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Decreasing sequences of compact absolute retracts and nonlinear problems. (English) Zbl 0667.47035

Let E be a Hilbert space and L: D(L)\(\to E\) a selfadjoint positive definite Fredholm operator of zero index with dim ker L\(=1\). Write \(E=E_ 1\oplus \ker L\) and call Q the orthogonal projection onto \(E_ 1\). Choose a unit vector \(\xi\in \ker L\). Assume that \(H:=(L| E_ 1)^{-1}\) is compact. Let B: \(E\to E\) be a continuous monotone operator with bounded range. The author is interested in the set S(h) of solutions to \(Lu+Bu=h.\) Choose a \(J>0\) such that \(\| Bx-h\| \leq J\) for all \(x\in H\). Assume that there exists an \(s>J\| HQ\|\) such that \(\liminf_{| a| \to \infty}(a,B(a\xi +u)-h)>0\) uniformly on \(\{u\in E_ 1|\) \(\| u\| \leq s\}\). If, in addition, one assumes that there exists a continuous bounded map G: \(E\to E\) which is strictly increasing then S(h) is shown to be an intersection of a decreasing sequence of compact absolute retracts. The result is applied to the solution set of second order differential equations with Dirichlet or von Neumann boundary conditions.
Reviewer: Chr.Fenske

MSC:

47J05 Equations involving nonlinear operators (general)
35G30 Boundary value problems for nonlinear higher-order PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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