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Higher monotonicity properties of special functions: application on Bessel case \(| \nu | <\). (English) Zbl 0721.34028
Consider an equation \((1)\quad y''+q(t)y=0\) and define \[ M_ i=\int^{t_{i+1}}_{t_ i}[p(t)]^{-\alpha}| y(t)|^{\lambda}dt,\quad i=0,1,..., \] where y(t) is an arbitrary nontrivial solution of (1) and \(\{t_ i\}^{\infty}_{i=0}\) denotes any sequence of consecutive zeros of any solution z(t) of (1), \(\lambda >-1\), \(\alpha <1+\lambda /2\). Using the symbol \(f={\mathcal O}(t^{- \alpha})\), \(t\to \infty\) for the case when \(f=O(t^{-\alpha})\) together with \(\lim_{t\to \infty}| f(t)| t^{\alpha +\xi}=\infty \text{ for } every\quad \xi >0,\) the author proves the following theorem:
Let \(q(\infty)>0\) and for \(k=0,1,...,n+2\) let hold \((-1)^ kq^{(k)}(t)\geq 0,\quad 0<t<\infty,\quad q^{(k)}={\mathcal O}(t^{- (k+\epsilon)}),\quad t\to \infty,\quad \epsilon >0.\) Then (1) has a pair of solutions \(y_ 1(t)\), \(y_ 2(t)\) such that the function \(p(t)=y^ 2_ 1(t)+y^ 2_ 2(t)\) satisfies p(t)\(\to 1\) for \(t\to \infty\), \((- 1)^ kp^{(k+1)}(t)\geq 0,\quad \mu_ k<t<\infty,\quad k=0,1,...,n,\) where \(\{\mu_ k\}^ n_ 1\) is a nondecreasing sequence and \(\mu_ k=\mu_{k+1}\) only if \(\mu_ k=0\); \(p^{(k)}={\mathcal O}(t^{- (k+\epsilon)}),\quad t\to \infty,\quad n\geq 4,\quad k=1,2,....,n-3,\) and the corresponding quantities \(M_ i\) satisfy \((-1)^ k\Delta^{k+1}M_ i\geq 0,\quad k=0,...,n-3,\quad i=l_ k,\quad l_ k+1,...,\) where \(l_ k=l(k)\) is integer, \(0=l_ 0\leq l_ 1\leq...\leq l_{n-3}\) and \(l_ k=l_{k+1}\) only if \(l_ k=0\).
Reviewer: J.Kalas (Brno)
MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
34A40 Differential inequalities involving functions of a single real variable
34A30 Linear ordinary differential equations and systems, general
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