Previous |  Up |  Next


differential inclusions; contraction multivalued map; fixed point; decomposable values; measurable
In this paper a fixed point theorem due to Covitz and Nadler for contraction multivalued maps, and the Schaefer’s theorem combined with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued operators with decomposables values, are used to investigate the existence of solutions for boundary value problems of fourth-order differential inclusions.
[1] Aftabizadeh, A. R.: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 116 (1986), 415–426. MR 0842808 | Zbl 0634.34009
[2] Agarwal, R.: On fourth-order boundary value problems arising in beam analysis. Differ. Integral Equ. 2 (1989), 91–110. MR 0960017 | Zbl 0715.34032
[3] Bressan, A. and Colombo, G.: Extensions and selections of maps with decomposable values. Studia Math. 90 (1988), 69–86. MR 0947921
[3] Cabada, A.: The method of lower and upper solutions for second, third, fourth, and higher order boundary value problem. J. Math. Anal. Appl. 248 (2000), 195–202. MR 1283059
[5] Castaing, C. and Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 0467310
[6] Covitz, H. and Nadler, S. B., Jr.: Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8 (1970), 5–11. MR 0263062
[7] De Coster, C., Fabry, C. and Munyamarere, F.: Nonresonance condition for fourth-order nonlinear boundary value problems. Int. J. Math. Math. Sci. 17 (1994), 725–740. MR 1298797
[8] Deimling, K.: Multivalued Differential Equations. De Gruyter, Berlin, 1992. MR 1189795 | Zbl 0820.34009
[9] Del Pino, M. A. and Manasevich, R. F.: Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition. Proc. Amer. Math. Soc. 112 (1991), 81–86. MR 1043407
[10] Frigon, M. and Granas, A.: Théorèmes d’existence pour des inclusions différentielles sans convexité. C. R. Acad. Sci. Paris, Ser. I Math. 310 (1990), 819–822. MR 1058503
[11] Gorniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings. Math. Appl. 495, Kluwer Academic Publishers, Dordrecht, 1999. MR 1748378 | Zbl 1107.55001
[12] Hu, Sh. and Papageorgiou, N.: Handbook of Multivalued Analysis, Volume I: Theory. Kluwer Academic Publishers, Dordrecht, Boston, London, 1997. MR 1485775
[13] Kisielewicz, M.: Differential Inclusions and Optimal Control. Kluwer, Dordrecht, The Netherlands, 1991. MR 1135796 | Zbl 0731.49001
[14] Korman, P.: A maximum principle for fourth-order ordinary differential equations. Appl. Anal. 33 (1989), 267–273. MR 1030113 | Zbl 0681.34016
[15] Ma, R. Y., Zhang, J. H. and Fu, M.: The method of lower and upper solutions for fourth-order two-point boundary value problem. J. Math. Anal. Appl. 215 (1997), 415–422. MR 1490759
[16] Schroeder, J.: Fourth-order two-point boundary value problems; estimates by two-sided bounds. Nonlinear Anal. 8 (1984), 107–114. MR 0734445 | Zbl 0533.34019
[17] Smart, D. R.: Fixed Point Theorems. Cambridge Univ. Press, Cambridge, 1974. MR 0467717 | Zbl 0427.47036
[18] Švec, M.: Periodic boundary value problem for fourth order differential inclusions. Arch. Math. (Brno) 33 (1997), 167–171. MR 1464311
Partner of
EuDML logo