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Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities. (English) Zbl 1197.34117

Summary: 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 are right-dense continuous with r>0 nondecreasing; τ k :𝕋𝕋 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= and 𝕋=. 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.

MSC:
34K11Oscillation theory of functional-differential equations
34N05Dynamic equations on time scales or measure chains
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