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

de Figueiredo, Djairo G.; Gupta, Chaitan P.

doi:10.1090/S0002-9939-1973-0318988-X

On the variational method for the existence of solutions of nonlinear equations of Hammerstein type
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by Djairo G. de Figueiredo and Chaitan P. Gupta PDF

Proc. Amer. Math. Soc. 40 (1973), 470-476 Request permission

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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