| Title:
|
Non-autonomous implicit integral equations with discontinuous right-hand side (English) |
| Author:
|
Anello, Giovanni |
| Author:
|
Cubiotti, Paolo |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
417-429 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We deal with the implicit integral equation $$ h(u(t))=f(\,t\,,\int_Ig(t,z)\,u(z)\,dz) \hbox{ for a.a. } t\in I, $$ where $I:=[0,1]$ and where $f:I\times [0,\lambda]\to{\Bbb R}$, $g:I\times I\to[0,+\infty[$ and $h:\,]\,0,+\infty\,[\,\to {\Bbb R}$. We prove an existence theorem for solutions $u\in L^s(I)$ where the contituity of $f$ with respect to the second variable is not assumed. (English) |
| Keyword:
|
implicit integral equations |
| Keyword:
|
discontinuity |
| Keyword:
|
lower semicontinuous multifunctions |
| Keyword:
|
operator inclusions |
| Keyword:
|
selections |
| MSC:
|
45P05 |
| MSC:
|
47G10 |
| MSC:
|
47J05 |
| MSC:
|
47N20 |
| idZBL:
|
Zbl 1099.45004 |
| idMR:
|
MR2103137 |
| . |
| Date available:
|
2009-05-05T16:46:14Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119470 |
| . |
| Reference:
|
[1] Artstein Z., Prikry K.: Caratheodory selections and the Scorza Dragoni property.J. Math. Anal. Appl. 127 (1987), 540-547. Zbl 0649.28011, MR 0915076 |
| Reference:
|
[2] Aubin J.P., Frankowska H.: Set-Valued Analysis.Birkhäuser, Boston, 1990. Zbl 1168.49014, MR 1048347 |
| Reference:
|
[3] Banas J., Knap Z.: Integrable solutions of a functional-integral equation.Rev. Mat. Univ. Complut. Madrid 2 (1989), 31-38. Zbl 0679.45003, MR 1012104 |
| Reference:
|
[4] Cammaroto F., Cubiotti P.: Implicit integral equations with discontinuous right-hand side.Comment. Math. Univ. Carolinae 38 (1997), 241-246. Zbl 0886.47031, MR 1455490 |
| Reference:
|
[5] Castaing C., Valadier M.: Convex Analysis and Measurable Multifunctions.Springer-Verlag, Berlin, 1977. Zbl 0346.46038, MR 0467310 |
| Reference:
|
[6] Cubiotti P.: Non-autonomous vector integral equations with discontinuous right-hand side.Comment. Math. Univ. Carolinae 42 (2001), 319-329. Zbl 1055.45004, MR 1832150 |
| Reference:
|
[7] Emmanuele G.: About the existence of integrable solutions of a functional-integral equation.Rev. Mat. Univ. Complut. Madrid 4 (1991), 65-69. Zbl 0746.45004, MR 1142550 |
| Reference:
|
[8] Emmanuele G.: Integrable solutions of a functional-integral equation.J. Integral Equations Appl. 4 (1992), 89-94. Zbl 0755.45005, MR 1160090 |
| Reference:
|
[9] Engelking R.: Theory of Dimensions, Finite and Infinite.Heldermann-Verlag, 1995. Zbl 0872.54002, MR 1363947 |
| Reference:
|
[10] Fečkan M.: Nonnegative solutions of nonlinear integral equations.Comment. Math. Univ. Carolinae 36 (1995), 615-627. MR 1378685 |
| Reference:
|
[11] Hewitt E., Stromberg K.: Real and Abstract Analysis.Springer-Verlag, Berlin, 1965. Zbl 0307.28001, MR 0367121 |
| Reference:
|
[12] Himmelberg C.J., Van Vleck F.S.: Lipschitzian generalized differential equations.Rend. Sem. Mat. Univ. Padova 48 (1973), 159-169. Zbl 0289.49009, MR 0340692 |
| Reference:
|
[13] Kantorovich L.V., Akilov G.P.: Functional Analysis in Normed Spaces.Pergamon Press, Oxford, 1964. Zbl 0127.06104, MR 0213845 |
| Reference:
|
[14] Klein E., Thompson A.C.: Theory of Correspondences.John Wiley and Sons, New York, 1984. Zbl 0556.28012, MR 0752692 |
| Reference:
|
[15] Kucia A.: Scorza Dragoni type theorems.Fund. Math. 138 (1991), 197-203. Zbl 0744.28011, MR 1121606 |
| Reference:
|
[16] Lang S.: Real and Functional Analysis.Springer-Verlag, New York, 1993. Zbl 0831.46001, MR 1216137 |
| Reference:
|
[17] Naselli Ricceri O., Ricceri B.: An existence theorem for inclusions of the type $\Psi(u)(t)\in $ $F(t,\Phi(u)(t))$ and application to a multivalued boundary value problem.Appl. Anal. 38 (1990), 259-270. MR 1116184 |
| Reference:
|
[18] Ricceri B.: Sur la semi-continuitè infèrieure de certaines multifonctions.C.R. Acad. Sci. Paris 294 (1982), 265-267. Zbl 0483.54010, MR 0653748 |
| Reference:
|
[19] Saint Raymond J.: Riemann-measurable selections.Set Valued Anal. 2 (1994), 481-485. Zbl 0851.54021, MR 1304050 |
| Reference:
|
[20] Scorza Dragoni G.: Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un'altra variabile.Rend. Sem. Mat. Univ. Padova 17 (1948), 102-106. Zbl 0032.19702, MR 0028385 |
| Reference:
|
[21] Villani A.: On Lusin's condition for the inverse function.Rend. Circ. Mat. Palermo 33 (1984), 331-335. Zbl 0562.26002, MR 0779937 |
| . |
