On Ulam stability of a partial differential operator in Banach spaces

2 years ago

Abstract

In this paper, we prove that, if $\inf \limits_{x\in A}\left \vert f\left(x\right) \right \vert =m>0$ , then the partial differential operator $D$ defined by $D\left( u\right) =\sum \limits_{k=1}^{n}f_{k}\frac{\partial u}{\partial x_{k}}-fu,$ where $f,f_{i}\in C\left( A,\mathbb{R}\right),uC^{1}\left( A,X\right) ,\ i=1,\ldots,n\subset \mathbb{R}$ is an interval, $A=I\times \mathbb{R}^{n-1}$ and $X$ is a Banach space, is Ulam stable with the Ulam constant $K=\frac{1}{m}$ . Moreover, if $\inf \limits_{x\in A}\left \vert f\left( x\right) \right \vert =0$ , we prove that $D$ is not generally Ulam stable.

Authors

Adela Novac
Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Diana Otrocol
Department of Mathematics, Technical University of Cluj-Napoca
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Dorian Popa
Department of Mathematics, Technical University of Cluj-Napoca, Romania

Keywords

Ulam stability; partial differential operator; gauge; Banach space

Paper coordinates

A. Novac, D. Otrocol, D. Popa, On Ulam stability of a partial differential operator in Banach spaces, Mathematics, 11 (2023) no. 11, art. no. 2488, https://doi.org/10.3390/math11112488

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About this paper

Journal

Mathematics

Publisher Name

MDPI

DOI

https://doi.org/10.3390/math11112488

Print ISSN

2227-7390

Online ISSN

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