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From chik@d165.icyb.kiev.ua Mon Jul 16 15:46:38 2001
Date: Mon, 14 May 2001 16:58:32 +0300 (EET)
From: Arkadii Chikrii <chik@d165.icyb.kiev.ua>
To: congress@imath.kiev.ua
Subject: ABSTRACT (13)


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\begin{document}
\pagestyle{empty}
\begin{center}
{\Large \bf Game Problems for
Evolutionary Equations of Fractional Order}
\end{center}
{\large
\begin{center}
{\bf Chikrii, Arkadii, Cybernetics Institute, Kyiv, Ukraine,}\\
e-mail:chik@d165.icyb.kiev.ua\\
{\bf Eidelman, Samuil, Mathematics Institute, Kyiv, Ukraine},\\
e-mail:seidelman@math.pp.kiev.ua
\end{center}
}
We present a general method for solving game problems of approach  for
dynamical systems with Volterra evolution. This method is based on the
method of resolving functions and essentially uses the technique of
set-valued mappings and the properties of their selections [1].
The properties of resolving functions are studied in detail.
The cases, when the resolving functions can be found in analytic form,
are singled out. The suggested scheme encompasses a wide range of
functional-differential systems, in particular, integral,
integral-differential and difference-differential systems of equations.
In more detail, we study game problems for systems with Riemann-Liouville
fractional derivatives and regularized derivatives of Dshrbashyan-Nersesyan.
Such game problems will be referred to as the fractal games.

An important role in presenting solutions of such systems belongs to the
introduced in the paper, generalized matrix functions of Mittag-Leffler
[2]. To find this matrix function Lagrange-Silvestre interpolation polynomials
are used. With help of asymptotic representation of the above mentioned
functions, in the frame work of the method's scheme, makes it feasible to
establish sufficient conditions for solvability of the game problems under
consideration. The rigorous definition of parallel approach is given, which
is illustrated by the game problems for systems with fractional derivatives
[3].



{\bf Keywords}: dynamical game, fractional derivative, Mittag-Leffler
function, set-value map.

\begin{thebibliography}{9}

\bibitem{B1}
A.A. Chikrii, Conflict-Controlled Processes, Kluwer Academic
Publishers, Dord\-recht-Bos\-ton-Lon\-don, 1997, 424 p.

\bibitem{B2}
A.A. Chikrii, S.D. Eidelman, Generalized Matrix Functions of Mittag-Leffler in
the Game Problems for Evolutionary Systems of
Fractional Order, J. Cybernetic and System Analysis, 2000, No. 3, pp. 3-32.

\bibitem{B3}
A.A. Chikrii, S.D. Eidelman, Game Problems for Systems with Volterra Evolution.
Fractal Games, Game Theory and Applications, 2000, No. 6, pp. 21-54.

\end{thebibliography}

\end{document}