New existence theorems for nonlinear equations of Hammerstein type.

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 $(\text {M} )$. 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 $(\text {M} )$. 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 $(\text {M} )$.