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)
On existence of integrable solutions of a functional integral equation under Carathéodory conditions. (English) Zbl 1168.45005
The authors consider the functional integral equation $$x(t)= f_1\left(t, \int^t_0k(t,x)f_2(s,x(s))\,ds\right),\quad t\in\bbfR_+.\tag{$*$}$$ which generalizes several equations arising in mechanics, physics, engineering etc. and have been studied in the literature. The functions $f_1,f_2$ and $k$ are supposed to satisfy Carathéodory conditions and some other technical assumptions. The authors prove by using the Schauder fixed point principle the existence of at least one solution of $(*)$ in $L^1 (\bbfR_+)$. The main tool of the proof is the measure of weak noncompactnes developed by {\it J. Banas} and {\it Z. Knap} [J. Math. Anal. Appl. 146, No. 2, 353--362 (1990; Zbl 0699.45002)]. To prove that the image of the operator associated to $(*)$ is relatively compact in $L_1(\bbfR_+)$ several considerations are necessary. Finally, two examples are given.

MSC:
45G10Nonsingular nonlinear integral equations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
WorldCat.org
Full Text: DOI
References:
[1] Appel, J.: Implicit functions, nonlinear integral equations and the measure of noncompactness of the superposition operator. J. math. Anal. appl. 83, 251-263 (1981) · Zbl 0495.45007
[2] Appel, J.; De Pascale, E.: Su alcuni parametri connesi con la misura di non compatezza di Hausdorff in spazi funzioni misarubili. Boll. unione. Mat. ital. (6) 3--13, 497-515 (1984)
[3] Appel, J.; Zabrejko, P. P.: Nonlinear superposition operators. Cambridge tracts in mathematics 95 (1990)
[4] Banaś, J.; El-Sayed, W. G.: Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variaton. J. math. Anal. appl. 167, 133-151 (1992) · Zbl 0754.45008
[5] Banaś, J.; Knap, Z.: Integrable solutions of a functional- integral equation. Revista mat. Univ. complutense de Madrid 2, 31-38 (1989) · Zbl 0679.45003
[6] Banaś, J.; Knap, Z.: Measures of weak noncompactness and nonlinear integral equations of convolution type. J. math. Anal. appl. 146, 353-362 (1990) · Zbl 0699.45002
[7] Banaś, J.; Rivero, J.: On measures of weak noncompactness. Ann. mat. Pure appl. 151, 213-224 (1988) · Zbl 0653.47035
[8] Burton, T. A.: Volterra integral and differential equations. (1983) · Zbl 0515.45001
[9] Carathéodory, K.: Vorlesungen über reele funktionen. (1918) · Zbl 46.0376.12
[10] Castaing, C.: Une nouvelle extension du théorème de dragoni-scorza. C. R. Acad. sci., Paris, sér. A 271, 396-398 (1970) · Zbl 0199.49302
[11] De Blasi, F. S.: On a property of the unit sphere in Banach spaces. Bull. math. Soc. sci. Math. R. S. roumanie. 21, 259-262 (1977) · Zbl 0365.46015
[12] Deimling, K.: Ordinary differential equations in Banach spaces. Lect. notes in mathematics 596 (1977) · Zbl 0361.34050
[13] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040
[14] Dieudonné, J.: Sur LES espaces de köthe. J. anal. Math. 1, 81-115 (1951) · Zbl 0044.11703
[15] Dunford, N.; Schwartz, J. T.: Linear operators. (1963)
[16] Emmanuele, G.: Integrable solutions of a functional-integral equation. J. integral equations 4, 89-94 (1992) · Zbl 0755.45005
[17] Hale, J. K.: Theory of functional differential equations. (1977) · Zbl 0352.34001
[18] Krasnosel’skii, M. A.: On the continuity of the operator $Fu(x)=f(x,u(x))$. Dokl. akad. Nauk SSSR 77, 185-188 (1951)
[19] Krasnosel’skii, M. A.; Zabrejko, P. P.; Pustyl’nik, J. I.; Sobolevskii, P. J.: Integral operators in spaces of summable functions. (1976)
[20] Lakshmikantham, V.; Leela, S.: Nonlinear differential equations in abstract spaces. (1981) · Zbl 0456.34002
[21] O’regan, D.; Meehan, M.: Existence theory for nonlinear integral and integradifferential equations. (1998)
[22] Pogorzelski, W.: Integral equations and their applications. (1966) · Zbl 0137.30502
[23] Dragoni, G. Scorza: Un teorema sulle funzioni continue rispetto ad une e misarubili rispetto ad un’altra variable. Rend. sem. Mat. univ. Padova 17, 102-106 (1948) · Zbl 0032.19702
[24] Zabrejko, P. P.; Koshelev, A. I.; Krasnosel’skii, M. A.; Mikhlin, S. G.; Rakovshchik, L. S.; Stecenko, V. J.: Integral equations. (1975)