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

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