Nonhomogeneous nonlinear oscillator with damping: asymptotic analysis in continuous and discrete time

where A is a maximal monotone and α-inverse strongly monotone operator in a real Hilbert space H. With suitable assumptions on γ and f (t)we show that A−1(0) = ̸ ∅, if and only if (SEE) has a bounded solution and in this casewe provide approximation results for elements of A−1(0) by provingweak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A−1(0). As a discrete version of (SEE), we consider the following second order difference equation


Introduction
Let H be a real Hilbert space with scalar product ⟨·, ·⟩, norm ‖ · ‖. We denote weak convergence in H by ⇀ and strong convergence by →. An operator A : D(A) ⊂ H → H (possibly multivalued) is called monotone (respectively strongly monotone) if ⟨y 2 − y 1 , x 2 − x 1 ⟩ ≥ 0, (respectively ⟨y 2 − y 1 , x 2 − x 1 ⟩ ≥ α‖x 2 − x 1 ‖ 2 for some α > 0) for all x i ∈ D(A), y i ∈ A(x i ), for i = 1, 2. A monotone operator A is maximal if R(I + A) = H, where I is the identity operator on H. For α > 0, we call a maximal monotone operator A : H → H α-inverse strongly monotone, if for all x and y in H, we have α‖Ax − Ay‖ 2 ≤ ⟨Ax − Ay, x − y⟩.
Obviously, every α-inverse strongly monotone operator is single-valued and Lipschitz with Lipschitz constant 1 α . We denote the first and the second order derivatives of a curve u respectively byu andü. By introducing the notion of almost nonexpansive sequences and curves in H, the asymptotic behavior of solutions to the first order evolution equation where A is a maximal monotone operator in H, was studied by Djafari Rouhani [1,2] without assuming the zero set of A to be nonempty. The asymptotic behavior of the dynamical system: u(0) = u 0 , sup t≥0 ‖u(t)‖ < +∞, was studied by Djafari Rouhani and Khatibzadeh [3][4][5], where A is a maximal monotone operator in H and ∈ R. In [6], Alvarez studied the initial value problem {︃ü (t) +u(t) + ∇ϕu(t) = 0, where > 0 and ϕ : H → R is differentiable. This system is called the Heavy Ball with Friction, (HBF) for short, which is a nonlinear oscillator with damping. Assuming that ϕ is convex and Argminϕ ≠ ∅, he proved that each trajectory of the system (HBF) converges weakly to some minimum point of ϕ, see [6,7]. In 2011, Attouch and Maingé [8] considered (HBF) when ∇ϕ is replaced by A = ∇ϕ + B, where ϕ : H → R is a convex and continuously differentiable function and B is a maximal monotone operator which is also α-inverse strongly monotone for some α > 0 with α 2 > 1. Assuming A −1 (0) ≠ ∅, they obtained the weak convergence of the solution to some point in A −1 (0). In the second section of this paper motivated by our previous results [1][2][3][4][5] and by the results in [6][7][8], we study the asymptotic behavior of the solutions to the following second order evolution equation where A is an α-inverse strongly monotone operator. More precisely, we show weak and strong convergence of u(t) to an element of A −1 (0), which is also the asymptotic center of u(t), if and only if sup t≥0 ‖u(t)‖ < +∞, therefore showing that A −1 (0) is nonempty if and only if sup t≥0 ‖u(t)‖ < +∞. We note that in [6][7][8], the existence of bounded solutions was proved by assuming that A −1 (0) is nonempty. Here we show that this assumption follows from the existence of bounded solutions. In Section 3, we consider the discrete version of (1.1), which was studied by Alvarez and Attouch [9] for the homogeneous case, and leads to the following inexact inertial proximal algorithm where A is a general maximal monotone operator. We prove ergodic, as well as convergence theorems for bounded solutions to (1.2) without assuming A −1 (0) ≠ ∅, and provide an answer to the open problem raised in [9]. Definition 1.1. Given a bounded curve u(t) in H, the asymptotic center c of u(t) is defined as follows (see [10]): for every q ∈ H, let ϕ(q) = lim sup t→+∞ ‖u(t)q‖ 2 . Then ϕ is a continuous and strictly convex function on H, satisfying ϕ(q) → +∞ as ‖q‖ → +∞. Therefore ϕ achieves its minimum on H at a unique point c, called the asymptotic center of the curve u(t).
For a bounded sequence un in H, its asymptotic center is defined in a similar way.
The following theorem establishes the weak convergence of the solutions to (1.1).
Since A is Lipschitz, Au(t) is bounded, and thereforeu is bounded too. Let Since u(t) is bounded, w(t) has a weak cluster point, say p. Let t k be a sequence such that t k → +∞ and w(t k ) ⇀ p as k → +∞. By the α-inverse strong monotonicity of A we have Integrating both sides of the inequality with respect to s on the interval [0, t k ], and then dividing by t k , we get: where M 1 := sup t≥0 ‖u(t)‖. Taking the lim sup as k → +∞ in the above inequality, we get: On the other hand Now, since α 2 > 1, f ∈ L 1 , and u andu are bounded, by taking the lim sup as k → +∞, we get lim sup Hence Multiplying (1.1) by u(t) − p and using (2.1), we geẗ where hp(t) := 1 2 ‖u(t) − p‖ 2 . Substituting from (1.1), yields Integrating the above inequality on the interval [0, t], we get: where C is some constant. The above inequality implies thatu,ü − f ∈ L 2 (0, +∞; H), which yields Au ∈ L 2 (0, +∞; H). Since A, as well as u are Lipschitz (becauseu is bounded), we deduce that Au is Lipschitz too. Therefore Au(t) → 0 as t → +∞. Using the maximality of A, we conclude that every weak cluster point of u(t) is in A −1 (0). Hence A −1 (0) ≠ ∅. Now, let q be a weak cluster point of u(t). Multiplying (1.1) by u(t) − q and using the monotonicity of A, we getḧ where hq(t) := 1 2 ‖u(t)−q‖ 2 and M 3 := sup t≥0 ‖u(t)−q‖. Applying Lemma 2.1 yields that lim t→+∞ ‖u(t)−q‖ 2 exists for each weak cluster point q of u(t). Let q 1 and q 2 be two weak cluster points of u(t). Then lim t→+∞ (‖u(t)− q 1 ‖ 2 − ‖u(t) − q 2 ‖ 2 ) exists. It follows that lim t→+∞ ⟨u(t), q 1 q 2 ⟩ exists. Therefore ⟨q 1 , q 1 − q 2 ⟩ = ⟨q 2 , q 1 − q 2 ⟩, and so q 1 = q 2 . This shows that u(t) converges weakly to some point in A −1 (0). Let u(t) ⇀ q. For an arbitrary x ∈ H, we have The above identity shows that q is the asymptotic center of u(t).
In the following theorem we provide a sufficient condition for the strong convergence of the solutions to (1.1). Theorem 2.3. Let A be an α-inverse strongly monotone and β-strongly monotone operator. If f ∈ L 1 (0, +∞; H), α 2 > 1 and u(t) satisfy (1.1), then u(t) converges strongly to some q ∈ A −1 (0), which is also the asymptotic center of u(t), if and only if u(t) is bounded.
Remark 2.4. We note that our assumptions on the operator A contains the case where A = I − T, with T nonexpansive (that is ‖Tx − Ty‖ ≤ ‖x − y‖, for every x, y ∈ H).

Inexact inertial proximal method
In [9], Alvarez and Attouch studied the iterative method where A is a general maximal monotone operator. Assuming that A −1 (0) ≠ ∅, and with the following conditions on the parameters (i) ∃λ > 0 such that ∀n ∈ N, λn ≥ λ, (ii) ∃α ∈ [0, 1) such that ∀n ∈ N, 0 ≤ αn ≤ α, (iii) ∑︀ +∞ n=1 αn‖u k − u k−1 ‖ 2 < +∞, they obtained the weak convergence of un. At the end of their paper, they raised the open problem to find convergence results for the inexact inertial proximal method, as well as to develop a general theory to guide the choices of the parameters λn and αn. In this section, we consider the following discrete counterpart of (1.1) which is also the inexact version of (3.1) u n+1 − un − αn(un − u n−1 ) + λn Au n+1 ∋ fn , (3.2) where A is a general maximal monotone operator. Motivated by [11], we note that by taking αn(un − u n−1 ) + fn as the nonhomogeneous term, the inexact inertial proximal algorithm (3.2) is reduced to the inexact proximal point algorithm where gn := αn(un − u n−1 ) + fn. Therefore, if un given by (3.2) is bounded, then by using the results in [11], we prove ergodic, as well as convergence theorems for the sequence un satisfying (3.2), and provide an answer to the open problem raised in [9], without assuming A −1 (0) ≠ ∅. Set We denote by ωw(wn) the set of all weak cluster points of the sequence wn. First, we state the following lemma from classical analysis. The following is a weak ergodic theorem for the sequence un.
Proof. By the monotonicity of A, for all k, n ∈ N, we have Multiplying both sides of the above inequality by λ k λn, then substituting λn Au n+1 and λ k Au k+1 from (3.2), we get: Since un is bounded, so is wn, and therefore ωw(wn) ≠ ∅. Let p ∈ ωw(wn). Then there exists a subsequence m j such that wm j ⇀ p. Summing up both sides of the above inequality from k = 1 to m j , dividing by ∑︀ m j k=1 λ k , and then taking the limit as j → +∞, we get: It follows that: Applying Lemma 3.1, we get limn→+∞ ‖un −p‖ exists. Hence for any p, q ∈ ωw(wn), limn→+∞ 1 2 (‖un−p‖ 2 −‖un− q‖ 2 ) and therefore limn→+∞⟨un , p−q⟩ exist. This implies that ⟨q, p−q⟩ = ⟨p, p−q⟩, and hence p = q. Therefore wn ⇀ p. Now, we are going to show that p ∈ A −1 (0), which implicitly implies that A −1 (0) is nonempty. For this, let x ∈ D(A) and y ∈ Ax. Then Letting n → +∞, this yields ⟨y, x − p⟩ ≥ 0. Now the maximality of A implies that p ∈ A −1 (0). To prove that p is the asymptotic center of the sequence un, choose an arbitrary element x ∈ H. Then we have Multiplying the above identity by λn, summing up from n = 1 to n = N, dividing by ∑︀ N n=1 λn, and then letting N → +∞, we get: if x ≠ p, and this concludes the result. Remark 3.3. By [12,Remark 14], the weak (resp.strong) convergence of un given by (3.3) (and so un given by (3.2)) in the case gn(= αn(un − u n−1 ) + fn) ≡ 0 implies the weak (resp. strong) convergence of un in the case where gn ≠ 0, provided that ∑︀ +∞ n=1 ‖gn‖ < +∞. Therefore for the study of weak (resp. strong) convergence of un given by (3.2), we may assume without loss of generality that αn(un − u n−1 ) + fn ≡ 0. We note that in this case the inexact inertial proximal algorithm (3.2) is reduced to the proximal point algorithm.
Hence limn→+∞ ‖Aun‖ = 0. Assume that un j ⇀ q. By the monotonicity of A, we have Letting j → +∞ in the above inequality, we get Substituting Au n+1 from (3.3) with gn ≡ 0, yields Therefore limn→+∞ ‖un − q‖ 2 exists. Now the result follows by applying the same argument as in the proof of Theorem 3.2.
In the next two theorems we give sufficient conditions for the strong convergence of un.

Theorem 3.5. Assume that (I + A) −1 is compact, and the conditions (i) and (ii) in Theorem 3.4 hold.
Then the sequence un given by (3.2) converges strongly to an element p ∈ A −1 (0), which is also the asymptotic center of un, if and only lim infn→+∞ ‖wn‖ < +∞.
Proof. Necessity is clear. To prove the sufficiency by Remark 3.3, it is enough to consider the case αn(un − u n−1 ) + fn ≡ 0. The proof of Theorem 3.4 implies that limn→+∞ ‖Aun‖ = 0, and un ⇀ p. Therefore (I + A)un is bounded. Since (I + A) −1 is compact, then un contains a strongly convergent subsequence to some p ∈ H, say un j . From the monotonicity of A, we have ⟨Aun − Aun j , un − un j ⟩ ≥ 0, and therefore by letting j → +∞, we get ⟨Aun , un − p⟩ ≥ 0.
Now the same argument as in the proof of Theorem 3.4 yields that limn→+∞ ‖un − p‖ 2 exists, which concludes the proof.

Theorem 3.6. Suppose that A is strongly monotone and the conditions (i) and (ii) in Theorem 3.2 hold.
Then the sequence un given by (3.2) converges strongly to an element p ∈ A −1 (0), which is also the asymptotic center of un, if and only if lim infn→+∞ ‖wn‖ < +∞.
Remark 3.7. Assuming that αn , fn ∈ l 1 , the condition (ii) in Theorems 3.2 and 3.4 is satisfied, if un is bounded.

Conclusions
In this paper, we studied the weak and strong convergence to a zero of the operator for the solutions to the nonlinear oscillator with damping with a maximal monotone and inverse strongly monotone operator A, without assuming the zero set of A to be nonempty. In particular, we showed that the zero set of A is nonempty if and only if bounded solutions exist. We also studied and proved similar results for the asymptotic behavior of the inexact inertial proximal algorithm obtained by its discretization, where the operator A is only assumed to be maximal monotone, without assuming its zero set to be nonempty, therefore extending and solving an open problem raised in [9]. As a future direction for research, it might be interesting to explore and extend these ideas to more general operators and settings, and with nonconstant damping.