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  "body": "\n## Summary\n\nA positive solution of an [[integral equation|R3SG1SN7JQ]] is a solution that is nonnegative and not identically zero, or in a stronger sense strictly positive.\nIn models of densities, concentrations, populations or probabilities only such solutions have a meaning, so existence of a positive solution is often the real question.\nThe main tools are order methods: the cone of nonnegative functions, [[positive operators|77020CY5ZW]] that map it into itself, and monotone iteration.\nFixed point and eigenvalue theorems on cones, such as the [[Krein–Rutman theorem|VADXDK3TKM]], complete the toolbox.\nFor linear equations with nonnegative kernels, positivity is closely tied to the spectral radius of the integral operator.\n\n## Explanations\n\n### L1 human\n\nIf the unknown is a density of particles, a negative value makes no sense, so one wants solutions that are never negative.\nIntegral operators with nonnegative kernels send nonnegative functions to nonnegative functions, and this helps.\nBut it is not enough: $\\varphi(x) = 1 + 2\\int_0^1 \\varphi(t) \\, dt$ on $[0, 1]$ has the unique solution $\\varphi = -1$, although all its data are positive.\nThe trouble is that the integral term is too strong; with the factor $\\frac12$ instead of $2$, the solution is $\\varphi = 2$.\n",
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