Non‐oscillation of linear and half‐linear differential equations with unbounded coefficients

Abstract

We deal with Euler-type half-linear second-order differential equations, and our intention is to derive conditions in order their non-trivial solutions are non-oscillatory. This paper connects to the article P. Hasil, J. Šišoláková, M. Veselý: Averaging technique and oscillation criterion for linear and half-linear equations, Appl. Math. Lett. 92 (2019), 62-69, where the corresponding oscillatory counterpart is studied and an oscillation criterion is established. The used effective technique for this investigation is the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This paper is completed by two corollaries and an example concerning the linear case when we obtain new results as well.

Nonoscillation of half‐linear dynamic equations on time scales

“…We conjecture that Theorem 3.1 remains valid also for more general coefficients (at least for quasiperiodic, limit or almost periodic coefficients; in contrast to the threshold case discussed in Remark 1). This conjecture follows mainly from known results about differential and difference equations, where the corresponding results are valid even for generalizations of asymptotically almost periodic coefficients if β ≤ 0 (see, e.g., previous studies 45–48 ). Nevertheless, it is an open problem and such a generalization of Theorem 3.1 cannot be proved using the presented process.…”

Oscillation and nonoscillation of perturbed nonlinear equations with p‐Laplacian

“…are conditionally oscillatory for some coefficients 𝑅, 𝑆. We refer to [13] (and also [29,55]). Nevertheless, there are not known any perturbations, which preserve the conditional oscillation of Equation (6.21) for 𝛼 ≠ 1.…”

“…It is known that the Euler‐type equations in the form $$$$\begin{equation} {\left[t^{\alpha -1} R^{1-p} (t) \Phi {\left(x^{\prime }(t)\right)}\right]}^{\prime }+\frac{S(t)}{t^{p-\alpha +1}}\, \Phi (x(t))=0, \qquad \alpha \ne p, \end{equation}$$$$are conditionally oscillatory for some coefficients $$R, S$$. We refer to [13] (and also [29, 55]). Nevertheless, there are not known any perturbations, which preserve the conditional oscillation of Equation (6.21) for $$\alpha \ne 1$$.…”

Integral Comparison Criteria for Half-Linear Differential Equations Seen as a Perturbation

“…Here, we refer to the basic literature, for example, [8] (Section 1.1.4), for introduction to the theory (see also [11]). It can be shown (as introduced by [12]) that among all non-oscillatory solutions of (5), there exists the minimal onew, for which any other solution of (5) satisfies the inequality w(t) >w(t) for large t. Then, the solution of (5) given bỹ…”

“…On the other hand, classification of solutions and equations in terms of oscillation remains the same-a solution is called oscillatory if it has got infinitely many zeros tending to infinity, and non-oscillatory otherwise; and since oscillatory and non-oscillatory solutions cannot coexist, equations are classified as oscillatory or non-oscillatory according to their solutions. To refer to the most current results of the oscillation theory of (1), let us mention, for example, papers [2][3][4][5][6]. Because we are interested in the qualitative behavior of solutions of (1), we study it on a neighborhood of infinity, that is, on intervals of the form t ≥ t 0 , where t 0 is a real constant.…”