Hi
(,) denotes scalar product.
I am trying to prove that the following system has a solution: $A$ is an nXn matrix and satisfies (Ax,x) < 0 for all non-zero vector x in R^n, b a vector in $R^n$, n finite, $x$ is the unknown vector, max(b-x,0) is the vector with components max(b^i-x^i,0) for i in {1,..,n}. The system is $-Ax = (1/E) max(b-x, 0)$. E is just a small number >0.
I am supposed to use the following fixed point theorem.
Theorem: (Fixed point lemma).
Let m ≥ 1 and let F : R^n -->R^n be acontinuous map such that for a suitable r> 0 one has
(F(ξ), ξ) ≥ 0, ∀ξ ∈ R^n with |ξ| = r.
Then there exists ξ_0, |ξ_0| ≤ r, such that F(ξ_0) = 0.
In this case my function F is F(x)=-Ax-(1/E)*max(b-x, 0), Now I can not find a suitable r such that (F(ξ), ξ) ≥ 0 ∀ξ ∈ R^n with |ξ| = r.