Unboundedness of the large solutions of some asymmetric oscillators at resonance

In this paper, we consider the unboundedness of solutions of the following differential equation ( φ p ( x ′))′ + ( p − 1)[ α φ p (x + ) − βφ p (x − )] = f ( x ) x ′ + g ( x ) + h ( x ) + e ( t ) where φ p ( u ) = | u | p − 2 u , p > 1, x ± = max {± x , 0}, α and β are positive constants satisfying $\alpha ^{-{{1} \over {p}}} + \beta ^{-{{1} \over {p}}} = {{2m} \over {n}}$ with m , n ∈ N and ( m , n ) = 1, f and g are continuous and bounded functions such that lim x →±∞ g ( x ) ≕ g (±∞) exists and h has a sublinear primitive, e ( t ) is 2π p ‐periodic and continuous. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)