According to https://www.encyclopediaofmath.org/index.php/Fredholm_alternative, if for $x\in [a,b]$, \begin{align} \phi(x)-\lambda \int_a^b K(x,s)\phi(s)\,\mathrm{d}s&=0 \\ \psi(x)-\overline{\lambda}\int_a^b \overline{K(s,x)}\psi(s)\,\mathrm{d}s&=0 \end{align}
have non-zero solutions, say of dimension $n$, the equation $$ \phi(x)-\lambda \int_a^b K(x,s)\phi(s)\,\mathrm{d}s=f(x) $$ has a solution iff $\int_a^b f(x)\overline{\psi_k(x)}\, \mathrm{d}x=0$, for $k=1,..,n$, where $\overline{\psi_k(x)}$ is in the null space of the second equation above.
My question: What can we say for $x\notin [a,b]$, but $x\in\mathbb{R}$? In other words, when is the mapping $$ \phi \mapsto \phi(x)-\lambda \int_a^b K(x,s)\phi(s)\,\mathrm{d}s $$ surjective for $f\in C_b(\mathbb{R})$? Is this idea even related to the Fredholm Alternative?