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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the variational method for the existence of solutions of nonlinear equations of Hammerstein type


Authors: Djairo G. de Figueiredo and Chaitan P. Gupta
Journal: Proc. Amer. Math. Soc. 40 (1973), 470-476
MSC: Primary 47H15
DOI: https://doi.org/10.1090/S0002-9939-1973-0318988-X
MathSciNet review: 0318988
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Abstract: Let $ X$ be a real Banach space and $ {X^ \ast }$ its conjugate Banach space. Let $ A$ be an unbounded monotone linear mapping from $ X$ to $ {X^ \ast }$ and $ N$ a potential mapping from $ {X^ \ast }$ to $ X$. In this paper we establish the existence of a solution of the equation $ u + ANu = v$ for a given $ v$ in $ {X^ \ast }$ using variational method. Our method consists in using a splitting of $ A$ via an auxiliary Hilbert space and solving an equivalent equation in this auxiliary Hilbert space. In §2, we prove the same result in the case when $ X$ is a Hilbert space using the natural splitting of $ A$ in terms of its square root. We do this to compare and contrast the proofs in the two cases.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0318988-X
Keywords: Variational method, integral equations of Hammerstein type, weak $ \ast $-gradient, Carleman kernels
Article copyright: © Copyright 1973 American Mathematical Society