Computation of topological degree of unilaterally asymptotically linear operators and its applications. (English) Zbl 1191.47076

Liu, Xiaoying; Sun, Jingxian

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 1-2, 96-106 (2009).

Suppose that \(E\) is a Banach space, \(P\) a cone in \(E\), and let \(w\in E\). A nonlinear operator \(A: E\to E\) is said to be unilaterally asymptotically linear along \(w+P\) if there is a bounded linear operator \(L\) in \(E\) such that
\[ \lim_{\|x\|\to\infty,\;x\geq w}\frac{\|Ax-Lx\|}{\|x\|}=0. \]
\(A\) is said to be a cone mapping if \(A(P)\subseteq P\).
The authors present several theorems on the computation of the topological degree and of fixed point indices for unilaterally asymptotically linear operators that are not cone mappings. These theorems are applied to a class of semilinear elliptic boundary value problems with asymptotically linear nonlinearity. Under a variety of conditions, the existence of one or three solutions is proved, including an existence result for a sign changing solution.

Reviewer: Nils Ackermann (México)

MSC:

47H11	Degree theory for nonlinear operators
35J61	Semilinear elliptic equations
35J25	Boundary value problems for second-order elliptic equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

asymptotically linear operator; topological degree; vector lattice; elliptic boundary value problem

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