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**Nagumo type condition for partial differential inclusions.**
*(English)*
Zbl 0654.49016

The viability problem for autonomous differential inclusions in Hilbert and Banach spaces and for the generalized equation \(0\in F(x)\) is studied. Let V, H be two Hilbert spaces such that \(V\subset H=H'\subset V'\), the inclusions being compact and dense, and let \(K\subset H\) be a closed set with the so-called internal approximation property. For the differential inclusion (1) \(\dot x+Ax\in G(x)\), \(x(0)=x_ 0\in K\), x(t)\(\in K\) for all \(t\in [0,T]\), where G: \(K\to V\) is an upper- semicontinuous multifunction with closed convex values such that G(K) is bounded and \(A\in L(V,V')\) is a V-elliptic operator. Under a simple generalization of the so-called tangential condition \([G(x)-Ax]\cap T_ K'(x)\neq \emptyset\) for all \(x\in K\cap V\), the existence of a \(W^{1,2}(0,T)\)-solution of problem (1) is proved for all \(x_ 0\in K\) and any \(T>0\). The proof is based on a finite-dimensional approximation and classical results. Next, under the same assumptions the existence of solutions of the equation Ax\(\in G(x)\) in \(K\cap V\) is proved.

At the end some finite-difference scheme for equation (1) is proposed, and the existence of a solution of problem (1) in the space \(W^{\infty}(0,T)\) is obtained.

As applications, the boundary and obstacle problems for parabolic differential inclusions, equations and inequalities and for variational inequalities are studied.

At the end some finite-difference scheme for equation (1) is proposed, and the existence of a solution of problem (1) in the space \(W^{\infty}(0,T)\) is obtained.

As applications, the boundary and obstacle problems for parabolic differential inclusions, equations and inequalities and for variational inequalities are studied.

Reviewer: Z.Wyderka

### MSC:

93B05 | Controllability |

49J45 | Methods involving semicontinuity and convergence; relaxation |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35K85 | Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators |

34A60 | Ordinary differential inclusions |

34G20 | Nonlinear differential equations in abstract spaces |

49J40 | Variational inequalities |

### Keywords:

viability; autonomous differential inclusions; tangential condition; obstacle problems; variational inequalities
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\textit{S. Shi}, Nonlinear Anal., Theory Methods Appl. 12, No. 9, 951--967 (1988; Zbl 0654.49016)

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