Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

Abstract

We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals:(p(t)xΔn-1(t)α-1xΔn-1(t))Δ+f(t,x(δ(t))) + ∫aσ(b)k(t,s)x(g(t,s))θ(s)sgn(x(g(t,s)))Δξ(s)=0,wheret∈[t0,∞)𝕋:=[t0,∞)∩𝕋,𝕋a time scale which is unbounded above,n⩾2is even,f(t,u)⩾q(t)uα,α>0is a constant, andθ:[a,b]𝕋1→ℝis a strictly increasing right-dense continuous function;p,q:[t0,∞)𝕋→ℝ,k:[t0,∞)𝕋×[a,b]𝕋1→ℝ,δ:[t0,∞)𝕋→[t0,∞)𝕋, andg:[t0,∞)𝕋×[a,b]𝕋1→[t0,∞)𝕋are right-dense continuous functions;ξ:[a,b]𝕋1→ℝis strictly increasing. Our results extend and supplement some known results in the literature.