The author deals with integral inclusions of the following form
where and are some continuous functions and is an -Carathéodory multivalued map. Under suitable assumptions the existence of at least one (Theorem 1) or at least two (Theorem 2) positive solutions to (*) is established. The proofs are based on an expansion and compression fixed point theorem due to R. P. Agarwal and D. O’Regan [J. Differ. Equations 160, No. 2, 389–403 (2000; Zbl 1008.47055)] for upper semicontinuous -set contractive (with ) multivalued maps defined on a subset of a cone in a Banach space, with values in the set of all convex, compact and nonempty subsets of .
Unfortunately, the paper is not easy to read. Some definitions are obscure; e.g. the definition of a -set contraction on page 21. If the space in question is simply (see page 20, below the formula (*), then this definition does not make sense. There are also some mistakes. For example, on page 25 there is written: “For any bounded and any given , there exist finitely many balls, say, for , such that .”
Here stands for the Hausdorff measure of noncompactness and is a cone in infinite dimensional Banach space, so in general the above statement is not true. And finally, there are some computational mistakes; e.g. in the Example 1 on page 28 there is written . Unfortunately, in such a case this example fails to illustrate Theorem 1, in which there is assumed that . A similar comment concerns Example 2 from this paper.