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

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