# Non-autonomous vector integral equations with discontinuous right-hand side

• Volume: 42, Issue: 2, page 319-329
• ISSN: 0010-2628

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## Abstract

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We deal with the integral equation $u\left(t\right)=f\left(t,{\int }_{I}g\left(t,z\right)u\left(z\right)\phantom{\rule{0.166667em}{0ex}}dz\right)$, with $t\in I:=\left[0,1\right]$, $f:I×{ℝ}^{n}\to {ℝ}^{n}$ and $g:I×I\to \left[0,+\infty \left[$. We prove an existence theorem for solutions $u\in {L}^{s}\left(I,{ℝ}^{n}\right)$, $s\in \phantom{\rule{0.166667em}{0ex}}\right]1,+\infty \right]$, where $f$ is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where $f$ does not depend explicitly on the first variable $t\in I$.

## How to cite

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Cubiotti, Paolo. "Non-autonomous vector integral equations with discontinuous right-hand side." Commentationes Mathematicae Universitatis Carolinae 42.2 (2001): 319-329. <http://eudml.org/doc/248787>.

@article{Cubiotti2001,
abstract = {We deal with the integral equation $u(t)=f(t,\int _I g(t,z)u(z)\,dz)$, with $t\in I:=[0,1]$, $f:I\times \mathbb \{R\}^n \rightarrow \mathbb \{R\}^n$ and $g:I\times I\rightarrow [0,+\infty [$. We prove an existence theorem for solutions $u\in L^s(I,\mathbb \{R\}^n)$, $s\in \,]1,+\infty ]$, where $f$ is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where $f$ does not depend explicitly on the first variable $t\in I$.},
author = {Cubiotti, Paolo},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {vector integral equations; discontinuity; multifunctions; operator inclusions; -solutions; nonlinearities; operator inclusion},
language = {eng},
number = {2},
pages = {319-329},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Non-autonomous vector integral equations with discontinuous right-hand side},
url = {http://eudml.org/doc/248787},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Cubiotti, Paolo
TI - Non-autonomous vector integral equations with discontinuous right-hand side
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 2
SP - 319
EP - 329
AB - We deal with the integral equation $u(t)=f(t,\int _I g(t,z)u(z)\,dz)$, with $t\in I:=[0,1]$, $f:I\times \mathbb {R}^n \rightarrow \mathbb {R}^n$ and $g:I\times I\rightarrow [0,+\infty [$. We prove an existence theorem for solutions $u\in L^s(I,\mathbb {R}^n)$, $s\in \,]1,+\infty ]$, where $f$ is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where $f$ does not depend explicitly on the first variable $t\in I$.
LA - eng
KW - vector integral equations; discontinuity; multifunctions; operator inclusions; -solutions; nonlinearities; operator inclusion
UR - http://eudml.org/doc/248787
ER -

## References

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1. Aubin J.P., Frankowska H., Set-Valued Analysis, Birkhäuser, Boston, 1990. Zbl1168.49014MR1048347
2. Banas J., Knap Z., Integrable solutions of a functional-integral equation, Rev. Mat. Univ. Complut. Madrid 2 (1989), 31-38. (1989) Zbl0679.45003MR1012104
3. Cammaroto F., Cubiotti P., Implicit integral equations with discontinuous right-hand side, Comment. Math. Univ. Carolinae 38 (1997), 241-246. (1997) Zbl0886.47031MR1455490
4. Cammaroto F., Cubiotti P., Vector integral equations with discontinuous right-hand side, Comment. Math. Univ. Carolinae 40 (1999), 483-490. (1999) Zbl1065.47505MR1732487
5. Emmanuele G., About the existence of integrable solutions of a functional-integral equation, Rev. Mat. Univ. Complut. Madrid 4 (1991), 65-69. (1991) Zbl0746.45004MR1142550
6. Emmanuele G., Integrable solutions of a functional-integral equation, J. Integral Equations Appl. 4 (1992), 89-94. (1992) Zbl0755.45005MR1160090
7. Fečkan M., Nonnegative solutions of nonlinear integral equations, Comment. Math. Univ. Carolinae 36 (1995), 615-627. (1995) MR1378685
8. Hewitt E., Stomberg K., Real and Abstract Analysis, Springer-Verlag, Berlin, 1965.
9. Himmelberg C.J., Van Vleck F.S., Lipschitzian generalized differential equations, Rend. Sem. Mat. Univ. Padova 48 (1973), 159-169. (1973) Zbl0289.49009MR0340692
10. Kantorovich L.V., Akilov G.P., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1964. Zbl0127.06104MR0213845
11. Klein E., Thompson A.C., Theory of Correspondences, John Wiley and Sons, New York, 1984. Zbl0556.28012MR0752692
12. Lang S., Real and Functional Analysis, Springer-Verlag, New York, 1993. Zbl0831.46001MR1216137
13. Naselli Ricceri O., Ricceri B., An existence theorem for inclusions of the type $\Psi \left(u\right)\left(t\right)\in F\left(t,\Phi \left(u\right)\left(t\right)\right)$ and application to a multivalued boundary value problem, Appl. Anal. 38 (1990), 259-270. (1990) MR1116184
14. 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. (1948) Zbl0032.19702MR0028385
15. Villani A., On Lusin's condition for the inverse function, Rend. Circ. Mat. Palermo 33 (1984), 331-335. (1984) Zbl0562.26002MR0779937

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