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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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