Practical Ulam-Hyers-Rassias stability for nonlinear equations. (English) Zbl 1438.47098

Wang, Jin Rong; Fečkan, Michal

Math. Bohem. 142, No. 1, 47-56 (2017).

Summary: 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 the homotopy method together with a 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.

MSC:

47J05	Equations involving nonlinear operators (general)
39B82	Stability, separation, extension, and related topics for functional equations

Keywords:

Ulam-Hyers-Rassias stability; nonlinear equation; homotopy method