Fredholm Solvability of Time-Periodic Boundary Value Hyperbolic Problems

TL;DR

This paper studies the conditions under which linear first-order hyperbolic boundary value problems with time-periodic data are solvable, establishing Fredholm properties and non-resonance criteria for specific systems.

Contribution

It provides broad natural conditions ensuring Fredholm solvability for hyperbolic systems and introduces a non-resonance condition crucial for the analysis.

Findings

Fredholm alternative holds under broad data conditions

Non-resonance condition formulated for hyperbolic systems

Criteria established for 2x2 systems with reflection boundary conditions

Abstract

We investigate a large class of linear boundary value problems for the general first-order one-dimensional hyperbolic systems in the strip $[0,1]\times\R$. We state rather broad natural conditions on the data under which the operators of the problems satisfy the Fredholm alternative in the spaces of continuous and time-periodic functions. A crucial ingredient of our analysis is a non-resonance condition, which is formulated in terms of the data responsible for the bijective part of the Fredholm operator. In the case of $2\times 2$ systems with reflection boundary conditions, we provide a criterium for the non-resonant behavior of the system.