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Zbl 0667.47035
Nieto, Juan J.
Decreasing sequences of compact absolute retracts and nonlinear problems. (English)
[J] Boll. Unione Mat. Ital., VII. Ser., B 2, No.3, 497-507 (1988). ISSN 0392-4041

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 $E=E\sb 1\oplus \ker L$ and call Q the orthogonal projection onto $E\sb 1$. Choose a unit vector $\xi\in \ker L$. Assume that $H:=(L\vert E\sb 1)\sp{-1}$ 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 $\Vert Bx-h\Vert \le J$ for all $x\in H$. Assume that there exists an $s>J\Vert HQ\Vert$ such that $\liminf\sb{\vert a\vert \to \infty}(a,B(a\xi +u)-h)>0$ uniformly on $\{u\in E\sb 1\vert$ $\Vert u\Vert \le s\}$. 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.
[Chr.Fenske]

MSC 2000:: *47J05 Equations involving nonlinear operators (general)
35G30 Boundary value problems for nonlinear higher-order PDE
35J65 (Nonlinear) BVP for (non)linear elliptic equations

Keywords: retract; selfadjoint positive definite Fredholm operator of zero index; continuous monotone operator with bounded range; solution set of second order differential equations with Dirichlet or von Neumann boundary conditions

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