Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities

Abstract

Listen

Translate article

Interval oscillation criteria are established for second-order forced delay dynamic equations on time scales containing mixed nonlinearities of the form ( r ( t ) Φ α ( x Δ ( t ) ) ) Δ + p 0 ( t ) Φ α ( x ( τ 0 ( t ) ) ) + ∑ i = 1 n p i ( t ) Φ β i ( x ( τ i ( t ) ) ) = e ( t ) , t ∈ [ t 0 , ∞ ) T where T is a time scale, t 0 ∈ T a fixed number; [ t 0 , ∞ ) T is a time scale interval; Φ ∗ ( u ) = | u | ∗ − 1 u ; the functions r , p i , e : [ t 0 , ∞ ) T → R are right-dense continuous with r > 0 nondecreasing; τ k : T → T are nondecreasing right-dense continuous with τ k ( t ) ≤ t , lim t → ∞ τ k ( t ) = ∞ ; and the exponents satisfy β 1 > ⋯ > β m > α > β m + 1 > ⋯ β n > 0 . All results are new even for T = R and T = Z . Analogous results for related advance type equations are also given, as well as extended delay and advance equations. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives. Two examples are provided to illustrate one of the theorems.