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Stability of solutions of delay functional integro-differential equations and their discretizations. (English) Zbl 1049.65150
The authors derive asymptotic stability results for solutions of delay functional integro-differential equations of the type $$ {d\over dt}\Big[y(t)-\int_0^t a(t-s)G(s,y(s-\tau))\,ds\Big] \, = \, F(t,y(t)), \quad t\geq 0. $$ The idea is based on an approach introduced by {\it M. Zennaro} [Numer. Math. 77, 549--563 (1997; Zbl 0886.65092)] who studied stability with resprect to the forcing term. Therefore, the initial integro-differential equation is reformulated into $$ {d\over dt}\big[ y(t)-\big(V_\tau y\big)(t)\big]= F(t,y(t)), \qquad t\geq 0, $$ using the delay Volterra integral operator $$ \big(V_\tau y\big)(t)= \int_0^t a(t-s)G(s,y(s-\tau))\, ds, \qquad t\geq 0. $$ Studying stability and contractivity properties of the latter equation allows to deduce similar properties for solutions generated by continuous Runge-Kutta or collocation methods.

65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations
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