On oscillatory solutions of quasilinear differential equations

Abstract

Necessary and sufficient conditions for the existence of at least one oscillatory solution of a second-order quasilinear differential equation are presented. These results yield also new conditions guaranteeing the coexistence of oscillatory and nonoscillatory solutions. Our approach is based on the asymptotic representation of solutions by means of a periodic function and of a suitable zerocounting function. (M. Cecchi), dosla@math.muni.cz (Z. Došlá), marini@ing.unifi.it (M. Marini).

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

“…Proof The theorem is a consequence of Došlý and Řehák 2, theorem 1.4.4 (see also directly Elbert 6,7 ). It is sufficient to consider that the non‐oscillation of the analyzed equations is equivalent to the boundedness from above of the generalized adapted Prüfer angle (we also refer to Došlý et al 5, lemma 3.1 or, for example, other studies 4,42‐45 ). □…”

On a variant of the maximum principle involving radial $p$-Laplacian with applications to~nonlinear eigenvalue problems and nonexistence results

“…Remark 3.4 (oscillatory properties: the case of entire space). Bartušek, Cecchi, Došlá and Marini in the paper [9] considered the quasilinear ODE:…”

“…With the notation of [9] functions a, b are continuous positive defined on R + , Φ s (u) = |u| s−2 u, s > 1, while function a 1/(p−1) b is continuously differentiable on R + . Authors investigated the existence theory for oscillatory solutions of the above ODE.…”

Monotonicity properties of oscillatory solutions of differential equation $(a(t)\vert y^{\prime }\vert ^{p-1}y^{\prime })^{\prime }+f(t,y,y^{\prime })=0$

“…The study of oscillatory solutions of the differential equation (1.2) is an interest subject of many papers also in our days, see e.g. [4,5,6,9,12,13,14].…”