Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 15, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMPTOTIC BEHAVIOR FOR A QUASI-AUTONOMOUS GRADIENT SYSTEM OF EXPANSIVE TYPE GOVERNED BY A QUASICONVEX FUNCTION BEHZAD DJAFARI ROUHANI, MOHSEN RAHIMI PIRANFAR Communicated by Jerome A Goldstein Abstract. We consider the quasi-autonomous first-order gradient system u̇(t) = ∇φ(u(t)) + f(t), t ∈ [0,+∞) u(0) = x0 ∈ H, where φ : H → R is a differentiable quasiconvex function such that ∇φ is Lipschitz continuous. We study the asymptotic behavior of solutions to this system in continuous and discrete time. We show that each solution either approaches infinity in norm or converges weakly to a critical point of φ. This further concludes that the existence of bounded solutions and implies that φ has a nonempty set of critical points. Some strong convergence results, as well as numerical examples, are also given in both continuous and discrete cases. 1. Introduction Let H be a real Hilbert space endowed with the scalar product 〈·, ·〉 and induced norm ‖ · ‖. By → and ⇀ we denote strong and weak convergence, respectively, in H. The study of existence and asymptotic behavior of solutions to first-order evolution equations of the form u̇(t) +Au(t) 3 0 a.e t ∈ [0,+∞), (1.1) where A : D(A) ⊂ H → H is a possibly multivalued maximal monotone operator, goes back to the 1970s; see [5]. In [1, 3], the authors proved that if A−1(0) 6= ∅, then the mean of solutions to (1.1) converges weakly to an element of A−1(0). But in general, the solutions to (1.1) are not strongly convergent [2]. Bruck [6] proved the weak convergence of solutions to (1.1) under an additional condition on the monotone operator A, which is called demipositivity. By introducing the notion of nonexpansive and almost nonexpansive curves in H, Djafari Rouhani [9, 8] studied the asymptotic behavior of the mean of bounded solutions to the following quasi- autonomous system of monotone type u̇(t) +Au(t) 3 f(t), t ∈ [0,+∞). (1.2) 2010 Mathematics Subject Classification. 34G20, 47J35, 39A30, 37N40. Key words and phrases. First-order evolution equation; expansive type gradient system; asymptotic behavior; quasiconvex function; minimization. c©2021 Texas State University. Submitted January 25, 2021. Published March 18, 2021. 1 2 B. D. ROUHANI, M. R. PIRANFAR EJDE-2021/15 An important example of a maximal monotone operator is the subdifferential of a proper, convex and lower semicontinuous function. Inspired by its applications in economics, the study of functions that are not convex but have convex sublevel sets have received a particular attention; see [7, 16] and the references therein. The functions with convex sublevel sets are called quasiconvex. Here is a formal definition: a function φ : H → (−∞,+∞] is called quasiconvex if φ(λx+ (1− λ)y) ≤ max{φ(x), φ(y)}, ∀x, y ∈ H, ∀λ ∈ [0, 1]. We say that φ is strongly quasiconvex if there exists α > 0 such that φ(λx+ (1− λ)y) ≤ max{φ(x), φ(y)} − αλ(1− λ)‖x− y‖2, ∀x, y ∈ H, ∀λ ∈ [0, 1]. There are have been many attempts to generalize the notion of subdifferential for non convex functions; see [14] and the references therein. However in any circum- stance, the subdifferential of a quasiconvex function is not monotone. On the other hand, if φ : H → R is Gâteaux differentiable, then the following characterization for a quasiconvex function φ holds: φ is quasiconvex on H if and only if for all x, y ∈ H: φ(y) ≤ φ(x) implies 〈∇φ(x), x− y〉 ≥ 0. The above characterization may prove to be useful given the lack of monotonicity. Applying this fact, Goudou and Munier [13] studied (1.2) for the case where A is replaced by ∇φ where φ : H → R is a differentiable quasiconvex function with a nonempty set of minimizers. Generally, even the monotone type systems of the form u̇(t) ∈ Au(t), t ∈ [0,+∞), (1.3) are “strongly ill-posed”. For example, consider the simple linear case of A = −∆ with Dirichlet boundary conditions, which yields the heat equation with a final Cauchy data and is not generally solvable. Djafari Rouhani [10, 11] introduced the notion of almost expansive curves, and studied their ergodic and asymptotic properties. Then he applied these results to study the asymptotic behavior of possible solutions to (1.3). In [12], by considering the explicit discretization of (1.3), the authors studied the asymptotic behavior and periodicity of the generated sequence. In this article, we consider the differential equation u̇(t) = ∇φ(u(t)) + f(t), t ∈ [0,+∞), (1.4) where φ : H → R is a differentiable quasiconvex function such that ∇φ is Lipschitz continuous and f ∈W 1,1((0,+∞);H). The Lipschitz continuity of ∇φ implies that the system (1.4) with an initial condition has a unique solution u(t). To study the asymptotic behavior of such a solution, we define L(u) = {y ∈ H : ∃T > 0 s.t. φ(y) ≤ φ(u(t))∀t ≥ T}. Since we are only concerned about the asymptotic behavior of the trajectories, for simplicity and without loss of generality, in our calculations we always take T = 0. The set of all global minimizers of φ is denoted by argminφ. Clearly, argminφ ⊂ L(u). In Section 2, we show that if (1.4) has a solution u such that lim inft→+∞ ‖u(t)‖ < +∞, then L(u) 6= ∅, and we prove that every such solution converges weakly to some element in (∇φ)−1(0) and if this element does not belong to argminφ, then the convergence is strong. We also show that if u(t) is an unbounded solution, EJDE-2021/15 QUASI-AUTONOMOUS GRADIENT SYSTEMS 3 then ‖u(t)‖ approaches infinity as t → +∞. Some strong convergence results are obtained as well. Section 3 is devoted to the study of the explicit discretization of (1.4). In that section, we prove similar convergence results as in Section 2. This provides an algorithm to approximate an element of (∇φ)−1(0). A numerical ex- ample shows that by choosing suitable step sizes, the convergence of the scheme can be fast. Our results in this paper extend and improve our previous results in [10, 11, 12]. Lemma 1.1 ([13]). Let φ : H → R be a continuously differentiable and quasiconvex function and u : [0,+∞)→ H be a curve such that there is some point x̃ in H and a number r > 0 and T ≥ 0 satisfying φ(y) ≤ φ(u(t)) ∀t ≥ T, ∀y ∈ B(x̃, r). Then for all t ≥ T , we have r‖∇φ(u(t))‖ ≤ 〈∇φ(u(t)), u(t)− x̃〉. 2. Continuous case Let φ : H → R be a quasiconvex function such that ∇φ is Lipschitz continuous. We consider the Cauchy problem u̇(t) = ∇φ(u(t)) + f(t), t ∈ [0,+∞), u(0) = x ∈ H, (2.1) where f ∈ W 1,1((0,+∞), H). Since ∇φ is Lipschitz continuous, the Cauchy- Lipschitz theorem guatentees the existence of a unique solution u ∈ C1([0,+∞);H) to (2.1). Proposition 2.1. Assume that u(t) is a solution to (2.1). For an arbitrary interval [a, b], where b ≥ a ≥ 0, and each y ∈ L(u), we have ‖u(a)− y‖ ≤ ‖u(b)− y‖+ ∫ b a ‖f(t)‖dt, (2.2) and therefore limt→+∞ ‖u(t)− y‖ exists (it may be infinite). Proof. Let y ∈ L(u) be arbitrary and fixed. Multiplying both sides of (2.1) by (u(t)− y), we obtain 〈u̇(t), u(t)− y〉 = 〈∇φ(u(t)), u(t)− y〉+ 〈f(t), u(t)− y〉, which together with the characterization of quasiconvex functions imply that 〈u̇(t), u(t)− y〉 − 〈f(t), u(t)− y〉 ≥ 0. (2.3) Since u(t) is absolutely continuous, ‖u(t) − y‖ is also absolutely continues, and hence d dt‖u(t)− y‖ exists almost everywhere on [a, b]. Thus, we have 〈u̇(t), u(t)− y〉 = 1 2 d dt ‖u(t)− y‖2 = ‖u(t)− y‖ d dt ‖u(t)− y‖ a.e. t ∈ [a, b]. (2.4) Combining (2.3) and (2.4) and using the Cauchy-Schwarz inequality, we obtain ‖u(t)− y‖ ( d dt ‖u(t)− y‖+ ‖f(t)‖ ) ≥ 0 a.e. t ∈ [a, b]. (2.5) If there exists t0 ∈ [a, b] such that d dt‖u(t)− y‖ ∣∣ t=t0 + ‖f(t0)‖ < 0, then d dt‖u(t)− y‖ ∣∣∣ t=t0 < 0, and by (2.5), u(t0) = y. Also, by the definition of the derivative, there 4 B. D. ROUHANI, M. R. PIRANFAR EJDE-2021/15 exists some sufficiently small h > 0, such that ‖u(t0 + h) − y‖ < ‖u(t0) − y‖ = 0 which is impossible. Hence d dt ‖u(t)− y‖+ ‖f(t)‖ ≥ 0 a.e. t ∈ [a, b]. (2.6) Integrating the above inequality on [a, b], we obtain ‖u(a)− y‖ ≤ ‖u(b)− y‖+ ∫ b a ‖f(t)‖dt. Now we conclude the result by taking lim inf as b→ +∞, and then taking lim sup as a→ +∞ in the above inequality. � Proposition 2.2. Let u(t) be a solution to (2.1) such that lim inft→+∞ ‖u(t)‖ < +∞. Then limt→+∞∇φ(u(t)) = 0 and limt→+∞ φ(u(t)) exists and is finite. Proof. Multiplying both sides of (2.1) by u̇(t), we obtain ‖u̇(t)‖2 = 〈∇φ(u(t)), u̇(t)〉+ 〈f(t), u̇(t)〉. (2.7) Applying the Cauchy-Schwarz inequality to the above equation, we obtain ‖u̇(t)‖2 ≤ d dt φ(u(t)) + ‖f(t)‖‖u̇(t)‖. (2.8) Since lim inft→+∞ ‖u(t)‖ < +∞ and φ is bounded on bounded sets, there is a sequence u(tn) such that the sequence φ(u(tn)) is bounded. Integrating the above inequality on [0, tn], we obtain∫ tn 0 ‖u̇(t)‖2dt ≤ φ(u(tn))− φ(u(0)) + ∫ tn 0 ‖f(t)‖‖u̇(t)‖dt. (2.9) Again by applying the Cauchy-Schwarz inequality, we have∫ tn 0 ‖u̇(t)‖2dt ≤ C1 + C2 (∫ tn 0 ‖u̇(t)‖2dt )1/2 , (2.10) where C1 = supn≥0{φ(u(tn)) − φ(0)}, and C2 = ( ∫ +∞ 0 ‖f(t)‖2dt )1/2 . This shows that u̇ ∈ L2((0,+∞);H). On the other hand, ‖u(t)− u(s)‖ ≤ ∫ t s ‖u̇(τ)‖dτ ≤ (t− s)1/2 (∫ t s ‖u̇(τ)‖2dτ )1/2 which implies that u is uniformly continuous. This, the Lipschitz continuity of ∇φ, and the fact that f ∈W 1,1((0,+∞);H) imply that u̇ is uniformly continuous. Now, since u̇(t) is uniformly continuous and belongs to L2((0,+∞), H), we have limt→+∞ u̇(t) = 0. Therefore, since f ∈W 1,1((0,+∞);H), we have lim t→+∞ ∇φ(u(t)) = 0. From (2.7), by applying the Cauchy-Schwarz inequality, we also have d dt φ(u(t)) ≤ ‖u̇(t)‖2 + ‖u̇‖‖f(t)‖. Integrating on [s, t], we obtain φ(u(t)) ≤ φ(u(s)) + ∫ t s ‖u̇(τ)‖2dτ + sup t≥0 ‖u̇(t)‖ ∫ t s ‖f(τ)‖dτ. Taking lim sup as t→ +∞, then taking lim inf as s→ +∞ in the above inequality, we conclude that limt→+∞ φ(u(t)) exists and is finite. � EJDE-2021/15 QUASI-AUTONOMOUS GRADIENT SYSTEMS 5 Proposition 2.3. If u(t) is a solution to (2.1) such that lim inft→+∞ ‖u(t)‖ < +∞, then L(u) 6= ∅ and u is bounded. Proof. Assume by contradiction that lim inft→+∞ ‖u(t)‖ < +∞, and L(u) = ∅. Then for each z ∈ H there exists a sequence tzn such that φ(z) > φ(u(tzn)). On the other hand, by Proposition 2.2, we know that limt→+∞ φ(u(t)) exists and is finite. Hence, we have lim t→+∞ φ(u(t)) = lim n→+∞ φ(u(tzn)) ≤ φ(z) ∀z ∈ H, which implies that limt→+∞ φ(u(t)) = inf φ. Since lim inft→+∞ ‖u(t)‖ < +∞, there exists a bounded subsequence of u(t), say u(tn). Since φ is bounded on bounded sets, then inf φ = limn→+∞ φ(u(tn)) > −∞. Also, the boundedness of u(tn) implies that there exist a subsequence of u(tn), which we denote again by u(tn), and some p ∈ H such that u(tn) ⇀ p. Now using the lower semicontinuity of φ for the weak topology, we obtain φ(p) ≤ lim n→+∞ φ(u(tn)) = lim t→+∞ φ(u(t)) = inf φ. This yields that p ∈ argminφ, which contradicts L(u) = ∅. Now Proposition 2.1 implies that u is bounded. � Theorem 2.4. Let u(t) be a solution to (2.1). If lim inft→+∞ ‖u(t)‖ < +∞, then there exists some p ∈ (∇φ)−1(0) such that u(t) ⇀ p as t → +∞, and if p /∈ argminφ, the convergence is strong. If u(t) is unbounded, then ‖u(t)‖ → +∞ as t→ +∞. Proof. If lim inft→+∞ ‖u(t)‖ < +∞, then by Proposition 2.3, u is bounded and L(u) 6= ∅. Let y ∈ L(u). By Proposition 2.1, we know that limt→+∞ ‖u(t) − y‖ exists and is finite, and by Proposition 2.2, we know that limt→+∞ φ(u(t)) exists and is finite too. Let q be a weak cluster point of u(t). There exists a sequence tn such that tn → +∞ as n→ +∞, and u(tn) ⇀ q as n→ +∞. If limt→+∞ φ(u(t)) = φ(y), using the fact that φ is lower semicontinuous for the weak topology, we have φ(q) ≤ lim n→+∞ φ(u(tn)) = lim t→+∞ φ(u(t)) = φ(y). Also since y ∈ L(u), we have q ∈ L(u). Now an easy application of the Opial lemma [15] shows that there exists p ∈ L(u) such that u(t) ⇀ p as t → +∞. If p ∈ argminφ, then the conclusion follows. Otherwise, there is an element in L(u) which we denote again by y, such that limt→+∞ φ(u(t)) > φ(y). Hence there exist t0 ≥ 0 and r > 0, such that φ(z) ≤ φ(u(t)), ∀t ≥ t0, ∀z ∈ B(y, r). Thus by Lemma 1.1, we have r‖∇φ(u(t))‖ ≤ 〈∇φ(u(t)), u(t)− y〉 t ≥ t0. (2.11) Replacing ∇φ(u(t)) from (2.1) on the right hand side of the above inequality and then using the Cauchy-Schwarz inequality, we obtain r‖∇φ(u(t))‖ ≤ 1 2 d dt ‖u(t)− y‖2 + ‖u(t)− y‖‖f(t)‖ t ≥ t0. Integrating the above inequality on [t0, t] and then letting t → +∞ implies that ∇φ(u(t)) ∈ L1((0,+∞), H). Therefore u̇(t) ∈ L1((0,+∞), H). Hence u(t) → p as t → +∞. Since ∇φ is continuous and by Proposition 2.2, limt→+∞∇φ(u(t)) = 6 B. D. ROUHANI, M. R. PIRANFAR EJDE-2021/15 0, we obtain p ∈ (∇φ)−1(0). If u(t) is unbounded and lim inft→+∞ ‖u(t)‖ < +∞, then by Proposition 2.3, u is bounded which is a contradiction. Therefore limt→+∞ ‖u(t)‖ = +∞, if u is unbounded. � Theorem 2.4 shows that if (∇φ)−1(0) = ∅, then for any solution to (2.1), we have limt→+∞ ‖u(t)‖ = +∞. We illustrate Theorem 2.4 by the following example. Example 2.5. Let φ : R → R be defined by φ(x) = arctan(x3). Then φ is a quasiconvex function such that ∇φ is Lipschitz continuous and (∇φ)−1(0) = {0} and argminφ = ∅. Considering (2.1), where φ(x) = arctan(x3), we have the following Cauchy problem u̇(t) = 3u(t)2 1 + u(t)6 , t ∈ [0,+∞), u(0) = u0 ∈ R. (2.12) By the Cauchy-Lipschitz theorem, the initial value problem (2.12) has a unique solution. Therefore if u(0) = 0, then u(t) ≡ 0 is the unique solution to (2.12). Hence if u(t0) = 0, for some t0 ≥ 0, then by the uniqueness of solutions to (2.12) u(t) = 0 for all t ≥ t0. Now we assume that u(t) 6= 0 for all t ≥ 0. Using (2.12), by a simple calculation, we obtain t+ c = −1 3u(t) + u(t)5 15 , t ≥ 0, (2.13) where c is a constant. Letting t → +∞, from (2.13), we see that u(t) → 0− or u(t) → +∞. This also shows that in each case L(u) 6= ∅ for (2.12). In fact, here L(u) contains the set {u(t) : t ≥ 0} because in our case φ(u(t)) is increasing. Remark 2.6. Theorem 2.4 shows that the existence of a bounded solution to (2.1) implies that (∇φ)−1(0) is nonempty. However, Example 2.5 shows that the converse is not true. In fact, this is because if for example we choose u(0) = 1 in (2.1), then since u(t) is increasing, the unique solution of (2.1) must tend to +∞, because it cannot tend to 0− as t→ +∞. Theorem 2.7. If either one of the following assumptions is satisfied, then bounded solutions to (2.1) converge strongly to some point in (∇φ)−1(0): (i) Sublevel sets of φ are compact. (ii) intL(u) 6= ∅. Proof. (i) By Theorem 2.4, it suffices to consider the case limt→+∞ φ(u(t)) = inf φ. That is when u(t) ⇀ p ∈ argminφ as t→ +∞. Let ψ(t) = φ(u(t))− inf φ. Clearly ψ(t) ≥ 0, and limt→+∞ ψ(t) = 0. If there exists a sequence tn ⊂ [0,+∞) such that tn ↑ +∞, and ψ(tn) = 0 for all n ≥ 0, then φ(u(tn)) = inf φ ≤ φ(u(t)) for all t ≥ 0 and in particular for all t ∈ [t0, tn] and for all n ≥ 0. Otherwise, there exists some t0 > 0 such that ψ(t) > 0 for all t ≥ t0. Let n ≥ t0. Since ψ is continuous and [t0, n] is compact, then ψ takes its minimum on [t0, n]. We define the sequence tn as the smallest element of argminψ|[t0,n]. Therefore for all n ∈ N if t ∈ [t0, tn], then ψ(tn) ≤ ψ(t) and hence φ(u(tn)) ≤ φ(u(t)). Now either tn has a subsequence tnk such that tnk ↑ +∞ as k → +∞, or there exists N ∈ N such that tn < N for all n. In the first case, by going to a subsequence, we conclude that tn ↑ +∞. In the second case, for all n ≥ N the minimizer of ψ on [t0, tn] is not greater than N . Therefore tn = tN for all n ≥ N . This implies that limn→+∞ ψ(tn) = ψ(tN ) 6= 0 EJDE-2021/15 QUASI-AUTONOMOUS GRADIENT SYSTEMS 7 which is a contradiction. Therefore the second case never happens. Thus we showed that we always have u(tn) ∈ {x : φ(x) ≤ φ(u(t0))} for all n. Since sublevel sets of φ are compact, then the sequence u(tn) has a strong limit point, say p, which by going to a subsequence we may assume u(tn)→ p as n→ +∞. Since φ is continuous and limt→+∞ φ(u(t)) = inf φ, we obtain p ∈ argminφ. By Proposition 2.1, we know that limt→+∞ ‖u(t)− p‖2 exists which implies that u(t)→ p as t→ +∞. (ii) Since intL(u) 6= ∅, there exists some y ∈ L(u) such that for some r > 0, we have B(y, r) ⊂ L(u). Let z = y + r ∇φ(u(t))‖∇φ(u(t))‖ . Since φ(z) ≤ φ(u(t)), using the characterization of quasiconvex functions, we have 〈∇φ(u(t)), u(t)− z〉 = 〈∇φ(u(t)), u(t)− y〉 − r‖∇φ(u(t))‖ ≥ 0. The above inequality is in fact (2.11) which holds for all t ≥ 0. The rest of the proof is similar to the proof of Theorem 2.4. � Theorem 2.8. Assume that φ : H → R is a strongly quasiconvex function and u(t) is a bounded solution to (2.1). Then argminφ is a singleton and u(t) converges strongly to the unique minimizer of φ. Proof. By Proposition 2.3, the boundedness of u(t) implies that L(u) 6= ∅. Let y ∈ L(u) and λ ∈ (0, 1). Since φ is strongly quasiconvex, there exists some α > 0 such that φ(u(t) + λ(y − u(t)))− φ(u(t)) ≤ −αλ(1− λ)‖u(t)− y‖2, where φ(u(t)) = max{φ(u(t)), φ(y)}. Dividing both sides of the above inequality by λ and letting λ tend to zero, we obtain α‖u(t)− y‖2 ≤ 〈∇φ(u(t)), u(t)− y〉. Replacing ∇φ(u(t)) from (2.1), we have α‖u(t)− y‖2 ≤ 〈u̇(t)− f(t), u(t)− y〉 ≤ 1 2 d dt ‖u(t)− y‖2 +M‖f(t)‖, where M = supt≥0 ‖u(t)− y‖. Integrating the above inequality on [0, t], we obtain α ∫ t 0 ‖u(τ)− y‖2dτ ≤ 1 2 ‖u(t)− y‖2 − 1 2 ‖u(0)− y‖2 +M ∫ t 0 ‖f(τ)‖dτ. Since f(t) ∈ L1((0,+∞), H), letting t→ +∞, we have ‖u(t)−y‖ ∈ L2((0,+∞), H). On the other hand, from Proposition 2.1, we know that limt→+∞ ‖u(t)− y‖ exists, hence limt→+∞ u(t) = y. The uniqueness of the limit implies that L(u) is a single- ton. Therefore argminφ = L(u). � 3. Discrete case Consider the following discrete version of (2.1), un+1 − un = λn∇φ(un) + fn, u0 = x ∈ H, (3.1) where fn ∈ l1, λn ≥ ε for some ε > 0, and φ : H → R is a differentiable quasiconvex function such that ∇φ is Lipschitz continuous with Lipschitz constant K. We start by recalling the following lemma from [4]. 8 B. D. ROUHANI, M. R. PIRANFAR EJDE-2021/15 Lemma 3.1. Let U be a nonempty, open and convex subset of H, K > 0, and φ : U → R be a Fréchet differentiable function such that ∇φ is Lipschitz continuous with Lipschitz constant K on U , and let x and y be in U . Then the following hold: |φ(y)− φ(x)− 〈∇φ(x), y − x〉| ≤ K 2 ‖y − x‖2. To study the asymptotic behavior of un, we define a discrete version of L(u), L(un) = {y ∈ H : ∃N > 0 s.t. φ(y) ≤ φ(un) ∀n ≥ N}. As in the continuous case, without loss of generality we assume that N = 0. The following proposition, which is a discrete version of Proposition 2.1, will be needed in the sequel. Proposition 3.2. Let un be the sequence generated by (3.1). For each y ∈ L(un), and k < m, we have ‖uk − y‖ ≤ ‖um − y‖+ m−1∑ n=k ‖fn‖, (3.2) and consequently limn→+∞ ‖un − y‖ exists (it may be infinite). Proof. Multiplying (3.1) by (un − y) and then using the characterization of quasi- convex functions, we obtain 〈un+1 − un, un − y〉 ≥ 〈fn, un − y〉. Applying the polarization identity and the Cauchy-Schwarz inequality, we obtain ‖un+1 − y‖2 − ‖un − y‖2 + 2‖un − y‖‖fn‖ ≥ 0. If ‖un+1 − y‖+ ‖un − y‖ 6= 0, then ‖un+1 − y‖ − ‖un − y‖+ 2 ‖un − y‖ ‖un+1 − y‖+ ‖un − y‖ ‖fn‖ ≥ 0, which implies that ‖un+1 − y‖ − ‖un − y‖+ 2‖fn‖ ≥ 0. (3.3) If ‖un+1 − y‖+ ‖un − y‖ = 0, the above inequality is clearly true. Summing (3.3) from n = k to n = m− 1, we obtain ‖uk − y‖ ≤ ‖um − y‖+ m−1∑ n=k ‖fn‖. Now by taking lim inf as m → +∞ and then taking lim sup as k → +∞ in the above inequality, we conclude the result. � Proposition 3.3. Let un be a solution to (3.1) such that lim infn→+∞ ‖un‖ < +∞. Then L(un) is nonempty if and only if limn→+∞ φ(un) exists, and in this case un is bounded. Proof. Let y ∈ L(un). Since lim infn→+∞ ‖un‖ < +∞, by (3.2), un is bounded. Multiplying both sides of (3.1) by (un − y) and then applying the polarization identity and the Cauchy-Schwarz inequality, we obtain ‖un+1 − un‖2 ≤ ‖un+1 − y‖2 − ‖un − y‖2 + 2‖un − y‖‖fn‖. (3.4) EJDE-2021/15 QUASI-AUTONOMOUS GRADIENT SYSTEMS 9 Summing both sides of (3.4) from n = 0 to n = m, and then letting m→ +∞, we find that (un+1 − un) ∈ l2. Hence lim n→+∞ ∇φ(un) = lim n→+∞ (un+1 − un) = 0. Now since ∇φ is Lipschitz continuous with Lipschitz constant K, by Lemma 3.1 we have φ(un+1)− φ(un) ≤ 〈∇φ(un+1), un+1 − un〉+ K 2 ‖un+1 − un‖2. (3.5) On the other hand, by the polarization identity, we have 〈∇φ(un+1), un+1 − un〉 = 1 2 ( ‖∇φ(un+1)‖2 + ‖un+1 − un‖2 − ‖un+1 − un −∇φ(un+1)‖2 ) . Substituting from the above identity in (3.5) and then using (3.1), we obtain φ(un+1)− φ(un) ≤ 1 2ε2 ‖un+2 − un+1 − fn+1‖2 + 1 +K 2 ‖un+1 − un‖2. (3.6) Summing the above inequality from n = k to n = m− 1, we obtain φ(um)− φ(uk) ≤ 1 2ε2 m∑ n=k+1 ‖un+1 − un − fn‖2 + 1 +K 2 m−1∑ n=k ‖un+1 − un‖2. Since fn and (un+1−un) belong to l2, by taking lim sup asm→ +∞ and then taking lim inf as k → +∞ in the above inequality, we conclude that limn→+∞ φ(un) exists. Moreover un is bounded by Proposition 3.2. Conversely, assume by contradiction that L(un) = ∅. Then for each z ∈ H there exists a subsequence uznk of the sequence un such that φ(z) > φ(uznk ) for all k ≥ 1. Since limn→+∞ φ(un) exists, we have lim n→+∞ φ(un) = lim k→+∞ φ(uznk ) ≤ φ(z) ∀z ∈ H, hence limn→+∞ φ(un) = inf φ. Since by assumption lim infn→+∞ ‖un‖ < +∞, Proposition 3.3 shows that un is bounded. This together with the boundedness of φ on bounded sets, and the above inequality imply that inf φ > −∞. Also, un has a nonempty set of weak cluster points. Hence there exists a subsequence unk of un that converges weakly to some point p ∈ H. By the lower semicontinuity of φ for the weak topology, we obtain φ(p) ≤ lim n→+∞ φ(unk ) = lim n→+∞ φ(un) = inf φ which implies that p ∈ argminφ, a contradiction with L(un) = ∅. � Remark 3.4. If φ is convex, we can omit the condition that ∇φ is Lipschitz continuous, because in this case, (3.5) is satisfied with K = 0. As we have seen in the continuous case, clearly argminφ ⊂ L(un). In the fol- lowing Proposition, we state some conditions which together with the assumption lim infn→+∞ ‖un‖ < +∞, imply that L(un) is nonempty. Proposition 3.5. Assume that un is a solution to (3.1) such that limn→+∞ ‖un‖ < +∞. If either one of the following conditions is satisfied, then L(un) is nonempty: (i) φ is convex and the sequence of step sizes λn is bounded above, (ii) lim supn→+∞ λn < 2/K. 10 B. D. ROUHANI, M. R. PIRANFAR EJDE-2021/15 Proof. Since ∇φ is Lipschitz continuous, by Lemma 3.1, we have 〈∇φ(un), un+1 − un〉 ≤ φ(un+1)− φ(un) + K 2 ‖un+1 − un‖2. (3.7) Note that if φ is convex, using the subdifferential inequality, we can omit the Lips- chitz continuity of ∇φ and take K = 0 in (3.7). Multiplying both sides of (3.1) by (un+1 − un), using (3.7), we obtain( 1 λn − K 2 ) ‖un+1 − un‖2 ≤ φ(un+1)− φ(un) + 1 ε ‖fn‖‖un+1 − un‖. (3.8) Since lim infn→+∞ ‖un‖ < +∞, the sequence un has a bounded subsequence, say unk . On the other hand, φ is bounded on bounded sets hence summing (3.8) from n = 0 to n = nk − 1, and then using the Cauchy-Schwarz inequality, if either one of the assumptions (i) or (ii) holds, we obtain c nk−1∑ n=0 ‖un+1−un‖2 ≤ φ(unk )−φ(u0) + 1 ε ( nk−1∑ n=0 ‖fn‖2 )1/2( nk−1∑ n=0 ‖un+1−un‖2 )1/2 , where c = infn≥0 ( 1 λn − K 2 ) is a positive constant. Dividing both sides of the above inequality by (∑nk−1 n=0 ‖un+1 − un‖2 )1/2 , and then letting k → +∞, we obtain (un+1 − un) ∈ l2. By an argument similar to the proof of the first part of Proposition 3.3, we conclude that limn→+∞ φ(un) exists. Now, Proposition 3.3 implies that L(un) 6= ∅. � Open problem. In the continuous case, Proposition 2.3 shows that condition lim inft→+∞ ‖u(t)‖ < +∞ implies L(u) 6= ∅. However, in the discrete case, we do not know whether without any additional assumptions, this implication holds. Theorem 3.6. Let un be the sequence generated by (3.1) and L(un) 6= ∅. If lim infn→+∞ ‖un‖ < +∞, then there exists some p ∈ (∇φ)−1(0) such that un ⇀ p as n → +∞ and if p /∈ argminφ the convergence is strong. If un is unbounded, then ‖un‖ → +∞, as n→ +∞. Proof. Let y ∈ L(un). From Proposition 3.2, we know that limn→+∞ ‖un−y‖ exists. Hence if un is unbounded then ‖un‖ goes to infinity as n→ +∞. If un is bounded, then limn→+∞ ‖un − y‖ is finite. Let q be a weak cluster point of un. There exists a subsequence unk such that unk ⇀ q as k → +∞. If limn→+∞ φ(un) = φ(y), then φ(q) ≤ lim inf k→+∞ φ(unk ) = lim n→+∞ φ(un) = φ(y), which yields q ∈ L(un). By Opial’s lemma [15], there is a p ∈ L(un) such that un ⇀ p. If p /∈ argminφ, then there exists an element in L(un) which we denote again by y such that limn→+∞ φ(un) > φ(y). In this case, there are n0 > 0 and r > 0, such that φ(z) ≤ φ(un), ∀n ≥ n0, ∀z ∈ B(y, r). Therefore a discrete version of Lemma 1.1 yields r‖∇φ(un)‖ ≤ 〈∇φ(un), un − y〉, ∀n ≥ n0. (3.9) Multiplying both sides of the above inequality by λn, replacing λn∇φ(un) from (3.1) with (un+1 − un − fn) and then applying the polarization identity and the EJDE-2021/15 QUASI-AUTONOMOUS GRADIENT SYSTEMS 11 Cauchy-Schwarz inequality, we obtain 2λnr‖∇φ(un)‖ ≤ ‖un+1 − y‖2 − ‖un − y‖2 + 2‖un − y‖‖fn‖. The above inequality implies that λn∇φ(un) ∈ l1. Hence (un+1 − un) ∈ l1, which implies that un is a Cauchy sequence for the strong topology, therefore un con- verges strongly to p. Since limn→+∞∇φ(un) = 0 and ∇φ is continuous, then p ∈ (∇φ)−1(0). � Example 3.7. Assume that φ is the same function as in Example 2.5 and consider (3.1) with λn = 2 3n and fn ≡ 0. Summing both sides of (3.1) from n = 0 to n = N − 1, we have uN = u0 + N−1∑ n=0 λn 3u2n 1 + u6n . (3.10) If the summation in (3.10) converges as N → +∞, then uN converges as N → +∞, and by the continuity of φ, limN→+∞ φ(uN ) exists. By Proposition 3.3, we see that L(un) is nonempty in this case. On the other hand, if the summation in (3.10) diverges as N → +∞, then uN → +∞ as N → +∞. This together with the fact that φ is nondecreasing, imply that L(un) is nonempty. Therefore all the assumptions of Theorem 3.6 are satisfied. Table 1 compares 1000 iterations of the sequence un given by (3.1) with two different initial values u0 = −0.5 and u0 = 1. The numerical results show that for u0 = −0.5, un → 0 ∈ (∇φ)−1(0) and for u0 = 1, un slowly goes to infinity. Table 1. Numerical results for (3.10) n un un 0 -0.5 1 1 -0.00769231 2 10 -0.00404869 3.63765 20 -0.00171074 4.68854 30 -0.0008858 5.46951 40 -0.000533135 6.11128 50 -0.000354164 6.66517 60 -0.000251763 7.15741 70 -0.000187942 7.60348 80 -0.000145564 8.01339 90 -0.000116023 8.39404 100 -0.0000946225 8.7504 1000 -9.94968×10−7 21.8786 Theorem 3.8. Assume that un is a bounded sequence which satisfies (3.1) and L(un) 6= ∅. If either one of the following assumptions is satisfied, then un converges strongly to some point in (∇φ)−1(0): (i) Sublevel sets of φ are compact, (ii) intL(un) 6= ∅. 12 B. D. ROUHANI, M. R. PIRANFAR EJDE-2021/15 Proof. (i) By Proposition 3.3, we know that limn→+∞ φ(un) exists. As we have already seen in the proof of Theorem 3.6, if limn→+∞ φ(un) > inf φ, then the sequence un strongly converges to some point in (∇φ)−1(0). Therefore we only need to consider the case where limn→+∞ φ(un) = inf φ. Let ψn = φ(un) − inf φ. Clearly ψn ≥ 0 and limn→+∞ ψn = 0. If there exists a subsequence nk such that nk ↑ +∞ and ψnk = 0 for all k ≥ 0, then φ(unk ) = inf φ ≤ φ(un) for all n ≥ 0 and in particular for all n0 ≤ n ≤ nk and for all k ≥ 0. Otherwise, there exists some n0 > 0 such that ψn > 0 for all n ≥ n0. Let k ≥ n0. Choose the subsequence nk such that ψnk = min{ψn0 , · · · , ψk}. Therefore if n0 ≤ n ≤ k, then ψnk ≤ ψn and hence φ(unk ) ≤ φ(un). Now either nk has a subsequence nkl such that nkl ↑ +∞ as l→ +∞, or there exists N ∈ N such that nk < N for all k. In the first case, by going to a subsequence, we conclude that nk ↑ +∞ as k → +∞. In the second case, the minimum of ψn0 , · · · , ψk is not greater than ψnN , for all k ≥ N . Therefore nk = nN for all k ≥ N . This implies that limk→+∞ ψnk = ψnN 6= 0 which is a contradiction. Therefore the second case never happens. We have unk ∈ {x : φ(x) ≤ φ(un0 )} for all k. Since sublevel sets of φ are compact, then the sequence unk has a strong cluster point, which is necessarily a weak cluster point of the sequence un as well. This together with Theorem 3.6 imply that the set of all strong cluster points of the sequence un is the singleton {p}, where p ∈ argminφ is the unique weak cluster point of the sequence un. (ii) Since intL(un) 6= ∅, there exist y ∈ L(un) and r > 0 such that B(y, r) ⊂ L(un). Therefore z = y + r ∇φ(un) ‖∇φ(un)‖ ∈ L(un). By the characterization of quasicon- vex functions, we have 〈∇φ(un), un − z〉 = 〈∇φ(un), un − y〉 − r‖∇φ(un)‖ ≥ 0. Therefore we have obtained (3.9) for all n ≥ 0. The rest of the proof follows from the proof of Theorem 3.6. � Theorem 3.9. Assume that φ : H → R is a strongly quasiconvex function and un is a bounded sequence generated by (3.1) with L(un) 6= ∅. Then the sequence un converges strongly to the unique minimizer of φ. Proof. Let y ∈ L(un). Since φ is strongly quasiconvexe, there is some α > 0 such that φ(un + λ(y − un))− φ(un) ≤ −αλ(1− λ)‖un − y‖2 ∀λ ∈ (0, 1). Dividing both sides of the above inequality by λ, and then letting λ→ 0, we obtain α‖un − y‖2 ≤ 〈∇φ(un), un − y〉. Multiplying both sides of the above inequality by λn and then replacing λn∇φ(un) from (3.1), we obtain αε‖un − y‖2 ≤ 〈un+1 − un − fn, un − y〉 ≤ 1 2 ‖un+1 − y‖2 − 1 2 ‖un − y‖2 +M‖fn‖, where M = supn≥0 ‖un − y‖. Summing the above inequality from n = 1 to n = N and then letting N → +∞, we obtain ‖un − y‖ ∈ l2. This shows that L(un) is a singleton and un converges strongly to the unique minimizer of φ, which completes the proof. � EJDE-2021/15 QUASI-AUTONOMOUS GRADIENT SYSTEMS 13 4. conclusions In this article, we studied the asymptotic behavior of solutions to a quasi- autonomous gradient system of expansive type governed by a differentiable quasi- convex function φ, both in continuous and discrete time. In particular, we showed that solutions either blow up and go to infinity in norm, or converge weakly to some critical point of φ. Since the gradient of a quasiconvex function is no longer a monotone operator, then compared to the convex case, new methods had to be developed to study this problem. Numerical examples are also given to illustrate our results. Besides the open problem stated in the paper, as a future direction for research, it would be interesting to investigate the possibility of extending the results of the paper to the case where φ is not assumed to be differentiable. 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Munier; The gradient and heavy ball with friction dynamical systems: the quasiconvex case. Math. Program. Ser. B 116 (2009) 173–191. [14] N. Hadjisavvas; Generalized convexity, generalized monotonicity and nonsmooth analysis, in N. Hadjisavvas, S. Komlósi and S. Schaible, Handbook of generalized convexity and general- ized monotonicity, Springer, New York (2005) 465–499. [15] Z. Opial; Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull. Am. Math. Soc. 73 (1967) 591–597. [16] J. P. Penot; Glimpses upon quasiconvex analysis, ESAIM: Proc. 20 (2007) 170–194. Behzad Djafari Rouhani Department of Mathematical Sciences, University of Texas at El Paso, 500 W. Univer- sity Ave., El Paso, TX 79968, USA Email address: behzad@utep.edu Mohsen Rahimi Piranfar Isfahan Mathematics House, Isfahan, Iran Email address: m.piranfar@gmail.com 1. Introduction 2. Continuous case 3. Discrete case 4. conclusions References