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Zbl 1151.45004
Hong, Shihuang
Multiple positive solutions for a class of integral inclusions.
(English)
[J] J. Comput. Appl. Math. 214, No. 1, 19-29 (2008). ISSN 0377-0427

The author deals with integral inclusions of the following form $$x(t) \in f\bigl(t,x(t)\bigr)\int_{0}^{1}k(s)U\bigl(t,s,x(s)\bigr)ds \quad \text{for }\, t \in [0,1],\tag*$$ 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 {\it R. P. Agarwal} and {\it D. O'Regan} [J. Differ. Equations 160, No.~2, 389--403 (2000; Zbl 1008.47055)] for upper semicontinuous $k$-set contractive (with $0\leq k <1$) 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 $\mathbb{R}^{n}$ (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 $\varepsilon >0$, there exist finitely many balls, say, $B_{\varepsilon}(x_{i})=\{x: \left\Vert x-x_{i}\right\Vert \leq \varepsilon\}$ for $i=1,2,\ldots,m$, such that $D \subset \bigcup_{i=1}^{m}B_{\varepsilon}(x_{i})$.'' 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.
[Dariusz Bugajewski (Baltimore)]
MSC 2000:
*45G10 Nonsingular nonlinear integral equations
47G10 Integral operators
45M20 Positive solutions of integral equations
47H04 Set-valued operators

Keywords: integral inclusions; positive solutions; fixed points; set contractive map

Citations: Zbl 1008.47055

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