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  "body": "\n## Definition\n\nLet $T \u003e 0$.\nLet $f \\colon \\R \\to \\R$ be continuous and $T$-periodic.\nLet $C$ be a continuous function on $\\set{(t, s) \\in \\R^2 : s \\le t}$ with $C(t + T, s + T) = C(t, s)$ for all $s \\le t$.\nLet $g \\colon \\R \\times \\R \\to \\R$ be continuous and $T$-periodic in its first variable.\nConsider the [[integral equation|R3SG1SN7JQ]]\n$$x(t) = f(t) + \\int_{-\\infty}^t C(t, s) \\, g(s, x(s)) \\, ds, \\qquad t \\in \\R.$$\nA *$T$-periodic solution* of this equation is a function $x \\colon \\R \\to \\R$ such that:\n\n1. $x$ is continuous;\n2. $x(t + T) = x(t)$ for all $t \\in \\R$, that is, $x$ is a $T$-[[periodic function|0ATF6Q0633]];\n3. for every $t \\in \\R$ the integral converges absolutely and the equation holds.\n\n## Remarks\n\nThe same notion applies to systems, where $x$ takes values in $\\R^n$, and to equations of Fredholm type over one period, such as $x(t) = f(t) + \\int_0^T k(t - s) \\, x(s) \\, ds$ with a $T$-periodic kernel $k$.\n\nExample: $T = 2\\pi$, $C(t, s) = e^{-(t - s)}$, $g(s, x) = x$ and $f(t) = \\frac{1}{2}(\\sin t + \\cos t)$.\nThe function $x(t) = \\sin t$ is continuous and $2\\pi$-periodic.\nThe integral $\\int_{-\\infty}^t e^{-(t - s)} \\sin s \\, ds$ converges absolutely and equals $\\frac{1}{2}(\\sin t - \\cos t)$, as the imaginary part of $\\int_{-\\infty}^t e^{-(t - s)} e^{i s} \\, ds = e^{i t} / (1 + i)$.\nAdding $f(t)$ gives $\\sin t$, so all three clauses hold.\n\nNon-example: periodic data do not force periodic solutions.\nThe equation $x(t) = 1 + \\int_0^t x(s) \\, ds$ on $[0, \\infty)$ has constant data, but its only continuous solution is $x(t) = e^t$, which fails clause 2 for every $T$.\n\n## Explanations\n\n### L1 human\n\nA periodic solution is a solution that comes back to the same value after each period and is valid for all times, past and future.\n",
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