Existence of fixed point on Banach space

Question

I need your help please: If $(X,\|\cdot\|_{X})$ is a Banach space, $A:X\times X\times X\rightarrow\mathbb{R}$ a trilinear form. Suppose that $A$ is continuous,i.e., there exists $M>0$ such that $$|A(w,u,v)|\leq M\|w\|_{X}\|u\|_{X}\|v\|_{X}\quad\forall w,u,v\in X$$ Besides for all $w\in X$, $A(w,\cdot,\cdot)$ is elliptical,i.e., there exists $\alpha>0$ independent of $w$ such that $$A(w,v,v)\geq\alpha\|v\|_{X}^{2}\quad\forall v\in X$$ If $F\in X'$ and $T:X\rightarrow X$ the operator define by $T(w)=u$ with $u\in X$ the only element that $A(w,u,v)=F(v)\quad\forall v\in X$.\ Also define $K:=\{w\in X /\|w\|_{X}\leq\frac{1}{\alpha}\|F\|_{X'}\}$ and suppose that $\frac{M}{\alpha^2}\|F\|_{X'}<1$. Prove that there exists a unique $u\in K$ such that $T(u)=u$