Opial inequalities for fractional derivatives (Q2770836)

The Opial inequality NEWLINE\[NEWLINE\int_0^a |f(x)f^\prime(x)|dx\leq \frac{a}{4}\int_0^a |f^\prime(x)|^2 dxNEWLINE\]NEWLINE (where \(f\in C^1([0,a])\) and \(f(0)=f(a)=0\)) is extended to the case of derivatives of fractional order. The main result is as follows. Let \(f(x)\) have an integrable fractional derivative \(D^\nu f\), \(\nu >0\) (in the sense adopted in fractional calculus) such that \(D^\nu f\in L^\infty(0,x)\) and \(D^{\nu -j}f(0)=0\) for \(j=1,\dots , [\nu]+1\). \ Let also \(0\leq \gamma\leq \nu-\frac{1}{q}\) with \(q>1\) and \(q\geq \frac{1}{\nu}\). Then NEWLINE\[NEWLINE\int_0^x|D^\gamma f(s)D^\nu f(s)|ds \leq cx^{\nu-\gamma+1-\frac 2p}\left(\int_0^x |D^\nu f(s)|^q\right)^\frac 2qNEWLINE\]NEWLINE with the constant \(c\) depending on \(\nu-\gamma\) and \(q\). Some modifications and corollaries are also given. A special attention is paid to the limiting cases \(q=1\) and \(q=\infty\). The case of negative \(q\) is also treated. Of special interest is the application of the above result to the proof of uniqueness of solution to a fractional differential equation of order \(\nu\) with given initial values \(D^{\nu-j} f(0)\), \(j=1,2,\dots [\nu]+1\).