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  "body": "\n## Definition\n\nLet $X$ be a real [[Banach space|06GZ956657]], let $K \\colon X \\to X$ be a [[compact linear operator|M2NEP1K5AE]], and let $R \\colon X \\to X$ be a continuous map with $\\norm{R(\\varphi)} / \\norm{\\varphi} \\to 0$ as $\\varphi \\to 0$.\nConsider the equation\n$$\\varphi = \\lambda\\bigl(K \\varphi + R(\\varphi)\\bigr), \\quad \\lambda \\in \\R,\\ \\varphi \\in X,$$\nwhich has the trivial solution $\\varphi = 0$ for every $\\lambda$.\nA number $\\lambda_0$ is a *bifurcation point* of the equation if every neighbourhood of $(\\lambda_0, 0)$ in $\\R \\times X$ contains a solution $(\\lambda, \\varphi)$ with $\\varphi \\ne 0$.\nOne then says that nonzero solutions *bifurcate from* $\\lambda_0$.\n\n## Remarks\n\nEvery bifurcation point is a [[characteristic value|DC7HMHVREP]] of $K$: a number $\\lambda_0$ with $\\varphi = \\lambda_0 K \\varphi$ for some $\\varphi \\ne 0$.\nIndeed, if $\\lambda_0$ is not one, then $I - \\lambda K$ is invertible with uniformly bounded inverse for $\\lambda$ near $\\lambda_0$, and $\\norm{\\varphi} \\le C \\abs{\\lambda} \\norm{R(\\varphi)}$ forces $\\varphi = 0$ for small $\\varphi$.\nKrasnosel'skii's theorem gives a converse: if $R$ is also compact, every characteristic value of $K$ of odd [[algebraic multiplicity|0FY42N02DV]] is a bifurcation point.\n\nExample: $X = \\R$, $K = 1$ and $R(\\varphi) = -\\varphi^3$.\nThe characteristic value is $\\lambda_0 = 1$, and for $\\lambda \u003e 1$ the nonzero solutions are $\\varphi = \\pm\\sqrt{1 - 1/\\lambda}$, which tend to $0$ as $\\lambda \\to 1$; so $1$ is a bifurcation point, a [[pitchfork bifurcation|P88X34MSVV]].\n\nNon-example: $X = \\R^2$, $K = I$ and $R(x, y) = (-y^3, x^3)$.\nThe characteristic value $1$ has multiplicity $2$, which is even.\nThe equations are $(1 - \\lambda) x + \\lambda y^3 = 0$ and $(1 - \\lambda) y - \\lambda x^3 = 0$; multiplying the first by $y$, the second by $x$ and subtracting gives $\\lambda(x^4 + y^4) = 0$.\nSo for $\\lambda$ near $1$ the only solution is $x = y = 0$, and $1$ is not a bifurcation point.\n",
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