Existence Theory in L p Spaces

Radu Precup
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Radu Precup: Babeş-Bolyai University, Department of Applied Mathematics

Chapter Chapter 5 in Methods in Nonlinear Integral Equations, 2002, pp 61-76 from Springer

Abstract: 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).

Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-015-9986-3_6

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DOI: 10.1007/978-94-015-9986-3_6

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