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Stability of solutions for an abstract Dirichlet problem. (English) Zbl 1097.47053

Summary: We consider continuous dependence of solutions on the right hand side for a semilinear operator equation \(Lx=\nabla G( x) \), where \(L:D(L) \subset Y\rightarrow Y\) (\(Y\) a Hilbert space) is self-adjoint and positive definite and \(G:Y\rightarrow Y\) is a convex functional with superquadratic growth. As applications, we derive some stability results and dependence on a functional parameter for a fourth order Dirichlet problem. Applications to P.D.E.are also given.

MSC:

47J05 Equations involving nonlinear operators (general)
35A15 Variational methods applied to PDEs
34D20 Stability of solutions to ordinary differential equations
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
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