| Title:
|
Equations containing locally Henstock-Kurzweil integrable functions (English) |
| Author:
|
Heikkilä, Seppo |
| Author:
|
Ye, Guoju |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
57 |
| Issue:
|
6 |
| Year:
|
2012 |
| Pages:
|
569-580 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions. (English) |
| Keyword:
|
integrability |
| Keyword:
|
Henstock-Kurzweil integral |
| Keyword:
|
ordered Banach space |
| Keyword:
|
order cone |
| Keyword:
|
chain |
| Keyword:
|
fixed point |
| Keyword:
|
functional integral equation |
| Keyword:
|
Volterra |
| Keyword:
|
Cauchy problem |
| Keyword:
|
ordered Banach space |
| Keyword:
|
fixed point |
| MSC:
|
26A39 |
| MSC:
|
28B15 |
| MSC:
|
34A36 |
| MSC:
|
34A37 |
| MSC:
|
45N05 |
| MSC:
|
46B40 |
| MSC:
|
47H07 |
| MSC:
|
47H10 |
| idZBL:
|
Zbl 1274.45017 |
| idMR:
|
MR3010237 |
| DOI:
|
10.1007/s10492-012-0034-7 |
| . |
| Date available:
|
2012-11-10T20:38:48Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143003 |
| . |
| Reference:
|
[1] Carl, S., Heikkilä, S.: On discontinuous implicit and explicit abstract impulsive boundary value problems.Nonlinear Anal., Theory Methods Appl. 41 (2000), 701-723. MR 1780640, 10.1016/S0362-546X(98)00305-8 |
| Reference:
|
[2] Federson, M., Bianconi, M.: Linear Fredholm integral equations and the integral of Kurzweil.J. Appl. Anal. 8 (2002), 83-110. Zbl 1043.45010, MR 1921473, 10.1515/JAA.2002.83 |
| Reference:
|
[3] Federson, M., Schwabik, Š.: Generalized ordinary differential equations approach to impulsive retarded functional differential equations.Differ. Integral Equ. 19 (2006), 1201-1234. MR 2278005 |
| Reference:
|
[4] Federson, M., Táboas, P.: Impulsive retarded differential equations in Banach spaces via Bochner-Lebesgue and Henstock integrals.Nonlinear Anal., Theory Methods Appl. 50 (2002), 389-407. Zbl 1011.34070, MR 1906469, 10.1016/S0362-546X(01)00769-6 |
| Reference:
|
[5] Guo, D., Cho, Y. J., Zhu, J.: Partial Ordering Methods in Nonlinear Problems.Nova Science Publishers, Inc. New York (2004). Zbl 1116.45007, MR 2084490 |
| Reference:
|
[6] Heikkilä, S., Kumpulainen, S., Kumpulainen, M.: On improper integrals and differential equations in ordered Banach spaces.J. Math. Anal. Appl. 319 (2006), 579-603. Zbl 1105.34037, MR 2227925, 10.1016/j.jmaa.2005.06.051 |
| Reference:
|
[7] Heikkilä, S., Lakshmikantham, V.: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations.Marcel Dekker, Inc. New York (1994). Zbl 0804.34001, MR 1280028 |
| Reference:
|
[8] Heikkilä, S., Seikkala, S.: On non-absolute functional Volterra integral equations and impulsive differential equations in ordered Banach spaces.Electron. J. Differ. Equ., paper No. 103 (2008), 1-11. Zbl 1168.45011, MR 2430900 |
| Reference:
|
[9] Heikkilä, S., Ye, G.: Convergence and comparison results for Henstock-Kurzweil and McShane integrable vector-valued functions.Southeast Asian Bull. Math. 35 (2011), 407-418. Zbl 1240.26025, MR 2856387 |
| Reference:
|
[10] Lu, J., Lee, P.-Y.: On singularity of Henstock integrable functions.Real Anal. Exch. 25 (2000), 795-797. Zbl 1015.26016, MR 1778532, 10.2307/44154035 |
| Reference:
|
[11] Satco, B.-R.: Nonlinear Volterra integral equations in Henstock integrability setting.Electron. J. Differ. Equ., paper No. 39 (2008), 1-9. Zbl 1169.45300, MR 2392943 |
| Reference:
|
[12] Schwabik, Š., Ye, G.: Topics in Banach Space Integration.World Scientific Hackensack (2005). Zbl 1088.28008, MR 2167754 |
| Reference:
|
[13] Sikorska-Nowak, A.: On the existence of solutions of nonlinear integral equations in Banach spaces and Henstock-Kurzweil integrals.Ann. Pol. Math. 83 (2004), 257-267. Zbl 1101.45006, MR 2111712, 10.4064/ap83-3-7 |
| Reference:
|
[14] Sikorska-Nowak, A.: Existence theory for integrodifferential equations and Henstock-Kurzweil integral on Banach spaces.J. Appl. Math., Article ID31572 (2007), 1-12. MR 2317885, 10.1155/2007/31572 |
| Reference:
|
[15] Sikorska-Nowak, A.: Existence of solutions of nonlinear integral equations in Banach spaces and Henstock-Kurzweil integrals.Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 47 (2007), 227-238. MR 2377959 |
| Reference:
|
[16] Sikorska-Nowak, A.: Nonlinear integrodifferential equations of mixed type in Banach spaces.Int. J. Math. Math. Sci., Article ID65947 (2007), 1-14. Zbl 1147.45009, MR 2336140, 10.1155/2007/65947 |
| Reference:
|
[17] Sikorska-Nowak, A.: Nonlinear integral equations in Banach spaces and Henstock-Kurzweil-Pettis integrals.Dyn. Syst. Appl. 17 (2008), 97-107. Zbl 1154.45011, MR 2433893 |
| . |
