# Article

 Title: On four-point boundary value problems for differential inclusions and differential equations with and without multivalued moving constraints (English) Author: Gomaa, Adel Mahmoud Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 62 Issue: 1 Year: 2012 Pages: 139-154 Summary lang: English . Category: math . Summary: We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri and B. Ricceri we prove existence results improving earlier theorems by Gupta and Marano. (English) Keyword: differential equations Keyword: differential inclusions Keyword: multipoint boundary value problems Keyword: bang-bang controls Keyword: Green functions MSC: 05C35 MSC: 34A60 MSC: 34B05 MSC: 34B27 MSC: 49J30 idZBL: Zbl 1249.34053 idMR: MR2899741 DOI: 10.1007/s10587-012-0002-0 . Date available: 2012-03-05T07:18:46Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/142047 . Reference: [1] Aubin, J.-P., Cellina, A.: Differential Inclusions. Set-Valued Maps and Viability Theory.Grundlehren der Mathematischen Wissenschaften, 264, Springer-Verlag, Berlin (1984). Zbl 0538.34007, MR 0755330 Reference: [2] Benamara, M.: "Point Extrémaux, Multi-applications et Fonctionelles Intégrales".Thése de 3éme Cycle, Université de Grenoble (1975). Reference: [3] Bressan, A., Colombo, G.: Extensions and selections of maps with decomposable values.Stud. Math. 90 (1988), 69-86. Zbl 0677.54013, MR 0947921 Reference: [4] Brown, L. D., Purves, R.: Measurable selections of extrema.Ann. Stat. 1 (1973), 902-912. Zbl 0265.28003, MR 0432846, 10.1214/aos/1176342510 Reference: [5] Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions.Lecture Notes in Mathematics, 580. Springer Verlag, Berlin-Heidelberg-New York (1977). Zbl 0346.46038, MR 0467310, 10.1007/BFb0087688 Reference: [6] Gomaa, A. M.: On the solution sets of four-point boundary value problems for nonconvex differential inclusions.Int. J. Geom. Methods Mod. Phys. 8 (2011), 23-37. Zbl 1220.34021, MR 2782872, 10.1142/S021988781100494X Reference: [7] Gupta, Ch. P.: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation.J. Math. Anal. Appl. 168 (1992), 540-551. Zbl 0763.34009, MR 1176010, 10.1016/0022-247X(92)90179-H Reference: [8] Ibrahim, A. G., Gomaa, A. M.: Extremal solutions of classes of multivalued differential equations.Appl. Math. Comput. 136 (2003), 297-314. Zbl 1037.34052, MR 1937933, 10.1016/S0096-3003(02)00040-1 Reference: [9] Klein, E., Thompson, A.: Theory of Correspondences. Including Applications to Mathematical Economic.Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. New York, John Wiley & Sons (1984). MR 0752692 Reference: [10] Marano, S. A.: A remark on a second-order three-point boundary value problem.J. Math. Anal. Appl. 183 (1994), 518-522. Zbl 0801.34025, MR 1274852, 10.1006/jmaa.1994.1158 Reference: [11] Tolstonogov, A. A.: Extremal selections of multivalued mappings and the "bang-bang" principle for evolution inclusions.Sov. Math. Dokl. 43 (1991), 481-485 Translation from Dokl. Akad. Nauk SSSR 317 (1991), 589-593. Zbl 0784.54024, MR 1121349 Reference: [12] Papageorgiou, N. S.: Convergence theorems for Banach space valued integrable multifunctions.Int. J. Math. Math. Sci. 10 (1987), 433-442. Zbl 0619.28009, MR 0896595, 10.1155/S0161171287000516 Reference: [13] Papageorgiou, N. S., Kravvaritis, D.: Boundary value problems for nonconvex differential inclusions.J. Math. Anal. Appl. 185 (1994), 146-160. Zbl 0817.34009, MR 1283047, 10.1006/jmaa.1994.1238 Reference: [14] Papageorgiou, N. S.: On measurable multifunction with applications to random multivalued equations.Math. Jap. 32 (1987), 437-464. MR 0914749 Reference: [15] Ricceri, O. N., Ricceri, B.: An existence theorem for inclusions of the type $\Psi (u)(t) \in F(t, \Phi (u)(t))$ and an application to a multivalued boundary value problem.Appl. Anal. 38 (1990), 259-270. MR 1116184, 10.1080/00036819008839966 .

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