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## 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

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.

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Accepted: September 14, 2021; Received: January 09, 2021 This is an open-access article distributed under the terms of the Creative Commons Attribution License