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{\bf Regular algorithms for solving the first kind equations with
finite-dimensional nonlinearity}
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{\bf  A.~L.~Ageev}
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Integral equations of the first kind
$$
A[\mu]x\equiv\int^b_a K(t,s,\mu)x(s)ds=y(t),\quad t\in [c,d],\eqno(1)
$$
which kernal depend on finite-dimensional vector of parametrs $\mu\in R^m$,
are considered. For fixed $\mu$ the operator $A[\mu]$ acts between
Hilbert spaces $X$ and $Y.$ Instead of the exact data $\mu^*$ and $y^*$
approximate vector $\hat\mu: \|\hat\mu-\mu^*\|\le\rho$ and
approximate function $y_{\delta}:
\|y_{\delta}-y^*\|\le\delta$ are known. The existence of the exact solution
of equations (1) is assumed (the exact solutions is a pair $\{\mu^*,x^*\}:
A[\mu^*]x^*=y^*$).

Different conditions on the function $K(t,s,\mu)$ and on class of the
functions $x^*$ are discussed. This conditions guaranttee a local uniqueness
of the pair $\{\mu^*,x^*\}$ and permit to construct regular iteration algorithms
for solving of non-linear equation (1).

Different applications of the iterative processes to integral equations
are discussed.

Supported by the Russian Foundation of Fundamental Research,
Grant 97-01-00520.

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