New monotonicity properties and oscillation of $n$-order functional differential equations with deviating argument

Abstract

In this paper, we offer new technique for investigation of the even order linear differential equations of the form ( E ) y ( n ) ( t ) = p ( t ) y ( τ ( t ) ) . We establish new criteria for bounded and unbounded oscillation of ( E ) which improve a number of related ones in the literature. Our approach essentially involves establishing stronger monotonicities for the positive solutions of ( E ) than those presented in known works. We illustrate the improvement over known results by applying and comparing our technique with the other known methods on the particular examples.