Nonoscillation for second order sublinear dynamic equations on time scales

Consider the Emden–Fowler sublinear dynamic equation (0.1) x Δ Δ ( t ) + p ( t ) f ( x ( σ ( t ) ) ) = 0 , where p ∈ C ( T , R ) , T is a time scale, f ( x ) = ∑ i = 1 m a i x β i , where a i > 0 , 0 < β i < 1 , with β i the quotient of odd positive integers, 1 ≤ i ≤ m . When m = 1 , and T = [ 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 ) to be negative for arbitrarily large values of t . 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 Δ 2 x ( n ) + b ( − 1 ) n n − c x α ( n + 1 ) = 0 , 0 < α < 1 , has a nonoscillatory solution, for b > 0 , c > α , and the sublinear q-difference equation x Δ Δ ( t ) + b ( − 1 ) n t − c x α ( q t ) = 0 , 0 < α < 1 , has a nonoscillatory solution, for t = q n ∈ T = q 0 N , q > 1 , b > 0 , c > 1 + α .