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Opial-type inequalities for differential operators. (English) Zbl 1122.26018
The paper establishes some generalized weighted Opial-type inequalities, including both the continuous and discrete versions. Applications of the continuous version on fractional derivatives are also given. The original Opial inequality $$ \int_0^h \vert f(x)\vert \left\vert \frac d{dx}f(x)\right\vert \,dx \le \frac h 4 \int_0^h \left\vert \frac d{dx}f(x)\right\vert ^2\, dx $$ where $f(0)=f(h)=0$, has been generalized to lots of situations and settings over decades. The paper gives some further generalizations to some recent results in the field. The main result is the following generalized weighted Opial-type inequality, $$ \left\vert \int_a^x u(t)\vert h(t)\vert ^q \prod_{i=1}^N \vert y_i(t)\vert ^{p_i}\, dt \right\vert \le A(x) \left\vert \int_a^x v(t)\vert h(t)\vert ^r\, dt \right\vert ^{(p+q)/r},$$ where $u,v$ are continuous positive weight functions on some closed interval and $a$ is a fixed point in the interval. The functions $y_i$ and $h$ satisfy $$ \vert y_i(t)\vert \le \left\vert \int_a^t K_i(t,s) \vert h(s)\vert\, ds \right\vert , $$ where $\{K_i\}$ are positive kernels. The function $A(x)$ is an integral function involving the weight functions, the kernels, and the indexes $p_i, q$ and $r$. Some corollaries and the discrete analogues are then provided. Finally the paper discusses applications of the main result on the Riemann-Liouville fractional derivative operators, and corresponding inequalities are established.

MSC:
26D15Inequalities for sums, series and integrals of real functions
26D10Inequalities involving derivatives, differential and integral operators
26A33Fractional derivatives and integrals (real functions)
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References:
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