Hyers-Ulam Stability of Euler’s Equation in the Calculus of Variations

Marian, Daniela; Ciplea, Sorina Anamaria; Lungu, Nicolaie

doi:10.3390/math9243320

Open AccessArticle

Hyers-Ulam Stability of Euler’s Equation in the Calculus of Variations

by

Daniela Marian

^1,*,†

,

Sorina Anamaria Ciplea

^2,† and

Nicolaie Lungu

^1,†

¹

Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania

²

Department of Management and Technology, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Mathematics 2021, 9(24), 3320; https://doi.org/10.3390/math9243320

Submission received: 21 November 2021 / Revised: 14 December 2021 / Accepted: 18 December 2021 / Published: 20 December 2021

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Abstract

In this paper we study Hyers-Ulam stability of Euler’s equation in the calculus of variations in two special cases: when

F = F (x, y^{'})

and when

F = F (y, y^{'}) .

For the first case we use the direct method and for the second case we use the Laplace transform. In the first Theorem and in the first Example the corresponding estimations for

|J [y (x)] - J [y_{0} (x)]|

are given. We mention that it is the first time that the problem of Ulam-stability of extremals for functionals represented in integral form is studied.

Keywords: Euler’s equation; calculus of variations; Ulam-Hyers stability; Laplace transform

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MDPI and ACS Style

Marian, D.; Ciplea, S.A.; Lungu, N. Hyers-Ulam Stability of Euler’s Equation in the Calculus of Variations. Mathematics 2021, 9, 3320. https://doi.org/10.3390/math9243320

AMA Style

Marian D, Ciplea SA, Lungu N. Hyers-Ulam Stability of Euler’s Equation in the Calculus of Variations. Mathematics. 2021; 9(24):3320. https://doi.org/10.3390/math9243320

Chicago/Turabian Style

Marian, Daniela, Sorina Anamaria Ciplea, and Nicolaie Lungu. 2021. "Hyers-Ulam Stability of Euler’s Equation in the Calculus of Variations" Mathematics 9, no. 24: 3320. https://doi.org/10.3390/math9243320

APA Style

Marian, D., Ciplea, S. A., & Lungu, N. (2021). Hyers-Ulam Stability of Euler’s Equation in the Calculus of Variations. Mathematics, 9(24), 3320. https://doi.org/10.3390/math9243320

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Hyers-Ulam Stability of Euler’s Equation in the Calculus of Variations

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