Nonoscillation for second order sublinear dynamic equations on time scales

Consider the Emden–Fowler sublinear dynamic equation equation(0.1)xΔΔ(t)+p(t)f(x(σ(t)))=0,xΔΔ(t)+p(t)f(x(σ(t)))=0, where p∈C(T,R)p∈C(T,R), TT is a time scale, f(x)=∑i=1maixβi, where ai>0ai>0, 0<βi<10<βi<1, with βiβi the quotient of odd positive integers, 1≤i≤m1≤i≤m. When m=1m=1, and T=[a,∞)⊂RT=[a,∞)⊂R, (0.1) is the usual sublinear Emden–Fowler equation which has attracted the attention of many researchers. In this paper, we allow the coefficient function p(t)p(t) to be negative for arbitrarily large values of tt. We extend a nonoscillation result of Wong for the second order sublinear Emden–Fowler equation in the continuous case to the dynamic equation (0.1). As applications, we show that the sublinear difference equation Δ2x(n)+b(−1)nn−cxα(n+1)=0,0<α<1, has a nonoscillatory solution, for b>0b>0, c>αc>α, and the sublinear q-difference equation xΔΔ(t)+b(−1)nt−cxα(qt)=0,0<α<1, has a nonoscillatory solution, for t=qn∈T=q0N, q>1q>1, b>0b>0, c>1+αc>1+α.