On Ulam stability of a partial differential operator in Banach spaces


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.


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


Ulam stability; partial differential operator; gauge; Banach space

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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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