GLOBAL BEHAVIOR OF BIFURCATION CURVES FOR THE NONLINEAR EIGENVALUE PROBLEMS WITH PERIODIC NONLINEAR TERMS

. We consider the bifurcation problem where g ( u ) 2 C 1 ( R ) is a periodic function with period 2 (cid:25) and (cid:21) > 0 is a bifurcation parameter. It is known that, under the appropriate conditions on g , (cid:21) is parameterized by the maximum norm (cid:11) = ∥ u (cid:21) ∥ 1 of the solution u (cid:21) associated with (cid:21) and is written as (cid:21) = (cid:21) ( (cid:11) ) . If g ( u ) is periodic, then it is natural to expect that (cid:21) ( (cid:11) ) is also oscillatory for (cid:11) ≫ 1 . We give a simple condition of g ( u ) , by which we can easily check whether (cid:21) ( (cid:11) ) is oscillatory and intersects the line (cid:21) = (cid:25) 2 = 4 infinitely many times for (cid:11) ≫ 1 or not.

The study of the structures of the bifurcation curves is one of the main topics in bifurcation analysis, and there are quite many works concerning the properties of bifurcation diagrams. We refer to [1-4, 6, 7, 10, 14, 16, 17] and the references therein. In particular, the qualitative properties of the oscillatory bifurcation diagrams have been studied intensively. We refer to [8,11,12,13,18,19] and the references therein. In this paper, we focus on the study whether λ(g, α) inherits the oscillatory properties of g (u) or not if g(u) is a periodic function.
Now we state our main results. where As a corollary of Theorem 1.1, we obtain a meaningful result for the asymptotic property of λ(g, α).
does not satisfy (OP).
We apply Corollary 1.2 to λ(g ϵ , α). In this case, we have The method to study the local behavior of λ(α) has been already established in [17] and [18], since the time-map method and Taylor expansion work very well in this case. To understand the total structure of λ(g, α), we show the following asymptotic formulas for completeness.
) . (1.11) The proof of Theorem 1.1 is given by the combination of time-map method, Fourier expansion and the asymptotic formulas for some special functions.
2. Proof of Theorem 1.1. In this section, let α ≫ 1. For simplicity, we write λ = λ(g, α). Furthermore, we denote by C the various positive constants independent of α. We put We construct the well known time-map (2.7) below (cf. [18]). By (1.1), we have By this and putting t = 0, we obtain ) .
3. Proof of Theorem 1.3. The local behavior of λ(α) is easy to calculate, since Taylor expansion and the time-map method work very well. We only prove Theorem 1.3 (i) for completeness.
This implies (1.9). Thus the proof is complete.