The question says
Let $A:X \to Y$ where $X,Y$ are normed spaces.
$(i)$ Discuss the uniqueness and existence of a solution of the operator equation $Ax=y$ where $x \in X$ and $y \in Y$ for different operators $A$
$(ii)$ Discuss for the different values of $\lambda$ the existence and uniqueness of this operator equation $Ax=\lambda x -y$
I think for $(i)$ the only way for the solution $x$ to exist is that $A$ becomes bijective in order to have an inverse st $x=A^{-1}y$ and in this case the solution is unique. I don't seem to have any other ideas concerning $(i)$.
for $(ii)$ I think we can never discuss it unless $A:X \to X$. Moreover, I assumed the following about $A$ if $A \in L(X)$ where $L(X)$ is the space of linear bounded operators and $\|A\| \leq \lambda$ then we can re-write the equation as $(I\lambda - A)x=-y$
where $x= -\frac{1}{\lambda} \sum_{n=0}^{\infty} \frac{A^n}{\lambda^n}(y)$ this proves the existence and for every choice of $\lambda$ we get a new solution.
I don't have any other ideas.
(**Note. I did not study compact operators **)