Positivity 13 (2009), 287–298
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010287-12, published online March 12, 2008
The essential spectral radius of Volterra
operators and an application to
Volterra-Hammerstein functional inclusions
Abstract. It is shown that for abstract linear Volterra operators the spectral
radius is the essential spectral radius. The result is applied to prove the
existence of a global solution of a Volterra-Hammerstein functional inclusion
when the nonlinearity has at most aﬃne growth.
Mathematics Subject Classiﬁcation (2000). 45D05, 45G10, 45N05, 47A10, 47B65,
Keywords. Abstract Volterra operator, essential spectrum, measure of non-
compactness, Leray-Schauder alternative, Volterra-Hammerstein functional
inclusion, nonlinear integral equation, multivalued maps.
1. Volterra operators
Let X be a Banach space, T be a totally ordered set, and let P
: X → X (t ∈ T )
be linear projections which are ordered in the sense that
Note that then
are also projections.
Example 1.1. Let T
(T )(1≤ p ≤∞) (or, more general, let X
be a Banach function space (i.e. ideal space) over T )and
x = χ
The paper was written in the framework of a Heisenberg fellowship (Az. VA 206/1-2); ﬁnancial
support by the DFG is gratefully acknowledged. The author wants to thank Vladimir Kadets for
valuable remarks about multivalued integration and Egor A. Alekhno for various remarks.