Existence Theory in L p Spaces

In this chapter we present three examples of continuous operators acting in Lp 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 Lp for Hammerstein and Volterra-Hammerstein integral equations in Rn. 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 Rn, 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).