On the asymptotic integration of a class of sublinear fractional differential equations

Details On the asymptotic integration of a class of sublinear fractional differential equations On the asymptotic integration of a class of sublinear fractional differential equations

2009-04-09

arxiv.org/abs/0904.1495v2

Journal of Mathematical Physics 50 (2009), 123520

Mathematics

Dynamical Systems

Scientific paper

We estimate the growth in time of the solutions to a class of nonlinear fractional differential equations $D_{0+}^{\alpha}(x-x_0) =f(t,x)$ which includes $D_{0+}^{\alpha}(x-x_0) =H(t)x^{\lambda}$ with $\lambda\in(0,1)$ for the case of slowly-decaying coefficients $H$. The proof is based on the triple interpolation inequality on the real line and the growth estimate reads as $x(t)=o(t^{a\alpha})$ when $t\to+\infty$ for $1>\alpha>1-a>\lambda>0$. Our result can be thought of as a non--integer counterpart of the classical Bihari asymptotic integration result for nonlinear ordinary differential equations. By a carefully designed example we show that in some circumstances such an estimate is optimal.

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