×

zbMATH — the first resource for mathematics

Existence theory for nonlinear Volterra and Hammerstein integral equations. (English) Zbl 0842.45003
Agarwal, R. P. (ed.), Dynamical systems and applications. Singapore: World Scientific. World Sci. Ser. Appl. Anal. 4, 601-615 (1995).
Existence results are given for two different but related scalar integral equations, namely for the Volterra equation \[ y(t)= h(t)+ \int_0^t k(t,s) f(s, y(s)) ds, \quad \text{a.e. }t\in [0,T], \] and for the corresponding Hammerstein equation \[ y(t)= h(t)+ \int_0^T k(t, s) f(s, y(s)) ds, \quad \text{a.e. }t\in [0,T ], \] where the upper bound \(t\) in the integral has been replaced by \(T\).
The proofs are based on fixed point arguments of compactness type and a priori boundedness estimates. One class of results requires that the operator \(y\mapsto f(t, y(t))\) maps \(L^\alpha [0,T ]\) into \(L^\beta [0,T]\) and that the integral operator with kernel \(k(t, s)\) maps \(L^\beta [0,T ]\) into \(L^\alpha [0,T ]\); here \(\alpha\) and \(\beta\) are conjugate exponents. Another class of results requires that the operator \(y\mapsto f(t, y(t))\) maps the space of bounded continuous functions into \(L^\beta [0,T ]\), and that the integral operator maps \(L^\beta [0,T ]\) into the space of bounded continuous functions.
For the entire collection see [Zbl 0834.00036].
Reviewer: O.Staffans (Espoo)

MSC:
45G10 Other nonlinear integral equations
PDF BibTeX XML Cite