Oscillatory Bifurcation for Semilinear Ordinary Differential Equations

We consider the nonlinear eigenvalue problem u ′′ (t) + λ f (u(t)) = 0, u(t) > 0, t ∈ I := (−1, 1), u(1) = u(−1) = 0, where f (u) = u + (1/2) sin k u (k ≥ 2) and λ > 0 is a bifurcation parameter. It is known that λ is parameterized by the maximum norm α = ∥u λ ∥ ∞ of the solution u λ associated with λ and is written as λ = λ(k, α). When we focus on the asymptotic behavior of λ(k, α) as α → ∞, it is natural to expect that λ(k, α) → π 2 /4, and its convergence rate is common to k. Contrary to this expectation, we show that λ(2n 1 + 1, α) tends to π 2 /4 faster than λ(2n 2 , α) as α → ∞, where n 1 ≥ 1, n 2 ≥ 1 are arbitrary given integers.

As for asymptotic behavior of λ(k, α) and it is quite natural to expect that the rate of convergence of λ(2n, α) to π 2 /4 as α → ∞ is the same as that of λ(2n + 1, α).However, contrary to our expectation, it will turn out that the following inequality holds.
By this and (1.12), for the constant ) π , we obtain This implies our assertion.
(iii) The study of bifurcation problems has a long history and there are so many topics.For the readers who are interested in this field, we refer to [2][3][4][5][6].
The proofs of Theorems 1.1 and 1.2 depend on the time map arguments used in [14].However, we understand easily from Theorems 1.0-1.2 that all the terms in Theorems 1.1 and 1.2 are extremely more complicated than those in Theorem 1.0.Therefore, we proceed all the steps of the calculation very carefully.
Proof of (1.13).By (2.19) and Taylor expansion, we obtain ) cos Then we have two cases to consider.
Then by integration by parts, we obtain By the same calculation as that just above, we obtain S r,m,2 = O(α −3 ).

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