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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Nonlinear operations and the solution of integral equations


Author: Jon C. Helton
Journal: Trans. Amer. Math. Soc. 237 (1978), 373-390
MSC: Primary 45N05; Secondary 46G99, 47H99
MathSciNet review: 479379
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Abstract: The letters S, G and H denote a linearly ordered set, a normed complete Abelian group with zero element 0, and the set of functions from G to G that map 0 into 0, respectively. In addition, if $ V \in H$ and there exists an additive function $ \alpha $ from $ S \times S$ to the nonnegative numbers such that $ \left\Vert {V(x,y)P - V(x,y)Q} \right\Vert \leqslant \alpha (x,y)\left\Vert {P - Q} \right\Vert$ for each $ \{ x,y,P,Q\} $ in $ S \times S \times G \times G$, then $ V \in \mathcal{O}\mathcal{S}$ only if $ \smallint _x^yVP$ exists for each $ \{ x,y,P\} $ in $ S \times S \times G$, and $ V \in \mathcal{O}\mathcal{P}$ only if $ _x{\Pi ^y}(1 + V)P$ exists for each $ \{ x,y,P\} $ in $ S \times S \times G$. It is established that $ V \in \mathcal{O}\mathcal{S}$ if, and only if, $ V \in \mathcal{O}\mathcal{P}$. Then, this relationship is used in the solution of integral equations of the form $ f(x) = h(x) + \smallint _c^x[U(u,v)f(u) + V(u,v)f(v)]$, where U and V are in $ \mathcal{O}\mathcal{S}$. This research extends known results in that requirements pertaining to the additivity of U and V are weakened.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0479379-3
PII: S 0002-9947(1978)0479379-3
Keywords: Sum integral, product integral, subdivision-refinement integral, nonlinear integral equation
Article copyright: © Copyright 1978 American Mathematical Society