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  "body": "\n## Statement\n\nLet $(X, d)$ be a nonempty [[complete metric space|0BY52JW6NK]].\nLet $\\varphi \\colon X \\to [0, \\infty)$ be a lower semicontinuous function.\nLet $T \\colon X \\to X$ be any map, not necessarily continuous, such that\n$$d(x, T(x)) \\le \\varphi(x) - \\varphi(T(x)) \\quad \\text{for all } x \\in X.$$\nThen $T$ has a fixed point: there is $x \\in X$ with $T(x) = x$.\n\n## Remarks\n\nThe theorem follows from [[Ekeland's variational principle|MJGPW5EVB3]].\nWith $\\varepsilon = 1$, it gives a point $x_0$ with $\\varphi(x_0) \u003c \\varphi(x) + d(x, x_0)$ for every $x \\ne x_0$.\nIf $T(x_0) \\ne x_0$, then taking $x = T(x_0)$ gives $\\varphi(x_0) \u003c \\varphi(T(x_0)) + d(T(x_0), x_0) \\le \\varphi(x_0)$, a contradiction.\n\nExample: the [[Banach fixed point theorem|0VK6VEH345]] is a special case.\nIf $d(T(x), T(y)) \\le k\\, d(x, y)$ with $0 \\le k \u003c 1$, put $\\varphi(x) = d(x, T(x))/(1 - k)$, which is continuous.\nThe Caristi condition becomes $d(T(x), T(T(x))) \\le k\\, d(x, T(x))$, which holds.\n\nCompleteness is needed: on $X = (0, 1]$ with the usual metric, $T(x) = x/2$ and $\\varphi(x) = x$ satisfy $d(x, T(x)) = x/2 = \\varphi(x) - \\varphi(T(x))$, but $T$ has no fixed point in $X$.\n\n## Explanations\n\n### L1 human\n\nThink of $\\varphi$ as a supply of fuel and of $T$ as a move.\nEach move costs at least as much fuel as the distance travelled.\nSince fuel cannot go below zero, the total distance of a run of moves is finite.\nIn a complete space the run settles down to a point that $T$ does not move.\n\n### L2 human\n\nThe theorem does not ask $T$ to be continuous, which makes it more flexible than the Banach fixed point theorem.\nIt is equivalent in strength to Ekeland's variational principle, and each can be derived from the other.\nIt is used in nonlinear analysis for maps that are only known to decrease a lower semicontinuous potential, such as some descent schemes in optimisation.\n",
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