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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.

45M10Stability theory of integral equations
45G10Nonsingular nonlinear integral equations
Full Text: EMIS EuDML