System of linear integral equations

Remarks

Example: on $\Omega = [0, 1]$ with $n = 2$ and $\lambda = 1$, take $K_{11} = K_{22} = 0$, $K_{12}(x, t) = 1$, $K_{21}(x, t) = x$, $f_1(x) = x$ and $f_2 = 0$. Put $A = \int_0^1 \varphi_2$ and $B = \int_0^1 \varphi_1$. The system says $\varphi_1(x) = x + A$ and $\varphi_2(x) = B x$, so $A = B/2$ and $B = \frac{1}{2} + A$, which give $B = 1$ and $A = \frac{1}{2}$. Check: $\varphi_1(x) = x + \frac{1}{2}$ and $\varphi_2(x) = x$ give $\varphi_1(x) - \int_0^1 \varphi_2 = x$ and $\varphi_2(x) - x \int_0^1 \varphi_1 = x - x = 0$.

Non-example: the pair $\varphi_1(x) - \int_0^1 \varphi_2(t)^2\, dt = x$, $\varphi_2(x) - \int_0^1 \varphi_1(t)\, dt = 0$ is a system of integral equations, but not a linear one, since $\varphi_2$ enters squared.

If $\Omega = [0, 1]$, the substitution $\Phi(x + i - 1) = \varphi_i(x)$ for $x \in [0, 1)$ turns the system into one linear integral equation for $\Phi$ on $[0, n)$ with a piecewise defined kernel.