Note on nonstability of the linear recurrence (original) (raw)
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Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica XIV ( 2015 )
2015
In this paper we prove the Hyers-Ulam stability of the following K-quadratic functional equation ∑ k∈K f(x + k · y) = Lf(x) + Lf(y), x, y ∈ E, where E is a real (or complex) vector space. This result was used to demonstrate the Hyers-Ulam stability on a set of Lebesgue measure zero for the same functional equation.
We obtain the general solution of the generalized mixed additive and quadratic functional equation ( + ) + ( − ) = 2 ( ) − 2 2 ( ) + 2 (2 ), is even; ( + ) + ( − ) − 2( 2 − 1) ( ) + ( 2 − 1) (2 ), is odd, for a positive integer . We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces when is an even positive integer or = 3.
The study of Hyers-Ulam-Rassias stability for several functional equations has been wildly spreaded in the context of different areas of mathematics and such type stability in real Banach spaces along with its several extensions has been examined by a number of mathematicians. In this paper we prove the Hyers-Ulam-Rassias stability of a Pexiderized functional equation in complex Banach spaces under suitable conditions.
In this paper, we prove the Hyers-Ulam-Rassias stability of the cubic functional equation
On the Stability of the Linear Functional Equation
Proceedings of The National Academy of Sciences, 1941
theory of q-difference equations to a comparable degree of completeness. This program includes in particular the complete theory of the convergence and divergence of formal series, the explicit determination of the essential transcendental invariants (constants in the canonical form), the inverse Riemann theory both for the neighborhood of x = o and in the complete plane (case of rational coefficients), explicit integral representation of the solutions, and finally the definition of q-sigma periodic matrices, so far defined essentially only in the case n = 1. Because of its extensiveness this material cannot be presented here.