Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE AND UNIQUENESS THEOREMS OF SECOND-ORDER EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

  • Bougoffa, Lazhar (Department of Mathematics Al Imam Mohammad Ibn Saud Islamic University) ;
  • Khanfer, Ammar (Department of Mathematics Al Imam Mohammad Ibn Saud Islamic University)
  • Received : 2017.04.25
  • Accepted : 2017.12.15
  • Published : 2018.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider the second-order nonlinear differential equation with the nonlocal boundary conditions. We first reformulate this boundary value problem as a fixed point problem for a Fredholm integral equation operator, and then present a result on the existence and uniqueness of the solution by using the contraction mapping theorem. Furthermore, we establish a sufficient condition on the functions ${\mu}$ and $h_i$, i = 1, 2 that guarantee a unique solution for this nonlocal problem in a Hilbert space. Also, accurate analytic solutions in series forms for this boundary value problems are obtained by the Adomian decomposition method (ADM).

Keywords

References

  1. G. Adomian, Modification of the decomposition approach to the heat equation, J. Math. Anal. Appl. 124 (1987), no. 1, 290-291. https://doi.org/10.1016/0022-247X(87)90040-0
  2. G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Mathematics and its Applications, 46, Kluwer Academic Publishers Group, Dordrecht, 1989.
  3. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Fundamental Theories of Physics, 60, Kluwer Academic Publishers Group, Dordrecht, 1994.
  4. L. Bougoffa and R. C. Rach, Solving nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the Adomian decomposition method, Appl. Math. Comput. 225 (2013), 50-61.
  5. L. Bougoffa, R. C. Rach, and A. Mennouni, An approximate method for solving a class of weakly-singular Volterra integro-differential equations, Appl. Math. Comput. 217 (2011), no. 22, 8907-8913. https://doi.org/10.1016/j.amc.2011.02.102
  6. L. Bougoffa, A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method, Appl. Math. Comput. 218 (2011), no. 5, 1785-1793. https://doi.org/10.1016/j.amc.2011.06.062
  7. L. Bougoffa, R. C. Rach, and A.-M. Wazwaz, Solving nonlocal initial-boundary value problems for the Lotka-von Foerster model, Appl. Math. Comput. 225 (2013), 7-15.
  8. S. A. Brykalov, A second order nonlinear problem with two-point and integral boundary conditions, Georgian Math. J. 225 (1994), no. 3, 243-249.
  9. Y. Cui, Solvability of second-order boundary-value problems at resonance involving integral conditions, Electron. J. Differential Equations 2012 (2012), No. 45, 9 pp.
  10. Y. Cui and Y. Zou, Existence of solutions for second-order integral boundary value problems, Nonlinear Anal. Model. Control 21 (2016), no. 6, 828-838. https://doi.org/10.15388/NA.2016.6.6
  11. Z. Du and J. Yin, A second order differential equation with generalized Sturm-Liouville integral boundary conditions at resonance, Filomat 28 (2014), no. 7, 1437-1444. https://doi.org/10.2298/FIL1407437D
  12. M. Feng, X. Li, and C. Xue, Multiple positive solutions for impulsive singular boundary value problems with integral boundary conditions, Int. J. Open Probl. Comput. Sci. Math. 2 (2009), no. 4, 546-561.
  13. G. Infante, Nonlocal boundary value problems with two nonlinear boundary conditions, Commun. Appl. Anal. 12 (2008), no. 3, 279-288.
  14. A. Lomtatidze and L. Malaguti, On a nonlocal boundary value problem for second order nonlinear singular differential equations, Georgian Math. J. 7 (2000), no. 1, 133-154.
  15. A.-M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science, Higher Education Press, Beijing, 2009.
  16. A.-M. Wazwaz, Linear and Nonlinear Integral Equations, Higher Education Press, Beijing, 2011.
  17. J. R. L. Webb, A unified approach to nonlocal boundary value problems, in Dynamic systems and applications. Vol. 5, 510-515, Dynamic, Atlanta, GA, 2006.
  18. J. R. L. Webb, Positive solutions of some higher order nonlocal boundary value problems, Electron. J. Qual. Theory Differ. Equ. 2009 (2009), Special Edition I, no. 29, 15 pp.
  19. J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc. (2) 74 (2006), no. 3, 673-693. https://doi.org/10.1112/S0024610706023179
  20. Z. Yang, Existence and nonexistence results for positive solutions of an integral boundary value problem, Nonlinear Anal. 65 (2006), no. 8, 1489-1511. https://doi.org/10.1016/j.na.2005.10.025
  21. Z. Yang, Positive solutions of a second-order integral boundary value problem, J. Math. Anal. Appl. 321 (2006), no. 2, 751-765. https://doi.org/10.1016/j.jmaa.2005.09.002
  22. Z. Yang, Existence and uniqueness of positive solutions for an integral boundary value problem, Nonlinear Anal. 69 (2008), no. 11, 3910-3918. https://doi.org/10.1016/j.na.2007.10.026
  23. X. Zhang, M. Feng, and W. Ge, Existence result of second-order differential equations with integral boundary conditions at resonance, J. Math. Anal. Appl. 353 (2009), no. 1, 311-319. https://doi.org/10.1016/j.jmaa.2008.11.082