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  "body": "\n## Definition\n\nLet $X$ be a complex [[Banach space|06GZ956657]] and let $A \\colon D(A) \\subseteq X \\to X$ be a [[closed linear operator|VM7B38PA87]].\nLet $a \\in L^1_{\\mathrm{loc}}(\\R_+)$ be a scalar [[locally integrable|X7GTF3D98F]] kernel on $\\R_+ = [0, \\infty)$, let $\\tau \u003e 0$, $J = [0, \\tau]$, and let $f \\in C(J; X)$.\nThe *abstract Volterra equation* with kernel $a$, operator $A$ and forcing $f$ is\n$$u(t) = f(t) + \\int_0^t a(t - s) \\, A u(s) \\, ds, \\qquad t \\in J.$$\nA *strong solution* is a function $u \\in C(J; X)$ such that $u(t) \\in D(A)$ for all $t \\in J$, $A u \\in C(J; X)$, and the equation holds for all $t \\in J$.\n\n## Remarks\n\nA *mild solution* is a function $u \\in C(J; X)$ such that $\\int_0^t a(t - s) u(s) \\, ds \\in D(A)$ for all $t \\in J$ and $u(t) = f(t) + A \\int_0^t a(t - s) u(s) \\, ds$.\nEvery strong solution is a mild solution, since $A$ is closed; compare the [[mild solution|3VBRS94V1G]] of an abstract Cauchy problem.\n\nSpecial cases.\nFor $a = 1$, a strong solution with $f \\in C^1(J; X)$ satisfies $u' = A u + f'$ and $u(0) = f(0)$.\nFor $a(t) = t$, differentiating twice gives $u'' = A u + f''$ with $u(0) = f(0)$ and $u'(0) = f'(0)$, when $f \\in C^2(J; X)$.\n\nExample: $X = \\C$, $A = 1$, $a = 1$ and $f = 1$.\nThe function $u(t) = e^t$ is a strong solution, since $1 + \\int_0^t e^s \\, ds = e^t$.\n\nNon-example: in the same setting, $u(t) = 1 + t$ is not a solution, since $1 + \\int_0^t (1 + s) \\, ds = 1 + t + t^2/2$.\n\n## Explanations\n\n### L1 human\n\nIt is a Volterra equation in which numbers are replaced by vectors of a Banach space and multiplication by a fixed number is replaced by an operator $A$.\nThe kernel $a$ weights the past, and the operator $A$ usually acts in the space variables, for instance as a Laplacian.\n",
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