zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
De la Vallée-Poussin theorem on the differential inequality for equations with an aftereffect. (English. Russian original) Zbl 0957.34064
Proc. Steklov Inst. Math. 211, 28-34 (1995); translation from Tr. Mat. Inst. Steklova 211, 32-39 (1995).
In this paper, the authors investigate the boundary value problem (BVP) $${\cal L}x=f,\ x(a)=\alpha \text{ and } x(b)= \beta,$$ where ${\cal L}={\cal L}_0-T, {\cal L}_0:W^2[a,b]\to L[a,b]$ and $T:C[a,b]\to L[a,b]$ are linear, bounded Volterra-type operators and $(Tx_1)(t)\le (Tx _2)(t)$ for $t\in [a,b]$, whenever $x_1(t)\ge x_2(t)$ on $[a,b]$. Defining the mapping $H:C[a,b]\to C[a,b],\ (Hx)(t)=\int_a^bG_0(t,s) (Tx) (s)ds$, $t\in[a,b]$, $x\in C([a,b])$, where $G_0$ is the Green function of (BVP), the authors prove that the following statements are equivalent: (i) There exists $\nu\in W^2[a,b]$, such that $\nu(t)\ge 0$, $({\cal L}\nu) (t)\le 0$ $(a\le t\le b)$, and $\nu(a)+ \nu(b)-\int_a^b({\cal L}\nu) (s)ds>0$; (ii) The spectral radius of operator $H$ is less than one; (iii) (BVP) has exactly one solution for any $f\in L[a, b]$ and any $\alpha,\beta$, and its Green operator is antitone; (iv) The fundamental system of ${\cal L}x=0$ is not oscillatory. For the entire collection see [Zbl 0863.00015].

34K10Boundary value problems for functional-differential equations
34A40Differential inequalities (ODE)