zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

MSC:
45G10Nonsingular nonlinear integral equations
47G10Integral operators
45M20Positive solutions of integral equations
47H04Set-valued operators
References:
[1]Agarwal, R. P.; O’regan, D.: A note on the existence of multiple fixed points for multivalued maps with applications, J. differential equation 160, 389-403 (2000) · Zbl 1008.47055 · doi:10.1006/jdeq.1999.3690
[2]Argyros, I. K.: Quadratic equations and applications to Chandrasekar’s and related equations, Bull. austral. Math. soc. 32, 275-292 (1985) · Zbl 0607.47063 · doi:10.1017/S0004972700009953
[3]Banaś, J.; Rzepka, B.: On existence and asymptotic stability of solutions of a nonlinear integral equation, J. math. Anal. appl. 284, 165-173 (2003) · Zbl 1029.45003 · doi:10.1016/S0022-247X(03)00300-7
[4]Belarbi, A.; Benchohra, M.: Existence theory for perturbed impulsive hyperbolic differential inclusions with variable times, J. math. Anal. appl. 327, 1116-1129 (2007) · Zbl 1122.35148 · doi:10.1016/j.jmaa.2006.05.003
[5]Cheng, X.; Zhong, C.: Existence of positive solutions for a second-order ordinary differential system, J. math. Anal. appl. 312, 14-23 (2005) · Zbl 1088.34016 · doi:10.1016/j.jmaa.2005.03.016
[6]Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[7]Deimling, K.: Multivalued differential equations, (1992) · Zbl 0760.34002
[8]Dhage, B. C.: Multivalued mapping and fixed points II, Nonlinear funct. Anal. appl. 10, 359-378 (2005) · Zbl 1100.47040
[9]Dunninger, D. R.; Wang, H.: Multiplicity of positive radial solutions for an elliptic system on an annulus, Nonlinear anal. 42, 803-811 (2000) · Zbl 0959.35051 · doi:10.1016/S0362-546X(99)00125-X
[10]Guo, D.; Lakshmikantham, V.; Liu, X.: Nonlinear integral equations in abstract spaces, (1996)
[11]Hong, C. H.; Yeh, C. C.: Positive solutions for eigenvalue problems on a measure chain, Nonlinear anal. 51, 499-507 (2002) · Zbl 1017.34018 · doi:10.1016/S0362-546X(01)00842-2
[12]Hong, S. H.; Wang, L.: Existence of solutions for integral inclusions, J. math. Anal. appl. 317, 429-441 (2006) · Zbl 1125.45006 · doi:10.1016/j.jmaa.2006.01.057
[13]Li, Y. X.: Positive solutions of fourth-order boundary value problems with two parameters, J. math. Anal. appl. 281, 477-484 (2003) · Zbl 1030.34016 · doi:10.1016/S0022-247X(03)00131-8
[14]M. Martelli, A Rothe-type theorem for non-compact acyclic valued maps, Boll. Unione Mat. Ital. 11 (Suppl. fasc. 3) (1975) 70 – 76. · Zbl 0314.47035
[15]Marino, G.: Nonlinear boundary value problems for multivalued differential equations in Banach spaces, Nonlinear anal. 14, 545-558 (1990) · Zbl 0692.34018 · doi:10.1016/0362-546X(90)90061-K
[16]Zecca, P.; Zezza, P. L.: Nonlinear boundary value problems in Banach spaces for multivalued differential equations in noncompact interval, Nonlinear anal. 3, 347-352 (1979) · Zbl 0443.34060 · doi:10.1016/0362-546X(79)90024-5