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)
Controllability of mixed Volterra-Fredholm-type integro-differential systems in Banach space. (English) Zbl 1119.93016

Summary: M. B. Dhakne and S. D. Kendre [Commun. Appl. Nonlinear Anal. 13, No. 4, 101–112 (2006; Zbl 1119.45005)] have proved the existence of the abstract nonlinear mixed Volterra-Fredholm integro-differential system of the type

x ' (t)=ft,x(t), 0 t k(t,s,x(s))ds, 0 T h(t,s,x(s))ds,x(0)=x 0 ,tJ=[0,T]·

In this short article, we have studied sufficient conditions for controllability of semi-linear mixed Volterra-Fredholm-type integro-differential systems in Banach space of the type

x ' (t)=Ax(t)+(Bu)(t)+ft,x(t), 0 t g(t,s,x(s))ds, 0 T h(t,s,x(s))ds,x(0)=x 0 ,tJ=[0,T],

where the state x(·) takes values in a Banach space X and the control function u(·) is given in L 2 (J,U), with U as a Banach space. Here A is the infinitesimal generator of a strongly continuous semigroup in a Banach space X. B is a bounded linear operator from U into X. The result is obtained by using the application of the topological transversality theorem known as Leray-Schauder alternative and rely on a priori bounds of solutions. An example is provided to illustrate the theory.

93C23Systems governed by functional-differential equations
45J05Integro-ordinary differential equations
45N05Abstract integral equations, integral equations in abstract spaces