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Multiple positive solutions for a class of integral inclusions. (English) Zbl 1151.45004

The author deals with integral inclusions of the following form

x(t)ft , x ( t ) 0 1 k(s)Ut , s , x ( s )dsfort[0,1],(*)

where f and k are some continuous functions and U is an L 1 -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 k-set contractive (with 0k<1) multivalued maps defined on a subset of a cone P in a Banach space, with values in the set of all convex, compact and nonempty subsets of P.

Unfortunately, the paper is not easy to read. Some definitions are obscure; e.g. the definition of a k-set contraction on page 21. If the space E in question is simply n (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 DP and any given ε>0, there exist finitely many balls, say, B ε (x i )={x:x-x i ε} for i=1,2,...,m, such that D i=1 m B ε (x i ).”

Here β stands for the Hausdorff measure of noncompactness and P 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 η=5 8. Unfortunately, in such a case this example fails to illustrate Theorem 1, in which there is assumed that 0<η<1 2. A similar comment concerns Example 2 from this paper.

45G10Nonsingular nonlinear integral equations
47G10Integral operators
45M20Positive solutions of integral equations
47H04Set-valued operators
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