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Study of a maximal monotone model with a delay term. (English) Zbl 1051.34064

Here, a mathematical problem of the form \[ \dot u(t)+A(u(t))+B(t,u(t))+G(u(t-r))\ni 0\quad \text{a.e. on } ]0.T[, \quad\forall t\in [-\tau,0], \quad u(t)=z(t), \] is considered, where \(\tau>0,\) \(u\in W^{1,\infty}(-\tau,0;H),\) \(H\) is a separable Hilbert space, \(A\) a maximal monotone multivalued operator on \(H,\) \(G\) is a mapping from \(H\) to \(H\) whose differential is locally bounded on \(H,\) and \(B\) a mapping from \([0,T]\times H\) to \(H,\) Lipschitz continuous with respect to its second argument and whose derivative maps the bounded sets of \(L^{2}(0,T;H)\) into bounded sets of \(L^{2}(0,T;H).\) By using results of existence and uniqueness for maximal monotone differential inclusions, the authors give theoretical results for the above model. An implicit numerical scheme and results related to the order of convergence are also provided from both a theoretical and a numerical point of view.

MSC:

34K30 Functional-differential equations in abstract spaces
65L99 Numerical methods for ordinary differential equations
34G25 Evolution inclusions
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