On the asymptotic behavior of solutions of certain differential functional equations Original Research Article

We investigate the asymptotic behavior, the oscillatory character and the periodic nature of solutions of the retarded and advanced equation View the MathML sourcex˙(t)=ax(t)+bx(t−r(t)), Turn MathJax on , where the argument deviation is given by View the MathML sourcer(t)=t−2[t+12]. Turn MathJax on . Here, [.] denotes the usual “greatest-integer map”. And, besides discussing explicit conditions for such behavior, as the ones obtained by Cooke and Wiener in [4] and [5], we also introduce a more recent result by Oliveira Filho and Carvalho (9 and 10). Also, we give a set of necessary and sufficient condictions for the existence of certain types of periodic solutions of the equation above to the particular case r(t) = 1, i.e., the parameter family of scalar differential-difference equation View the MathML sourcex˙(t)=ax(t)+bx(t−1), Turn MathJax on , where we restrict attention to solutions wich integral period: τ = 3, 4, 5, · · ·. AMS subject classification: 34K15, 34K20, 39A11, 39A12