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  "body": "\n## Definition\n\nLet $a \u003c b$, let $K \\colon [a, b] \\times [a, b] \\to \\C$ and $f \\colon [a, b] \\to \\C$ be given [[functions|0EPAT51DXQ]], and let $\\lambda \\in \\C$.\nConsider an [[integral equation|R3SG1SN7JQ]] of one of the forms\n$$\\int_a^b K(x, s) \\varphi(s) \\, ds = f(x), \\qquad \\varphi(x) - \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds = f(x),$$\nor the same forms with the upper limit $b$ replaced by $x$.\nThe function $K$ is the *kernel* of the equation.\nThe map $\\varphi \\mapsto \\int_a^b K(\\cdot, s) \\varphi(s) \\, ds$ is the *integral operator* with kernel $K$.\n\n## Remarks\n\nExample: in $\\varphi(x) - \\int_0^1 e^{x - s} \\varphi(s) \\, ds = x$ the kernel is $K(x, s) = e^{x - s}$ and $\\lambda = 1$.\n\nNon-example: in $\\varphi(x) = x + \\int_0^1 \\varphi(s)^2 \\, ds$ the unknown does not enter linearly under the integral, so the equation has no kernel in this sense.\nFor a nonlinear equation $\\varphi(x) = f(x) + \\int_a^b H(x, s, \\varphi(s)) \\, ds$, the function $H$ of three variables is sometimes also called the kernel.\n\nFor a Volterra equation, with upper limit $x$, only the values of $K$ with $s \\le x$ matter, and one may set $K(x, s) = 0$ for $s \u003e x$.\n",
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