Let $A:X→X$ be a compact linear operator on a normed space $X$. If the homogeneous equation $φ−Aφ=0$ only has the trivial solution $φ=0$, then for each $f∈X$ the inhomogeneous equation $φ−Aφ=f$ has a unique solution $φ∈X$ and this solution depends continuously on $f$ . If the homogeneous equation has a nontrivial solution, then it has only a finite number $m∈\mathbb{N}$ of linearly independent solutions $φ_1,...,φ_m$ and the inhomo-geneous equation is either unsolvable or its general solution is of the form $$\phi =\overline{\phi}+\sum_{k=1}^{m}\alpha_k \phi _k,$$ where $α_1,...,α_m$ are arbitrary complex numbers and $\overline{\phi}$ denotes a particular solutionof the inhomogeneous equation.
my work : I am get for the trivial case but i am getting any idea that how to proceed for for non trivial case.