Infinitely Many Solutions for a Generalized Periodic Boundary Value Problem without the Evenness Assumption

where λ > 0, B0ðtÞ, B1ðtÞ ∈ L∞ð1⁄20, 1 ,L sðRÞÞ = fBðtÞ = ðbjkÞn×n ∣ bjkðtÞ = bkjðtÞ, t ∈ 1⁄20, 1 , bjkðtÞ ∈ L∞ð1⁄20, 1 Þg, M,N ∈GLðnÞ = fA = ðajkÞn×n ∣ ajk ∈ R and det ðAÞ ≠ 0g with M NT = In, In is the unit matrix of order n, ∇Vðt, xÞ denotes the gradient of Vðt, xÞ for x ∈ Rn, and Vðt, xÞ satisfies (H0), that is, Vðt, xÞ is continuously differentiable in x for a.e. t ∈ 1⁄20, 1 and measurable in t for every x ∈ Rn, and there exist aðxÞ ∈ CðR+, R+Þ and bðtÞ ∈ L1ð1⁄20, 1 , R+Þ such that

Note that if M = N = I n , B 0 ðtÞ ≡ 0, B 1 ðtÞ is 1-periodic and Vðt, xÞ is 1-periodic in t; then, the solutions of problem (1) are the 1-periodic solutions of second-order Hamiltonian systems.
For second-order Hamiltonian systems, under various conditions, the authors in [3-5, 7, 8, 14-17] obtained infinitely many periodic solutions under the evenness assumption of Vðt, xÞ. Without the evenness assumption of Vðt, xÞ, the authors in [2,6,[9][10][11][12][13] also obtained infinitely many periodic solutions for first-(or second-) order Hamiltonian systems under the potential function Vðt, xÞ ∈ C 2 ðR × R n , RÞ. In this paper, we are interested in the potential function Vðt, xÞ ∈ C 1 ðR × R n , RÞ, and without the evenness assumption. We study the existence of infinitely many solutions of problem (1) via the multiple critical point theorem established in [21,22] under Vðt, xÞ ∈ C 1 ðR × R n , RÞ. Now, we use the index ði M ðBÞ, ν M ðBÞÞ ∈ N × N defined in [23] (see Section 2) to state our results.  Then, for each λ ∈ ðλ 1 , λ 2 Þ, problem (1) possesses infinitely many solutions, where and Theorem 2. The conclusion of Theorem 1still holds if we replace(H 2 ) with (H′ 2 ): We postpone the proofs to the next section and turn to applications to second-order Hamiltonian systems. For systematic researches of second-order Hamiltonian systems, we refer to the excellent books (see [18][19][20]).
Next, an example of problem (7) is given below.

Variational Setting and Proof of the Main Result
In this section, we first recall the multiple critical point theorem due to [21,22] and some conclusions of index theory due to [23,24], respectively. 3
For any BðtÞ ∈ L ∞ ð½0, 1, L s ðR n ÞÞ, we define a bilinear form as follows: such that q B is positive definite, null, and negative definite on Z + ðBÞ, Z 0 ðBÞ, and Z − ðBÞ, respectively. Moreover, Z 0 ðBÞ and Z − ðBÞ are finite-dimensional. For anyBðtÞ ∈ L ∞ ð½0, 1, L s ðR n ÞÞ, we define We call ν M ðBÞ and i M ðBÞ the nullity and index of B with respect to the bilinear form q B ð·, · Þ, respectively. Let a 1 ≤ a 2 ≤⋯≤a n be the eigenvalues of a constant n × n symmetric matrix B. For ζ ∈ R \ f0g with ξ 0 = arccos ð2/ζ −1 + ζÞ, we have where S # denotes the number of elements in set S. In particular, formulae (23) and (24) when ζ = 1 were given first by Mawhin and Willem in [19].
Next, we establish the variational setting for problem (1). It is known that the operator Λ − B 0 is also self-adjoint and σðΛ − B 0 Þ = σ d ðΛ − B 0 Þ is bounded from below. Noticing that i M ðB 0 Þ = 0 and ν M ðB 0 Þ ≠ 0, by Definition 9 and Proposition 10, we know that the operator Λ − B 0 has a sequence of eigenvalues: and the system of eigenfunctions fe n : n ∈ Ng corresponding to fλ n : n ∈ Ng forming an orthogonal basis in L 2 = X. Hence, we can define another inner product: with the corresponding norm kxk = ðq B 0 ðx, xÞ + kxk 2 L 2 Þ 1/2 , ∀x ∈ Z: Clearly, k·k is equivalent to k·k Z . Put Since B 1 ðtÞ is positive definite, there exist b 0 > b 0 > 0 such that Journal of Function Spaces for all x ∈ Z. So, we have for all x ∈ Z.
Noticing the compactness of the embedding Z↪L ∞ , from (29), we know that there is an embedded constant δ 0 > 0 such that for all x ∈ Z, where k 0 = δ 0 ðmin f1, b 0 gÞ −ð1/2Þ and kxk ∞ is the norm of L ∞ ð½0, 1, R n Þ. Now, we define From the assumption (H 0 ) and Theorem 1.2 in [19], it is easy to verify that I λ ∈ C 1 ðZ, RÞ is weakly lower semicontinuous on Z and I ′ λ is weakly continuous with for all x, y ∈ Z. If I ′ λ ðxÞ = 0, we easily find that the critical points of I correspond to the solutions of problem (1) and omit the details. Finally, we give the proofs of Theorems 1 and 2. For convenience, put Since Vðt, θÞ = 0, we have Γ ≥ 0. So, if Γ = 0, we put λ 2 = +∞, and if Y = +∞, we put λ 1 = 0.
Proof of Theorem 1. Set that Obviously, Φ is (strongly) continuous, coercive, and Gâteaux differentiable, Ψ is sequentially weakly upper semicontinuous and Gâteaux differentiable, and inf Z Φ = 0. On the other hand, the critical points of Φ − λΨ in Z are the solutions of problem (1).