Fixed points and generalized stability for functional equations in abstract spaces

Abstract

We use a fixed point method, initiated in [V. Radu, Fixed Point Theory 4(2003), No.1, 91-96], to prove the generalized Ulam-Hyers stability of functional equations in single variable for mappings with values in random normed spaces. This result is then used to obtain the stability for Cauchy, quadratic and monomial functional equations. (2000): 39B52, 39B62, 39B82, 47H09. Mathematics subject classification

On the probabilistic stability of the monomial functional equation

“…In view of Remark 2.8, our two hypotheses on Φ could have been replaced with Φ 2x,2y (αt) ≥ Φ x,y (t), which is the condition that appears in [4].…”

The General Solution and Approximations of a Decic Type Functional Equation in Various Normed Spaces

“…On the other hand, Cȃdariu and Radu noticed that a fixed point alternative method is very important for the solution of the Ulam problem. In other words, they employed this fixed point method to the investigation of the Cauchy functional equation [20] and for the quadratic functional equation [19]. The fixed point method was used for the first time by Baker [9] who applied a variant of Banachs fixed point theorem to obtain the Hyers-Ulam stability of a functional equation in a single variable (for more applications of this method, see [5,6,13,14,15,16,42,66]).…”

Ulam-Hyers stability of Euler-Lagrange additive functional equation in intuitionistic fuzzy Banach spaces: Direct and fixed point methods

“…On the other hand, Cȃdariu and Radu noticed that a fixed point alternative method is very important for the solution of the Ulam problem. In other words, they employed this fixed point method to the investigation of the Cauchy functional equation [13] and for the quadratic functional equation [12] (for more applications of this method, see [7] and [10]). The generalized Hyers-Ulam stability of different functional equations in intuitionistic fuzzy normed spaces has been studied by a number of the authors (see [3-6, 9, 11, 25-27, 35]).…”