Nontrivial solutions of nonlinear equations from simple eigenvalues and their stability (Q1194540)

The authors write: ``The purpose of this paper is to study stability and instability of steady-state solutions of the dynamical system \(du/dt=F(\lambda,u)\) depending on a parameter, where the nonlinear mapping \(F\) is of the form \(F(\lambda,u):=T(u)-L(\lambda,u)-H(\lambda,u)- K(\lambda,u)\), \(T\) and \(L(\lambda,\cdot)\) are continuous linear mappings'', \(H(\lambda,\cdot)\) is homogeneous of degree \(a\geq 2\), and \(K\) is higher order. ``We assume that \((\bar\lambda,0)\) is a solution of \(F(\lambda,u)=0\) and that the mapping \(T-L(\bar\lambda,0)\) is a Fredholm operator with index zero and nullity one. We establish theorems on the existence and stability of nontrivial solutions under some additional hypotheses on the mappings''.