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Oscillation theorems and existence criteria of asymptotically monotone solutions for second order differential equations. (English) Zbl 0840.34021
The authors study oscillation and asymptotic properties of nonlinear differential equations of the form \[ (a(t)(y')^\sigma)'+ Q(t, y)= P(t, y, y')\tag{1} \] and \[ (a(t)(y')^\sigma)'= b(t)(y')^\sigma+ h(t, y)= 0,\tag{2} \] where \(\sigma\) is a positive rational number and \(a(t)\) is an eventually positive function.
Conditions on the nonlinearities \(P\), \(Q\) and on the functions \(a\), \(b\) are given which guarantee that the equations under consideration have all solutions oscillatory or possess a monotonically decreasing positive solution. The principal method used in the proofs of oscillation criteria for (1) consists, roughly speaking, in comparing this equation with a (simpler) equation \((a(t)(y')^\sigma)'+ [q(t)- p(t)] f(y)= 0\). The criteria for existence of a positive monotone solution of (2) are proved using a similar method. A typical result is the following criterion for (1).
Theorem. Suppose that \(Q(t, u)/f(u)\geq q(t)\), \(P(t, u, v)/f(u)\geq p(t)\) for \(u, v\neq 0\), where the function \(f\) satisfies \(f'(u)\geq 0\) and \(uf(u)> 0\) for \(u\neq 0\). If \(\sigma\) is of the form \(\sigma= {\text{even}\over \text{odd}}\) and \(\int^\infty [q(t)- p(\;)]dt< \infty\) then all solutions of (1) are oscillatory.
Reviewer: O.Došlý (Brno)

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations