Fixed Points and Stability for Functional Equations in Probabilistic Metric and Random Normed Spaces

Abstract

We prove a general Ulam-Hyers stability theorem for a nonlinear equation in probabilistic metric spaces, which is then used to obtain stability properties for different kinds of functional equations linear functional equations, generalized equation of the square root, spiral generalized gamma equations in random normed spaces. As direct and natural consequences of our results, we obtain general stability properties for the corresponding functional equations in deterministic metric and normed spaces.

On Ulam’s Type Stability of the Linear Equation and Related Issues

“…Let us mention here that an analogous result for the complete probabilistic metric spaces has been obtained in [42].…”

Two methods of generalized Ulam-Hyers stability of a quattuorvigintic functional equation in various Banach spaces

“…During the last seven decades, the stability problems of a variety of functional equations in quite a lot of spaces have been broadly investigated by number of mathematicians [3,5,8,12,15,17,22,27,32,34,36,39,42].…”

Approximate mixed type additive and quartic functional equation

“…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 [10] and for the quadratic functional equation [9] (for more applications of this method, refer to [6], [8] and [20]). …”