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)
On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation. (English) Zbl 1175.45002
The authors study the integral equation $$x(t) = p(t) +f(t, x(t))\int_0^t v(t, s, x(s))ds.$$ They write the following: “We will study the solvability of this equation in the space $BC (\mathbb{R}_{+})$ consisting of all real functions defined, continuous and bounded on the interval $\mathbb{R}_{+} = [0; \infty).$ More precisely, we will look for assumptions concerning the functions involved in this equation which guarantee that this equation has solution belonging to $BC (\mathbb{R}_{+})$ and being locally attractive or asymptotic stable on $\mathbb{R}_{+}$.” As example the authors consider the equation $$x(t) =t\exp(-2t) +\frac{1}{2\pi}\arctan(\sqrt{t} +tx(t)) \int_{0}^{t} \left( \frac{2x(s)^{2/3}+x(s)}{(s+1)(t^2+1)} +\frac{1}{10(t^2+1)} \right)\,ds .$$

45G10Nonsingular nonlinear integral equations
47H30Particular nonlinear operators
45M05Asymptotic theory of integral equations
45M10Stability theory of integral equations
47H09Mappings defined by “shrinking” properties
Full Text: DOI
[1] Agarwal, R. P.; O’regan, D.; Wong, P. I. Y.: Positive solutions of differential, difference and integral equations, (1999)
[2] Agarwal, R. P.; O’regan, D.: Infinite interval problems for differential, difference and integral equations, (2001)
[3] Banaś, J.; Goebel, K.: Measures of noncompactness in Banach spaces, Lecture notes in pure and applied mathematics 60 (1980) · Zbl 0441.47056
[4] Banaś, J.; Martinon, A.: Monotonic solutions of a quadratic integral equation of Volterra type, Comput. math. Appl. 47, 271-279 (2004) · Zbl 1059.45002 · doi:10.1016/S0898-1221(04)90024-7
[5] Banaś, J.; O’regan, D.: On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order, J. math. Anal. appl. 345, 573-582 (2008) · Zbl 1147.45003 · doi:10.1016/j.jmaa.2008.04.050
[6] Banaś, J.; Rzepka, B.: An application of a measure of noncompactness in the study of asymptotic stability, Appl. math. Lett. 16, 1-6 (2003) · Zbl 1015.47034 · doi:10.1016/S0893-9659(02)00136-2
[7] 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
[8] Burton, T. A.; Zhang, B.: Fixed points and stability of an integral equation: nonuniqueness, Appl. math. Lett. 17, 839-846 (2004) · Zbl 1066.45002 · doi:10.1016/j.aml.2004.06.015
[9] Caballero, J.; Mingarelli, A. B.; Sadarangani, K.: Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electronic J. Diff. eq. 57, 1-11 (2006) · Zbl 1113.45004 · emis:journals/EJDE/Volumes/2006/57/abstr.html
[10] Corduneanu, C.: Integral equations and applications, (1991) · Zbl 0714.45002
[11] Chandrasekhar, S.: Radiative transfer, (1960) · Zbl 0037.43201
[12] Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[13] Hu, S.; Khavanin, M.; Zhuang, W.: Integral equations arising in the kinetic theory of gases, Appl. anal. 34, 261-266 (1989) · Zbl 0697.45004 · doi:10.1080/00036818908839899
[14] Hu, X.; Yan, J.: The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. math. Anal. appl. 321, 147-156 (2006) · Zbl 1108.45006 · doi:10.1016/j.jmaa.2005.08.010
[15] Kelly, C. T.: Approximation of solutions of some quadratic integral equations in transport theory, J. integral eq. 4, 221-237 (1982) · Zbl 0495.45010
[16] O’regan, D.; Meehan, M.: Existence theory for nonlinear integral and integrodifferential equations, (1998)
[17] Väth, M.: Volterra and integral equations of vector functions, Pure and applied math (2000)
[18] Zabrejko, P. P.; Koshelev, A. I.; Krasnosel’skii, M. A.; Mikhlin, M. A.; Rakovschik, S. G.; Stetsenko, V. J.: Integral equations, (1975)