4 MIECZYSLAW ALTMAN
The general iterative process which converges to a solution of (5.2) is
given by the formula
(5.3) xn+1(t) = xn(t) - [yn(t)
+
J~r(s,xn(s))yn(s)dsJ,
(5.4) yn(t) = xn(t) -
J~F(s,xn(s))ds-x 0 ,
n=0,1, ...
Note that Kuo's proof is different from that in A[1J, Remark 5.2, p. 82 which
involves a transfinite induction argument but holds true in discrete case
(5.2), (5.3) as well. The above method which is more general is actually the
method of contractor directions and will be discussed later.
Lee and Padgett [1] have applied the concept of integral contractors to
study random nonlinear integral equations in population growth problems. This
approach allows to obtain very general existence theorems for the above
equations.
Padgettand Rao [2] have also applied the concept of integral contractor
to obtain more general results in the theory of stochastic integral equations.
An integral contractor appears also in paoer A[10J which presents a
method of solving quasi-linear evolution equations of "hyperbolic" type in
nonreflexive Banach spaces. The evolution operator for solving the corre-
sponding linear equations at each iteration step plays there the r part.
Let us mention that Kate's theory of quasilinear evolution equations applies
to reflexive Banach spaces, and his proof is based on the Banach contraction
principle.
6. RANDOM CONTRACTORS. W.
J.
Padgett has introduced the probabilistic
analogue of the contractor concept, and in Padgett and Lee [1J the authors
apply the contractor method to the solution of random nonlinear equations. An
application to random integral equations is also given there.
Lee [1] has also applied random contractors to random nonlinear operator
equations.
Padgett and Rao [1] have applied the contractor concept in their studies
of existence and stability of solutions for generalized McShane systems.
7. GENERALIZED CONTRACTORS. A new contribution to the contractor concept has
recently been made by K. Balakrishna Reddy and P. V. Subrahmanyam [1J in their
paper "Altman's contractors and Matkowski's fixed point theorem," they made an
interesting observation that the apparently disconnected theorems of
Matkowski [1,2] and Altman [1] can be unified on the basis of a contractor
concept extended to a product of Banach spaces. A special matrix introduced by
Matkowski plays the part of a majorant function for the extended contractor
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