Abstract
In this chapter we present three examples of continuous operators acting in L p spaces, namely: the Nemytskii superposition operator; the Fredholm linear integral operator; and the Hammerstein nonlinear integral operator. As applications we shall prove via the Leray-Schauder principle several existence results in L p for Hammerstein and Volterra-Hammerstein integral equations in R n. We show that these results immediately yield existence theorems of weak solutions (in Sobolev spaces) to the initial value and two-point boundary value problems for ordinary differential equations in R n, under some more general conditions than the continuity. Notice the weak solutions are functions which satisfy the differential equations almost everywhere (a.e., that is, except a set of measure zero).
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