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 is an element of(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 ->+infinity for 1>alpha>1-a>lambda>0. Our result can be thought of as a noninteger 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