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Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations. (English) Zbl 1177.45010
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. 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.

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