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.
In @cite_13 we applied our results from @cite_1 to prove a Hopf bifurcation theorem for semilinear hyperbolic systems.
@cite_1 @cite_13
2023763411 2045361520 1994588735 2006592838
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. Abstract Elementary proofs are given for theorems of Bapat and Raghavan on the scaling of nonnegative multidimensional matrices. Theorems of Sinkhorn and of Brualdi, Parter, and Schneider are derived as corollaries. For positive two-dimensional matrices, Hilbert's projective metric and a theorem of G. Birkhoff are used to prove that Sinkhorn's original iterative procedure converges geometrically; the ratio of convergence is estimated from the given data. We define infinitary Combinatory Reduction Systems (iCRSs), thus providing the first notion of infinitary higher-order rewriting. The systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary @l-calculus are special cases. Furthermore, we generalise a number of known results from first-order infinitary rewriting and infinitary @l-calculus to iCRSs. In particular, for fully-extended, left-linear iCRSs we prove the well-known compression property, and for orthogonal iCRSs we prove that (1) if a set of redexes U has a complete development, then all complete developments of U end in the same term and that (2) any tiling diagram involving strongly convergent reductions S and T can be completed iff at least one of S T and T S is strongly convergent. We also prove an ancillary result of independent interest: a set of redexes in an orthogonal iCRS has a complete development iff the set has the so-called finite jumps property. We prove that the hyperbolicity cones of elementary symmetric polynomials are spectrahedral, i.e., they are slices of the cone of positive semidefinite matrices. The proof uses the matrix-tree theorem, an idea already present in