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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

$x\left(t\right)\in f\left(t,x\left(t\right)\right){\int }_{0}^{1}k\left(s\right)U\left(t,s,x\left(s\right)\right)ds\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}t\in \left[0,1\right],\phantom{\rule{2.em}{0ex}}\left(*\right)$

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 $0\le k<1$) 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 ${ℝ}^{n}$ (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 $D\subset P$ and any given $\epsilon >0$, there exist finitely many balls, say, ${B}_{\epsilon }\left({x}_{i}\right)=\left\{x:∥x-{x}_{i}∥\le \epsilon \right\}$ for $i=1,2,...,m$, such that $D\subset {\bigcup }_{i=1}^{m}{B}_{\epsilon }\left({x}_{i}\right)$.”

Here $\beta$ 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 $\eta =\frac{5}{8}$. Unfortunately, in such a case this example fails to illustrate Theorem 1, in which there is assumed that $0<\eta <\frac{1}{2}$. A similar comment concerns Example 2 from this paper.

##### MSC:
 45G10 Nonsingular nonlinear integral equations 47G10 Integral operators 45M20 Positive solutions of integral equations 47H04 Set-valued operators
##### References:
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