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Transactions of the American Mathematical Society

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

 
 

 

A resolvent for an iteration method for nonlinear partial differential equations


Author: J. W. Neuberger
Journal: Trans. Amer. Math. Soc. 226 (1977), 321-343
MSC: Primary 47H99; Secondary 35A35, 35R20
DOI: https://doi.org/10.1090/S0002-9947-1977-0425705-X
MathSciNet review: 0425705
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Abstract: For each of m and n a positive integer denote by $ S(m,i)$ the space of all real-valued symmetric i-linear functions on $ {E_m},i = 1,2, \ldots ,n$. Denote by L a nonzero linear functional on $ S(m,n)$, denote by f a real-valued analytic function on $ {E_m} \times R \times S(m,1) \times \cdots \times S(m,[n/2])$ and denote by $ \alpha $ a member of $ D(f)$. Denote by H the space of all real-valued functions U, analytic at the origin of $ {E_m}$, so that $ \alpha = (0,U(0),U'(0), \ldots ,{U^{([n/2])}}(0))$. For $ U \in H,{f_U}(x) \equiv f(x,U(x),U'(x), \ldots ,{U^{([n/2])}}(x))$ for all x for which this is defined. A one-parameter semigroup (nonlinear if $ f \ne 0$) K on H is constructed so that if $ U \in K$, then $ K(\lambda )U$ converges, as $ \lambda \to \infty $, to a solution Y to the partial differential equation $ L{Y^{(n)}} = {f_Y}$. A resolvent j for this semigroup is determined so that $ J(\lambda )U$ also converges to y as $ \lambda \to \infty $ and so that $ J{(\lambda /n)^n}U$ converges to $ K(\lambda )U$ as $ n \to \infty $. The solutions $ Y \in H$ of $ L{Y^{(n)}} = {f_Y}$ are precisely the fixed points of the semigroup K.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0425705-X
Keywords: Semigroup of nonlinear transformations, resolvent, nonlinear partial differential equations
Article copyright: © Copyright 1977 American Mathematical Society

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