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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Omega$ be a measurable subset of $\\R^m$, for example an interval $[a, b]$.\nLet $K_{ij} \\colon \\Omega \\times \\Omega \\to \\C$, for $1 \\le i, j \\le n$, be [[kernels|GSMBXT7FPK]], let $f_1, \\ldots, f_n$ be given functions on $\\Omega$, and let $\\lambda \\in \\C$.\nA *system of linear integral equations of the second kind* for unknown functions $\\varphi_1, \\ldots, \\varphi_n$ on $\\Omega$ is\n$$\\varphi_i(x) - \\lambda \\sum_{j=1}^{n} \\int_\\Omega K_{ij}(x, t)\\, \\varphi_j(t)\\, dt = f_i(x), \\quad x \\in \\Omega, \\quad i = 1, \\ldots, n.$$\nA system *of the first kind* omits the terms $\\varphi_i(x)$.\nIn vector form, with $\\varphi = (\\varphi_1, \\ldots, \\varphi_n)$ and the [[matrix|0VD9ZSWWJG]] kernel $K = (K_{ij})$, it reads $\\varphi(x) - \\lambda \\int_\\Omega K(x, t)\\, \\varphi(t)\\, dt = f(x)$.\n\n## Remarks\n\nExample: 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 $f_2 = 0$.\nPut $A = \\int_0^1 \\varphi_2$ and $B = \\int_0^1 \\varphi_1$.\nThe 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 $A = \\frac{1}{2}$.\nCheck: $\\varphi_1(x) = x + \\frac{1}{2}$ and $\\varphi_2(x) = x$ give $\\varphi_1(x) - \\int_0^1 \\varphi_2 = x$ and $\\varphi_2(x) - x \\int_0^1 \\varphi_1 = x - x = 0$.\n\nNon-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.\n\nIf $\\Omega = [0, 1]$, the substitution $\\Phi(x + i - 1) = \\varphi_i(x)$ for $x \\in [0, 1)$ turns the system into one [[linear integral equation|AYZCKG34QK]] for $\\Phi$ on $[0, n)$ with a piecewise defined kernel.\n",
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