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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)
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
Full Text: EuDML