CUBO, A Mathematical Journal Vol. 24, no. 02, pp. 227–237, August 2022 DOI: 10.56754/0719-0646.2402.0227 Variational methods to second-order Dirichlet boundary value problems with impulses on the half-line Meriem Djibaoui 1 Toufik Moussaoui 1, B 1Laboratory of Fixed Point Theory and Applications, École Normale Supérieure, Kouba, Algiers. Algeria. djibaouimeriem@gmail.com toufik.moussaoui@g.ens-kouba.dz B ABSTRACT In this paper, the existence of solutions for a second-order impulsive differential equation with a parameter on the half- line is investigated. Applying Lax-Milgram Theorem, we deal with a linear Dirichlet impulsive problem, while the non- linear case is established by using standard results of critical point theory. RESUMEN En este art́ıculo, se investiga la existencia de soluciones de una ecuación diferencial de segundo orden impulsiva con un parámetro en la semi-recta. Aplicando el Teorema de Lax- Milgram, tratamos un problema lineal impulsivo de Dirich- let, mientras que el caso no lineal es establecido usando re- sultados estándar de teoŕıa de punto cŕıtico. Keywords and Phrases: Dirichlet boundary value problem, half-line, Lax-Milgram theorem, critical points, impulsive differential equation. 2020 AMS Mathematics Subject Classification: 34B37, 34B40, 35A15, 35B38. Accepted: 30 March, 2022 Received: 04 August, 2021 c©2022 M. Djibaoui et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ https://doi.org/10.56754/0719-0646.2402.0227 https://orcid.org/0000-0002-0917-0394 mailto:toufik.moussaoui@g.ens-kouba.dz https://orcid.org/0000-0001-7495-0269 mailto:djibaouimeriem@gmail.com mailto:toufik.moussaoui@g.ens-kouba.dz 228 M. Djibaoui & T. Moussaoui CUBO 24, 2 (2022) 1 Introduction In recent years, many researchers have extensively applied variational methods to study boundary value problems (BVPs) for impulsive differential equations on the finite intervals. More precisely, employing critical point theory, Nieto and O’Regan [8] studied a linear Dirichlet boundary value problem with impulses        −u′′(t) + λu(t) = σ(t), a.e. t ∈ [0, T ], △u′(tj) = dj, j ∈ {1, 2, . . . , l}, u(0) = u(T ) = 0, (1.1) and a nonlinear impulsive problem        −u′′(t) + λu(t) = f(t, u(t)), a.e. t ∈ [0, T ], △u′(tj) = Ij(u(t − j )), j ∈ {1, 2, . . . , l}, u(0) = u(T ) = 0, (1.2) where λ is a positive parameter. Moreover, the study of solutions for impulsive BVPs on the infinite intervals by using variational methods has received considerably more attention, see for example [1, 2, 3, 9, 10], and the references therein. In the present paper, our aim is to improve some assumptions made in [8] in order to extend problems (1.1) and (1.2) on the half-line via variational approach. This paper is organized as follows. In Section 2 we state some preliminaries. In Section 3 we consider the linear Dirichlet problem with impulses in the derivative. Due to the Lax-Milgram Theorem, we show the existence of weak solutions that are precisely the critical points of some functionals. The last section is to deal with the nonlinear Dirichlet problem. To investigate the existence of solutions, we use standard results of critical point theory. Also, some examples are given to illustrate our main results. 2 Preliminaries We cite some basic and celebrated theorems from critical point theory which are crucial tools in the proof of our main results. Let H be a Hilbert space. Theorem 2.1 (Lax-Milgram [4, 5]). Let a : H × H → R be a bounded bilinear form. If a is coercive, i.e., there exists α > 0 such that a(u, u) ≥ α‖u‖2 for every u ∈ H, then for any σ ∈ H′ (the conjugate space of H) there exists a unique u ∈ H such that a(u, v) = (σ, v), for every v ∈ H. CUBO 24, 2 (2022) Variational methods to second-order Dirichlet boundary value... 229 Moreover, if a is also symmetric, then the functional ϕ : H → R defined by ϕ(v) = 1 2 a(v, v) − (σ, v) attains its minimum at u. Theorem 2.2 ([7]). If ϕ is weakly lower semi-continuous (w.l.s.c.) on a reflexive Banach space X and has a bounded minimizing sequence, then ϕ has a minimum on X. Now, let us recall some necessary concepts that will be needed in our argument. Let us define the following reflexive Banach space H10 (0, ∞) = { u : [0, ∞) → R is absolutely continuous, u, u′ ∈ L2(0, ∞), u(0) = u(∞) = 0 } , equipped with the norm ‖u‖ =   +∞ ∫ 0 |u(t)|2dt + +∞ ∫ 0 |u′(t)|2dt   1 2 . Set the space Cl,p[0, +∞) = {u ∈ C([0, +∞), R) : lim t→∞ p(t)u(t) exists} with the norm ‖u‖∞,p = sup t∈[0,+∞) p(t)|u(t)|, where the function p : [0; +∞) → (0, +∞) is continuously differentiable and bounded, satisfying C = 2 max(‖p‖L2, ‖p ′‖L2) < +∞. Concerning the above spaces, we get the following vital embeddings. Lemma 2.3 ([6]). The space H10 (0, ∞) embeds continuously in Cl,p[0, ∞), more precisely ‖u‖∞,p ≤ C‖u‖ for every u ∈ H10 (0, ∞). Lemma 2.4 ([6]). The embedding H10 (0, ∞) →֒ Cl,p[0, ∞) is compact. 3 Impulsive linear problem We consider the following linear Dirichlet boundary value problem with impulses in the derivative at the prescribed instants tj, j ∈ N ∗ = {1, 2, 3, . . .}        −u′′(t) + λu(t) = σ(t), a.e. t ∈ [0, ∞), t 6= tj, △u′(tj) = d(tj), j ∈ N ∗, u(0) = u(+∞) = 0, (3.1) 230 M. Djibaoui & T. Moussaoui CUBO 24, 2 (2022) where λ ∈ R, σ ∈ L2(0, ∞), 0 = t0 < t1 < t2 < · · · < tj < · · · < tm → ∞, as m → ∞, are the impulse points, d : [0, ∞) → R satisfies ∞ ∑ j=1 d(tj) p(tj) < ∞ and △u′(tj) = u ′(t+j ) − u ′(t−j ) for u′(t±j ) = limt→t± j u′(t). Now, multiply the equation in problem (3.1) by v ∈ H10 (0, ∞), and then integrate over (0, +∞), we obtain − +∞ ∫ 0 u ′′ v + λ +∞ ∫ 0 uv = +∞ ∫ 0 σv. We have − +∞ ∫ 0 u′′v = − ∞ ∑ j=0 tj+1 ∫ tj u′′v, and tj+1 ∫ tj u′′v = u′(t−j+1)v(t − j+1) − u ′(t+j )v(t + j ) − tj+1 ∫ tj u′v′. Consequently, − +∞ ∫ 0 u ′′ v = ∞ ∑ j=1 △u′(tj)v(tj) + u ′(0)v(0) − u′(∞)v(∞) + +∞ ∫ 0 u ′ v ′ = ∞ ∑ j=1 d(tj)v(tj) + +∞ ∫ 0 u′v′. This leads to define the bilinear form a : H10 (0, ∞) × H 1 0 (0, ∞) → R, by a(u, v) = +∞ ∫ 0 u ′ v ′ + λ +∞ ∫ 0 uv, (3.2) and the linear operator l : H10 (0, ∞) → R by l(v) = +∞ ∫ 0 σv − ∞ ∑ j=1 d(tj)v(tj). (3.3) Definition 3.1. We say that a function u is a weak solution of the impulsive problem (3.1) if u ∈ H10 (0, ∞) such that a(u, v) = l(v) is valid for any v ∈ H 1 0 (0, ∞). In what follows we refer to problem (3.1) as (LP). It is easily verified that a and l defined by (3.2), (3.3) respectively are continuous, and a is coercive if λ > 0. Consider the functional ϕ : H10 (0, ∞) → R, defined by ϕ(u) = 1 2 +∞ ∫ 0 u′2 + λ 2 +∞ ∫ 0 u2 − +∞ ∫ 0 σu + ∞ ∑ j=1 d(tj)u(tj). (3.4) CUBO 24, 2 (2022) Variational methods to second-order Dirichlet boundary value... 231 It is clear that ϕ is differentiable at any u ∈ H10 (0, ∞) and ϕ′(u)v = +∞ ∫ 0 u ′ v ′ + λ +∞ ∫ 0 uv − +∞ ∫ 0 σv + ∞ ∑ j=1 d(tj)v(tj) = a(u, v) − l(v). Thus, a critical point of (3.4) gives us a weak solution of the problem (LP). Definition 3.2. We mean by a classical solution of the problem (LP) a function u ∈ H2(tj, tj+1) for all j ∈ N∗, where H2(tj, tj+1) = { u : [0, ∞) → R is absolutely continuous, u′, u′′ ∈ L2(tj, tj+1) } , and u satisfies the first equation of (3.1) a.e. on [0, ∞) with u(0) = u(∞) = 0, the limits u′(t+j ), u′(t−j ), j ∈ N ∗ exist and the impulse conditions hold. Lemma 3.3. If u ∈ H10 (0, ∞) is a weak solution of (LP), then u is a classical solution of (LP). Proof. Since u ∈ H10 (0, ∞), it is evident that u(0) = u(∞) = 0. For j ∈ {1, 2, . . .}, choose any v ∈ H10 (0, ∞) such that v(t) = 0 for t ∈ [0, tj] ∪ [tj+1, +∞). Then tj+1 ∫ tj u′v′ + λ tj+1 ∫ tj uv = tj+1 ∫ tj σv. Hence, −u′′ + λu = σ a.e. on (tj, tj+1). So, u ∈ H 2(tj, tj+1) and satisfies the previous equation a.e. on [0, ∞). Multiplying −u′′ + λu = σ by v ∈ H10 (0, ∞) and integrating over [0, ∞), we get ∞ ∑ j=1 △u′(tj)v(tj) = ∞ ∑ j=1 d(tj)v(tj). Therefore, △u′(tj) = d(tj) for every j ∈ N ∗, and the impulsive conditions are satisfied. Lemma 3.4. If u ∈ H10 (0, ∞) is a critical point of ϕ defined by (3.4), then u is a weak solution of the impulsive Dirichlet problem (LP). Proof. Let u ∈ H10 (0, ∞). The assumption that u is a critical point of ϕ means that ϕ ′(u)v = 0, for all v ∈ H10 (0, ∞). Thus, +∞ ∫ 0 u′v′ + λ +∞ ∫ 0 uv − +∞ ∫ 0 σv + ∞ ∑ j=1 d(tj)v(tj) = 0, ∀v ∈ H 1 0 (0, ∞). Hence, +∞ ∫ 0 u′v′ + λ +∞ ∫ 0 uv = +∞ ∫ 0 σv − ∞ ∑ j=1 d(tj)v(tj), ∀v ∈ H 1 0 (0, ∞). This implies that a(u, v) = l(v) is valid for any v ∈ H10 (0, ∞). As a result, u is a weak solution of the (LP). 232 M. Djibaoui & T. Moussaoui CUBO 24, 2 (2022) In view of Lax-Milgram theorem, we formulate the following main result. Theorem 3.5. If λ > 0, then the Dirichlet impulsive problem (LP) has a weak solution u ∈ H10 (0, ∞) for any σ ∈ L 2(0, ∞). Moreover, u ∈ H2(0, ∞) and u is a classical solution and minimizes the functional (3.4) and hence it is a critical point of (3.4). Proof. For λ > 0, it follows that the bilinear a is coercive. The fact that a is continuous, by applying Theorem 2.1, for any σ ∈ L2(0, ∞), there exists a unique u ∈ H10 (0, ∞) such that a(u, v) = l(v) for all v ∈ H10 (0, ∞). So, the problem (LP) has a weak solution u ∈ H 1 0 (0, ∞). Owing to Lemma 3.3, a weak solution of (LP) is a classical solution. In addition, a is symmetric, then the functional ϕ attains its minimum at u which is exactly a critical point of ϕ since it is differentiable. Example 3.6. As an example, let λ = 1 and p(t) = 1 1+t2 · This impulsive boundary value problem        −u′′(t) + u(t) = 1 1+t , a.e. t ∈ [0, ∞), △u′(j) = e−j, j ∈ N∗, u(0) = u(+∞) = 0, (3.5) has a solution. 4 Impulsive nonlinear problem In the nonlinear situation we consider the following impulsive boundary value problem        −u′′(t) + λu(t) = f(t, u(t)), a.e. t ∈ [0, ∞), t 6= tj, △u′(tj) = g(tj)Ij(u(t − j )), j ∈ N ∗, u(0) = u(+∞) = 0, (4.1) where λ is a positive parameter, the functions f : [0, ∞) × R → R, Ij : R → R, j ∈ N ∗, and g : [0, ∞) → [0, ∞) are continuous with ∞ ∑ j=1 g(tj) < ∞. We refer to problem (4.1) as (NP). Definition 4.1. A weak solution of (NP) is a function u ∈ H10 (0, ∞) such that +∞ ∫ 0 u′v′ + λ +∞ ∫ 0 uv + ∞ ∑ j=1 g(tj)Ij(u(tj))v(tj) − +∞ ∫ 0 f(t, u(t))dt = 0, for every v ∈ H10 (0, ∞). CUBO 24, 2 (2022) Variational methods to second-order Dirichlet boundary value... 233 Setting F(t, u) = u ∫ 0 f(t, s)ds, we define the functional ϕ : H10 (0, ∞) → R by ϕ(u) = 1 2 +∞ ∫ 0 u′2(t)dt + λ 2 +∞ ∫ 0 u2(t)dt + ∞ ∑ j=1 g(tj) u(tj) ∫ 0 Ij(s)ds − +∞ ∫ 0 F(t, u(t))dt. (4.2) Now we present our principal results for this part. Theorem 4.2. Suppose that the following conditions hold: (H1) There exists a positive bounded function M ∈ L 1(0, +∞) with M p ∈ L1(0, +∞) such that |f(t, u)| ≤ M(t) for (t, u) ∈ [0, +∞) × R. (I1) There exist Mj > 0, j ∈ N ∗, satisfying ∞ ∑ j=1 Mjg(tj) < ∞ and ∞ ∑ j=1 Mjg(tj) p(tj) < ∞, such that the impulsive functions Ij are bounded i.e., |Ij(u)| ≤ Mj for every u ∈ R, j ∈ {1, 2, . . .}. Then there is a critical point of ϕ, and (NP) has at least one solution. Proof. Claim 1. ϕ is weakly lower semi-continuous (w.l.s.c). Let (un) ⊂ H 1 0 (0, ∞) be a sequence such that un ⇀ u in H 1 0 (0, ∞), when n → ∞. Then, ‖u‖ ≤ lim inf n→∞ ‖un‖, and by Lemma 2.4 we have that (un) converges to u in Cl,p[0, ∞), hence un(t) converges to u(t) for all t ∈ [0, ∞). From (H1) and (I1), using the continuity of f and Ij, j ∈ N ∗, together with the Lebesgue Dominated Convergence Theorem, we obtain lim inf n→+∞ ϕ(un) = lim inf n→+∞    1 2 +∞ ∫ 0 u ′2 n + λ 2 +∞ ∫ 0 u2n + ∞ ∑ j=1 g(tj) un(tj) ∫ 0 Ij(s)ds − +∞ ∫ 0 F(t, un(t))dt    ≥ 1 2 +∞ ∫ 0 u ′2 + λ 2 +∞ ∫ 0 u 2 + ∞ ∑ j=1 g(tj) u(tj) ∫ 0 Ij(s)ds − +∞ ∫ 0 F(t, u(t))dt = ϕ(u). Thus, ϕ is w.l.s.c. Claim 2. ϕ is coercive. For any u ∈ H10 (0, ∞), the fact that λ > 0, there exists α > 0 such that ϕ(u) ≥ α‖u‖2 + ∞ ∑ j=1 g(tj) u(tj ) ∫ 0 Ij(s)ds − +∞ ∫ 0 F(t, u(t))dt. 234 M. Djibaoui & T. Moussaoui CUBO 24, 2 (2022) Using conditions (H1), (I1) and Lemma 2.3, we have ϕ(u) ≥ α‖u‖2 − ∞ ∑ j=1 Mjg(tj) p(tj) p(tj)|u(tj)| − +∞ ∫ 0 M(t) p(t) p(t)|u(t)|dt ≥ α‖u‖2 − ‖u‖∞,p ∞ ∑ j=1 Mjg(tj) p(tj) − ‖u‖∞,p +∞ ∫ 0 M(t) p(t) dt ≥ α‖u‖2 − C‖u‖ ∞ ∑ j=1 Mjg(tj) p(tj) − C‖u‖ ∥ ∥ ∥ ∥ M p ∥ ∥ ∥ ∥ L1 ≥ α‖u‖2 − C   ∞ ∑ j=1 Mjg(tj) p(tj) + ∥ ∥ ∥ ∥ M p ∥ ∥ ∥ ∥ L1  ‖u‖, for some C > 0. Then, the above inequality implies that lim ‖u‖→+∞ ϕ(u) = +∞. Hence, ϕ is coercive. Applying Theorem 2.2, ϕ possesses a minimum which is a critical point of ϕ. Finally, by (H1) and (I1), it is easy to check that ϕ is continuous and differentiable for any u ∈ H 1 0 (0, ∞) and that ϕ′(u)v = +∞ ∫ 0 u ′ v ′ + λ +∞ ∫ 0 uv + ∞ ∑ j=1 g(tj)Ij(u(tj))v(tj)dt − +∞ ∫ 0 f(t, u(t))v(t)dt. (4.3) Therefore, a critical point of ϕ is a weak solution of the problem (NP). Remark 4.3. Assume M ∈ L2(0, ∞) in (H1), then it is easy to see that a weak solution u is in H2(0, ∞). Example 4.4. Take λ = 1, p(t) = e−t, M(t) = e−2t, g(t) = e−2t, Mj = 1 j and Ij(s) = 1 j + s2 , j ∈ N∗. The following IBVP:          −u′′(t) + u(t) = e−3t, a.e. t ∈ [0, ∞), △u′(j) = e−2j j + u2(j) , j ∈ N∗, u(0) = u(+∞) = 0, has at least one solution. (See Figure 1) CUBO 24, 2 (2022) Variational methods to second-order Dirichlet boundary value... 235 Figure 1 Theorem 4.5. Assume the following conditions are satisfied: (H2) The function f is sublinear i.e., there exist a constant γ ∈ [0, 1) and positive functions a, b ∈ L1(0, ∞) with a p , b pγ , b pγ+1 ∈ L1[0, ∞) such that |f(t, u)| ≤ a(t) + b(t)|u|γ for (t, u) ∈ [0, +∞) × R. (I2) There exist constants δ ∈ [0, 1) and aj, bj > 0, j ∈ {1, 2, . . .} with ∞ ∑ j=1 ajg(tj), ∞ ∑ j=1 ajg(tj) p(tj) , ∞ ∑ j=1 bjg(tj) pδ(tj) , ∞ ∑ j=1 bjg(tj) pδ+1(tj) are convergent series, such that the impulsive functions Ij have sublinear growths i.e., |Ij(u)| ≤ aj + bj|u| δ for every u ∈ R, j ∈ {1, 2, . . .}. Then there is a critical point of ϕ, and (NP) has at least one solution. Proof. Claim 1. ϕ is weakly lower semi-continuous. Under (H2) and (I2), arguing analogously to the proof of Theorem 4.2, we find the weak lower semi-continuity of ϕ. Claim 2. ϕ is coercive. In view of conditions (H2), (I2) and (4.2), for any u ∈ H 1 0 (0, ∞), we have ϕ(u) = 1 2 +∞ ∫ 0 u ′2 + λ 2 +∞ ∫ 0 u 2 + ∞ ∑ j=1 g(tj) u(tj) ∫ 0 Ij(s)ds − +∞ ∫ 0 F(t, u(t))dt ≥ α‖u‖2 − ∞ ∑ j=1 g(tj) u(tj) ∫ 0 (aj + bj|s| δ)ds − +∞ ∫ 0 ( a(t)|u(t)| + b(t) γ + 1 |u(t)|γ+1 ) dt 236 M. Djibaoui & T. Moussaoui CUBO 24, 2 (2022) ϕ(u) ≥ α‖u‖2 − ∞ ∑ j=1 g(tj) ( aj p(tj) p(tj)|u(tj)| + bj (δ + 1)pδ+1(tj) |p(tj)u(tj)| δ+1 ) − +∞ ∫ 0 a(t) p(t) p(t)|u(t)|dt − 1 (γ + 1) +∞ ∫ 0 b(t) pγ+1(t) |p(t)u(t)|γ+1dt ≥ α‖u‖2 − ‖u‖∞,p ∞ ∑ j=1 ajg(tj) p(tj) − ‖u‖δ+1∞,p ∞ ∑ j=1 bjg(tj) pδ+1(tj) − ‖u‖∞,p ∥ ∥ ∥ ∥ a p ∥ ∥ ∥ ∥ L1 − ‖u‖γ+1∞,p ∥ ∥ ∥ ∥ b pγ+1 ∥ ∥ ∥ ∥ L1 . Hence, by Lemma 2.3, we get ϕ(u) ≥ α‖u‖2 − C‖u‖ ∞ ∑ j=1 ajg(tj) p(tj) − Cδ+1‖u‖δ+1 ∞ ∑ j=1 bjg(tj) pδ+1(tj) − C‖u‖ ∥ ∥ ∥ ∥ a p ∥ ∥ ∥ ∥ L1 − Cγ+1‖u‖γ+1 ∥ ∥ ∥ ∥ b pγ+1 ∥ ∥ ∥ ∥ L1 ≥ α‖u‖2 − C   ∥ ∥ ∥ ∥ a p ∥ ∥ ∥ ∥ L1 + ∞ ∑ j=1 ajg(tj) p(tj)  ‖u‖ − Cδ+1   ∞ ∑ j=1 bjg(tj) pδ+1(tj)  ‖u‖δ+1 − Cγ+1 ∥ ∥ ∥ ∥ b pγ+1 ∥ ∥ ∥ ∥ L1 ‖u‖γ+1. Since δ, γ ∈ [0, 1), then lim ‖u‖→+∞ ϕ(u) = +∞. This means, ϕ is coercive. Using Theorem 2.2, ϕ has a minimum, which is a critical point of ϕ. Finally, from (H2) and (I2), we get the differentiability of ϕ such that its differentiable is defined by (4.3). Consequently, (NP) has at least one solution. Remark 4.6. In (H2), assume a, b pγ ∈ L2(0, ∞), then a weak solution u is in H2(0, ∞). Example 4.7. Consider the following problem            −u′′(t) + u(t) = e−2t √ |u(t)| + e−3t, a.e. t ∈ [0, ∞), △u′(j) = e−2j ( 1 j2 + |s| 1 4 j ) , j ∈ N∗, u(0) = u(+∞) = 0, where λ = 1, p(t) = e−t, g(t) = e−2t, aj = 1 j2 , bj = 1 j and Ij(s) = 1 j2 + |s| 1 4 j , j ∈ N∗. By simple calculations, all conditions in Theorem 4.5 are satisfied, then (4.1) has at least one solution. CUBO 24, 2 (2022) Variational methods to second-order Dirichlet boundary value... 237 References [1] L. Bai and J. J. Nieto, “Variational approach to differential equations with not instantaneous impulses”, Appl. Math. Lett., vol. 73, pp. 44–48, 2017. [2] V. Barutello, R. Ortega and G. Verzini, “Regularized variational principles for the perturbed Kepler problem”, Adv. Math., vol. 383, Paper No. 107694, 64 pages, 2021. [3] D. Bouafia and T. Moussaoui, “Existence results for a sublinear second order Dirichlet bound- ary value problem on the half-line”, Opuscula Math., vol. 40, no. 5, pp. 537–548, 2020. [4] H. 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Wei, “Existence and uniqueness of solutions for a second-order delay differential equation boundary value problem on the half-line”, Bound. Value Probl., Art. ID 752827, 14 pages, 2008. Introduction Preliminaries Impulsive linear problem Impulsive nonlinear problem