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}
Previous Page Next Page