SciELO - Scientific Electronic Library Online

 
vol.23 issue3Foundations of generalized Prabhakar-Hilfer fractional calculus with applicationsSome integral inequalities related to Wirtinger’s result for p-norms author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.23 no.3 Temuco Dec. 2021

http://dx.doi.org/10.4067/S0719-06462021000300441 

Articles

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

Saleh S. Almuthaybiri1 
http://orcid.org/0000-0002-9399-3253

Jagan Mohan Jonnalagadda2 
http://orcid.org/0000-0002-1310-8323

Christopher C. Tisdell3 
http://orcid.org/0000-0002-3387-2505

1Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah, Kingdom of Saudi Arabia. s.almuthaybiri@qu.edu.sa

2Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad, 500078, Telangana, India. j.jaganmohan@hotmail.com

3School of Mathematics and Statistics, The University of New South Wales, UNSW, Sydney, NSW, 2052, Australia. cct@unsw.edu.au

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 and Phrases: Forward difference; boundary value problem; Green’s function; contraction; fixed point; existence, uniqueness

RESUMEN

El propósito de este artículo es avanzar hacia un entendimiento más completo de las propiedades cualitativas de las soluciones a problemas discretos de valor en la frontera. En particular, introducimos y desarrollamos condiciones suficientes bajo las cuales se garantiza la existencia de una única solución para una ecuación en diferencias de tercer orden sujeta a condiciones de borde en tres puntos. Nuestras contribuciones son de dos tipos. En primer lugar, construimos las funciones de Green correspondientes para el problema y formulamos nuevas cotas para su suma. En segundo lugar, aplicamos estas propiedades al problema de valor en la frontera usando el teorema del punto fijo de Banach junto con métricas interesantes y desigualdades apropiadas. Discutimos varios ejemplos para ilustrar la naturaleza de nuestros avances.

Texto completo disponible sólo en PDF

Full text available only in PDF format

References

[1] 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. [ Links ]

[2] 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. [ Links ]

[3] 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. [ Links ]

[4] 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. [ Links ]

[5] 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. [ Links ]

[6] 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. [ Links ]

[7] 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. [ Links ]

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

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

[10] C. Goodrich and A. C. Peterson, Discrete fractional calculus, Cham: Springer, 2015. [ Links ]

[11] 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. [ Links ]

[12] J. Ji andB. 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. [ Links ]

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

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

[15] 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. [ Links ]

[16] 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. [ Links ]

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

[18] 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. [ Links ]

[19] 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. [ Links ]

[20] 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. [ Links ]

[21] 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. [ Links ]

[22] 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. [ Links ]

[23] 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. [ Links ]

[24] Y. Xu, W. Tian and Ch. Gao , “Existence of positive solutions of discrete third-order threepoint BVP with sign-changing Green’s function”, Adv. Difference Equ. , vol. 2019, no. 206, 19 pages, 2019. [ Links ]

[25] 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. [ Links ]

Accepted: September 14, 2021; Received: January 09, 2021

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License