# A derivative-type operator and its application to the solvability of a nonlinear three point boundary value problem

- René Erlín Castillo recastillo@unal.edu.co
- Babar Sultan babarsultan40@yahoo.com

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https://doi.org/10.56754/0719-0646.2403.0521## Abstract

In this paper we introduce an operator that can be thought as a derivative of variable order, i.e. the order of the derivative is a function. We prove several properties of this operator, for instance, we obtain a generalized Leibniz‘s formula, Rolle and Cauchy‘s mean theorems and a Taylor type polynomial. Moreover, we obtain its inverse operator. Also, with this derivative we analyze the existence of solutions of a nonlinear three-point boundary value problem of “variable order”.

## Keywords

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*CUBO*, vol. 24, no. 3, pp. 521–539, Dec. 2022.