Oscillatory Behavior of Certain Class of Mixed Nonlinear Sub-Elliptic Equations in the Heisenberg Group

This paper investigates the oscillatory behavior of solutions to a class of nonlinear differential equations on the Heisenberg group, where the dynamics are governed by the interplay between superlinear and sublinear terms. The Heisenberg group, as a fundamental example of a non-commutative Lie group with sub-Riemannian geometry, offers a natural and rich framework for analyzing such equations, which frequently arise in mathematical physics and geometric analysis. To establish oscillatory criteria, we employ a combination of the Riccati transformation and the integral averaging method. The Riccati technique enables the reformulation of the original equation into a form suitable for qualitative analysis, while the integral averaging approach helps derive sufficient conditions by capturing the average behavior of the nonlinear terms and coefficients over certain intervals. This dual approach allows us to rigorously examine how the interaction between superlinear and sublinear growth influences the oscillatory nature of solutions. We derive new sufficient conditions that guarantee the oscillation of all solutions under appropriate structural assumptions. These results generalize and extend classical oscillation criteria to the subelliptic equation of the Heisenberg group. To demonstrate the applicability and sharpness of our results, several illustrative examples are presented. The findings contribute to the broader understanding of oscillation theory in non-Euclidean space and pave the way for future research in nonlinear analysis on Lie groups.

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