The author combines the method of upper and lower solutions with the topological degree and the Miranda fixed point theorem to prove the existence of at least one solution for second order systems: x’(t)=f(t,x(t)) with periodic boundary conditions x(0)=x(1), x’(0)=x’(1), and first order systems: x’(t)=f(t,x(t)), with periodic boundary condition x(0)=x(T), assuming that there exist lower and upper solutions, and that the nonlinearity f satisfies a Lipschitz condition.
Research-oriented and intended for mathematicians and engineers, this paper contains interesting ideas, and is of good physical form. The statement on page 417 that there is no proof in English of Miranda’s fixed point theorem is incorrect, however. The interested reader can consult the book Ordinary differential equations in Rn [1].
