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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Nonlinear operations and the solution of integral equations
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by Jon C. Helton PDF
Trans. Amer. Math. Soc. 237 (1978), 373-390 Request permission

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 $\left \| {V(x,y)P - V(x,y)Q} \right \| \leqslant \alpha (x,y)\left \| {P - Q} \right \|$ 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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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 237 (1978), 373-390
  • MSC: Primary 45N05; Secondary 46G99, 47H99
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0479379-3
  • MathSciNet review: 479379