Opial's type inequalities on time scales and some applications (Q2880821)

Some Opial-type inequalities on time scales are established (see Section 2) and applied (see Section 3). This result provides a new dynamic analogue of the original inequality \textit{Z. Opial} [Ann. Pol. Math. 8, 65--69 (1960; Zbl 0090.06602)], compare with \textit{M. Bohner} and \textit{B. Kaymakçalan} [Ann. Pol. Math. 77, No. 1, 11--20 (2001; Zbl 0992.34008)] and \textit{B. Karpuz}, \textit{B. Kaymakçalan}, and \textit{Ö. Öcalan} [Differ. Equ. Dyn. Syst. 18, No. 1--2, 11--18 (2010; Zbl 1210.34135)]. In Section 3, these inequalities are utilized to give sufficient conditions for the disfocality of the second order Sturm-Liouville dynamic equation NEWLINE\[NEWLINE(r(t)\,y^{\Delta})^{\Delta}+q(t)\,y^{\sigma}=0,\quad t\in[\alpha,\beta]\cap\mathbb{T},NEWLINE\]NEWLINE to determine a lower bound for the distance between consecutive generalized zeros of solutions of the latter equation (see Theorem 17), and also to find a lower bound for the smallest eigenvalue of the following Sturm-Liouville eigenvalue problem (see Theorem 21) NEWLINE\[NEWLINE-y^{\Delta\Delta}+q(t)\,y^{\sigma}=\lambda y^{\sigma},\quad y(\alpha)=0=y(\beta).NEWLINE\]