Superlinear damped vibration problems on time scales with nonlocal boundary conditions

. This paper studies a class of superlinear damped vibration equations with nonlocal boundary conditions on time scales by using the calculus of variations. We consider the Cerami condition, while the nonlinear term does not satisfy Ambrosetti–Rabinowitz condition such that the critical point theory could be applied. Then we establish the variational structure in an appropriate Sobolev’s space, obtain the existence of inﬁnitely many large energy solutions. Finally, two examples are given to prove our results.


Introduction
Vibration is a common form of motion in daily life and engineering technology, such as the reciprocating swing of a pendulum, the vibration of a spring, the vibration of a string in a musical instrument, the vibration of the spindle of a machine tool, the electromagnetic oscillation in a circuit, and so on. The study of vibration problem can be reduced to the second-order constant coefficient differential equation under certain conditions. Assuming that the pendulum is oscillating in a viscous medium, considering the air resistance and the constant external force F (t) acting on the pendulum along its direction of motion, the pendulum's action called forced vibration, its damped forced vibration equation is In recent papers, many authors studied the damped forced vibration problems with local conditions, especially, two-point boundary conditions. Nieto and Xiao [16,26]  where p ∈ C 1 ([0, σ(T )] T , (0, ∞)), q ∈ C([0, T ] T , (0, ∞)), h ∈ C([0, T ] T × R, R), α 1 , α 2 , α 3 , α 4 , α 1 + α 2 0, α 1 + α 3 , α 3 + α 4 > 0. They got the existence of solutions by variational methods on time scales. More literatures we can see [2,19]. However, we find it is better to impose nonlocal conditions in some problems [9,14,20,21] because the measurements required by nonlocal conditions may be more accurate than those given by local conditions. As example, the following second-order differential equation: y (s) + h s, y(s), y (s) = 0 with Robin boundary condition y(0) = 0, y (1) = 0. Let the nonlocal condition y(1) = y(η) replace the local condition y (1) = 0, then the problem can be transfer to a nonlocal problem. Moreover, because in the actual conditions and the process of numerical calculation, the value of (y(η) − y(1))/(η − 1) is easier to determine than the value of y (1) = 0, the effect of nonlocal problems is better than that of local problems. Therefore, nonlocal problems can be considered as boundary value problems that include continuous equations and one or more discrete multipoint boundary conditions. The origin of nonlocal boundary value problems of differential equations is related to the mathematical models of nonlinear problems in mathematics, physics, and other disciplines. Many problems in elastic stability theory can also be treated as nonlocal boundary value problems. Therefore, the damped forced vibration problems will be more accurate with nonlocal boundary conditions. However, there are few literatures on solving nonlocal boundary value problems by using variational method. So far, we only know what we have done recently [12,[22][23][24].
In order to find the existence of infinitely many solutions, the following superquadratic conditions proposed by Ambrosetti and Rabinowitz [3] are needed: (AR) there exists the constant µ > 2 such that 0 < µH(s, y) yh(s, y) for all y ∈ R \ {0} and s ∈ Ω, where h is the nonlinear term, H(s, y) = y 0 h(s, τ ) dτ , and Ω ⊂ R. The role of Ambrosetti-Rabinowitz (AR) condition ensure the compactness of Palais-Smale (PS) condition. However, condition (AR) eliminates many nonlinearities and has certain limitations. In fact, some examples show that the nonlinearity h(s, y) could not satisfy the (AR) condition such as h(s, y) = 2y 2 log(1 + |y|). Thus it is meaningful to study problem (1) without (AR) condition.
The interesting points of this paper are the followings: (i) we establish the variational structure of the superlinear damped vibration problem with nonlocal boundary conditions; (ii) we consider the Cerami condition such that although the Palais-Smale sequence is unbounded, the critical point theory could be applied; (iii) the existence of infinitely many large energy solutions for the problem is obtained without (AR) condition by using the mountain pass theorem and fountain theorem.

Preliminaries
In order to solve problem (1), the following definitions and lemmas are needed.
Say that property P holds for -almost all ( -a.a.) s ∈ X if there is a -null set A ⊂ X such that P holds for all s ∈ X \ A. We say that property P holds -almost everywhere ( -a.e.) on X.  Then we give the following notations throughout this paper: We denote by where y ∈ AC(J, R), that is, y is a absolutely continuous function. If we integrate y ∈ AC(J, R) by parts, we can get V 1,q (J, R) ⊂ H 1,q (J, R). In fact, sets V 1,q (J, R) and H 1,q (J, R) are equivalent for a class of functions.
1, and y : T i → R, i = 1, 2. We say that y ∈ H 1,q (T i , R) if and only if y ∈ L q (T 0 i , R), and there exists g : where J = T 1 ∪ T 2 .
For q ∈ R, q 1, we set the space   [17].) Let X be a real Banach space, and let Ψ ∈ C 1 (X, R) satisfies (PS) condition and the following conditions: Then there exists a critical value c α, which can be characterized as where Γ = y y ∈ C [0, 1], X , y(0) = 0, y(1) = r .

Lemma 11 [Fountain theorem].
(See [6].) (i) The compact group G acts isometrically on the Banach space X = ⊕ j∈N X j and is invariant; there exists a finite dimensional space V such that, for every j ∈ N, X j V ; and the action of G on V is admissible.
Under assumption (i), let Ψ ∈ C 1 (X, R) be an invariant functional and satisfies (PS) condition. Let for every k ∈ N, there exits ρ k > r k > 0 such that (ii) a k := max y∈Y k : y =ρ k Ψ (y) 0 for Y k = ⊕ k j=0 X j ; (iii) b k := inf y∈Z k : y =r k Ψ (y) → ∞, k → ∞, for Z k = ⊕ ∞ j=k X j . Then Ψ has an unbounded sequence of critical values. (1) (1) (1) For this superlinear damped three-point boundary problem (1), the variational structure due to the presence of the damped term g(s)y (σ(s)) is not apparent. However, we will be able to transform it into a variational formulation.

Variational formulation of problem
We assume that λ > −λ 1 m/M , where λ 1 is the first eigenvalue of system (3) for s ∈ J and 0 ω y or y ω 0.

Remark 2.
Since the deformation theorem is still valid under the Cerami condition, we see that the mountain pass theorem and the fountain theorem are true under the Cerami condition, more details can be found in [11,18]. After integrating the above formula, we get Thus the norm · 1 is equivalent to · 2 . Then the norm · is equivalent to · 1 . Besides, there is constant c > 0 such that We can assume that δ = (c|b − a|)/m > 0, then the proof is completed.
Lemma 16. If y ∈ H 1,2 (J, R) is a critical point of the functional Ψ , then y = y(s) is a solution of problem (1).
Proof. By Lemma 15 the functional Ψ is continuously differentiable, and the assumption that y is a critical point of Ψ means that Ψ (y), ω = 0 for all ω ∈ H 1 . Obviously, there is Proof. By the functional Ψ defined in (5) let Since Ψ 1 is continuous and convex, Ψ 1 is weakly lower semicontinuous, and By condition (A), Ψ 2 is a weakly continuous functional. Thus, Ψ is weakly lower semicontinuous. For any y ∈ H 1 , there exists a enough large constant c > 0 such that This implies that lim y →∞ Ψ (y) = +∞, that is, Ψ is coercive. Hence Ψ has minimum by Lemma 9, which is also the critical point of Ψ . Therefore, problem (1) has at last one solution. Proof. First, suppose a subsequence {y i } of sequence {y n } such that y i y in H 1 , then y i → y in C(J, R). Thus when i → ∞, there are That is, sequence {y n } has a convergent subsequence. Second, we show that the sequence {y n } ⊂ H 1 is bounded. If {y n } is unbounded, then for some c ∈ R, we have Ψ (y n ) → c, y n → ∞, Ψ (y n ) · y n → 0 as n → ∞, We consider v n := y n / y n , then up to the subsequence {v n } in For any m > 0, definev n := (4k) 1/2 v n . There isv n → 0 in L 2 (J), and by condition (A2) there exists D > 0 such that |H(s, y)| D(|y| + |y| p ). We see H(·,v n ) → 0 in L 1 (J). Thus lim n→∞ J H(s,v n ) ∆s = 0. So for n large enough, which implies that lim n→∞ Ψ (t n y n ) = +∞. Then by Ψ (0) = 0 and Ψ (y n ) → c get t n ∈ (0, 1). Thus, if n big enough, then J e G(s) (t n y n ) (s) 2 + λ t n y σ n (s) then there is h(σ(s), y σ n (s))y σ n (s) |y n | 2 |v n | 2 ∆s. h(σ(s), y σ n (s))y σ n (s) |y n | 2 |v n | 2 ∆s → +∞ as n → ∞, and there exists ϑ > −∞ such that h(σ(s), y σ n (s))y σ n (s) |y n | 2 ϑ for -a.e. s ∈ J.
Remark 3. The technology we used in Theorem 2 to eliminate the case v = 0, is derived from Jeanjean [10]. We proved that although there may be unbounded (PS) sequence, every Cerami sequence of the functional Ψ is bounded. To prove the boundedness of Cerami sequence, we refer to Zou [29].
We choose to define X j := Rτ j as j ∈ N, and let X = ⊕ j∈N X j , Y k = ⊕ k j=0 X j , and Z k = ⊕ ∞ j=k X j .
Thus by Lemma 11 there is the unbounded sequence {y k } such that Ψ (y k ) → +∞ as k → ∞. The proof is completed.  where h(s, y) = 2|y| 2 y log(1 + s 2 |y|) and λ > −λ 1 e −3/8 . Then all assumptions in Theorem 4 are fulfilled, and we can obtain the primitive function of h, but it is almost impossible to check (AR) condition.