6

MIECZYSLAW ALTMAN

Reich [1]. Moreover, their results also contain an extension to multivalued

mappings of their own results from their paper Balakrishna Reddy and

Subrahmanyan [1].

Let bik'

cik~O

(i,k=1, ... ,n) and let aik=bik+cik' where (aik) is

the matrix defined by Matkowski [1,2]. Put

n

q=max(r:

1

r a.krk).

i

1

k= 1

1

Then Oq1 and

n

r a.krkqr., (i=1, ... ,n).

k=1

1 - 1

The following is the main result of Balakrishna Reddy and Subrahmanyan

[2].

THEOREM. Suppose that the closed nonlinear transformations

P.:DcX1

x ••• x

X

-+

CL(Y.), (i=1, ... ,n)

1

n

1

fulfill the following:

\here exist bounded linear operators

r;(x;):Y;-+ X;, X;EX;

(i=1, ... ,n) such that

llr;(x;lll

~

B, (x;, ... ,xn)ED, (i=1, ... ,n)

(a) (x1 + r1(x1)y1, ... ,xn+rn(xn)yn)ED whenever

(x 1, ... ,xn)ED and yiEY;, (i=l, ... ,n);

H;CP;(x;+r1(x1)y1, ... ,xn+rn(xn)yn),P;(x1, ... ,xn)-Y; J

n n

~

E

b.ki!Ykll+ r c.k0k[yk,yk-Pk(x 1, ... ,x

)J

k=1

1

k=1

1

n

+ cD;Cx;+Yi' xi+yi-ri(xi)Pi(x1+r1(x1)y1, ... ,xn+rn(xn)ynJ

for (x1, ... ,xn)ED, Y;EY;, (i=1, ... ,n), where c is a constant such that

O~cB~1-q.

Then there exists (x1, ... ,xn)ED such that 6iEP;(xi, ... ,xn), (i=1, ... ,n),

where 6; is the zero element of the Banach space Vi (i=l, ... ,n).

In the above the following notation is used (see Nadler [1,2J).

CL(X)={C:C is a nonempty closed subset of X} N(£,C)={xcX: II x-c II£ for some

CEC}

MIECZYSLAW ALTMAN

Reich [1]. Moreover, their results also contain an extension to multivalued

mappings of their own results from their paper Balakrishna Reddy and

Subrahmanyan [1].

Let bik'

cik~O

(i,k=1, ... ,n) and let aik=bik+cik' where (aik) is

the matrix defined by Matkowski [1,2]. Put

n

q=max(r:

1

r a.krk).

i

1

k= 1

1

Then Oq1 and

n

r a.krkqr., (i=1, ... ,n).

k=1

1 - 1

The following is the main result of Balakrishna Reddy and Subrahmanyan

[2].

THEOREM. Suppose that the closed nonlinear transformations

P.:DcX1

x ••• x

X

-+

CL(Y.), (i=1, ... ,n)

1

n

1

fulfill the following:

\here exist bounded linear operators

r;(x;):Y;-+ X;, X;EX;

(i=1, ... ,n) such that

llr;(x;lll

~

B, (x;, ... ,xn)ED, (i=1, ... ,n)

(a) (x1 + r1(x1)y1, ... ,xn+rn(xn)yn)ED whenever

(x 1, ... ,xn)ED and yiEY;, (i=l, ... ,n);

H;CP;(x;+r1(x1)y1, ... ,xn+rn(xn)yn),P;(x1, ... ,xn)-Y; J

n n

~

E

b.ki!Ykll+ r c.k0k[yk,yk-Pk(x 1, ... ,x

)J

k=1

1

k=1

1

n

+ cD;Cx;+Yi' xi+yi-ri(xi)Pi(x1+r1(x1)y1, ... ,xn+rn(xn)ynJ

for (x1, ... ,xn)ED, Y;EY;, (i=1, ... ,n), where c is a constant such that

O~cB~1-q.

Then there exists (x1, ... ,xn)ED such that 6iEP;(xi, ... ,xn), (i=1, ... ,n),

where 6; is the zero element of the Banach space Vi (i=l, ... ,n).

In the above the following notation is used (see Nadler [1,2J).

CL(X)={C:C is a nonempty closed subset of X} N(£,C)={xcX: II x-c II£ for some

CEC}