Research Article A Fixed Point Approach to the Stability of a VolterraIntegral Equation

In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms. Let G1 be a group and let G2 be a metric group with the metric d(·,·). Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈G1, then there exists a homomorphismH :G1 →G2 with d(h(x), H(x)) < ε for all x ∈G1? The case of approximately additive functions was solved by Hyers [2] under the assumption that G1 and G2 are Banach spaces. Indeed, he proved that each solution of the inequality ‖ f (x+ y)− f (x)− f (y)‖ ≤ ε, for all x and y, can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, f (x+ y)= f (x) + f (y), is said to have the Hyers-Ulam stability. Rassias [3] attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows: