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
??
About this paper
Print ISSN
2227-7390
Online ISSN
google scholar link
[1] Ulam, S.M. A Collection of Mathematical Problems; Interscience: New York, NY, USA, 1960. [Google Scholar]
[2] Hyers, D.H. On the stability of the linear functional equation. Proc. Matl. Acad. Sci. USA 1941, 27, 222–224. [Google Scholar] [CrossRef] [PubMed]
[3] Brzdek, J.; Popa, D.; Raşa, I.; Xu, B. Ulam Stability of Operators; Academic Press: Cambridge, MA, USA, 2018. [Google Scholar]
[4] Jung, S.-M. Hyers–Ulam Rassias Stability of Functional Equations in Mathematical Analysis; Hadronic Press: Palm Harbour, FL, USA, 2001. [Google Scholar]
[5] Baias, A.R.; Popa, D. On the best Ulam constant of the second order linear differential operator. Rev. De La Real Acad. De Cienc. Exactas 2020, 114, 23. [Google Scholar] [CrossRef]
[6] Fukutaka, R.; Onitsuka, M. Best constant in Hyers-Ulam stability of first order homogeneous linear differential equations with a periodic coefficient. J. Math. Anal. 2019, 473, 1432–1446. [Google Scholar] [CrossRef]
[7] Rus, I.A. Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 2009, 10, 305–320. [Google Scholar]
[8] Jung, S.M. Hyers-Ulam stability of linear partial differential equations of first order. Appl. Math. Lett. 2009, 22, 70–74. [Google Scholar] [CrossRef]
[9] Lungu, N.; Popa, D. Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 2012, 385, 86–91. [Google Scholar] [CrossRef]
[10] Marian, D.; Ciplea, S.A.; Lungu, N. Ulam-Hyers stability of a parabolic partial differential equation. Demonstr. Math. 2019, 52, 475–481. [Google Scholar] [CrossRef]
[11] Prastaro, A.; Rassias, T.M. Ulam stability in geometry of PDE’s. Nonlinear Funct. Anal. Appl. 2003, 8, 259–278. [Google Scholar]
[12] Brzdek, J.; Popa, D.; Raşa, I. Hyers–Ulam stability with respect to gauges. J. Math. Anal. Appl. 2017, 453, 620–628. [Google Scholar] [CrossRef]
[13] Vrabie, I.I. Differential Equations: An Introduction to Basic Concepts, Results and Applications; Word Scientific: Singapore, 2004. [Google Scholar]
[14] Corduneanu, C. Principles of Differential and Integral Equations; The Bronx: New York, NY, USA, 1977. [Google Scholar]
[15] Carter, S. On the Cobb-Douglas and all that’: The Solow-Simon correspondence over the neoclassical aggregate production function. J. Post Keynes. Econ. 2011, 34, 255–274. [Google Scholar] [CrossRef]
[16] Popa, D.; Ilca, M. On approximate Cobb-Douglas production functions. Carpathian J. Math. 2014, 30, 87–92. [Google Scholar]