SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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On Properties of Integral Algebraic Equations with Rectangular Coefficient Matrices

V. Chistyakov

In this talk we consider systems of Volterra integral equations $ (\Lambda _0+{\cal V})x=A(t)y+\int \limits _{\alpha }^{t}p(t,s)K(t,s)x(s)ds=\psi (t), t\in T=[\alpha ,\beta ], \eqno {(1)} $ where $A(t), K(t,s)$ are $(m\times n)$ matrices, $p(t,s)=1$ or $p(t,s)=(t-s)^{-\gamma },\gamma\in (0,1)$, $x\equiv x(t), \psi (t)$ are the desirable and given vector-functions, correspondingly. It is assumed that the following condition holds $ {\rm rank}A(t)<{\rm min}\{m, n\} \forall t\in T. \eqno {(2)} $ If $m=n$, then (2) is equivalent to ${\det}A(t)=0 \forall t\in T$. Such systems are commonly called integral algebraic equations (IAEs). They appear when modeling developing dynamical systems (based on the Glushkov model) and analyzing game problems. We give sufficient conditions for solvability of (1) satisfying (2). The methods of investigation are based on the results from [1, 2]. This work has been supported by RFBR, Projects No. 11-01-00639 and No. 13-01-93002.

[1] V.F. Chistyakov, On solvability for systems of Volterra equations of the fourth kind, Differential equations, 2002, vol 38, N5, pp. 698-707. (in Russian)
[2] V.F. Chistyakov, On some properties of the fourth kind Volterra systems with a kernel of the convolution type, Mathematical Notes, 2006, vol 80, issue 1, pp. 115-118. (in Russian)

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