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 and asymptotic behaviour of solutions of a functional integral equation. (English) Zbl 1128.45004
The authors deal with the following functional integral equation: $$ x(t)=f(t,\;\int_0^t x(s)\,ds,\;\int_0^t x(h(s,x(s)))\,ds),\quad t\geq 0. \tag1$$ Under reasonable hypotheses on $f$ and $h$ the authors show that problem (1) has at least one solution and specify its asymptotic behaviour. To achieve their goal they use the classical Schauder fixed point principle and the concept of measure of noncompactness. Concluding, they provide two examples to illustrate the applicability of their result.

MSC:
45G10Nonsingular nonlinear integral equations
45M05Asymptotic theory of integral equations
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
WorldCat.org
Full Text: DOI
References:
[1] Agarwal, R. P.; O’regan, D.: Infinite interval problems for differential, difference and integral equations. (2001)
[2] Appell, J.; Zabrejko, P. P.: Nonlinear superposition operators. Cambridge tracts in mathematics 95 (1990) · Zbl 0701.47041
[3] Azbelev, N. V.; Maksimov, V. P.; Rakhmatullina, L. F.: Introduction to theory of functional-differential equations. (1990) · Zbl 0725.34071
[4] Banaś, J.: The existence of solutions of some functional-differential and functional-integral equations. Funct. approx. 13, 3-12 (1982)
[5] J. Banaś, Applications of measures of noncompactness to various problems, Zeszyty Nauk. Politech. Rzeszowskiej Mat.-Fiz., z. 5, Rzeszów, 1987
[6] Banaś, J.; Goebel, K.: Measures of noncompactness in Banach spaces. Lecture notes in pure and applied mathematics 60 (1980)
[7] Burton, T. A.: Volterra integral and differential equations. (1983) · Zbl 0515.45001
[8] Corduneanu, C.: Integral equations and stability of feedback systems. (1973) · Zbl 0273.45001
[9] Corduneanu, C.: Integral equations and applications. (1991) · Zbl 0714.45002
[10] Czerwik, S.: The existence of global solutions of a functional-differential equations. Colloq. math. 36, 121-125 (1976) · Zbl 0356.34077
[11] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040
[12] Dhage, B. C.; Ntouyas, S. K.: Existence results for nonlinear functional integral equations via a fixed point theorem of Krasnoselskii type. Nonlinear stud. 9, 307-317 (2002) · Zbl 1009.47054
[13] Domoshnitsky, A.; Goltser, Ya.: An approach to study of stability of integral--differential equations. Nonlinear anal. 47, 3885-3896 (2001) · Zbl 1042.34570
[14] Grimm, L. J.: Existence and uniqueness for nonlinear neutral-differential equations. Bull. amer. Math. soc. 77, 374-376 (1971) · Zbl 0214.09803
[15] Infante, G.; Webb, J. R. L.: Nonzero solutions of Hammerstein integral equations with discontinuous kernels. J. math. Anal. appl. 272, 30-42 (2002) · Zbl 1008.45004
[16] O’regan, D.; Meehan, M.: Existence theory for nonlinear integral and integrodifferential equations. (1998)
[17] Santanilla, J.: Nonnegative solutions to boundary value problems for nonlinear first and second order differential equations. J. math. Anal. appl. 126, 394-408 (1987) · Zbl 0629.34017
[18] Zima, M.: A certain fixed point theorem and its applications to integral-functional equations. Bull. austral. Math. soc. 46, 179-186 (1992) · Zbl 0761.34048