Nonlinear and semilinear spectrum for asymptotically linear or positively homogeneous operators Original Research Article

We prove some properties for the nonlinear spectrum σ(f) and semilinear spectrum σ(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, we prove 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.