Even-order linear dynamic equations with mixed derivatives

We consider the time scale kkth-order differential operators DkΔy≔{yΔ∇…Δ∇k even ,yΔ∇…∇Δk odd ,D˜kΔy≔{y∇Δ…∇Δk even ,yΔ∇…∇Δk odd ,Dk∇y≔{y∇Δ…∇Δk even ,y∇Δ…Δ∇k odd ,D˜k∇y≔{yΔ∇…Δ∇k even ,y∇Δ…Δ∇k odd , and the higher-order dynamic equations L(y)≔∑ν=0n(−1)νD˜ν∇(rν(t)DνΔy)=0,M(y)≔∑ν=0n(−1)νD˜νΔ(rν(t)Dν∇y)=0. We will show that these equations can be investigated as special cases of the so-called (delta or nabla) symplectic dynamic systems zΔ=S(t)z,z∇=S(t)z, whose qualitative theory is well developed. We also suggest further perspectives of the investigation of the qualitative properties of higher-order equations with mixed derivatives.