Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems

We consider the regularized Tikhonov-like dynamical equilibrium problem: find \begin{document} $u: [0, +∞ [\to\mathcal H$ \end{document} such that for a.e. \begin{document} $t \ge 0$ \end{document} and every \begin{document} $y∈K$ \end{document} , \begin{document} $\langle \dot{u}(t), y-u(t)\rangle +F(u(t), y)+\varepsilon(t) \langle u(t), y-u(t)\rangle \ge 0$ \end{document} , where \begin{document} $F:K×K \to \mathbb{R}$ \end{document} is a monotone bifunction, \begin{document} $K$ \end{document} is a closed convex set in Hilbert space \begin{document} $\mathcal H$ \end{document} and the control function \begin{document} $\varepsilon(t)$ \end{document} is assumed to tend to 0 as \begin{document} $t \to +∞$ \end{document} . We first establish that the corresponding Cauchy problem admits a unique absolutely continuous solution. Under the hypothesis that \begin{document} $\int_{0}^{+∞} \varepsilon (t) dt , we obtain weak ergodic convergence of \begin{document} $u(t)$ \end{document} to \begin{document} $x∈K$ \end{document} solution of the following equilibrium problem \begin{document} $F(x, y) \ge 0, \;\forall y∈K$ \end{document} . If in addition the bifunction is assumed demipositive, we show weak convergence of \begin{document} $u(t)$ \end{document} to the same solution. By using a slow control \begin{document} $\int_{0}^{+∞} \varepsilon (t) dt = ∞$ \end{document} and assuming that the bifunction \begin{document} $F$ \end{document} is 3-monotone, we show that the term \begin{document} $\varepsilon (t)u(t)$ \end{document} asymptotically acts as a Tikhonov regularization, which forces all the trajectories to converge strongly towards the element of minimal norm of the closed convex set of equilibrium points of \begin{document} $F$ \end{document} . Also, in the case where \begin{document} $\varepsilon $ \end{document} has a slow control property and \begin{document} $\int_{0}^{+∞}\vert \dot{\varepsilon} (t) \vert dt , we show that the strong convergence property of \begin{document} $u(t)$ \end{document} is satisfied. As applications, we propose a dynamical system to solve saddle-point problem and a neural dynamical model to handle a convex programming problem. In the last section, we propose two Tikhonov regularization methods for the proximal algorithm. We firstly use the prox-penalization algorithm \begin{document} $(ProxPA)$ \end{document} by iteration \begin{document} $ x_{n+1} = J^{F_n}_{λ_n}(x_n)$ \end{document} where \begin{document} $F_n(x, y) = F(x, y)+\varepsilon_n \langle x, y-x\rangle$ \end{document} , and \begin{document} $\varepsilon_n$ \end{document} is the Liapunov parameter; afterwards, we propose the descent-proximal (forward-backward) algorithm \begin{document} $(DProxA)$ \end{document} : \begin{document} $x_{n+1} = J^F_{λ_n} ((1 - λ_n\varepsilon_n)x_n)$ \end{document} . We provide low conditions that guarantee a strong convergence of these algorithms to least norm element of the set of equilibrium points.


(Communicated by Hedy Attouch)
Abstract. We consider the regularized Tikhonov-like dynamical equilibrium problem: find u : [0, +∞[→ H such that for a.e. t ≥ 0 and every y ∈ K, u(t), y − u(t) + F (u(t), y) + ε(t) u(t), y − u(t) ≥ 0, where F : K × K → R is a monotone bifunction, K is a closed convex set in Hilbert space H and the control function ε(t) is assumed to tend to 0 as t → +∞. We first establish that the corresponding Cauchy problem admits a unique absolutely continuous solution. Under the hypothesis that +∞ 0 ε(t)dt < ∞, we obtain weak ergodic convergence of u(t) to x ∈ K solution of the following equilibrium problem F (x, y) ≥ 0, ∀y ∈ K. If in addition the bifunction is assumed demipositive, we show weak convergence of u(t) to the same solution. By using a slow control +∞ 0 ε(t)dt = ∞ and assuming that the bifunction F is 3-monotone, we show that the term ε(t)u(t) asymptotically acts as a Tikhonov regularization, which forces all the trajectories to converge strongly towards the element of minimal norm of the closed convex set of equilibrium points of F . Also, in the case where ε has a slow control property and +∞ 0 |ε(t)|dt < +∞, we show that the strong convergence property of u(t) is satisfied. As applications, we propose a dynamical system to solve saddle-point problem and a neural dynamical model to handle a convex programming problem. In the last section, we propose two Tikhonov regularization methods for the proximal algorithm. We firstly use the prox-penalization algorithm (P roxP A) by iteration x n+1 = J Fn λn (xn) where Fn(x, y) = F (x, y) + εn x, y − x , and εn is the Liapunov parameter; afterwards, we propose the descent-proximal (forward-backward) algorithm (DP roxA): x n+1 = J F λn ((1 − λnεn)xn). We provide low conditions that guarantee a strong convergence of these algorithms to least norm element of the set of equilibrium points.
1. Introduction. Throughout the paper, H is a real Hilbert space which is endowed with the scalar product ·, · , and the norm x = x, x for x ∈ H. Let K be a nonempty closed convex subset of H and F : K × K → R be a bifunction satisfying F (x, x) = 0, for every x ∈ K. Recall that an equilibrium problem in the sense of Blum-Oettli [6] is the following Ky Fan minimax inequality [20]: Findx ∈ K such that F (x, y) 0, ∀y ∈ K.
(EP ) Many authors had contributed to the study of existence for equilibrium problems when K is assumed to be compact or some coerciveness assumptions are imposed on the bifunction F , we refer to [20,6,12,10,11,1] and the bibliography therein. We denote by S F the solution set of (EP ), which is supposed nonempty and not necessarily reduced to a single point. To explain the interest in equilibrium problems, we can consider a number of particular cases, both classical and recently discovered, as optimization problems (minimization and saddle point), Nash equilibria, fixed points and variational inequalities (see [6,12]). The interest of the equilibrium problem is that it unifies all above mentioned problems in a convenient way. Moreover, many methods devoted for solving one of these problems can be extended, with suitable modifications, to an equilibrium problem. The Tikhonov regularization method [35] is a powerful tool in convex optimization to handle discrete or continuous ill-posed problems. It has been recently applied to equilibrium problems (see [27,17,29,18,23]). In this paper, we deal with Tikhonov regularization methods for dynamic monotone equilibrium problems. The basic idea of this method is to approach the bifunction of the following evolution differential equilibrium problem:    find u ∈ C 0 ([0, +∞[; H) such that for a.e. t ≥ 0, u(t), y − u(t) + F (u(t), y) ≥ 0, ∀y ∈ K, u(t) ∈ K for each t ≥ 0, (DEP ) by a family of strongly monotone bifunctions F ε (u, v) := F (u, v) + ε u, v − u depending on a regularization real parameter ε > 0 to obtain a perturbed evolution equilibrium problem (DEP ε ).
The resulting regularized evolution problem, see [15], has a unique solution u ∈ C 0 ([0, +∞[; H). We show in Lemma 5.1 that this solution converges strongly, while t goes to +∞, to the unique solution x ε (that depends on the regularization parameter ε) of the following equilibrium problem: find x ε ∈ K such that F (x ε , y) + ε x ε , y − x ε 0, ∀y ∈ K.
In the preliminaries section, we set up the abstract formulation of the resolvent and the associated Yosida approximate for the bifunction F ε and then summarize and prove some basic associate properties which are useful for further discussion. Section 3 is devoted to existence and uniqueness of a strong solution for the evolution equilibrium problem (DEP ε(·) ) with an initial condition, i.e., an absolutely continuous function u : [0, +∞[→ H, such that (DEP ε(·) ) holds for almost every t ≥ 0 and u(0) = u 0 .
In section 4, we discuss an introduced kind of rate of convergence for values of the dynamic equilibrium problems (DEP ) and (DEP ε(·) ). More precisely, under suitable conditions in Theorems 4.2, 4.3, we obtain F (z, u(t)) = • 1 t near the infinite value for t, and moreover lim t→+∞ tε(t) u(t) 2 = 0.
In the main section of this paper, we establish that the asymptotic convergence properties of trajectories for the dynamical equilibrium problem (DEP ε(·) ) depend on whether ε(·) is in L 1 ([0, +∞[) or not. For ε(·) ∈ L 1 ([0, +∞[), the authors in [9,33] established that the following regularized dynamical monotone inclusion: reinforce the convergence properties of the initial problemu(t) ∈ Au(t). This result remains true for our problem (DEP ε(·) ). In Theorem 5.4, we establish the weak ergodic convergence of u(t) to some x ∞ ∈ S for general maximal monotone bifunctions. In Theorem 5.7, we prove weak convergence of u(t) to some x ∞ ∈ S, when more generally the monotone bifunction F is only assumed to be demipositive. When ε(·) / ∈ L 1 ([0, +∞[), the strong convergence of the unique solution for the inclusion (1) to the least norm element of A −1 (0) were discussed in [8,34] under the additional condition ε(·) is nonincreasing to 0. We also refer to [16], where this strong convergence still holds without assuming ε(·) to be nonincreasing. In Theorem 5.10 of this paper, we prove strong convergence of u : [0, +∞[→ H, the unique solution of (DEP ε(·) ), to the least norm element of S F , as t → +∞, when moreover F is 3-monotone. In Theorem 5.15, this strong convergence of u(t) remains true for general monotone bifunctions when moreover +∞ 0 |ε(t)|dt < +∞. Using [3] we establish the link between solutions of (DEP ε(·) ) and those of the following evolution problem: where β tends to +∞ as t goes to +∞. Then, according to Theorems 5.10 and 5.15 we prove, under corresponding assumptions on β, i.e., lim t→+∞ β(t) = +∞ and +∞ 0 |β(t)| β(t) 2 dt < +∞, strong convergence of the solution of (DEP β(·) ) to the least norm element of S F whenever t goes to +∞, see Theorem 6.2. The remaining part of this section deals with a comparison of Theorem 6.2 and existing results in [3,4] and [13] where the Fitzpatrick transform, adapted for bifunctions in [2], is used and uniform control condition involving both F and S F has been imposed, see (31) and (32).
Finally, in the last section, we provide the time discretization of this system (DEP ε(·) ). As we can see in [25], the authors proposed an approximation method which combines the Tikhonov method with the proximal point algorithm for a maximal monotone operator. This technique can be used in the case of a monotone bifunction to prove the strong convergence to least norm element of S F . We firstly approximate the bifunction F by the Tikhonov parametrized family of bifunctions where x, y ∈ K and {ε n } is a positive sequence. Observe that an implicit discretization of the dynamical system (DEP ε(·) ) gives, for x 0 ∈ K, {λ n } and {ε n } are two positive sequences: (ProxPA) which can be rewritten as the prox-penalization algorithm x n+1 = J Fn λn (x n ), where J Fn λn is the proximal point associated to the perturbed bifunction F n . For many applications, however, evaluating the resolvent J Fn λn of the sum bifunction F n is much harder than evaluating the resolvent of F and G n (x, y) := ε n x, y − x individually. In fact, we propose the -proximal (forward-backward) algorithm: which can be rewritten as the forward-backward algorithm x n+1 = J F λn ((1 − λ n ε n ) x n ). When ∞ n=0 λ n ε n = ∞, we give conditions (more general than those required in [25] for the operators), to ensure strong convergence of the generated sequence {x n } by algorithms (ProxPA) and (DProxA) to the least norm element of the solutions set S F .

2.
Resolvent and Yosida approximate of monotone bifunctions. We assume that K is a non empty closed convex subset of H, and F : K × K → R satisfies the following usual assumptions: (H 2 ) F is a monotone bifunction, i.e.: F (x, y) + F (y, x) ≤ 0 ∀x, y ∈ K; (H 3 ) For any y ∈ K, x → F (x, y) is upper hemicontinuous, i.e., upper semicontinuous on each line segment of K; (H 4 ) for each x ∈ K, y → F (x, y) is convex and lower semicontinuous.
Lemma 2.1. [14] Suppose that F : K × K → R satisfies (H 1 ) − (H 4 ). Then, for each x ∈ H and λ > 0, there exists a unique z λ = J F λ (x) ∈ K, called the resolvent of F at x such that Moreover, x is an equilibrium point of Proof. The proof of the first step (see [12]) is based upon the generalized KKM-Fan's lemma since F , satisfying (H 1 ) − (H 4 ), is a maximal monotone bifunction. The second step relies on (2). Lemma 2.2. Suppose that F satisfies (H 1 ) − (H 4 ), and set, for ε > 0, F ε (x, y) = F (x, y) + ε x, y − x . Then for all λ > 0 and for all x, y ∈ H, and if x ε is the unique equilibrium point of F ε , then x ε = J Fε λ (x ε ) = J F 1/ε (0).
Proof. By characterization of the resolvent mapping J F λ , we have for each x, y ∈ H λF (J Fε By summing these two inequalities and using monotonicity of F , we obtain . which leads to (3).
If x ε is the equilibrium point of F ε , then x ε = J Fε λ (x ε ), which means: (2) for y = z can be expressed as Then summing and using monotonicity of F we get −λ J F λ (x) − z 2 ≥ 0 and thus J F λ (x) = z. Definition 2.4. Let F : K × K → R. For each λ > 0, the associated Yosida λ-approximate to F over K is defined, by B F λ := 1 λ (I − J F λ ). Remark 1. [12,28,15] For each λ > 0, we have the resolvent J F λ is firmly nonexpansive, namely As consequence, the Yosida λ-approximate is 1 λ -Lipschitz continuous, that is The following lemmas play an important role in the analysis of existence of solutions of (DEP ε(·) ). Lemma 2.5. Suppose that F satisfies (H 1 ) − (H 4 ), and set for ε ≥ 0 and x, y ∈ K, Then for each λ > 0 and x ∈ K we have B Fε λ x ≤ z x,ε . Proof. Fix ε ≥ 0, x ∈ K and λ > 0, there exists z x,ε ∈ H satisfying (5). Also, by Lemma 2.1, we have By setting y = J Fε λ x in (5) (respectively y = x in (6)), summing and using monotonicity of F ε , we get Lemma 2.6. Suppose F satisfies conditions (H 1 ) − (H 4 ). Then, for every x, y ∈ K, λ, β > 0 and ε, ε ≥ 0, we have We refer to [15,Lemma 2.5] for the assertion (i). For (ii), we use the associated λ-Yosida approximate to F ε to have, for each ε, ε ≥ 0 and each By adding the two last inequalities and using monotonicity of F , we obtain Also using the definition of J Fε λ x and J Fε β x, and summing the associated two inequalities, the desired relation is obtained.
3. Existence and uniqueness result for Tikhonov regularized dynamical equilibrium problem. Our main theorem on the existence and uniqueness for the Tikhonov regularized dynamical equilibrium problem (DEP ε(·) ) will exploit the following results.
Lemma 3.2. Assume that K is a nonempty, closed and convex subset of H and J : [0, +∞[×K → K satisfies Lipschitz condition: Then, for every λ > 0 and every u 0 ∈ K, the following differential equation: admits a unique solution u λ : [0, +∞[→ H, which is an absolutely continuous function, i.e., for each t > 0, we have Proof. This Lemma is a direct consequence of the Cauchy-Lipschitz-Picard Theorem (see [7,Theorem I.4]).
λ is Lipschitz continuous, and according to Lemma 3.2, the following differential equation admits a unique absolutely continuous solution, denoted by u λ , i.e., satisfying: for each t > 0, Step 1. We first prove, for each T > 0, the net {u λ : 0 < λ ≤ λ 0 } is Cauchy in C 0 ([0, T ], H), as λ → 0. For this, let t ∈ (0, T ), then for h > 0 sufficiently small, and by setting t and t + h in (9), we have : By using Lemma 2.6 (i), we get: So by integrating on [0, t] and using Lemma 2.6 (ii), we deduce Dividing by h and letting h → 0, we get By integrating on [0, t] the norm of relation (10), we obtain and then, we get By assumption (5) and Lemma 2.5, there exists z u0 ∈ H, such that B F0 λ u 0 ≤ z u0 , and so u λ (0) = ≤ z u0 which leads to We get By writing u λ = λB Ft λ u λ + J Ft λ u λ and likewise for u β , we get, for a.
Step 2. Strong convergence of {u λ } to a solution u of (DEP ε(·) ), as λ → 0. Since Return to equation (9), we have for every and by using (2), we obtain for a.e. t > 0 and all y ∈ K Since F is monotone and F (y, .) is convex and lower semicontinuous for every y ∈ K, passing to the limit in (14), we obtain By Lemma 3.1, we finally get for a.e. t > 0 and then u is a solution (DEP ε(·) ).

4.
Rate of convergence of the optimal values of the dynamic equilibrium systems. In this section, we discuss the rate of convergence of values of the dynamic equilibrium problems (DEP ) and (DEP ε(·) ). To prove our results, we need the following lemma. Proof. We have +∞ 0 ϕ(s)ds < +∞, which implies that lim inf t→+∞ ϕ(t) = 0, and since ϕ is nonincreasing, then lim t→+∞ ϕ(t) = 0.
Theorem 4.2. Suppose that F : K ×K → R satisfies the following condition: there exists δ ≥ 0 and α > 1 such that Then, for each solution u of (DEP ) and each x ∈ S F , we have Proof. Fix x ∈ S F and, for t ≥ 0, set ϕ(t) = F (x, u(t)). By using assumption (17), and u is solution to (DEP ), we have for each h ≥ 0 and a.e. t ≥ 0, Dividing this inequality by h (positive and negative) and letting h −→ 0, we obtain which means also that ϕ is nonincreasing. Return to (DEP ), we have Therefore, Lemma 4.1 ensures (17). Let u be a solution of (DEP ε(·) ), where ε is nonincreasing and satisfying +∞ 0 Proof. By assumption (17), since u is a solution of (DEP ε(·) ), we have for a.e. t > 0 each h > 0 and each By setting ϕ(t) = F (x, u(t)) and dividing this inequality by h and letting h −→ 0, we get Since ε is nonincreasing, then and therefore d dt We conclude that the real function ϕ + ε u 2 is nonincreasing. Return to u solution of (DEP ε(·) ), then After integrating, since +∞ 0 which yields the results.

Remark 3.
• In the first example of bifunctions satisfying condition (17), we consider the bifunction F (x, y) T maps K into H and f : H → R is a subadditive real function on H.
We have for x, y, z ∈ K, and then condition (17) is satisfied for δ = 0.
5. Asymptotic behavior of solutions. We assume that the equilibrium points set S F = {x ∈ K : F (x, y) ≥ 0, ∀y ∈ K} is nonempty and we denote by x * the least norm element of S F , i.e. x * = P S F (0) is the orthogonal projection of the point 0 onto S F . In the case where ε is a constant parameter, we have Lemma 5.1. For any fixed ε > 0 the perturbed bifunction F ε (x, y) = F (x, y) + ε x, y −x is strongly monotone and each solution u ε (·) of the dynamical equilibrium problem (DEP ε ) converges strongly, as t → +∞, to the unique equilibrium point x ε of (EP ε ).
Proof. The first assertion is justified as in Lemma 2.1, so for any fixed ε > 0, the equilibrium problem (EP ε ) has a unique equilibrium point x ε . If we suppose that u ε (t) is a solution of (DEP ε ) with u ε (0) = u 0 , we have and then d dt Integration of this differential inequality gives u ε (t) − x ε 2 ≤ u 0 − x ε 2 e −εt , and therefore u ε (t) converges strongly to x ε , when t → +∞.
Afterwards, see Lemma 5.8, when passing to the limit as the parameter ε tends to zero, the unique equilibrium point x ε converges strongly to the element of minimum norm of the closed convex nonempty set S F .
To move to the least norm solution x * of S F in a single step, we consider the dynamic perturbation parameter setting (DEP ε(·) ). We will see that the convergence depends whether ε(t) is in L 1 (]0, +∞[) or not.
In this section, when ε ∈ L 1 (R + ), we study the weak ergodic convergence of general maximal monotone bifunctions.

Weak ergodic convergence of general maximal monotone bifunctions.
At first, we give some lemmas which we need in the sequel.
Lemma 5.2 (Jensen's inequality ). Let H be a Hilbert space and K be a nonempty closed convex set. Let u : [0, +∞[ → K be an integrable function and let f : K → R be convex and lower semi-continuous. If f • u is integrable, then for any t > 0, ii) If t n → +∞ and U (t n ) U ∞ weakly in H, then U ∞ ∈ S. Then, the whole weak limit w − lim t→+∞ U (t) exists and belongs to S.
For Condition (i), fix z ∈ S F and set, for every t ≥ 0, θ(t) = 1 2 u(t) − z 2 . By monotonicity of F we have : from which, it follows that Thus the real function θ(t) − z 2 2 t 0 ε(s)ds is nonencreasing and hence converges to a limit as t → +∞. Since +∞ 0 ε(t)dt < ∞, we conclude that θ(t) has a limit as t → +∞, and then lim t→+∞ z − u(t) exists.
For Condition (ii), set U (t n ) = 1 tn tn 0 u(s)ds and suppose U (t n ) U ∞ , for a sequence t n → +∞. Since u(t) is solution of (DEP ε(·) ), for almost every t > 0 and y ∈ K we have, which implies that After integrating on [0, t n ] and dividing by t n , we obtain Since F (y, ·) is convex and lower semi-continuous, then by Lemma 5.2, we get Passing to the limit as t n → +∞ and using +∞ 0 ε(t)dt < ∞, we can easy conclude that for every y ∈ K, and so U ∞ ∈ S F . Therefore, all conditions of Opial-Passty Lemma 5.3 hold, and then the unique solution u(t) of (DEP ε(·) ) converges weakly in average to some x ∞ ∈ S F .

Case of demipositive bifunctions.
In this subsection, we study the Tikhonov regularization for a special class of demipositive monotone bifunctions.
Definition 5.5. A monotone bifunction F : K × K → R is called demipositive if there exists z ∈ S such that for every sequence {u n } ⊂ K converging weakly to u, we have: F (u n , z) → 0 implies that u ∈ S.
The following proposition gives an important example for demipositive bifunctions, which ensure the strong convergence.
Proof. For S = ∅, consider some z ∈ S and a sequence {u n } ⊂ K such that u n u and lim n−→+∞ F (u n , z) = 0. According to Definition 5.5, it suffices to show that u ∈ S. Fix y ∈ K, since F is a 3-monotone bifunction then F (u n , z) + F (z, y) + F (y, u n ) ≤ 0; and as z ∈ S, we find F (y, u n ) ≤ −F (u n , z).
By passing to the limit when n → +∞, and using lower semicontinuity of F (y, ·) we get F (y, u) ≤ lim inf Remark 5. This class of demipositive monotone bifunctions (see [15]) contains as particular cases, more than 3−monotone ones, strongly monotone bifunctions, those such that int(S F = ∅) and µ-cocoercive differentiable for the second variable bifunctions.
Lemma 5.6 (Opial [31]). Let H be a Hilbert space, let S be a nonempty subset of H and let u : [0, +∞[→ H be a function. if we suppose that i) ∀z ∈ S, lim t→+∞ u(t) − z exists. ii) If t n → +∞ and u(t n ) u ∞ weakly in H, then u ∞ ∈ S. Then, the whole weak limit w − lim t→+∞ u(t) exists and belongs to S.
Proof. For every z ∈ S F , set θ(t) = 1 2 u(t) − z 2 for t > 0. Following the proof of Theorem 5.4, we get the same inequality (20), which means, since +∞ 0 ε(t)dt < ∞, that the limit lim t→+∞ u(t) − z exists. Invoking Opial's Lemma 5.6, one concludes if we show that every weak accumulation point of u(t) belongs to S F . Consider t n → +∞ for which u(t n ) y. The fact that u is absolutely continuous implies that θ(t) is also absolutely continuous, and then Henceθ(t) ∈ L 1 [0, +∞[ and thus there is a subsequence (t n k ) of (t n ) such thaṫ θ(t n k ) → 0 as k → ∞.
Since u is the solution of (DEP ε(·) ) and z ∈ S F , we have: Passing to the limit when k → ∞, we obtain lim n→∞ F (u(t n k ), z) = 0. Since u(t n k ) y and F is demipositive, we conclude that y ∈ S F , and thus every weak accumulation point of u(t) belongs to S F .
To prove our result of strong asymptotic convergence when ε / ∈ L 1 (R + ), we need the following two lemmas. Proof. Since x * ∈ S F and x ε resolves (EP ε ), monotonicity of F gives and then (x ε ) remains bounded: x ε ≤ x * for every ε > 0. If we take x ∞ a weak limit point of x ε , then there exists a sequence ε k → 0 such that x ε k x ∞ . Since x ε k is an equilibrium point of F ε , then for any y ∈ K we have F (x ε k , y) + ε x ε k , y − x ε k ≥ 0, which implies by monotonicity of F , F (y, x ε k ) ≤ ε k x ε k , y − x ε k . Going to the limit when ε k → 0, we obtain By Minty's Lemma, we conclude that x ∞ belong to S F , and using the weak lower semicontinuity of the norm, the inequality x ε k ≤ x * for every k gives x ∞ ≤ x * , and then x ∞ = x * . We conclude x * is the unique weak cluster point of the bounded family {x ε }, and then x ε x * . Since the norm is weakly lower semicontinuous, we also have which implies that x ε → x * , and then the (x ε ) converges strongly to x * .  [0, +∞[→ H be the unique solution of (DEP ε(·) ) with ε(t) → 0 as t → +∞. If +∞ 0 ε(t)dt = ∞, then (u(t)) strongly converges to x * as t → +∞.
Let f : K → R be a convex and lower semicontinuous function, and consider the minimization problem min where the optimal solution set S = {x ∈ K : f (x) ≤ f (y), ∀y ∈ K} is assumed nonempty. The Tikhonov regularization concerning this problem has been studied in paper [16], by formulating the problem (MP) as an equilibrium problem (EP ) by and is 3-monotone; also the problem (DEP ε(·) ) becomes Using Theorems 4.3, 5.7 and 5.10, we get : Corollary 1. Suppose f : K → R is convex lsc, the solution set of (MP) is nonempty, and consider u(t) the unique solution of (MP ε(t) ).
Remark 6. Notice that in this framework, we generalize the result obtained by [16] about the strong convergence when ε / ∈ L 1 (R + ). We get this convergence not only for the Fenchel-Moreau subdifferential of convex functions but also for all 3-monotone bifunctions; this is one benefit of using bifunction's method.
In Example 5.13, we treat a 3-monotone bifunction which satisfies conditions (H 1 ) − (H 4 ) and is not cyclically monotone.
Definition 5.12. For n = 2, 3, . . . , we say that a bifunction F : K × K → R is n-cyclically monotone if for every x 1 , x 2 , . . . , x n ∈ K and for x n+1 = x 1 , one has n k=1 F (x k , x k+1 ) ≤ 0. F is said to be cyclically monotone if it is n-cyclically monotone for each integer n ≥ 2.

Remark that a bifunction F is monotone if it is 2-cyclically monotone, and F is 3-monotone if it is 3-cyclically monotone.
Remark 7. Suppose F is n-cyclically monotone, then according to Theorem 5.7 (for n = 2) and Theorem (5.10) (for n = 3), we obtain respectively the weak convergence and the strong convergence to a solution of (M P ) of the Tikhonov evolution problem (M P ε(t) ).
Proof. Consider θ(t) = 1 2 u(t) − x * 2 and using the monotonicity of F we obtaiṅ By the same argument as in [16,Proposition 6], we get equivalence between the properties (a), (b) and (c).
By exploiting Lemma 5.14 we can prove the following strong convergence result. converges strongly, as ε → 0, to the element of minimum norm x * of the solution set S F of the problem (EP ).
Proof. We adopt the scheme used by [16,Theorem 9] Step 1. There is a zero Lebesgue measure set outside of whichu(t) → 0 as t → +∞. Set, for h > 0, θ h (t) = 1 2 u(t + h) − u(t) 2 with t > 0. By monotonicity of F we haveθ Multiplying this inequality by e α h (t) , where α h (t) := t 0 (ε(s + h) + ε(s)) ds, and integrating on [t , t], we obtain (25) Since the absolutely continuous function u is a solution of (DEP ε(·) ), there exists a set N ⊂]0, +∞[ of zero Lebesgue measure such that u is differentiable on the dense set D =]0, +∞[\N , and then for M = sup s≥0 u(s) 2 , passing to the limit in (25) if h → 0, we obtain for each t, t ∈ D with t > t By Gronwall's Lemma, we get and thusu(t) strongly converges to 0 as t → +∞ in D.
Step 2. Every weak cluster point of u(t) lies in S F . Letx be a weak cluster point of u(t), and choose t k → +∞ such that u(t k ) x. Since u(.) is continuous and D is dense in [0, +∞[, one may findt k ∈ D close enough to t k so that |u(t k ) − u(t k )| ≤ 1 k , and therefore the sequence (u(t k )) also weakly converges tox.
Using the first step above, we affirm thatu(t k ) → 0, and since ε(t) → 0 and u(t) is bounded, it follows thatu(t k ) + ε(t k )u(t k ) converges strongly to 0. Now, using conditions (H 2 ) and (H 4 ) on F , we obtain : for each y ∈ K F (y,x) ≤ lim inf k→+∞ F (y, u(t k )) By using the Minty's Lemma, we conclude thatx ∈ S.
Step 3. Combining Step 2 with (a) ⇒ (c) in Lemma 5.14, we deduce the strong convergence of the solution u(t) of (DEP ε(·) ) to x * .
Remark 8. The demipositivity is an important condition to get the weak convergence of the solution u(t) of (DEP ), and the solution u(t) of (DEP ε(·) ). Theorem 5.15 offers a way for dealing with a non necessarily demipositive monotone bifunction to get strong convergence of u(t) to the least norm element of S F , whenever only ε satisfies the conditions: We consider the saddle-point problem: By setting the bifunction we can see [12] that the problems (SP ) and (SP ) are equivalent, i.e. (ū,v) is a solution of (SP ) iff it is a solution of (EP ). Then the regularization problem (DEP ε(·) ) of (SP ) becomes : for each (x, y) As in [15], when we take in (DSP ε(·) ) respectively y = v and x = u, we deduce the nonlinear dynamical system Conversely, one can easily go back from (DS) to (DSP ε(·) ), and then these two problems are equivalent. Thus, according to Theorem 5.15, we conclude Remark 9. In the following example, we justify that the strong convergence of the trajectory x(t) in Corollary 2 may be obtained without imposing the condition +∞ 0 |ε(t)|dt < +∞. Consider L(u, v) = uv on R 2 , then the unique saddle point of L is (ū,v) = (0, 0). The associated bifunction F L defined on R 2 × R 2 by F L ((u, v), (u , v )) = u v − uv satisfies all conditions (H 1 ) − (H 4 ) but is not demipositive (see [15]).

5.3.2.
Neural model for convex programming. The linear Programming Problems has received considerable research attention from the neural networks community. The first solution of the linear programming problem was proposed by Tank and Hopfield wherein they used the continuous-time Hopfield network [22], afterwards, many researchers were inspired by their work, see [36,26,19] for a historical and bibliographical study of these problems. We consider here the convex programming problem: where f and g i , for i = 1 · · · , m, are convex lower semicontinuous real-valued functions on the non-negative orthan R n + . In [15] Chbani-Riahi proposed a neural dynamical model for solving (CP ) and study the asymptotic behavior of the solution of an associated dynamical equilibrium problem generated by the associate bifunction F L where L is the Lagrangian saddle-function defined by: Their proof is based on the fact that (x,λ) is a saddle point of L, i.e. an equilibrium point of F L , impliesx is an optimum vector for (CP ). By using a Tikhonov regularization we consider the following neural dynamical model for solving (CP ): (M N ε(·) ) Since L is a closed convex-concave saddle function on R n + × R m + , Corollary 2 ensures Corollary 3. Suppose f, g i , for i = 1 · · · , m, are convex lower semicontinuous realvalued functions on R n + , +∞ 0 ε(s)ds = +∞ and conditions +∞ 0 |ε(s)|ds < +∞. The the unique solution of the nonlinear dynamical system (M N ε(·) ) converges to the least norm element of the set of the solutions of (CP ).

Remark 10.
In contrary to [15,Theorem 6.3.], we need no strict convexity condition on the functions f and g i to reach a solution of (CP ) via the asymptotic behavior of a solution of the dynamical system (M N ε(·) ).
6. Interchange of the penalty setting (Multiscale aspects). The purpose of this section is to establish the link between solutions of the following two Tikhonov's regularization evolution problems: where the positive function β(t) (respectively ε(t)) converges to +∞ (respectively to zero) as t → +∞. Let us first mention that any solution of one of the two following problems F (u, y) + ε(t) u, y − u ≥ 0 ∀y ∈ K and β(t)F (v, y) is also a solution of the other; simply use ε(t)β(t) = 1. For the link between solutions of the problems (DEP ε(·) ) and (DEP β(·) ), consider two C 1 functions β : (ii) the evolution problems (DEP ε(·) ) and (DEP β(·) ) are equivalent, i.e., if u is a solution of (DEP ε(·) ) then v = u • t ε is a solution of (DEP β(·) ), and conversely, if v is a solution of (DEP β(·) ), then u = v • t β is a solution of (DEP ε(·) ).
Proof. This is a consequence of Theorems 5.10 and 5.15, Lemma 6.1 and Remarks 11 and 12. Let us consider the differential inclusion: where ψ : H → R ∪ {+∞} is a convex lsc function with domaine K a closed convex set in H. This problem can be seen as (DEP β(·) ) with F (x, y) = ψ(y) − ψ(x). Then according to Theorem 6.2, the solution of equation (29) converges strongly to the least norm element of argmin ψ, the minimum set of ψ. We note that (29) is a particular case of the following asymptotic monotone inclusionv (t) + β(t)∂ψ(v(t)) + Av(t) 0 (30) where A is a strongly monotone operator.
In order to ensure the strong convergence of the solution of (30) to the unique solution of the hierarchical problem 0 ∈ (A + N C )(x * ), the authors in [3] use the following condition: for every p belonging to the range of N C , where C = argmin ψ, For bifunctions, using results of [2] and [4], the authors in [13] used the Fitzpatrick transform of parametrized family of bifunctions to study the strong convergence of trajectories of a more general dynamical equilibrium system than (DEP β(·) ), where the quadratic form (x, y) → x, y − x , is replaced by a more general strongly monotone bifunction. More precisely, Theorem 4.4 in [13] use the following condition dt < +∞. (32) In our setting, without condition (32), we present the same result in Theorem 6.2 by only assuming lim t→+∞ β(t) = +∞ and either F is 3-monotone or +∞ 0 Example 6.4. In the case when F (x, y) = ϕ(y) − ϕ(x), for x, y ∈ K and ϕ is a closed convex function on K, F is a 3-monotone bifunction, and according to Theorem 6.2, without assumption (32), the solution of (29) strongly converges to the element of minimal norm of S F = argmin K ϕ.
Consider the function β(t) = (1 + t) α , then our condition +∞ 0 |β(t)| β(t) 2 dt < +∞ is verified for each α ≥ 0, while the assumption (32) is satisfied iff α > 1, which means that conditions in our Theorem 6.2 are wider than those used in [13,Theorem 4.4]. 7. Strong convergence of the Prox-Tikhonov and forward-backward algorithms. A time discretization of dynamical systems is used to link between algorithms and continuous dynamical systems, and their asymptotic analysis. We firstly use an implicit discretization (the prox-penalization algorithm) (ProxPA) of the dynamical system (DEP ε(·) ) by starting from point x 0 ∈ K and iterating to go from x n ∈ K to x n+1 = J Fn λn (x n ), where {ε n } are Tikhonov parameters, and {λ n } is a positive sequence of proximal parameters. Afterwards, we propose the descentproximal (forward-backward) algorithm (DProxA): x n+1 = J F λn ((1 − λ n ε n )x n ). We consider the sequence {x εn } n∈N where x εn is the unique equilibrium point of F n , which existence and uniqueness is ensured by conditions (H 1 ) − (H 4 ) and strong monotonicity of F n . Then from Lemma 5.8, we have lim n→+∞ x εn − x * = 0, where x * the least norm element of the set of equilibrium points S F .
To study the strong convergence of the algorithms (ProxPA) and (DProxA), we need the following classical convergence lemma for real sequences.
If ∞ n=0 λ n ε n = ∞, then the sequence {x n }, generated by (ProxPA), strongly converges to x * the orthogonal projection of 0 onto S F .
It is important to note that, if {λ n ε n } is bounded above, then the condition In the next theorems, we avoid 3-monotonicity of F employed in Theorem 7.2 and reinforce the hypotheses on the parameters λ n and ε n , in order to ensure the strong convergence of x n to x * . Then the sequence {x n } generated by (ProxPA) strongly converges to the least norm element x * of S F .