| Title:
|
On the Volterra integral equation with weakly singular kernel (English) |
| Author:
|
Szufla, Stanisław |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
131 |
| Issue:
|
3 |
| Year:
|
2006 |
| Pages:
|
225-231 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give sufficient conditions for the existence of at least one integrable solution of equation $x(t)=f(t)+\int _{0}^{t} K(t,s)g(s,x(s))\mathrm{d}s$. Our assumptions and proofs are expressed in terms of measures of noncompactness. (English) |
| Keyword:
|
integral equation |
| Keyword:
|
integrable solution |
| Keyword:
|
measure of noncompactness |
| MSC:
|
45G05 |
| MSC:
|
45N05 |
| idZBL:
|
Zbl 1110.45003 |
| idMR:
|
MR2248592 |
| DOI:
|
10.21136/MB.2006.134139 |
| . |
| Date available:
|
2009-09-24T22:25:59Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134139 |
| . |
| Reference:
|
[1] N. V. Azbieliev, Z. B. Caliuk: Ob integralnych nieravienstvach.Matem. Sbornik 56 (1962), 325–342. MR 0140907 |
| Reference:
|
[2] J. Banaś, K. Goebel: Measure of Noncompactness in Banach Spaces.Marcel Dekker, New York, 1980. MR 0591679 |
| Reference:
|
[3] G. Gripenberg: Unique solutions of some Volterra integral equations.Math. Scand. 48 (1981), 59–67. Zbl 0463.45002, MR 0621417, 10.7146/math.scand.a-11899 |
| Reference:
|
[4] H. P. Heinz: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions.Nonlinear Anal., Theory Methods Appl. 7 (1983), 1351–1371. Zbl 0528.47046, MR 0726478, 10.1016/0362-546X(83)90006-8 |
| Reference:
|
[5] J. Kurzweil: Ordinary Differential Equations.Elsevier, Amsterdam-Oxford, 1986. Zbl 0667.34002, MR 0929466 |
| Reference:
|
[6] H. Mönch: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces.Nonlinear Anal., Theory Methods Appl. 4 (1980), 985–999. MR 0586861, 10.1016/0362-546X(80)90010-3 |
| Reference:
|
[7] W. Mydlarczyk: The existence of nontrivial solutions of Volterra equations.Math. Scand. 68 (1991), 83–88. Zbl 0701.45002, MR 1124821, 10.7146/math.scand.a-12347 |
| Reference:
|
[8] S. Szufla: Appendix to the paper An existence theorem for the Urysohn integral equation in Banach spaces.Commentat. Math. Univ. Carol. 25 (1984), 763–764. Zbl 0578.45018, MR 0782024 |
| . |
