Index at infinity and bifurcations of twice degenerate vector fields (Q1959006)

The paper deals with various equations and vector fields that contain nonlinear superposition operators \(x(t)\to \text{f}x= f(t,x(t))\) acting in functional spaces of scalar-valued functions \(x:\Omega \to\mathbb R\), the main case being \(\Omega=[0,T]\). The functions \(f\) are always bounded and continuous w.r.t. all the variables and are always of the form \[ f(t,x)=b(t)+a(t) \text{sign} (x) +g(t,x),\quad t\in\Omega,\;x\in\mathbb R. \] The nonlinearity \(g\) is asymptotically small and has a jump at zero to compensate the jump of the function \(\text{sign}(\cdot)\). By means of a completely continuous linear operator \(A\) in \(L^2\), the Hammerstein vector field \[ \Phi x= x-A(x+\text{f}x) \] is generated. The paper is devoted to the case, where vector fields are twice degenerate -- its principal linear part is degenerate (\(1\in \sigma(A)\)) as well as the projections of the leading homogeneous term. Results on vector fields concern the index at infinity computation and asymptotic bifurcation points. The main assumptions are expansions for projections of the nonlinearity. The most applications and examples are formulated for the solvability of the Dirichlet BVP. Applications to bifurcations at infinity are discussed here. Various examples of BVPs with degenerate linear part independent of the parameter are considered, bifurcation diagrams are defined by bounded nonlinearities. The possibilities to apply the presented methods for some other BVPs are also discussed.