Vector integral equations with discontinuous right-hand side

Filippo Cammaroto; Paolo Cubiotti

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 3, page 483-490
  • ISSN: 0010-2628

Abstract

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We deal with the integral equation u ( t ) = f ( I g ( t , z ) u ( z ) d z ) , with t I = [ 0 , 1 ] , f : 𝐑 n 𝐑 n and g : I × I [ 0 , + [ . We prove an existence theorem for solutions u L ( I , 𝐑 n ) where the function f is not assumed to be continuous, extending a result previously obtained for the case n = 1 .

How to cite

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Cammaroto, Filippo, and Cubiotti, Paolo. "Vector integral equations with discontinuous right-hand side." Commentationes Mathematicae Universitatis Carolinae 40.3 (1999): 483-490. <http://eudml.org/doc/248419>.

@article{Cammaroto1999,
abstract = {We deal with the integral equation $u(t)=f(\int _Ig(t,z)\,u(z)\,dz)$, with $t\in I=[0,1]$, $f:\mathbf \{R\}^n\rightarrow \mathbf \{R\}^n$ and $g:I\times I\rightarrow [0,+\infty [$. We prove an existence theorem for solutions $u\in L^\infty (I,\mathbf \{R\}^n)$ where the function $f$ is not assumed to be continuous, extending a result previously obtained for the case $n=1$.},
author = {Cammaroto, Filippo, Cubiotti, Paolo},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {vector integral equations; bounded solutions; discontinuity; vector integral equations; bounded solutions; discontinuity; multifunction},
language = {eng},
number = {3},
pages = {483-490},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Vector integral equations with discontinuous right-hand side},
url = {http://eudml.org/doc/248419},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Cammaroto, Filippo
AU - Cubiotti, Paolo
TI - Vector integral equations with discontinuous right-hand side
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 3
SP - 483
EP - 490
AB - We deal with the integral equation $u(t)=f(\int _Ig(t,z)\,u(z)\,dz)$, with $t\in I=[0,1]$, $f:\mathbf {R}^n\rightarrow \mathbf {R}^n$ and $g:I\times I\rightarrow [0,+\infty [$. We prove an existence theorem for solutions $u\in L^\infty (I,\mathbf {R}^n)$ where the function $f$ is not assumed to be continuous, extending a result previously obtained for the case $n=1$.
LA - eng
KW - vector integral equations; bounded solutions; discontinuity; vector integral equations; bounded solutions; discontinuity; multifunction
UR - http://eudml.org/doc/248419
ER -

References

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  1. Aubin J.P., Cellina A., Differential Inclusions, Springer-Verlag, Berlin, 1984. Zbl0538.34007MR0755330
  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. 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 Ψ ( u ) ( t ) F ( t , Φ ( u ) ( t ) ) 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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