Non-autonomous implicit integral equations with discontinuous right-hand side

Giovanni Anello; Paolo Cubiotti

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 3, page 417-429
  • ISSN: 0010-2628

Abstract

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We deal with the implicit integral equation where and where , and . We prove an existence theorem for solutions where the contituity of with respect to the second variable is not assumed.

How to cite

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Anello, Giovanni, and Cubiotti, Paolo. "Non-autonomous implicit integral equations with discontinuous right-hand side." Commentationes Mathematicae Universitatis Carolinae 45.3 (2004): 417-429. <http://eudml.org/doc/249338>.

@article{Anello2004,
abstract = {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 $f:I\times [0,\lambda ]\rightarrow \{\mathbb \{R\}\}$, $g:I\times I\rightarrow [0,+\infty [$ and $h:\,]\,0,+\infty \,[\,\rightarrow \{\mathbb \{R\}\}$. 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.},
author = {Anello, Giovanni, Cubiotti, Paolo},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {implicit integral equations; discontinuity; lower semicontinuous multifunctions; operator inclusions; selections; implicit integral equations; discontinuity; lower semicontinuous multifunction; operator inclusion; selection; existence},
language = {eng},
number = {3},
pages = {417-429},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Non-autonomous implicit integral equations with discontinuous right-hand side},
url = {http://eudml.org/doc/249338},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Anello, Giovanni
AU - Cubiotti, Paolo
TI - Non-autonomous implicit integral equations with discontinuous right-hand side
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 3
SP - 417
EP - 429
AB - 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 $f:I\times [0,\lambda ]\rightarrow {\mathbb {R}}$, $g:I\times I\rightarrow [0,+\infty [$ and $h:\,]\,0,+\infty \,[\,\rightarrow {\mathbb {R}}$. 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.
LA - eng
KW - implicit integral equations; discontinuity; lower semicontinuous multifunctions; operator inclusions; selections; implicit integral equations; discontinuity; lower semicontinuous multifunction; operator inclusion; selection; existence
UR - http://eudml.org/doc/249338
ER -

References

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  1. Artstein Z., Prikry K., Caratheodory selections and the Scorza Dragoni property, J. Math. Anal. Appl. 127 (1987), 540-547. (1987) Zbl0649.28011MR0915076
  2. Aubin J.P., Frankowska H., Set-Valued Analysis, Birkhäuser, Boston, 1990. Zbl1168.49014MR1048347
  3. Banas J., Knap Z., Integrable solutions of a functional-integral equation, Rev. Mat. Univ. Complut. Madrid 2 (1989), 31-38. (1989) Zbl0679.45003MR1012104
  4. Cammaroto F., Cubiotti P., Implicit integral equations with discontinuous right-hand side, Comment. Math. Univ. Carolinae 38 (1997), 241-246. (1997) Zbl0886.47031MR1455490
  5. Castaing C., Valadier M., Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin, 1977. Zbl0346.46038MR0467310
  6. Cubiotti P., Non-autonomous vector integral equations with discontinuous right-hand side, Comment. Math. Univ. Carolinae 42 (2001), 319-329. (2001) Zbl1055.45004MR1832150
  7. 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
  8. Emmanuele G., Integrable solutions of a functional-integral equation, J. Integral Equations Appl. 4 (1992), 89-94. (1992) Zbl0755.45005MR1160090
  9. Engelking R., Theory of Dimensions, Finite and Infinite, Heldermann-Verlag, 1995. Zbl0872.54002MR1363947
  10. Fečkan M., Nonnegative solutions of nonlinear integral equations, Comment. Math. Univ. Carolinae 36 (1995), 615-627. (1995) MR1378685
  11. Hewitt E., Stromberg K., Real and Abstract Analysis, Springer-Verlag, Berlin, 1965. Zbl0307.28001MR0367121
  12. Himmelberg C.J., Van Vleck F.S., Lipschitzian generalized differential equations, Rend. Sem. Mat. Univ. Padova 48 (1973), 159-169. (1973) Zbl0289.49009MR0340692
  13. Kantorovich L.V., Akilov G.P., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1964. Zbl0127.06104MR0213845
  14. Klein E., Thompson A.C., Theory of Correspondences, John Wiley and Sons, New York, 1984. Zbl0556.28012MR0752692
  15. Kucia A., Scorza Dragoni type theorems, Fund. Math. 138 (1991), 197-203. (1991) Zbl0744.28011MR1121606
  16. Lang S., Real and Functional Analysis, Springer-Verlag, New York, 1993. Zbl0831.46001MR1216137
  17. Naselli Ricceri O., Ricceri B., An existence theorem for inclusions of the type and application to a multivalued boundary value problem, Appl. Anal. 38 (1990), 259-270. (1990) MR1116184
  18. Ricceri B., Sur la semi-continuitè infèrieure de certaines multifonctions, C.R. Acad. Sci. Paris 294 (1982), 265-267. (1982) Zbl0483.54010MR0653748
  19. Saint Raymond J., Riemann-measurable selections, Set Valued Anal. 2 (1994), 481-485. (1994) Zbl0851.54021MR1304050
  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. (1948) Zbl0032.19702MR0028385
  21. 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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