A Knobloch-type result for p(t)-Laplacian systems

We consider the periodic boundary value problem of ordinary differential systems with p(t)- Laplacian of the form |u |p(t )−2u = f (t,u), u(0) −u(T ) = u (0) −u (T ) = 0, where p ∈ C(R,R) is a T -periodic function and p(t) > 1 for t ∈ R, f ∈ C(R×RN,RN) and f (t,u) is T -periodic with respect to t . We prove that, if there exists some r > 0 such that f (t,u),u 0 for t ∈ R and u ∈ RN with |u| = r, then the problem has at least one solution u satisfying |u(t )| r for t ∈ R. This is a generalization of the results obtained by Knobloch and Mawhin under the case of p(t) ≡ 2 and p(t) ≡ p ∈ (1,∞) respectively.  2003 Elsevier Inc. All rights reserved.