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)
A mixed neutral system. (English) Zbl 0553.34042
Consider a mixed type neutral system (1) $x'(t)=f(t,x)+\sum\sp{m}\sb{j=1}F\sb j(t,x)x'(t+P\sb j)$, (2) $x(0)=x\sb 0$, where f is a continuous n-vector-valued functional, each $F\sb j$ is a continuous $n\times n$ matrix-valued function defined on $R\times C(R,R\sp n)$, each $P\sb j$ is a constant real number and $x\sb 0\in R\sp n$. It is also assumed that $\vert \cdot \vert$ is a norm in $R\sp n$, $\Vert \cdot \Vert$ the induced matrix norm and P, $M\sb f,M\sb F,K\sb f$ and $K\sb F$ are positive constants such that $\vert f\vert \le M\sb f$, each $\Vert F\sb j\Vert \le M\sb F$ on $R\times C(R,R\sp n)$, $P=\max\sb{j} \vert P\sb j\vert$ and for all $t\in R$, with x, $\tilde x\in C(R,R\sp n)$, $\vert f(t,x)-f(t,\tilde x)\vert \le K\sb f\max\sb{t- p\le s\le t+p}\vert x(s)-\tilde x(s)\vert$ and $\Vert F(t,x)-F(t,\tilde x)\Vert \le K\sb F\max\sb{t-p\le s\le t+p}\vert x(s)-\tilde x(s)\vert.$ The author proves that if $P,M\sb f,M\sb F,K\sb f$ and $K\sb F$ are sufficiently small and for any constant $a>0$ $$ e\sp{ap}[(1/a)(K\sb f+(mK\sb FM\sb f)/(1-mM\sb F))+mM\sb F]<1 $$ then (1) and (2) have a unique solution such that $\int\sp{t+1}\sb{t}\vert x'(s)\vert ds$ is bounded for all t. An example is given, to illustrate the theory.
Reviewer: O.Akinyele

34K05General theory of functional-differential equations
Full Text: DOI
[1] Czerwik, S.: On the global existence of solutions of functional-differential equations. Period. math. Hungar. 6, 347-351 (1975) · Zbl 0322.34050
[2] Driver, R. D.: Existence and continuous dependence of solutions of a neutral functional-differential equation. Archs ration. Mech. analysis 19, 149-166 (1965) · Zbl 0148.05703
[3] Driver, R. D.: Point data problems for functional differential equations. Dynamical systems, an international symposium 2, 115-121 (1976)
[4] Driver, R. D.: Erratum. Phys. rev. D 20, 2639 (1979)
[5] Fite, W. B.: Properties of the solutions of certain functional differential equations. Trans. am. Math. soc. 22, 311-319 (1921) · Zbl 48.0534.01
[6] Polossuchin, O.: Über eine besondere klasse von differentialen funktionalgleichungen. Inaugural dissertation (1910)
[7] Robertson, M. L.: Concerning siu’s method for solving y’$(t) = $F$(t,y(g(t)))$. Pacif. J. Math. 59, 223-227 (1975) · Zbl 0325.34078
[8] Smítalová, K.: Existence of solutions of functional-differential equations. Časopis pést. Mat. 100, 261-264 (1975)