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

Abstract

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.

References in corpus (1)

• Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives