Existence and uniqueness of solutions to discrete, third-order three-point boundary value problems

Downloads

DOI:

https://doi.org/10.4067/S0719-06462021000300441

Abstract

The purpose of this article is to move towards a more complete understanding of the qualitative properties of solutions to discrete boundary value problems. In particular, we introduce and develop sufficient conditions under which the existence of a unique solution for a third-order difference equation subject to three-point boundary conditions is guaranteed. Our contributions are realized in the following ways. First, we construct the corresponding Green‘s function for the problem and formulate some new bounds on its summation. Second, we apply these properties to the boundary value problem by drawing on Banach‘s fixed point theorem in conjunction with interesting metrics and appropriate inequalities. We discuss several examples to illustrate the nature of our advancements.

Keywords

Forward difference , boundary value problem , Green‘s function , contraction , fixed point , existence , uniqueness
  • Saleh S. Almuthaybiri Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah, Kingdom of Saudi Arabia – School of Mathematics and Statistics, The University of New South Wales, UNSW, Sydney, NSW, 2052, Australia.
  • Jagan Mohan Jonnalagadda Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad, 500078, Telangana, India.
  • Christopher C. Tisdell School of Mathematics and Statistics, The University of New South Wales, UNSW, Sydney, NSW, 2052, Australia.
  • Pages: 441–455
  • Date Published: 2021-12-01
  • Vol. 23 No. 3 (2021)

R. P. Agarwal, Difference equations and inequalities. Theory, methods, and applications, Second edition, Monographs and Textbooks in Pure and Applied Mathematics, vol. 228. New York: Marcel Dekker, 2000.

R. P. Agarwal and J. Henderson, “Positive solutions and nonlinear eigenvalue problems for third-order difference equations”, Comput. Math. Appl., vol. 36, nos. 10–12, pp. 347–355, 1998.

R. P. Agarwal, M. Meehan and D. O‘Regan, Fixed point theory and applications, Cambridge Tracts in Mathematics, vol. 141, Cambridge: Cambridge University Press, 2001.

S. S. Almuthaybiri and C. C. Tisdell, “Sharper existence and uniqueness results for solutions to third-order boundary value problems”, Math. Model. Anal., vol. 25, no. 3, pp. 409–420, 2020.

D. R. Anderson, “Discrete third-order three-point right-focal boundary value problems”, Comput. Math. Appl., vol. 45, nos. 6–9, pp. 861–871, 2003.

D. R. Anderson and R. I. Avery, “Multiple positive solutions to a third-order discrete focal boundary value problem”, Comput. Math. Appl., vol. 42, nos. 3–5, pp. 333–340, 2001.

D. R. Anderson and C. C. Tisdell, “Discrete approaches to continuous boundary value problems: existence and convergence of solutions”, Abstr. Appl. Anal., vol. 2016, Article ID 3910972, 6 pages, 2016.

M. Bohner and A. Peterson, Dynamic equations on time scales. An introduction with applications, Boston: Birkhäuser Boston-Springer, 2001.

S. Elaydi, An introduction to difference equations, Third edition, Undergraduate Texts in Mathematics, New York: Springer, 2005.

C. Goodrich and A. C. Peterson, Discrete fractional calculus, Cham: Springer, 2015.

J. Ji and B. Yang, “Positive solutions of discrete third-order three-point right focal boundary value problems”, J. Difference Equ. Appl., vol. 15, no. 2, pp. 185–195, 2009.

J. Ji and B. Yang, “Computing the positive solutions of the discrete third-order three-point right focal boundary-value problems”, Int. J. Comput. Math., vol. 91, no. 5, pp. 996–1004, 2014.

I. Y. Karaca, “Discrete third-order three-point boundary value problem”, J. Comput. Appl. Math., vol. 205, no. 1, pp. 458–468, 2007.

W. G. Kelley and A. C. Peterson, Difference equations. An introduction with applications, Second edition, San Diego-CA: Harcourt/Academic Press, 2001.

S. Smirnov, “Green‘s function and existence of a unique solution for a third-order three-point boundary value problem”, Math. Model. Anal., vol. 24, no. 2, pp. 171–178, 2019.

C. P. Stinson, S. S. Almuthaybiri and C. C. Tisdell, “A note regarding extensions of fixed point theorems involving two metrics via an analysis of iterated functions”, ANZIAM J. (EMAC 2019), vol. 61 (2019), pp. C15–C30, 2020.

C. C. Tisdell, “On first-order discrete boundary value problems”, J. Difference Equ. Appl., vol. 12, no. 12, pp. 1213–1223, 2006.

C. C. Tisdell, “A note on improved contraction methods for discrete boundary value problems”, J. Difference Equ. Appl., vol. 18, no. 10, pp. 1173–1777, 2012.

C. C. Tisdell, “Rethinking pedagogy for second-order differential equations: a simplified approach to understanding well-posed problems”, Internat. J. Math. Ed. Sci. Tech., vol. 48, no. 5, pp. 794–801, 2017.

C. C. Tisdell, “Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions”, Internat. J. Math. Ed. Sci. Tech., vol. 48, no. 7, pp. 1087–1095, 2017.

C. C. Tisdell, “Critical perspectives of pedagogical approaches to reversing the order of integration in double integrals”, Internat. J. Math. Ed. Sci. Tech., vol. 48, no. 8, pp. 1285–1292, 2017.

C. C. Tisdell, “On Picard‘s iteration method to solve differential equations and a pedagogical space for otherness”, Internat. J. Math. Ed. Sci. Tech., vol. 50, no. 5, pp. 788–799, 2019.

J. Wang and Ch. Gao, “Positive solutions of discrete third-order boundary value problems with sign-changing Green‘s function”, Adv. Difference Equ., vol. 2015, 10 pages, 2015.

Y. Xu, W. Tian and Ch. Gao, “Existence of positive solutions of discrete third-order three- point BVP with sign-changing Green‘s function”, Adv. Difference Equ., vol. 2019, no. 206, 19 pages, 2019.

Ch. Yang and P. Weng, “Green functions and positive solutions for boundary value problems of third-order difference equations”, Comput. Math. Appl., vol. 54, no. 4, pp. 567–578, 2007.

Downloads

Download data is not yet available.

Published

2021-12-01

How to Cite

[1]
S. S. Almuthaybiri, J. M. Jonnalagadda, and C. C. Tisdell, “Existence and uniqueness of solutions to discrete, third-order three-point boundary value problems”, CUBO, vol. 23, no. 3, pp. 441–455, Dec. 2021.

Issue

Section

Articles