Existence Of Nonoscillatory Solutions For The High Order Nonlinear Neutral Dynamic Equations With Positive And Negative Terms

In this paper, we consider the existence of nonoscillatory solutions for the high order nonlinear dynamic equation with positive and negative terms and its corresponding with forced termsAlso, consider the neutral equation with oscillatory coefficients:We assume that,(H₁) p,g∈C_rd(T,R),Ï„,σ_i∈C_rd(T,T), (?)σ_i(t) =∞;(H₂) (?)Ï„(t) =∞,Ï„(t)is strictly increasing , and(Ï„^-1(t))^â–³is bounded ;(H₃) f_i∈C_rd(T×R,R)and uf_i(t, u) > 0 for u≠0, where 2 = 1,2;(H₄) A_i∈C_rd(T, R)is allowed changing signs all the time (i.e.oscillatory around zero).By define mappings, using the Banach contraction mapping principle and Kransnoselskii's fixed theorem, we obtain the existence of nonoscillatory solutions for the above equations.