A new approach to the stability of linear functional operators

Abstract

In the present work we discuss a new approach to the stability problem for an arbitrary linear functional operator P : C(I, B) → C(D, B) of the form PF : = Σc j (x)F a j (x) , x ∈ D, with D a compact or noncompact subset in R n , I ⊂ R an interval, and B a Banach space. We define strong stability of the operator P as an arbitrary nearness of a function F to the kernel of the operator P under condition of the smallness of PF (x) at points of some one-dimensional submanifold Γ ⊂ D. Such a stability turns out to be equivalent to some nonstandard a priori estimate for the P. This estimate is obtained in the work by functional analytic methods for an extensive class of operators P which has never been studied earlier.

Untitled

“…Let us mention that Hyers-Ulam stability is related to the notions of shadowing and controlled chaos (see, e.g., [Pilyugin 1999;Palmer 2000;Hayes and Jackson 2005;Stević 2008]) as well as the theories of perturbation (see, e.g., [Chang and Howes 1984;Lin and Zhou 1995]) and optimization. At the moment it is a very popular subject of investigation (for more details, references and examples of some recent results, see, e.g., [Hyers 1941;Ulam 1964;Forti 1995;Hyers et al 1998;Jung 2001;Agarwal et al 2003;Popa 2005a;Jabłoński and Reich 2006;Bahyrycz 2007;Jung and Rassias 2007;Moszner 2009;Paneah 2009;Ciepliński 2010;2012b;Sikorska 2010;Forti and Sikorska 2011;Piszczek 2013a;2013b]).…”

Remarks on the Cauchy functional equation and variations of it

“…The issue of stability of functional equations (and in particular, of the Cauchy functional equation) and various problems related to it have become quite popular during the last decades. This is somewhat illustrated in the following very short (and semi random) list of related works: [8,9,10,11,17,22,26,41,43,44,72,75,76,78,79,84,89,101]. A few reviews related to the issue of stability are [38,53,54,100].…”

On stability of the general linear equation

“…[9]). Since then many authors have studied this subject dealing with a lot of functional equations, for example: where a, b, A, B, p are given (cf., e.g., [1,2,5,7,8,10,11,[13][14][15][16][17]). We deal with the general linear equation, in the class of functions mapping a linear space X into a normed space Y (both over the field F ∈ {R, C}), namely where A, a ij ∈ F, A i ∈ F \ {0}, i ∈ {1, .…”