Geodesic
Practical Ulam-Hyers-Rassias stability for nonlinear equations
Mathematica Bohemica, Tome 142 (2017) no. 1, pp. 47-56
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we offer a new stability concept, practical Ulam-Hyers-Rassias stability, for nonlinear equations in Banach spaces, which consists in a restriction of Ulam-Hyers-Rassias stability to bounded subsets. We derive some interesting sufficient conditions on practical Ulam-Hyers-Rassias stability from a nonlinear functional analysis point of view. Our method is based on solving nonlinear equations via homotopy method together with Bihari inequality result. Then we consider nonlinear equations with surjective asymptotics at infinity. Moore-Penrose inverses are used for equations defined on Hilbert spaces. Specific practical Ulam-Hyers-Rassias results are derived for finite-dimensional equations. Finally, two examples illustrate our theoretical results.
In this paper, we offer a new stability concept, practical Ulam-Hyers-Rassias stability, for nonlinear equations in Banach spaces, which consists in a restriction of Ulam-Hyers-Rassias stability to bounded subsets. We derive some interesting sufficient conditions on practical Ulam-Hyers-Rassias stability from a nonlinear functional analysis point of view. Our method is based on solving nonlinear equations via homotopy method together with Bihari inequality result. Then we consider nonlinear equations with surjective asymptotics at infinity. Moore-Penrose inverses are used for equations defined on Hilbert spaces. Specific practical Ulam-Hyers-Rassias results are derived for finite-dimensional equations. Finally, two examples illustrate our theoretical results.
DOI : 10.21136/MB.2017.0058-14
Classification : 39B82, 46T20, 47H10, 47H99, 47J05
Keywords: practical Ulam-Hyers-Rassias stability; nonlinear equation 
@article{10_21136_MB_2017_0058_14,
     author = {Wang, Jin Rong and Fe\v{c}kan, Michal},
     title = {Practical {Ulam-Hyers-Rassias} stability for nonlinear equations},
     journal = {Mathematica Bohemica},
     pages = {47--56},
     year = {2017},
     volume = {142},
     number = {1},
     doi = {10.21136/MB.2017.0058-14},
     mrnumber = {3619986},
     zbl = {06738569},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2017.0058-14/}
}
TY  - JOUR
AU  - Wang, Jin Rong
AU  - Fečkan, Michal
TI  - Practical Ulam-Hyers-Rassias stability for nonlinear equations
JO  - Mathematica Bohemica
PY  - 2017
SP  - 47
EP  - 56
VL  - 142
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2017.0058-14/
DO  - 10.21136/MB.2017.0058-14
LA  - en
ID  - 10_21136_MB_2017_0058_14
ER  - 
%0 Journal Article
%A Wang, Jin Rong
%A Fečkan, Michal
%T Practical Ulam-Hyers-Rassias stability for nonlinear equations
%J Mathematica Bohemica
%D 2017
%P 47-56
%V 142
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2017.0058-14/
%R 10.21136/MB.2017.0058-14
%G en
%F 10_21136_MB_2017_0058_14
Wang, Jin Rong; Fečkan, Michal. Practical Ulam-Hyers-Rassias stability for nonlinear equations. Mathematica Bohemica, Tome 142 (2017) no. 1, pp. 47-56. doi: 10.21136/MB.2017.0058-14

[1] Allgower, E. L., Georg, K.: Numerical Continuation Methods. An Introduction. Springer Series in Computational Mathematics 13 Springer, Berlin (1990). | DOI | MR | JFM

[2] András, S., Mészáros, A. R.: Ulam-Hyers stability of dynamic equations on time scales via Picard operators. Appl. Math. Comput. 219 (2013), 4853-4864. | DOI | MR | JFM

[3] Ben-Israel, A., Greville, T. N. E.: Generalized Inverses. Theory and Applications. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 15 Springer, New York (2003). | DOI | MR | JFM

[4] Berger, M. S.: Nonlinearity and Functional Analysis. Lectures on Nonlinear Problems in Mathematical Analysis. Pure and Applied Mathematics 74 Academic Press (Harcourt Brace Jovanovich, Publishers), New York (1977). | MR | JFM

[5] Bihari, I.: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hung. 7 (1956), 81-94. | DOI | MR | JFM

[6] Burger, M., Ozawa, N., Thom, A.: On Ulam stability. Isr. J. Math. 193 (2013), 109-129. | DOI | MR | JFM

[7] Cădariu, L.: Stabilitatea Ulam-Hyers-Bourgin pentru ecuatii functionale. Univ. Vest Timişoara Timişoara (2007).

[8] Chow, S.-N., Hale, J. K.: Methods of Bifurcation Theory. Grundlehren der Mathematischen Wissenschaften 251. A Series of Comprehensive Studies in Mathematics Springer, New York (1982). | DOI | MR | JFM

[9] Cimpean, D. S., Popa, D.: Hyers-Ulam stability of Euler's equation. Appl. Math. Lett. 24 (2011), 1539-1543. | DOI | MR | JFM

[10] Hegyi, B., Jung, S.-M.: On the stability of Laplace's equation. Appl. Math. Lett. 26 (2013), 549-552. | DOI | MR | JFM

[11] Hyers, D. H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27 (1941), 222-224. | DOI | MR | JFM

[12] Hyers, D. H., Isac, G., Rassias, T. M.: Stability of Functional Equations in Several Vari- ables. Progress in Nonlinear Differential Equations and Their Applications 34 Birkhäuser, Boston (1998). | DOI | MR | JFM

[13] Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. The Hadronic Press Mathematics Series. Hadronic Press, Palm Harbor (2001). | MR | JFM

[14] Lakshmikantham, V., Leela, S., Martynyuk, A. A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1990). | MR | JFM

[15] Lungu, N., Popa, D.: Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 385 (2012), 86-91. | DOI | MR | JFM

[16] Park, C.: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. Bull. Sci. Math. 132 (2008), 87-96. | DOI | MR | JFM

[17] Popa, D., Raşa, I.: On the Hyers-Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381 (2011), 530-537. | DOI | MR | JFM

[18] Rassias, T. M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72 (1978), 297-300. | DOI | MR | JFM

[19] Rus, I. A.: Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J. Math. 26 (2010), 103-107. | MR | JFM

[20] Seydel, R.: Practical Bifurcation and Stability Analysis. Interdisciplinary Applied Mathematics 5 Springer, New York (2010). | DOI | MR | JFM

[21] Taylor, A. E., Lay, D. C.: Introduction to Functional Analysis. John Wiley & Sons, New York (1980). | MR | JFM

[22] Ulam, S. M.: A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathematics 8 Interscience Publishers, New York (1960). | MR | JFM

[23] Wang, J., Fečkan, M.: Ulam-Hyers-Rassias stability for semilinear equations. Discontin. Nonlinearity Complex 3 (2014), 379-388. | DOI | JFM

[24] Wang, J., Fečkan, M., Zhou, Y.: Ulam's type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395 (2012), 258-264. | DOI | MR | JFM

[25] Wei, Y., Ding, J.: Representations for Moore-Penrose inverses in Hilbert spaces. Appl. Math. Lett. 14 (2001), 599-604. | DOI | MR | JFM

Cité par Sources :