144 К Ritter

j= 1, ...,mi, such that

mi m

7=1 j = mi + l

Tx^O for all xeK,

Tjy^O whenever уе{Ц-д^{хо))пЩд'Рс,)) for ; = mi + l,..,,m

and either

ТхфО for all хеК^ or

T ; ° g'Mo) M Ф 0 whenever д'^{хо) (x) + ^/xo) > 0 for at least one j e {m^ + 1,..., m}.

Furthermore , if ^(v4^) is closed, then Tj e Ljo and T^j ^ 0 for all уеЩ- ^,(xo)). For any x^eS there is an X2 e X such that x^ = Xq + X2. Hence, Txj ^ Txq. Therefore, if Tj is a linear extension of f) to 7^ for j = mi + 1,. ., m and To = 0, then T, To, ...,T^ is a set of linear operators which has all the properties required in the theorem.

Case IV. Suppose g'jixo) ф 0 for j = 1,..., mj, the mapping A = {g\ (xo), , g'mMo)) OÎX into Y=m(g[(xo)) X"'X ^{g'mMo)) has range Y and the system (1) has a solution, say Xj.

For y = mj + 1,..., m, define the convex cones K/m У), respectively Sj in X as follows

Kj= { yeYj\y = y^-Xgj{xQ) for some у^еЩ and some Я^О},

S= { xeX\g ] { xç , ) { x ) eK^ } ,

m + 1

Furthermore , let M = f] Sj and

iV = {xGX|^;(xo)(x) = 0 for ;==!,..,mi).

First , it is shown that for any x g MnAT, F(xo)(x)^0 holds. To this end, we assume that there is an seMnN such that Р'{хд){з)фК^ and derive a contradiction from this assumption.

Since F'(xo) (5) ф K^ and K^ is closed, there is a positive number a such that

F ( xo ) ( s + aXl)^iC,^

where x^ is an arbitrary but fixed solution of (1).

By assumption, A = {g[(xo), ...,g'm{xo)) maps X onto У = ^(^;(хо)) x ••. •••X ^fem,(^o))- Since Xo is regular, ^{д){хо)) is a Banach space and, therefore, Y is also a Banach space in its product topology. Furthermore,