Periodic solutions of semilinear hyperbolic equations at resonance (Q921263)

The author considers the equation \(Ax+Nx=f\) (x\(\in D(A)\), \(f\in H)\), where H is a separable Hilbert space, A: D(A)\(\subset H\to H\) is a densely defined linear map with closed range R(A) and N: \(H\to H\) is a nonlinear map. The author establishes the pseudo A-properness of \(A+N\) for several new types of nonlinearities, and using the pseudo A-proper mapping approach proves several new solvability results for the above problem when there is a resonance at one or two eigenvalues of A. The author then gives applications of the theory to the weak solvability of periodic-boundary value problems for semilinear wave equations \[ U_{tt}-U_{xx}- F(t,x,u)=f(t,x),\quad t\in {\mathbb{R}},\quad x\in (0,\pi), \] \[ u(t,0)=u(t,\pi)=0,\quad t\in {\mathbb{R}},\quad u(t+T,x)=u(t,x),\quad t\in {\mathbb{R}},\quad x\in (0,\pi) \] involving nonmonotone inequalities.