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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h(t)+ \int_0^T k(t, s) f(s, y(s)) ds, \quad \text{a.e. }t\in [0,T ],$ where the upper bound $$t$$ in the integral has been replaced by $$T$$.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