On the Best Ulam Constant of the Linear Differential Operator with Constant Coefficients

Alina Ramona Baias and Dorian Popa
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Alina Ramona Baias: Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania
Dorian Popa: Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania

Mathematics, 2022, vol. 10, issue 9, 1-14

Abstract: The linear differential operator with constant coefficients D ( y ) = y ( n ) + a 1 y ( n − 1 ) + … + a n y , y ∈ C n ( R , X ) acting in a Banach space X is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. We prove that if the characteristic equation of D has distinct roots r k satisfying Re r k > 0 , 1 ≤ k ≤ n , then the best Ulam constant of D is K D = 1 | V | ∫ 0 ∞ | ∑ k = 1 n ( − 1 ) k V k e − r k x | d x , where V = V ( r 1 , r 2 , … , r n ) and V k = V ( r 1 , … , r k − 1 , r k + 1 , … , r n ) , 1 ≤ k ≤ n , are Vandermonde determinants.

Keywords: linear differential operator; Ulam stability; best constant; Banach space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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