MATHEMATICA BOHEMICA, Vol. 142, No. 1, pp. 47-56, 2017
Practical Ulam-Hyers-Rassias stability for nonlinear equations
Jin Rong Wang, Michal Fečkan
Received August 15, 2014. First published October 31, 2016.
Abstract: 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.
Affiliations: Jin Rong Wang, Department of Mathematics, Guizhou University, Xueshi Rd, Huaxi, Guiyang, Guizhou 550025, China, e-mail: email@example.com; Michal Fečkan, Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina 12, 842 48 Bratislava, Slovakia, e-mail: Michal.Feckan@fmph.uniba.sk