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This work is devoted to the study of the first order operator$x'(t)+m\,x(-t)$ coupled with periodic boundary value conditions. We describethe eigenvalues of the operator and obtain the expression of its relatedGreen's function in the non resonant case. We also obtain the range of thevalues of the real parameter $m$ for which the integral kernel, which providesthe unique solution, has constant sign. In this way, we automatically establishmaximum and anti-maximum principles for the equation. Some applications to theexistence of nonlinear periodic boundary value problems are showed.

Comparison results for first order linear operators with reflection and periodic boundary value conditions