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On four-point boundary value problems for differential inclusions and differential equations with and without multivalued moving constraints. (English) Zbl 1249.34053
The author deals with several problems with four-point boundary conditions for both differential inclusions and differential equations with and without moving constraints. Firstly, the existence of solutions is investigated for the second order differential inclusion with boundary conditions $$u''(t) \in \text{ext}\, F(t, u(t), u'(t))$$ a.e. on $$I$$, $$u(0) = 0$$, $$u(\eta )=u(\theta )=u(T)$$, where $$I=[0,T]$$, $$0 < \eta < \theta < 1$$ and $$\text{ext}\, F(.,u(.),u'(.))$$ is the set of extremal points of $$F(.,u(.),u'(.))$$. The author proves the existence of at least one generalized solution of this problem in $$W^{2,1}(I, \mathbb {R}^{n})$$ using Schauder’s fixed point theorem. This result improves known results by A. G. Ibrahim and A. M. Gomaa in [Appl. Math. Comput. 136, No. 2–3, 297–314 (2003; Zbl 1037.34052)] and by D. Kravvaritis and N. S. Papageorgiou in [J. Math. Anal. Appl. 185, No. 1, 146–160 (1994; Zbl 0817.34009)]. Secondly, the author obtains the existence of solutions in $$C^{1}(I,\mathbb {R}^{n})$$ for the second order differential inclusion with the above boundary conditions, which also improves results of A. G. Ibrahim and A. M. Gomaa. Thirdly, the author considers the following single valued boundary value problem with multivalued moving constraints $$u''(t) = b(t,u(t),u'(t),x(t))$$ a.e. on $$I$$, $$u(0) = 0$$, $$u(\eta )=u(\theta )=u(T)$$, $$x(t)\in K(t,u(t),u'(t))$$ a.e. on $$I$$, where $$b\:I\times \mathbb {R}^{n}\times \mathbb {R}^{n}\times \mathbb {R}^{m}\to \mathbb {R}^{n}$$ and $$K\: I\times \mathbb {R}^{n}\times \mathbb {R}^{n}\to P_{k}(\mathbb {R}^{m})$$, $$P_{k}(\mathbb {R}^{m})$$ is the set of all compact subsets of $$\mathbb {R}^{m}$$. Using the existence theorem due to O. N. Ricceri and B. Ricceri [Appl. Anal. 38, No. 4, 259–270 (1990; Zbl 0687.47044)], the existence of the “state-control” pairs of this problem in $$W^{2,1}(I,\mathbb {R}^{n})\times L^{1}(I, \mathbb {R}^{n})$$ is proved, which extents known results by Ch. P. Gupta in [J. Math. Anal. Appl. 168, No. 2, 540–551 (1992; Zbl 0763.34009)] and by S. A. Marano in [J. Math. Anal. Appl. 183, No. 3, 518–522 (1994; Zbl 0801.34025)]. Finally, the last part of the paper is devoted to the second order differential equation with four-point boundary conditions $$u''(t)=f(t,u(t),u'(t))$$ a.e. on $$[0, 1]$$, $$u(0) = 0$$, $$u(\eta )=u(\theta )=u(1)$$, where $$f$$ is a real function on $$[0, 1]\times \mathbb {R}\times \mathbb {R}$$. The existence of generalized solutions in $$W^{2,k}(I,\mathbb {R})$$ is proved, which improves known results by Ch. P. Gupta and S. A. Marano.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34A60 Ordinary differential inclusions 34B27 Green’s functions for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations 34H05 Control problems involving ordinary differential equations
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##### References:
 [1] J.-P. Aubin, A. Cellina: Differential Inclusions. Set-Valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984. [2] M. Benamara: ”Point Extrémaux, Multi-applications et Fonctionelles Intégrales”. Thése de 3éme Cycle, Université de Grenoble, 1975. [3] A. Bressan, G. Colombo: Extensions and selections of maps with decomposable values. Stud. Math. 90 (1988), 69–86. · Zbl 0677.54013 [4] L. D. Brown, R. Purves: Measurable selections of extrema. Ann. Stat. 1 (1973), 902–912. · Zbl 0265.28003 · doi:10.1214/aos/1176342510 [5] C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, 580. Springer Verlag, Berlin-Heidelberg-New York, 1977. · Zbl 0346.46038 [6] A. M. Gomaa: 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 · doi:10.1142/S021988781100494X [7] Ch. P. Gupta: 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 · doi:10.1016/0022-247X(92)90179-H [8] A. G. Ibrahim, A. M. Gomaa: Extremal solutions of classes of multivalued differential equations. Appl. Math. Comput. 136 (2003), 297–314. · Zbl 1037.34052 · doi:10.1016/S0096-3003(02)00040-1 [9] E. Klein, A. Thompson: 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. [10] S. A. Marano: A remark on a second-order three-point boundary value problem. J. Math. Anal. Appl. 183 (1994), 518–522. · Zbl 0801.34025 · doi:10.1006/jmaa.1994.1158 [11] A. A. Tolstonogov: 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 [12] N. S. Papageorgiou: Convergence theorems for Banach space valued integrable multifunctions. Int. J. Math. Math. Sci. 10 (1987), 433–442. · Zbl 0619.28009 · doi:10.1155/S0161171287000516 [13] N. S. Papageorgiou, D. Kravvaritis: Boundary value problems for nonconvex differential inclusions. J. Math. Anal. Appl. 185 (1994), 146–160. · Zbl 0817.34009 · doi:10.1006/jmaa.1994.1238 [14] N. S. Papageorgiou: On measurable multifunction with applications to random multivalued equations. Math. Jap. 32 (1987), 437–464. · Zbl 0634.28005 [15] O. N. Ricceri, B. Ricceri: An existence theorem for inclusions of the type $$\Psi$$(u)(t) F(t, $$\Phi$$(u)(t)) and an application to a multivalued boundary value problem. Appl. Anal. 38 (1990), 259–270. · Zbl 0687.47044 · doi:10.1080/00036819008839966
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