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)
Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. (English) Zbl 1080.45005

The following Volterra type integral equation is considered

u(t)=h(t)+ 0 t G(t,s)f(s,u(s),Tu(s),Su(s))ds,tJ,(1)

where J=[0,a],

Tu(t)= 0 t k(t,s)u(s)ds,Tu(t)= 0 a h(t,s)u(s)ds,tJ,

and k, h, f are continuous kernels. Existence of a global solution is studied with the help of a fixed point theorem, generalizing Darbo’s fixed point theorem. As a particular application, the authors establish the existence of a global solution to the following initial value problem for a nonlinear ordinary differential equation

x ''' =f(t,x '' ,x),0t1,α 1 x(0)+α 2 x ' (0)=β 1 x(1)+β 2 x ' (1),x '' (0)=x 0 ,(2)

and a similar problem for x ''' =f(t,x '' ,x ' ,x) when the first initial value condition in (2) is replaced by the following one x ' (0)=β 1 x(1)+β 2 x ' (1).


MSC:
45G10Nonsingular nonlinear integral equations
34A34Nonlinear ODE and systems, general
45N05Abstract integral equations, integral equations in abstract spaces
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47N20Applications of operator theory to differential and integral equations