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In this paper we study a class of abstract quasi-variational inequalities with nonlocal constraints depending on the unknown and establish an existence result. Further we give its applications to parabolic systems of partial differential inequalities with nonlocal obstacles depending on the unknowns.

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 ${\phi }^t(v;·)$ 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 $u^{\text{'}}\left(t\right)+\partial {\phi }^t(u;u\left(t\right))\ni f\left(t\right)$, 0 < t < T, in H. Our...