsolution result for boundary value problem by assuming 'f' to be bounded.

2020 Mathematics Subject Classification:

34B15, 34C25, 34K05, 34K10

Keywords

boundary value problem, Differential Equation with constant delay, compact operator, contraction mapping, Lipschitz condition

Full text

Author Details

Heramb B. Aiya

Heramb Aiya
Department of Mathematics
Government College of Arts Science and Commerce
Quepem – Goa – India
e-mail: heramb.aiya@gmail.com

Yeshwant Shivrai Valaulikar

Department of Mathematics
Goa University
Taleigao Plateau – Goa – India, 403206
e-mails: ysv@unigoa.ac.in, valaulikarys@gmail.com

References

  1. M. Bachar, M. A. Khamsi. Delay differential equations: a partially ordered set approach in vectorial metric spaces. Fixed Point Theory Appl. 193 (2014), 9 pp.
  2. D. Bai, Y. Xu. Existence of positive solutions for boundary value problems of second order delay differential equations, Appl. Math. Lett. 18, 6 (2005), 621–630.
  3. N. Bello, A. J. Alkali, A. Roko. A fixed point iterative method for the solution of two-point boundary value problems for a second order differential equation. Alexandria Engineering Journal, 57, 4 (2018), 2515–2520, https://doi.org/10.1016/j.aej.2017.09.010,
  4. A. Cabada, R. Enguiça, L. López-Somoza. Positive solutions for second order boundary value problems with sign changing green's functions. arXiv:1706.07395 [math.CA], 2017, https://doi.org/10.48550/arXiv.1706.07395.
  5. E. A. Coddington, N. Levinson. Theory of Ordinary Differential Equations. New York–Toronto–London, McGraw-Hill Book Co., Inc., 1955.
  6. S. G. Deo, V. Raghavendra, R. Kar, V. Lakshmikantham. Textbook of ordinary differential equations. McGraw-Hill Education, 2015.
  7. R. D. Driver. Ordinary and delay differential equations. Appl. Math. Sci., vol. 20 New York–Heidelberg, Springer-Verlag, 1977.
  8. D. Franco, J. J. Nieto, D. O'Regan. Anti-periodic boundary value problem for nonlinear first order ordinary differential equations. Math. Ineq. Appl. 6, 3 (2003), 477–485.
  9. D. Franco, J. J. Nieto, D. O'Regan. Upper and lower solutions for first order problems with nonlinear boundary conditions. Extracta Math. 18, 2 (2003), 153–160.
  10. H.-D. Gau. Existence of ω-periodic solutions for second order delay differential equation in Banach spaces. Filomat 36, 16 (2022), 5347–5358, https://doi.org/10.2298/FIL2216347G.
  11. H. Huang, X. Fu. Optimal feedback control results for second order evolution system with finite delay. Evol. Equ. Control Theory 12, 6 (2023), 1577–1601.
  12. Z. He, J. Shen. The boundary value problems of quadratic type of delay differential equations with eigenvalues. Math. Slovaca 64, 4 (2014), 893–912.
  13. T. Jankowski. On delay differential equations with boundary conditions. Dynam. Systems Appl. 16, 3 (2007), 425–432.
  14. D. Jiang. Multiple positive solutions for boundary value problems of second order delay differential equations. Appl. Math. Lett. 15, 5 (2002), 575–583.
  15. M. C. Joshi, R. K. Bose. Some topics in nonlinear functional analysis. Halsted Press Book, New York, John Wiley & Sons, Inc., 1985.
  16. V. Lakshmikantham. Periodic boundary value problem for first and second order differential equations. J. Appl. Math. Simulation 2, 3 (1989), 131–138.
  17. E. N. Mahmudov. Optimal control of second order delay-discrete and delay differential inclusions with state constraints. Evol. Equ. Control Theory 7, 3 (2018), 501–529.
  18. S. Ohkohchi. A boundary value problem for delay differential equations. Hiroshima Math. J., 7, 1 (1977), 379–385.
  19. L. Skóra. Boundary value problems for second order delay differential equations. Opuscula Math. 32, 3 (2012), 551–558.