ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION OF A DIFFERENTIAL EQUATION WITH INTERACTION GOVERNED BY GENERALIZED FUNCTION IN ABSTRACT WIENER SPACE

VOLODYMYR B. BRAYMAN

Theory of Stochastic Processes Vol. 11 (27), no. 3–4, 2005, pp. 29–41

We consider the following differential equation with interaction governed by a generalized function $\varkappa_0$
$$
\frac{dx(u,t)}{dt}=a(x(u,t),\varkappa_t), \ x(u,0)=u, \ \varkappa_t=\varkappa_0\circ x(\cdot,t)^{-1}.
$$

The conditions that guarantee the existence and uniqueness of a solution when mapping a belongs to some Sobolev space are obtained.



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