On a class of doubly nonlinear nonlocal evolution equations
The abstract nonlinear evolution equation \(f\in (Au)' + Bu\) is investigated, where \(A\) is a (possibly degenerate) maximal monotone operator from a Hilbert space \(V\) to its dual \(V^*\), while \(B\) is a maximal monotone operator from \(L^2(0,T; V)\) to \(L^2(0,T;V^*)\) (\(T\) is a fixed positive number). Existence for this equation, with the Cauchy condition \(v_0\in Au(0)\), is proved by an approximation procedure. Applications to various integro-differential equations are also discussed. The paper includes a good description of the history of this topic. The main novelty here is the generality of the operator \(B\). In addition, the approximation procedure is a new approach to this kind of problems.
scientific article; zbMATH DE number 1839911