Let $H=\ell_2$, the real Hilbert space whose elements are the square-summable sequences of real scalars, i.e., $$ H=\left\{u=(u_1,u_2,\ldots,u_i,\ldots): \sum_{i=1}^{\infty}|u_i|^2<+\infty\right\}\;. $$ Let $F: H\rightarrow H$ be a mapping given by $$ F(u)=(0, u_1, u_2, \ldots, u_n, \ldots ) \quad \forall u = (u_1, u_2, \ldots, u_n, \ldots)\in H. $$ Finding a linear mapping $A: H\rightarrow H$ satisfying the following conditions:
There exists $L>0$ such that $\|Au\|\leq L \|u\|$ for all $u \in H;$
$\langle Au, u\rangle\geq 0 \quad \forall u \in H;$
There exists $\alpha \in (0, 1/L)$ such that $$ I-\alpha A+\alpha^2 A^2=F, $$ where $I:H\rightarrow H$ is an identity map.