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Continuous dependence on data for quasiautonomous nonlinear boundary value problems. (English) Zbl 1086.34054

The author studies the continuous dependence on \(A, a, b, f\) of the solution of the second-order evolution equation \[ p u^{\prime \prime}(t) + r u^\prime(t) \in A u(t) +f, \text{ for a.e. }t \in (0, T) \] with the boundary conditions \( u(0)=a, u(T)=b\), where \(A : D(A) \subset H \to H\) is a maximal monotone operator in a real Hilbert space \(H\), \(a, b \in D(A)\), \(f \in L^2(0, T; H)\) and \(p, r : [0, T] \to \mathbb{R}\) are two continuous functions. Employing nonlinear analytic techniques, the author considers the continuous dependence both on the operator and on the boundary values. The main result is applied to the numerical approximation of the solution of an evolution equation by the solution of an internal scheme of approximation.

MSC:

34G25 Evolution inclusions
47H05 Monotone operators and generalizations
34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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