The author deals with the existence of multiple solutions to the problem
when the parameter runs near , the th eigenvalue of the operator under the periodic conditions. The proofs are based on the topological degree and bifurcation theory. The results complement those obtained by J. Mawhin and K. Schmitt [Result. Math. 14, No. 1/2, 138-146 (1988; Zbl 0780.35043)] for the case when is near , and in [Ann. Polon. Math. 51, 241-248 (1990; Zbl 0724.34025)] for some similar twopoint boundary value problems, when a simple eigenvalue is acrossed.