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Zbl 1177.45010
Castro, L.P.; Ramos, A.
Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations. (English)
[J] Banach J. Math. Anal. 3, No. 1, 36-43, electronic only (2009). ISSN 1735-8787/e

The paper deals with the Hyers-Ulam-Rassias stability for nonlinear Volterra equations of the form $$ y(x)=\int_a^x f(x,\tau,y(\tau))\,d\tau \quad(-\infty<a\le x\le b<\infty),\tag1$$ where $f$ is a continuous function satisfying a Lipschitz condition with respect to the third variable. The integral equation (1) possesses the Hyers-Ulam-Rassias stability if for each function $y$ satisfying the condition $$\left|y(x)-\int_a^x f(x,\tau,y(\tau))d\tau\right|\le\psi(x)\quad(x\in[a,b]),\tag2$$ where $\psi$ is a non-negative function, there exist a solution $y_0$ of (1) and a constant $C_1>0$ independent of $y$ and $y_0$ such that $$\big|y(x)-y_0(x)\big|\le\psi(x) \quad\text{for all}\;\;x\in[a,b].\tag3$$ The Hyers-Ulam stability means that $\psi$ in (2)--(3) is a constant function. \par Sufficient conditions for the Hyers-Ulam-Rassias and Hyers-Ulam stabilities of the integral equation (1) on a finite interval are established. The Hyers-Ulam-Rassias stability conditions are also obtained for the nonlinear Volterra equations of the form (1) in the case of infinite intervals.
[Yuri I. Karlovich (Cuernavaca)]

MSC 2000:: *45M10 Stability theory of integral equations
45G10 Nonsingular nonlinear integral equations

Keywords: Hyers-Ulam-Rassias stability; nonlinear Volterra integral equation; fixed-point theorem

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