Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
vector integral equations; bounded solutions; discontinuity
vector integral equations; bounded solutions; discontinuity
Summary:
We deal with the integral equation $u(t)=f(\int_Ig(t,z)\,u(z)\,dz)$, with $t\in I=[0,1]$, $f:\bold R^n\to\bold R^n$ and $g:I\times I\to[0,+\infty[$. We prove an existence theorem for solutions $u\in L^\infty(I,\bold R^n)$ where the function $f$ is not assumed to be continuous, extending a result previously obtained for the case $n=1$.
References:
[1] Aubin J.P., Cellina A.: Differential Inclusions. Springer-Verlag, Berlin, 1984. MR 0755330 | Zbl 0538.34007
[2] Aubin J.P., Frankowska H.: Set-Valued Analysis. Birkhäuser, Boston, 1990. MR 1048347 | Zbl 1168.49014
[3] Banas J., Knap Z.: Integrable solutions of a functional-integral equation. Rev. Mat. Univ. Complut. Madrid 2 (1989), 31-38. MR 1012104 | Zbl 0679.45003
[4] Cammaroto F., Cubiotti P.: Implicit integral equations with discontinuous right-hand side. Comment. Math. Univ. Carolinae 38 (1997), 241-246. MR 1455490 | Zbl 0886.47031
[5] Emmanuele G.: About the existence of integrable solutions of a functional-integral equation. Rev. Mat. Univ. Complut. Madrid 4 (1991), 65-69. MR 1142550 | Zbl 0746.45004
[6] Emmanuele G.: Integrable solutions of a functional-integral equation. J. Integral Equations Appl. 4 (1992), 89-94. MR 1160090 | Zbl 0755.45005
[7] Fečkan M.: Nonnegative solutions of nonlinear integral equations. Comment. Math. Univ. Carolinae 36 (1995), 615-627. MR 1378685
[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. MR 0340692 | Zbl 0289.49009
[10] Kantorovich L.V., Akilov G.P.: Functional Analysis in Normed Spaces. Pergamon Press, Oxford, 1964. MR 0213845 | Zbl 0127.06104
[11] Klein E., Thompson A.C.: Theory of Correspondences. John Wiley and Sons, New York, 1984. MR 0752692 | Zbl 0556.28012
[12] Lang S.: Real and Functional Analysis. Springer-Verlag, New York, 1993. MR 1216137 | Zbl 0831.46001
[13] 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
[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. MR 0028385 | Zbl 0032.19702
[15] Villani A.: On Lusin's condition for the inverse function. Rend. Circ. Mat. Palermo 33 (1984), 331-335. MR 0779937 | Zbl 0562.26002
Search
Partner of
