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Abstract quasi-variational inequalities of elliptic type and applications

Yusuke Murase — 2009

Banach Center Publications

A class of quasi-variational inequalities (QVI) of elliptic type is studied in reflexive Banach spaces. The concept of QVI was earlier introduced by A. Bensoussan and J.-L. Lions [2] and its general theory has been developed by many mathematicians, for instance, see [6, 7, 9, 13] and a monograph [1]. In this paper we give a generalization of the existence theorem established in [14]. In our treatment we employ the compactness method along with a concept of convergence of nonlinear multivalued operators...

Nonlinear evolution equations generated by subdifferentials with nonlocal constraints

Risei KanoYusuke MuraseNobuyuki Kenmochi — 2009

Banach Center Publications

We consider an abstract formulation for a class of parabolic quasi-variational inequalities or quasi-linear PDEs, which are generated by subdifferentials of convex functions with various nonlocal constraints depending on the unknown functions. In this paper we specify a class of convex functions φ t ( v ; · ) on a real Hilbert space H, with parameters 0 ≤ t ≤ T and v in a set of functions from [-δ₀,T], 0 < δ₀ < ∞, into H, in order to formulate an evolution equation of the form u ' ( t ) + φ t ( u ; u ( t ) ) f ( t ) , 0 < t < T, in H. Our...

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