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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 $W^{2,1}(I, \Bbb {R}^{n})$ using Schauder’s fixed point theorem. This result improves known results by {\it A. G. Ibrahim} and {\it A. M. Gomaa} in [Appl. Math. Comput. 136, No. 2--3, 297--314 (2003; Zbl 1037.34052)] and by {\it D. Kravvaritis} and {\it 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 $C^{1}(I,\Bbb {R}^{n})$ for the second order differential inclusion with the above boundary conditions, which also improves results of {\it A. G. Ibrahim} and {\it 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 $b\:I\times \Bbb {R}^{n}\times \Bbb {R}^{n}\times \Bbb {R}^{m}\to \Bbb {R}^{n}$ and $K\: I\times \Bbb {R}^{n}\times \Bbb {R}^{n}\to P_{k}(\Bbb {R}^{m})$, $P_{k}(\Bbb {R}^{m})$ is the set of all compact subsets of $\Bbb {R}^{m}$. Using the existence theorem due to {\it O. N. Ricceri} and {\it B. Ricceri} [Appl. Anal. 38, No. 4, 259--270 (1990; Zbl 0687.47044)], the existence of the “state-control” pairs of this problem in $W^{2,1}(I,\Bbb {R}^{n})\times L^{1}(I, \Bbb {R}^{n})$ is proved, which extents known results by {\it Ch. P. Gupta} in [J. Math. Anal. Appl. 168, No. 2, 540--551 (1992; Zbl 0763.34009)] and by {\it 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 $[0, 1]\times \Bbb {R}\times \Bbb {R}$. The existence of generalized solutions in $W^{2,k}(I,\Bbb {R})$ is proved, which improves known results by {\it Ch. P. Gupta} and {\it S. A. Marano}.

34B10Nonlocal and multipoint boundary value problems for ODE
34A60Differential inclusions
34B27Green functions
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
34H05ODE in connection with control problems
Full Text: DOI EuDML
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