Non-Monotonic Behavior of One-Signed Solutions of Quasilinear Differential Equations

We consider a class of the second-order quasilinear differential equations. By deriving relations between certain types of monotonic solutions of the quasilinear equation and corresponding reciprocal half-linear equation on a finite interval ( a , b ), we obtain criteria for all solutions of the main equation, which do not change sign in ( a , b ), to be non-monotonic in ( a , b ). This work is also extended to a perturbed half-linear equation as well as to the half-line ( a , ∞ ) .