| Abstract: |
| In this talk some results are presented about the existence and the precise asymptotic behavior for positive global solutions of a system of two differential equation of fractional order. The aims is both to generalize some of the results known for the ordinary case to the fractional case, and to analyze the role of fractional order, putting in evidence purely fraactional phenomena. Under the assumption that the coefficients are \textit{regularly varying} functions at infinity, an extension of the Karamata integration theorem to fractional integration enables the use of a fixed-point approach to prove the existence of positive unbounded solutions satisfying given asymptotic conditions, since it provides sharp two-sided estimates for the solutions. Furthermore, conditions are found under which all solutions with nonnegative initial conditions are regularly varying at infinity with a prescribed index. We then apply these results to the special case of a scalar equation of order $1+\alpha$, highlighting both the analogies and discrepancies with the integer-order case. |
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