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  "body": "\n## Definition\n\nLet $a \u003c b$, let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[kernel|GSMBXT7FPK]], let $f \\colon [a, b] \\to \\C$ be given, and let $\\lambda \\in \\C$.\nLet $a_0 \\colon [a, b] \\to \\C$ be a [[continuous function|0X3WGX95FE]] that has at least one zero in $[a, b]$ and is not identically zero.\nAn *integral equation of the third kind* is a [[linear integral equation|AYZCKG34QK]] of the form\n$$a_0(x) \\varphi(x) = f(x) + \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds \\quad \\text{for all } x \\in [a, b],$$\nfor the unknown function $\\varphi \\colon [a, b] \\to \\C$.\n\n## Remarks\n\nExample: $x \\varphi(x) = x + \\int_{-1}^{1} s \\, \\varphi(s) \\, ds$ on $[-1, 1]$, with $a_0(x) = x$, $f(x) = x$, $K(x, s) = s$ and $\\lambda = 1$.\nThe coefficient $a_0$ is continuous, vanishes at $0$, and is not identically zero.\nThe function $\\varphi = 1$ is a solution, since $\\int_{-1}^{1} s \\, ds = 0$.\n\nNon-example: $(2 + x) \\varphi(x) = f(x) + \\int_{-1}^{1} \\varphi(s) \\, ds$ on $[-1, 1]$ is not of the third kind, since $2 + x$ has no zero there.\nDividing by $2 + x$ turns it into an equation of the second kind.\n\nSome authors also allow the upper limit $x$ in place of $b$.\n",
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