Solution of integral equations by product integration

Helton, Jon C.

doi:10.1090/S0002-9939-1975-0405048-4

Solution of integral equations by product integration
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by Jon C. Helton PDF

Proc. Amer. Math. Soc. 49 (1975), 401-406 Request permission

Abstract:

Suppose $f,h$ and $G$ are functions with values in a normed complete ring. With suitable restrictions on these functions, it is established that \[ f(x) = h(x) + \int _a^x {f(u)G(u,v)} \] for $a \leq x \leq b$ only if $\int _a^x {h(u)G{{(u,v)}_v}{\Pi ^x}} (1 + G)$ exists and is $f(x) - h(x)$ for $a \leq x \leq b$, and that \[ f(x) = h(x) + \int _a^x {G(u,v)f(u)} \] for $a \leq x \leq b$ only if $\int _a^x {_x{\Pi ^v}} (1 + \mathcal {G})G(u,v)h(u)$ exists and is $f(x) - h(x)$ for $a \leq x \leq b$, where $\mathcal {G}(s,r) = G(r,s)$.

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