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

• 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.
Keywords: Forward difference, boundary value problem, Green’s function, contraction, fixed point, existence, uniqueness

### 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.

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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.
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