Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

Online
ISSN
1572-9176
See all formats and pricing
More options …
Ahead of print

Issues

Quasilinearization and boundary value problems at resonance

Kareem Alanazi / Meshal Alshammari / Paul EloeORCID iD: https://orcid.org/0000-0002-6590-9931
Published Online: 2019-10-06 | DOI: https://doi.org/10.1515/gmj-2019-2058

Abstract

A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

Keywords: Boundary value problem at resonance; shift method; upper and lower solutions; quasilinearization

MSC 2010: 34B15; 34A45; 47H05

References

  • [1]

    R. P. Agarwal, B. Ahmad and A. Alsaedi, Method of quasilinearization for a nonlocal singular boundary value problem in weighted spaces, Bound. Value Probl. 2013 (2013), Article ID 261. Web of ScienceGoogle Scholar

  • [2]

    E. Akin-Bohner and F. Merdivenci Atici, A quasilinearization approach for two point nonlinear boundary value problems on time scales, Rocky Mountain J. Math. 35 (2005), no. 1, 19–45. CrossrefGoogle Scholar

  • [3]

    S. Al Mosa and P. Eloe, Upper and lower solution method for boundary value problems at resonance, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Paper No. 40. Web of ScienceGoogle Scholar

  • [4]

    R. Bellman, Methods of Nonlinear Analysis. Vol. II, Math. Sci. Eng. 61, Academic Press, New York, 1973. Google Scholar

  • [5]

    R. E. Bellman and R. E. Kalaba, Quasilinearization and Nonlinear Boundary-value Problems, Modern Anal. Comput. Methods Sci. Math. 3, American Elsevier, New York, 1965. Google Scholar

  • [6]

    A. Cabada, P. Habets and S. Lois, Monotone method for the Neumann problem with lower and upper solutions in the reverse order, Appl. Math. Comput. 117 (2001), no. 1, 1–14. Google Scholar

  • [7]

    A. Cabada and L. Sanchez, A positive operator approach to the Neumann problem for a second order ordinary differential equation, J. Math. Anal. Appl. 204 (1996), no. 3, 774–785. CrossrefGoogle Scholar

  • [8]

    M. Cherpion, C. De Coster and P. Habets, A constructive monotone iterative method for second-order BVP in the presence of lower and upper solutions, Appl. Math. Comput. 123 (2001), no. 1, 75–91. Google Scholar

  • [9]

    P. W. Eloe and Y. Gao, The method of quasilinearization and a three-point boundary value problem, J. Korean Math. Soc. 39 (2002), no. 2, 319–330. CrossrefGoogle Scholar

  • [10]

    P. W. Eloe and Y. Zhang, A quadratic monotone iteration scheme for two-point boundary value problems for ordinary differential equations, Nonlinear Anal. 33 (1998), no. 5, 443–453. CrossrefGoogle Scholar

  • [11]

    G. Infante, P. Pietramala and F. A. F. Tojo, Non-trivial solutions of local and non-local Neumann boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 2, 337–369. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    R. A. Khan and R. R. Lopez, Existence and approximation of solutions of second-order nonlinear four point boundary value problems, Nonlinear Anal. 63 (2005), no. 8, 1094–1115. CrossrefGoogle Scholar

  • [13]

    I. Kiguradze, The Neumann problem for the second order nonlinear ordinary differential equations at resonance, Funct. Differ. Equ. 16 (2009), no. 2, 353–371. Google Scholar

  • [14]

    I. T. Kiguradze and N. R. Ležava, On the question of the solvability of nonlinear two-point boundary value problems, Mat. Zametki 16 (1974), 479–490; translation in Math. Notes 16 (1974), 873–880. Google Scholar

  • [15]

    G. A. Klaasen, Differential inequalities and existence theorems for second and third order boundary value problems, J. Differential Equations 10 (1971), 529–537. CrossrefGoogle Scholar

  • [16]

    V. Lakshmikantham, S. Leela and F. A. McRae, Improved generalized quasilinearization (GQL) method, Nonlinear Anal. 24 (1995), no. 11, 1627–1637. CrossrefGoogle Scholar

  • [17]

    V. Lakshmikantham, S. Leela and S. Sivasundaram, Extensions of the method of quasilinearization, J. Optim. Theory Appl. 87 (1995), no. 2, 379–401. CrossrefGoogle Scholar

  • [18]

    V. Lakshmikantham, N. Shahzad and J. J. Nieto, Methods of generalized quasilinearization for periodic boundary value problems, Nonlinear Anal. 27 (1996), no. 2, 143–151. CrossrefGoogle Scholar

  • [19]

    J. J. Nieto, Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2599–2604. CrossrefGoogle Scholar

  • [20]

    K. W. Schrader, Existence theorems for second order boundary value problems, J. Differential Equations 5 (1969), 572–584. CrossrefGoogle Scholar

  • [21]

    N. Shahzad and A. S. Vatsala, Improved generalized quasilinearization method for second order boundary value problem, Dynam. Systems Appl. 4 (1995), no. 1, 79–85. Google Scholar

  • [22]

    N. Sveikate, Resonant problems by quasilinearization, Int. J. Differ. Equ. 2014 (2014), Article ID 564914. Google Scholar

  • [23]

    I. Yermachenko and F. Sadyrbaev, Quasilinearization and multiple solutions of the Emden–Fowler type equation, Math. Model. Anal. 10 (2005), no. 1, 41–50. Google Scholar

About the article

Received: 2017-05-15

Revised: 2018-01-20

Accepted: 2018-02-14

Published Online: 2019-10-06


Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2019-2058.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in