Existence and approximation of a solution for a two point nonlinear Dirichlet problem

Pasquale Candito; Roberto Livrea; Luís Sanchez

doi:10.3934/dcdss.2024094

Article Contents

2025, Volume 18, Issue 6: 1540-1549. Doi: 10.3934/dcdss.2024094

This issue Previous Article Multiplicity of solutions for a higher $ m $-polyharmonic Kirchhoff type equation on unbounded domains Next Article Multiple solutions for fourth-order ordinary differential inclusions

Existence and approximation of a solution for a two point nonlinear Dirichlet problem

1.
Department DICEAM, University of Reggio Calabria, 89122 Reggio Calabria, Italy
2.
Department of Mathematics and Informatics, University of Palermo, Via Archirafi, Palermo, Italy
3.
CMAFcIO – Faculdade de Ciências, Universidade de Lisboa, Portugal

^*Corresponding author
^*Corresponding author

Received: March 2024

Revised: May 2024

Early access: June 2024

Published: June 2025

This work was started while the first-named author was visiting CMAFcIO, and he is grateful for the kind hospitality of the host institute.

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Abstract

The existence of at least one positive solution to a second-order nonlinear two-point boundary value problem, is established. Combining difference methods with Brouwer fixed point and Ascolì-Arzelà theorems, we get a solution as the limit of an appropriate sequence of piecewise linear interpolations. Furthermore, a priori bounds on the infinite norm of a solution and its derivatives are pointed out. Some examples are also discussed to illustrate our results.

Keywords:

Difference methods,
a priori bounds,
fixed point methods,
nonlinear two point boundary value problems.

Mathematics Subject Classification: Primary: 39A27; Secondary: 34B15.

Citation:

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