IN THIS WORK WE STUDY THE DIFFERENTIABILITY PROPERTIES OF THE CONJUGATION IN THE BARREIRA-VALLS VERSION OF THE HARTMAN-GROBMAN THEOREM FOR NON-AUTONOMOUS AND DELAYED SYSTEMS. INDEED, WE SHOW THAT THE CONJUGACY IN THE BARREIRA-VALLS THEOREM IS A DIFFEOMORPHISM IF WE IMPOSE SOME EXTRA HYPOTHESIS RELATED WITH THE DECAY OF THE PERTURBATION.

WE PROVIDE A RESULT ON THE EXISTENCE OF A POSITIVE PERIODIC SOLUTION FOR THE FOLLOWING CLASS OF DELAY EQUATIONS 0´(T) -0(T) + F (0(T - R)) = 0. IN PARTICULAR, WE FIND AN INFINITE FAMILY OF DISJOINT INTERVALS HAVING THE FOLLOWING PROPERTY: IF THE DELAY IS WITHIN ONE OF THESE INTERVALS, THEN THE EQUATION ADMITS A NON-TRIVIAL AND EVEN 2R-PERIODIC SOLUTION. FURTHERMORE, THE LENGTH OF THESE INTERVALS IS CONSTANT AND DEPENDS ON THE SIZE OF THE TERM |F (?)|, WHERE ? IS THE UNIQUE POSITIVE EQUILIBRIUM POINT OF THE EQUATION. CONSEQUENTLY, WE CAN FIND PERIODIC SOLUTIONS FOR ARBITRARILY LARGE DELAYS.

IN THIS PAPER WE STUDY A NEW CLASS OF FUNCTIONS, WHICH WE CALL $(\OMEGA,C)$-PERIODIC FUNCTIONS. THIS COLLECTION INCLUDES PERIODIC, ANTI-PERIODIC, BLOCH AND UNBOUNDED FUNCTIONS. WE PROVE THAT THE SET CONFORMED BY THESE FUNCTIONS IS A BANACH SPACE WITH A SUITABLE NORM. FURTHERMORE, WE SHOW SEVERAL PROPERTIES OF THIS CLASS OF FUNCTIONS AS THE CONVOLUTION INVARIANCE. WE PRESENT SOME EXAMPLES AND A COMPOSITION RESULT. AS AN APPLICATION, WE ESTABLISH SOME SUFFICIENT CONDITIONS FOR THE EXISTENCE AND UNIQUENESS OF $(\OMEGA,C)$-PERIODIC MILD SOLUTIONS TO A FRACTIONAL EVOLUTION EQUATION.