Existence of solutions for subquadratic convex operator equations at resonance and applications to Hamiltonian systems

This paper investigates the existence of solutions to subquadratic operator equations with convex nonlinearities and resonance by means of the index theory for self-adjoint linear operators developed by Dong and dual least action principle developed by Clarke and Ekeland. Applying the results to subquadratic convex Hamiltonian systems satisfying several boundary value conditions including Bolza boundary value conditions, generalized periodic boundary value conditions and Sturm–Liouville boundary value conditions yield some new theorems concerning the existence of solutions or nontrivial solutions. In particular, some famous results about solutions to subquadratic convex Hamiltonian systems by Mawhin and Willem and Ekeland are special cases of the theorems.


Introduction and main results
Periodic solutions of Hamiltonian systems are important in applications. In recent years, the existence of periodic solutions for Hamiltonian systems has been studied extensively by means of critical point theory and the least action principle, and many interesting results have been obtained. Different solvability hypotheses on the potential are given, such as: the convexity conditions (see [2, 5, 7-9, 15, 16, 20]); the subquadratic conditions (see [5, 7, 10-13, 15-18, 20]) and the sublinear conditions (see [19]). In particular, Mawhin and Willem [15,16] and Ekeland [7] investigated the existence of periodic solutions to convex Hamiltonian systems under subquadratic growth assumptions on the Hamiltonian. In this paper, we will generalize their results to abstract operator equations. Let X be a real infinite-dimensional separable Hilbert space with norm · and inner product (·, ·). Let A : D(A) ⊂ X → X be an unbounded self-adjoint operator with σ (A) = σ d (A) = {λ ∈ R|λ belongs to the point spectrum of A}. We consider the existence of solutions of the following equation: where B 1 ∈ L s (X) and L s (X) denotes the usual space consisting of bounded symmetric operators in X, ν A (B 1 ) = 0 (see the Appendix for the notation ν A (B 1 )), : X → R is continuous and convex, and ∂ denotes the subdifferential of . Set * (y) = sup x∈X {(y, x) -(x)} and (x) = (x) + 2 x 2 for > 0. Now we use the index (i A (B), ν A (B)) ∈ Z × N defined in [5,6] (see the Appendix) for all B ∈ L s (X) to formulate our main results.
Theorem 1. 1 Assume that ∈ C(X, R) and satisfies ( 1 ) is convex; ( 2 ) there exists l 0 ∈ X such that for all x ∈ X one has Then (1.1) has at least one nontrivial solution.
Remark 1 In [5], Theorem 3.1.7, one only assumed that ( 1 ) and ( 3 ) hold. The following BVP can serve as a counterexample: (1.5) In fact, it is easy to check that Eq. (1.5) satisfies the conditions of Theorem 3.1.7 in [5], but it has no solution, so Theorem 3.1.7 is also incorrect. These show that our results improve Theorem 3.1.7 in [5].
The paper is organized as follows. The proofs of Theorems 1.1-1.2 are given in Sect. 2 and in Sect. 3 we investigate their applications to convex Hamiltonian systems satisfying several boundary value conditions including Bolza boundary value conditions, generalized periodic boundary value conditions and Sturm-Liouville boundary value conditions, and we obtain some new theorems on the existence of solutions or nontrivial solutions. In addition, we give some remarks to illustrate that some famous results about solutions to subquadratic convex Hamiltonian systems by Mawhin and Willem [16] and Ekeland [7] are special cases of these results. In the Appendix we recall some useful results concerning the index theory for unbounded linear self-adjoint operator equations from [5,6] used in other sections.

Proofs of the theorems
In order to prove Theorems 1.1-1.2 we need some lemmas from [8,16]. Let X be a Hilbert space, we shall denote by (X) the set of all convex lower semicontinuous functions F : We define the subdifferential of a function F ∈ (X) at a point u ∈ X to be the set and say that F is subdifferentiable at u if ∂F(u) = ∅.
has a convergent subsequence in X and so does {B 1 x j }. This implies that B 1 : (D(A), · A ) → X is compact and B 1 is A-compact. By Weyl's theorem ([1], Theorem 6.5.21), we can see that A -B 1 is also an unbounded linear self-adjoint operator in X and σ ess (A - Consider the dual functional ψ : and for the existence of -1 , we refer to the Appendix. It is then easy to see that - , and there is ξ ∈ ker(A -B 1 ) such that --1 y + ∇F * (y) = ξ .
Proof of Theorem 1. 1 We divide the proof into three steps.
Step 2. A posteriori estimates on x . From the obvious inequality (x) ≤ (x) we deduce * (y) ≤ * (y). Thus, we can obtain for all y ∈ R(A -B 1 ), which implies that ψ (y ) ≤ ψ 1 (y 0 ) = c 1 < +∞ and y ≤ c 2 with c 2 independent of . By y = x , we know that . Proposition 2.5 implies that x ≤ c 3 . From the convexity of , hypothesis ( 3 ) and (2.5) we have due to the boundedness of B 1 , B 2 and -1 . By assumption ( 4 ), x ≤ c 5 . Then Step 3. Existence of a solution for the original problem. Since Moreover, by the compactness of -1 , we havẽ Again note that x = y and x ≤ c 2 , we have -1 y = x -x and y n y in R(A -B 1 ), so that -1 y n converges to (2.7) From Proposition 2.6, we can see that Taking the limit yields which shows that x ∈ X is a solution of (1.1).
Now, by the duality between x n and y n , we have It follows from (1.1) and (2.7) that Letting n → ∞ in (2.9), by (2.10) and Lemma 2.3, we obtain Thus ψ(y) ≤ ψ(h) for all h ∈ R(A -B 1 ) and the proof is complete.

Applications to convex Hamiltonian systems
In this section, we consider the applications of the main results to convex Hamiltonian systems. For systematic research of Hamiltonian systems, we refer to the excellent books [7,14].

First order Hamiltonian systems satisfying Bolza boundary value conditions
As a first example, we now consider a BVP for a the nonlinear Hamiltonian systems (3.1)-(3.3): (3.1) From Theorems 1.1-1.2, we have the following theorem.

First order Hamiltonian systems satisfying generalized periodic boundary value conditions
As a second example, consider the nonlinear Hamiltonian systems:    Remark 3 Noticing that Theorem III.2.1 in [7] considers the eigenvalue of operator -A 2 + B 1 , it is easy see that Corollary 3.7 reduces to ( [7], Theorem III.2.1) as P = I 2n , T = 1.
Similarly to Theorem 3.4, we have the following.
Similarly to Theorem 3.4, we have the following.
Theorem 3.14 Under the hypotheses of Theorem 3.12, assume, in addition, that Let B 0 ∈ L s (X) be given. From Lemma 3.2.1 in [5], we know that A -B 0 : Y → X is continuous, ker(A -B 0 ) is finite-dimensional, R(A -B 0 ) is closed and X = ker(A -B 0 ) ⊕ R (A -B 0 ), and the operator 0 : is compact and self-adjoint. For any B ∈ L s (X) with B -B 0 ≥ for some constant > 0, we define a bilinear form: By Proposition 3.2.2 in [5] we have such that q A,B|B 0 is positive definite, null and negative definite on E + A (B|B 0 ), E 0 A (B|B 0 ) and E -A (B|B 0 ), respectively. Moreover, E 0 A (B|B 0 ) and E -A (B|B 0 ) are finite-dimensional.

Definition A.5 ([5], Definition 3.2.3) For any
This relative index is a kind of Morse index. It plays an important role in the relationship between Morse index and the index (i A (B), ν A (B)).