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Existence theory for nonlinear Volterra integrodifferential and integral equations. (English) Zbl 0891.45004
The authors study parameter identification problems for the nonlinear delay system with state dependent delays $$x'(t)= f\Biggl(t,x(t), \int^0_{-r} d_s\mu(s,t,x_t,\sigma) x(t+ s),\theta\Biggr),$$ where $0\le t\le T$, and $x(t)= \phi(t)$, $-r\le t\le 0$. Under assumptions, too complicated to be stated here, an identification method is outlined. Some numerical examples are given.

45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations
Full Text: DOI
[1] Lee, J. W. and O’Regan, D., Existence principles for nonlinear integral equations on semiinfinite and half-open intervals. Advances in Nonlinear Dynamics. Gordon & Breach. (To appear.)
[2] Gripenberg, G.; Londen, S. O.; Staffans, O.: Volterra integral and functional equations. (1990) · Zbl 0695.45002
[3] Nohel, J. A.; Shea, D. F.: Frequency domain methods for Volterra equations. Adv. math. 22, 278-304 (1976) · Zbl 0349.45004
[4] Brezis, H.; Browder, F. E.: Existence theorems for nonlinear integral equations of Hammerstein type. Bull. amer. Math. soc. 81, 73-78 (1975) · Zbl 0298.47031
[5] O’regan, D.: Existence theory for nonlinear Volterra and Hammerstein integral equations. Dynamic systems and applications 4, 601-615 (1995)
[6] O’regan, D.: Existence results for nonlinear integral equations. J. math. Anal. appl. 192, 705-726 (1995)
[7] Pachpatte, B. G.: Applications of the Leray-Schauder alternative to some Volterra integral and integrodifferential equations. Indian J. Pure appl. Math. 26, 1161-1168 (1995) · Zbl 0852.45012