CONTRACTORS AND FIXED POINTS 5
concept. In this way, they were able to obtain a number of new results. The
following is their main result which generalizes Matkowski's fixed point ·
theorem (Theorem 1.4, Matkowski [1], see also Matkowski [2J) and Altman's [1],
Theorem 5.1, p. 13.
THEOREM. Let X;, Y;(i=1, ... ,n) be Banach spaces and P;:DcX
1 x ••• x
xn~vi
(i=l, ... ,n) be closed nonlinear operators. Suppose that there exist bounded
linear operators
r;(x;):Y;~X;
corresponding to each XiEX; (i=1, ... ,n) such
that
(a) (x;+r1(x1)y1, ... ,xn+rn(xn)yn)ED, whenever (x1, ... ,xn)ED, Y;EYi
(i=1, ... ,n); n
(b) IIP;(x1+r1(x1)y1, ... ,xn+rn(xn)yn)-P;(x1, ... ,xn)-y;ll
~ k~laik
IIYkll
(c)
llr;(x;)II~B,
(i=1, ... ,n), (x1, ... ,xn)ED,
where the real matrix as defined by Matkowski [1,2] has the following property:
aik~O
for i,k=1, ... ,n, n2. The system of inequalities
n
I:
a.krkr., (i=1, ... ,n),
k=1
1 1
has a positive solution r 1, ... ,rn.
Then the system of operator equations
P;(x 1, ... ,xn)=y; (i=1, ... ,n)
has a solution in D for arbitrary Y;EY;(i=1, ... ,n).
It is clear that various types of fixed points can be obtained on the
basis of the above theorem.
8. CONTRACTORS AND FIXED POINTS OF MULTIVALUED MAPPINGS. Let
Pi:DcXi
x ••• x xn~CL(Yi)
be multivalued mappings, where X;, Vi are Banach
spaces and CL(Yi) is the set of all nonempty closed subsets of Y;,
(i=1, ... ,n). Solving a multivalued operator equation means to find some
(x1, ... ,xn)ED such that eiEPi(x1, ... ,xn)' where ei is the zero element of
the Banach space Y;(i=1, ... ,n).
Balakrishna Reddy and Subrahmanyan [2] extended the contractor.concept to
multivalued mappings and proved in this way a general existence theorem for
multivalued operator equations e E Px on subsets of a Banach space. Their
result also contains a comprehensive fixed point theorem proved by Czervik [1]
for multivalued mappings. Czervik's fixed point theorem generalized the
earlier fixed point theorems obtained by Nadler [1], Covitz and Nadler [1J and
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