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Three boundary value problems for second order differential inclusions in Banach spaces. (English) Zbl 1111.34303

The subject of the paper is the investigation of a three-point boundary value problem for differential equations and inclusions of second order of the following form

$\stackrel{¨}{u}\left(t\right)\in F\left(t,u\left(t\right),\stackrel{˙}{u}\left(t\right)\right)\phantom{\rule{1.em}{0ex}}\text{a.e.}\phantom{\rule{4.pt}{0ex}}t\in \left[0,1\right],\phantom{\rule{1.em}{0ex}}u\left(0\right)=0;\phantom{\rule{2.em}{0ex}}u\left(\theta \right)=u\left(1\right)·$

The authors present an existence and uniqueness result for a second-order differential equation in a finite-dimensional space whose right-hand side satisfies Carathéodory-type conditions. As an application, a Bolza-type optimal control problem is considered.

The next section begins with a density result on a second-order differential inclusion. Further, it is proved that the solutions set of a three-point boundary value problem whose right-hand side may be “embedded” into a compact-valued measurable multifunction is a nonempty set which is compact in the space of continuous functions. As an example, the authors consider the differential inclusion whose right-hand side can be represented as the multivalued Carathéodory-type perturbation of an m-accretive operator.

In the last section, the authors consider the extension of the problem to the case when the “envelopping” multifunction is assumed to be scalarly Pettis uniformly integrable. In this situation, the existence of a solution in a new Sobolev-type space is obtained, as well as the compactness property of the solution set.

##### MSC:
 34A60 Differential inclusions 34B10 Nonlocal and multipoint boundary value problems for ODE 34G25 Evolution inclusions 49J24 Optimal control problems with differential inclusions (existence) (MSC2000)