The paper concerns the general linear one-dimensional second-order hyperbolic equation ( _t^2 u - a^2 ( x, ,t ) _x^2 u + a_1 ( x, ,t ) _t u + a_2 ( x, ,t ) _x u + a_3 ( x, ,t )u = f ( x, ,t ), ,x ( 0, ,1 ) ) with periodic conditions in time and Robin boundary conditions in space. Under a non-resonance condition (formulated in terms of the coefficients a, a1, and a2) ruling out the small divisors effect, we prove the Fredholm alternative. Moreover, we show that the solutions have higher regularity if the data have higher regularity and if additional non-resonance conditions are fulfilled. Finally, we state a result about smooth dependence on the data, where perturbations of the coefficient a lead to the known loss of smoothness while perturbations of the coefficients a1, a2, and a3 do not.
This paper concerns linear first-order hyperbolic systems in one space dimension of the type @math with periodicity conditions in time and reflection boundary conditions in space. We state a non-resonance condition (depending on the coefficients @math and @math and the boundary reflection coefficients), which implies Fredholm solvability of the problem in the space of continuous functions. Further, we state one more non-resonance condition (depending also on @math ), which implies @math -solution regularity. Moreover, we give examples showing that both non-resonance conditions cannot be dropped, in general. Those conditions are robust under small perturbations of the problem data. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity. Abstract We consider boundary value problems for semilinear hyperbolic systems of the type ∂ t u j + a j ( x , λ ) ∂ x u j + b j ( x , λ , u ) = 0 , x ∈ ( 0 , 1 ) , j = 1 , … , n , with smooth coefficient functions a j and b j such that b j ( x , λ , 0 ) = 0 for all x ∈ [ 0 , 1 ] , λ ∈ R , and j = 1 , … , n . We state conditions for Hopf bifurcation, i.e., for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence on λ of time-periodic solutions bifurcating from the zero stationary solution. Furthermore, we derive a formula which determines the bifurcation direction. The proof is done by means of a Liapunov–Schmidt reduction procedure. For this purpose, Fredholm properties of the linearized system and implicit function theorem techniques are used.