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Existence of solutions for a class of systems governed by nonlinear differential inclusions. (English) Zbl 0941.34062
This nice and short paper extends a previous work by N. U. Ahmed [Discuss. Math., Differ. Incl. 15, No. 1, 21-28 (1995; Zbl 0828.46003) and ibid. 75-94 (1995; Zbl 0824.49007)]. It states and proves an existence theorem for a class of semilinear differential inclusions with a monotone and hemicontinuous principal map and a sequentially upper hemicontinuous multifunction (with respect to inclusion). The system is forced by a time-dependent function, arising from a control term. The functional framework is the usual triplet defined by a separable Hilbert space together with a densely embedded subspace (a Banach space) and its dual. Solutions are seeked in a space of class \(W_{p,q}(0,T)\) \((T<\infty\), \(p^{-1}+ q^{-1}=1)\). The proof is done using a Galerkin approach and a priori estimates deduced readily from the hypotheses.
Reviewer: O.Arino (Pau)
34G20 Nonlinear differential equations in abstract spaces
47H04 Set-valued operators
34A60 Ordinary differential inclusions
49J24 Optimal control problems with differential inclusions (existence) (MSC2000)