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.References
- William D. L. Appling, Interval functions and real Hilbert spaces, Rend. Circ. Mat. Palermo (2) 11 (1962), 154–156. MR 154081, DOI 10.1007/BF02843951
- Burrell W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297–322. MR 188731
- Jon C. Helton, An existence theorem for sum and product integrals, Proc. Amer. Math. Soc. 39 (1973), 149–154. MR 317048, DOI 10.1090/S0002-9939-1973-0317048-1
- Jon C. Helton, Mutual existence of sum and product integrals, Pacific J. Math. 56 (1975), no. 2, 495–516. MR 405098
- Jon C. Helton, Product integrals and the solution of integral equations, Pacific J. Math. 58 (1975), no. 1, 87–103. MR 385480
- Jon C. Helton, Existence of integrals and the solution of integral equations, Trans. Amer. Math. Soc. 229 (1977), 307–327. MR 445245, DOI 10.1090/S0002-9947-1977-0445245-1
- Jon Helton and Stephen Stuckwisch, Numerical approximation of product integrals, J. Math. Anal. Appl. 56 (1976), no. 2, 410–437. MR 428688, DOI 10.1016/0022-247X(76)90053-6
- J. V. Herod, Coalescence of solutions for nonlinear Stieltjes equations, J. Reine Angew. Math. 252 (1972), 187–194. MR 291908, DOI 10.1515/crll.1972.252.187
- Alvin J. Kay, Nonlinear integral equations and product integrals, Pacific J. Math. 60 (1975), no. 1, 203–222. MR 399781
- J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148–173. MR 144179
- J. S. MacNerney, A nonlinear integral operation, Illinois J. Math. 8 (1964), 621–638. MR 167815
- P. R. Masani, Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc. 61 (1947), 147–192. MR 18719, DOI 10.1090/S0002-9947-1947-0018719-6
- J. W. Neuberger, Toward a characterization of the identity component of rings and near-rings of continuous transformations, J. Reine Angew. Math. 238 (1969), 100–104. MR 250144, DOI 10.1515/crll.1969.238.100
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