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On oscillatory behaviour of third-order half-linear dynamic equations on time scales

Grace Said R., Chhatria Gokula Nanda

Opuscula Mathematica

2022

Vol. 42, no. 6

849--865

In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case T = R or T = Z.

Multiple solutions of boundary value problems on time scales for φ-laplacian operator

Amster Pablo, Kuna Mariel Paula, Santos Dionicio Pastor

Opuscula Mathematica

2020

Vol. 40, no. 4

405--425

We establish the existence and multiplicity of solutions for some boundary value problems on time scales with a y-Laplacian operator. For this purpose, we employ the concept of lower and upper solutions and the Leray-Schauder degree. The results extend and improve known results for analogous problems with discrete p-Laplacian as well as those for boundary value problems on time scales.