Nonlinear and semilinear spectrum for asymptotically linear or positively homogeneous operators (Q1881331)

The author proves some properties for the nonlinear spectrum \(\sigma(f)\) and semilinear spectrum \(\sigma(L,N)\) when \(f\) and \(N\) are asymptotically linear or positively homogeneous, thus close to a linear operator. The results generalize a previous result which required \(N\) to be a linear operator and \(L\) to be the identity map. Applying the theorems, the author proves a result on the existence of a positive eigenvalue and eigenvector for a compact, positive operator. Examples of applications to the study of a three-point boundary value problem are also given. This extends and complements previous work by the author [ibid. 30, No.~8, 5369--5374 (1997; Zbl 0895.34014), Nonlinear Funct. Anal. Appl. 8, No.~4, 51--533 (2003; Zbl 1061.47046)], by the author and \textit{J. Webb} [Prog. Nonlinear Differ. Equ. Appl. 40, 149--163 (2000; Zbl 0952.47048)], and by \textit{E. De Pasade, A. Vignoli} and the reviewer [Z. Anal. Anwend. 20, No.~3, 565--577 (2001; Zbl 1002.47041)].