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Transactions of the American Mathematical Society

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



New existence theorems for nonlinear equations of Hammerstein type.

Authors: W. V. Petryshyn and P. M. Fitzpatrick
Journal: Trans. Amer. Math. Soc. 160 (1971), 39-63
MSC: Primary 47.80
MathSciNet review: 0281065
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Abstract: Let $ X$ be a real Banach space, $ {X^ \ast }$ its dual, $ A$ a linear map of $ X$ into $ {X^ \ast }$ and $ N$ a nonlinear map of $ {X^ \ast }$ into $ X$. Using the recent results of Browder and Gupta, Brezis, and Petryshyn, in this paper we study the abstract Hammerstein equation, $ w + ANw = 0$. Assuming suitable growth conditions on $ N$, new existence results are obtained under the following conditions on $ X,A$ and $ N$. In §1: $ X$ is reflexive, $ A$ bounded with $ f(x) = (Ax,x)$ weakly lower semicontinuous, $ N$ bounded and of type $ ($M$ )$. In §2: $ X$ is a general space, $ A$ angle-bounded, $ N$ pseudo-monotone. In §3: $ X$ is weakly complete, $ A$ strictly (strongly) monotone, $ N$ bounded (unbounded) and of type $ ($M$ )$. In §4: $ X$ is a general space, $ A$ is monotone and symmetric, $ N$ is potential. In §5: $ X$ is reflexive and with Schauder basis, $ {X^ \ast }$ strictly convex, $ N$ quasibounded and either monotone, or bounded and pseudo-monotone, or bounded and of type $ ($M$ )$.

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Keywords: Existence theorems, nonlinear operators, Hammerstein equations, weakly lower semicontinuous functionals, monotone and pseudo-monotone operators, operators of type $ ($M$ )$, pseudo-$ A$-proper operators, angle-bounded operators, potential operators, weakly and hemicontinuous operators
Article copyright: © Copyright 1971 American Mathematical Society

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