A Fixed Point Approach to the Stability of a Volterra Integral Equation
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Research Article A Fixed Point Approach to the Stability of a Volterra Integral Equation Soon-Mo Jung Received 13 April 2007; Accepted 23 May 2007 Recommended by Jean Mawhin
We will apply the fixed point method for proving the Hyers-Ulam-Rassias stability of a Volterra integral equation of the second kind. Copyright © 2007 Soon-Mo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction 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 homomorphism H : 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: f (x + y) − f (x) − f (y) ≤ ε x p + y p
(1.1)
2
Fixed Point Theory and Applications
and proved the Hyers theorem. That is, Rassias proved the Hyers-Ulam-Rassias stability of the Cauchy additive functional equation. Since then, the stability of several functional equations has been extensively investigated [4–10]. The terminologies Hyers-Ulam-Rassias stability and Hyers-Ulam stability can also be applied to the case of other functional equations, differential equations, and of various integral equations. For a given continuous function f and a fixed real number c, the integral equation y(x) =
x c
f τ, y(τ) dτ
(1.2)
is called a Volterra integral equation of the second kind. If for each function y(x) satisfying x y(x) − f τ, y(τ) dτ ≤ ψ(x), c
(1.3)
where ψ(x) ≥ 0 for all x, there exists a solution y0 (x) of the Volterra integral equation (1.2) and a constant C > 0 with y(x) − y0 (x) ≤ Cψ(x)
(1.4)
for all x, where C is independent of y(x) and y0 (x), then we say that the integral equation (1.2) has the Hyers-Ulam-Rassias stability. If ψ(x) is a constant function in the above inequalities, we say that the integral equation (1.2) has the Hyers-Ulam stability. For a nonempty set X, we introduce the definition of the generalized metric on X. A function d : X × X → [0, ∞] is called a generalized metric on X if and only if d satisfies the following: (
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