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  "body": "\n## Definition\n\nLet $(\\Omega, \\mu)$ be a measure space.\nLet $X$ be a space of real-valued functions on $\\Omega$, such as $C(\\Omega)$ for compact $\\Omega$ or $L^p(\\Omega, \\mu)$.\nIts [[positive cone|Z344RY5XX1]] is $X_+ = \\set{\\varphi \\in X : \\varphi \\ge 0}$, with the inequality almost everywhere in the case of $L^p$.\nConsider an [[integral equation|R3SG1SN7JQ]] $\\varphi = T\\varphi$ on $X$, for example $T\\varphi = f + \\int_\\Omega k(\\cdot, y) \\varphi(y) \\, d\\mu(y)$, or a nonlinear one such as a [[Hammerstein equation|T7E2EW2T0G]] $T\\varphi = \\int_\\Omega k(\\cdot, y) \\, g(y, \\varphi(y)) \\, d\\mu(y)$.\n\n1. A *positive solution* is a solution $\\varphi \\in X_+$ with $\\varphi \\ne 0$.\n2. A *strictly positive solution* is a solution with $\\varphi(x) \u003e 0$ for all $x \\in \\Omega$ (for almost all $x$, in the case of $L^p$).\n\n## Remarks\n\nExample: on $X = C[0, 1]$ take $\\varphi(x) = 1 + \\frac12 \\int_0^1 \\varphi(t) \\, dt$.\nThe right side is constant, so $\\varphi = c$ with $c = 1 + \\frac{c}{2}$, that is $c = 2$, a strictly positive solution.\nMore generally, if $k \\ge 0$, $f \\ge 0$, and the integral operator $K$ has [[spectral radius|9V7YQA85PC]] $r(K) \u003c 1$, the [[Liouville–Neumann series|H1HWSWME6B]] $\\varphi = \\sum_{n \\ge 0} K^n f$ is a sum of nonnegative terms, so the solution is nonnegative.\n\nNon-example: $\\varphi(x) = 1 + 2 \\int_0^1 \\varphi(t) \\, dt$ on $[0, 1]$ gives $c = 1 + 2c$, so its only solution is $\\varphi = -1$, which is not in $X_+$.\nHere $r(K) = 2 \u003e 1$, and the Neumann series argument fails.\n\nFor the homogeneous eigenvalue problem $\\lambda\\varphi = K\\varphi$, the [[Krein–Rutman theorem|VADXDK3TKM]] and [[Jentzsch's theorem|P8CAK0FHGJ]] give a positive eigenfunction for the eigenvalue $r(K)$ when $K$ is compact with nonnegative kernel and $r(K) \u003e 0$.\n",
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