Fixed-point theorem in classes of function with values in a dq-metric space

Abstract

We prove a fixed point result for nonlinear operators, acting on some classes of functions with values in a dq-metric space, and show some applications of it. The result has been motivated by some issues arising in Ulam stability. We use a restricted form of a contraction condition.

Stability in the Sense of Hyers–Ulam–Rassias for the Impulsive Volterra Equation

“…Numerous methodologies have been used to explore stability in general, including the direct method (see [20]), the fixed-point theory (FPT) (see [21,22]), the invariant means method (see [23]), Grönwall's inequality (GI), and many more (see, e.g., [24]). The FPT is the second most frequent technique for showing that functional equations (FEs) are stable.…”

On Hyers-Ulam-Rassias stability of fractional differential equations with Caputo derivative

“…In general terms, stability has been studied using each of the following methods. The method Hyers put forward quite frequently called the direct one (see [17]), the method of invariant means (introduced in [29]), the fixed point method (see [4,12]), the method based on sandwich theorems [26], and the method using the concept of shadowing (see [30]). The fixed point method is well-known as the second most popular method in proving the stability of functional equations.…”