Two solutions to superlinear Hamiltonian systems with impulsive effects

Journal Pre-proof Two solutions to superlinear Hamiltonian systems with impulsive effects

Jian Liu, Wenguang Yu

PII: DOI: Reference:

S0893-9659(19)30488-4 https://doi.org/10.1016/j.aml.2019.106162 AML 106162

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Applied Mathematics Letters

Received date : 24 October 2019 Revised date : 27 November 2019 Accepted date : 27 November 2019 Please cite this article as: J. Liu and W. Yu, Two solutions to superlinear Hamiltonian systems with impulsive effects, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106162. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.

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Two solutions to superlinear Hamiltonian systems with impulsive effects Jian Liua , Wenguang Yub a School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, China b School of Insurance, Shandong University of Finance and Economics, Jinan, 250014, China Abstract: As continuing research of our paper [11], in this article the existence of two solutions to superlinear second-order Hamiltonian systems that have damped and impulsive effects is proved. The main tools used are variational methods and two recent two-critical-points theorems. An example is presented to prove the value of the obtained theorems.

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MSC: 34B15; 58E30 Keywords: Two non-zero solutions; Hamiltonian systems; Variational methods; Impulsive effects

1. Introduction

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In the past several years, new progress has been made in the research of solutions to Hamiltonian systems by variational methods in combination with critical-point theorems (see [1-3,6-12] and its literature). In [6], Hamiltonian systems that have non-coercive potential were studied. Based on critical-point theorems, sufficient conditions of the existence of two periodic solutions were obtained. In [10], to study the existence of at least one periodic solution to some second-order Hamiltonian systems with impulsive effects, variational methods and the saddle-point theorem were primarily applied when the potential F and the nonlinearity satisfy certain growth conditions. In [11], the main objective is to obtain triple solutions to the following Hamiltonian systems   u ¨(t) + g(t)u(t) ˙ − A(t)u(t) = −λb(t)∇H(u), a.e. t ∈ [0, T ],    (1.1) △u˙ i (tj ) = Iij (ui (tj )), i = 1, 2, ..., N ; j = 1, 2, ..., l,    u(0) − u(T ) = u(0) ˙ − u(T ˙ ) = 0, RT in which λ is a positive parameter, T is a positive constant, g ∈ L1 [0, T ] satisfying 0 g(s)ds = 0, A : [0, T ] → RN ×N , A(t) = (aij (t)) is a symmetric matrix with aij (t) ∈ L∞ ([0, T ]), and there exists a constant γ > 0 such that A(t)ξ · ξ ≥ γ|ξ|2 , ∀ξ ∈ RN , H ∈ C 1 (RN , R), 0 < b ∈ L1 ([0, T ]), and Iij (s) ∈ C(R, R)(i = 1, 2, ..., N ; j = 1, 2, ..., l). As continuing research of [11], the objective of this paper is to acquire at least two solutions to Hamiltonian systems (1.1). 2. Preliminaries

Define a Sobolev space HT1 := {u : [0, T ] → RN |u is absolutely continuous, u˙ ∈ L2 (0, T ; RN ) and u(0) = u(T )} with the norm Z T Z T 1 2 kuk0 = ( |u(t)| ˙ dt + |u(t)|2 dt) 2 . Let G(s) =

Rs 0

0

0

g(t)dt, mins∈[0,T ] eG(s) = α, maxs∈[0,T ] eG(s) = β.

1Supported by Shandong Provincial Natural Science Foundation,China(ZR2017MA048,ZR2018MG002), the Taishan

Scholars Program of Shandong Province(No.tsqn20161041), the Fostering Project of Dominant Discipline and Talent Team of Shandong Province Higher Education Institutions(1716009). E-mail addresses:[email protected](J. Liu)

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2

Put

Z

hu, vi =

T

eG(t) (u(t), ˙ v(t))dt ˙ +

0

and the corresponding norm is

Z kuk = (

0

T

Z

T

eG(t) (A(t)u(t), v(t))dt, ∀u, v ∈ HT1 ,

0

2 eG(t) |u(t)| ˙ dt +

Z

T

1

eG(t) (A(t)u(t), u(t))dt) 2 .

0

By a similar calculation as that in [9] presents √ √ ρ1 kuk0 ≤ kuk ≤ ρ2 kuk0,

where ρ1 = min{γ, 1}, ρ2 = max{ΣN i,j=1 kaij k∞ , 1}, kaij k∞ := maxs∈[0,T ] |aij (s)|, thus kuk is equivalent to kuk0 . Noticing that (HT1 , k · k) ֒→֒→ C([0, T ], RN ), then there exists a positive constant k such that kuk∞ := max |u(s)| ≤ kkuk, ∀u ∈ HT1 ,

(2.2)

s∈[0,T ]

RT 0

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Define the functional Kλ (u) = −λΨ(u) + Φ(u), in which Ψ(u) = l P N R ui (t ) P eG(tj ) 0 j Iij (t)dt.

j=1 i=1

eG(t) b(t)H(u)dt, Φ(u) = 12 kuk2 +

The definition of a weak solution to Hamiltonian systems (1.1) is listed below. Definition 2.1([11]). u ∈ HT1 is defined as a weak solution of Hamiltonian systems (1.1) if Z T Z T l X N X eG(tj ) Iij (ui (tj ))v i (tj ) = 0, eG(t) (u(t), ˙ v(t))dt ˙ + eG(t) (A(t)u(t) − λb(t)∇H(u), v(t))dt + 0

0

HT1 .

j=1 i=1

v∈ The following lemma and two theorems are to be employed in section 3 of the present paper. Lemma 2.2([11]). Kλ is Gˆ ateaux differentiable at any u ∈ HT1 and Z T l X N X ′ eG(tj ) Iij (ui (tj ))v i (tj ), eG(t) [(u(t), ˙ v(t)) ˙ + (A(t)u(t), v(t))]dt + hΦ (u), vi = 0

′

hΨ (u), vi =

Z

j=1 i=1

(2.3)

T

e

G(t)

b(t)(∇H(u(t)), v(t))dt,

0

∀v ∈

HT1 .

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Thus, we get that a critical point of the functional Kλ is also a weak solution to Hamiltonian systems (1.1). Theorem 2.3([4,Theorem 3.2]). Φ, Ψ : X → R are two continuously Gˆ ateaux differentiable functionals, where X is a real Banach space, and Φ is bounded from below and satisfies Φ(0) = Ψ(0) = 0. Choose a positive constant r satisfying supΦ(u) , Φ(¯ u) r Φ(¯ u) and Jλ = −λΨ + Φ satisfies the Palais-Smale condition and it is unbounded from below when Ψ(¯ u) < λ < r . Then J has at least two nontrivial critical points u , u such that J (u ) < 0 < J λ 1 2 λ 1 λ (u2 ). sup Ψ(u) Φ(u)≤r

3. Main Results The main results are the following two theorems on the existence of at least two distinct solutions to Hamiltonian systems (1.1). Theorem 3.1. Providing that there exist µ > 2, δ > 0 such that Z s N 0 < µH(u) ≤ ∇H(u)u, ∀ u ∈ R \ {0}, 0 < Iij (s)s ≤ µ Iij (t)dt ≤ µδ|s|2 , ∀s ∈ R \ {0}. 0

Then for λ ∈ (0,

c2 2k2 kbk1 max|u|≤c H(u) ),

Hamiltonian systems (1.1) have at least two distinct solutions.

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Proof. In view of 0 <

Rs 0

3

Iij (t)dt, ∀ s ∈ R, one has Φ(u) = 12 kuk2 + ≥ 12 kuk2 ,

l P N P

eG(tj )

j=1 i=1

R ui (tj ) 0

Iij (t)dt

thus, Φ is bounded from below. Let {un } ∈ HT1 such that {Kλ (un )} is bounded and satisfies Kλ′ (un ) → 0, when R u n → +∞, then we claim that {un } is bounded. In fact, by 0 < µH(u) ≤ ∇H(u)u and Iij (u)u ≤ µ 0 Iij (s)ds, taking into account (2.3), one has −Kλ′ (un )un + µKλ (un )

= ( µ2 − 1)kun k2 + µ −λµ

0

e

G(t)

eG(tj )

j=1 i=1

R uin (tj ) 0

b(t)H(un (t))dt + λ

− 1)kun k2 .

RT 0

e

Iij (t)dt −

G(t)

l P N P

j=1 i=1

eG(tj ) Iij (uin (tj ))uin (tj )

b(t)∇H(un (t))un (t))dt

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≥

( µ2

RT

l P N P

As a consequence, we draw that {un } is bounded because of µ > 2. As a result, there is a subsequence {unm } of {un } and some u ∈ HT1 such that {unm } ⇒ u in the interval [0, T ]. Consequently, when m → +∞, one has (Kλ′ (unm ) − Kλ′ (u))(unm − u) → 0, Z T eG(t) b(t)(∇H(unm ) − ∇H(u))(unm − u)dt → 0, 0

Therefore,

(Iij (uinm (tj )) − Iij (ui (tj )))(uinm (tj ) − ui (tj )) → 0, i = 1, 2, ..., N ; j = 1, 2, ..., l.

(Kλ′ (unm ) − Kλ′ (u))(unm − u) = Kλ′ (unm )(unm − u) − Kλ′ (u)(unm − u) RT RT = 0 eG(t) (u′nm (t) − u′ (t))2 dt + λ 0 eG(t) (unm (t) − u(t))2 dt +

l P N P

j=1 i=1

−

RT 0

eG(tj ) [Iij (uinm (tj )) − Iij (ui (tj ))](uinm (tj ) − ui (tj ))

eG(t) b(t)(∇H(unm ) − ∇H(u))(unm − u)dt

= kunm − uk2 +

l P N P

j=1 i=1

−

RT 0

eG(tj ) [Iij (uinm (tj )) − Iij (ui (tj ))](uinm (tj ) − ui (tj ))

eG(t) b(t)(∇H(unm ) − ∇H(u))(unm − u)dt,

ur

which leads to kunm − uk → 0, as m → +∞. Then Kλ satisfies the Palais-Smale condition. Noticing ¯ Taking ∀u ∈ H 1 \{0}, and that 0 < µH(u) exist α, ¯ β¯ > 0 such that H(u) ≥ α|u| ¯ µ − β. T R s ≤ ∇H(u)u, there 2 noticing that 0 Iij (t)dt ≤ δ|s| , ∀s ∈ R \ {0}, one has RT

Jo

Kλ (qu) =

≤

1 2

0

βq2 T 2

eG(t) A(t)|qu|2 dt +

l P N P

eG(tj )

j=1 i=1 l P N P

j=1 i=1

R qu 0

Iij (t)dt − λ

RT 0

eG(t) b(t)H(qu)dt

¯ kaij k∞ |u|2 + lq 2 N β|u|2 − λq µ ααkbk ¯ 1 |u|µ + λkbk1 β β,

thus we get Kλ (qu) → −∞(q → +∞),thus the unboundedness √ from below of Kλ is obtained. Choose u ∈ HT1√such that Φ(u) ≤ r, thus one has kuk ≤ 2r. Take into account of (2.2), thus we get kuk∞ ≤ kkuk ≤ k 2r := c. Therefore, Z T sup Ψ(u) ≤ β b(t) max H(u)dt = βkbk1 max H(u). (3.4) Φ(u)≤r

0

2

|u|≤c

|u|≤c

c ), Kλ has at least two distinct critical points according to Thus, for any λ ∈ (0, 2k2 βkbk1 max |u|≤c H(u) Theorem 2.3. Thus Hamiltonian systems (1.1) have at least two distinct solutions.

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4

Remark 3.2. Taking into account that H ∈ C 1 (RN , R), b ∈ L1 ([0, T ]), the condition 0 < µH(u) ≤ ∇H(u)u, ∀ u ∈ RN \ {0} in Theorem 3.1 can be replaced by 0 < µH(u) ≤ ∇H(u)u, ∀ |u| ≥ C1 , in which C1 is a positive constant. Let p = k2 T ΣN 1 kaij k∞ , the following result is on the existence of at least two nontrivial solutions. i,j=1

Theorem 3.3. Assumeqthat the conditions of Theorem 3.1 are satisfied. Moreover, there exist c > 0 p ¯
2

¯2

2

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δlN )ξ 1 c ( β (1+2pk then for every 0 < b ∈ L1 ([0, T ]), when λ ∈ 2k2 βkbk , max|ξ|≤c ¯ H(ξ) ), Hamiltonian systems pαH(ξ) 1 (1.1) have at least two nontrivial solutions. R ¯ according to the condition 0 < s Iij (t)dt, ∀ s ∈ R, one has Proof. Choose v¯(t) = ξ, 0 Z v¯i (tj ) l N XX 1 Φ(¯ v ) = k¯ eG(tj ) Iij (s)ds v k2 + 2 0 j=1 i=1 (3.5) γαT ¯ 2 ≥ |ξ| > 0, 2 and Z v¯i (tj ) l X N X 1 eG(tj ) Iij (s)ds v k2 + Φ(¯ v ) = k¯ 2 0 j=1 i=1

≤

l X N X β ¯2 ¯2 |ξ| | ξ| + βδ 2k 2 p j=1 i=1

(3.6)

2 β ¯ 2 ≤ c = r. + βδlN )|ξ| 2 2k p 2k 2 In view of (3.5) and (3.6), we secure 0 < Φ(¯ v ) < r. R T G(t) ¯ Simple calculations show Ψ(¯ v) = 0 e b(t)H(¯ v )dt ≥ βH(ξ)kbk 1 , in combination with (3.4) and (3.6), one has supΦ(u)≤r Ψ(u) βkbk1 max|u|≤c H(u) ≤ r r 2k 2 βkbk1 max|u|≤c H(u) = c2 ¯ pα H(ξ) ≤ 2k 2 βkbk1 2 2 2 ¯ β (1 + 2pk δlN ) |ξ| Ψ(¯ v) , ≤ Φ(¯ v)

ur

≤(

Jo

thus the conditions of Theorem 2.4 are all satisfied. Consequently, Kλ has at least two nontrivial critical points, i.e. Hamiltonian systems (1.1) have at least two nontrivial solutions. Example 3.4.(Take T = 1, N = 1, A(t) ≡ 1/400, t1 = 12 , g(t) ≡ 1, b(t) ≡ 1, H(u) = |u|2 (|u|2 − 1)(|u|2 − 1

u 7 , |u| > 1, u7 , |u| ≤ 1. Take Hamiltonian systems (1.1) as the objective of study, and choose c = 10, δ = 87 , µ = 3, by sim¯ < 1 such that max|ξ|≤10 H(ξ) = ple calculations, one has k = 2, p = 100, and there exists 0 < |ξ| ¯ ¯ = 24.88, max|ξ|≤1 H(ξ) = H(ξ), furthermore, by Newton-iterative method we obtain ξ¯ = 0.706, H(ξ) thus 100), I11 (u) =

¯ ¯ max|ξ|≤10 H(ξ) H(ξ) H(ξ) 100 pα pα ¯ = ¯ < = ≤ H(ξ) H(ξ) . 2 2 2 2 2 2 2 2 ¯ 10 10 15e β (1 + 2pk δlN ) β (1 + 2pk δlN ) |ξ| In addition, λ∈

β 2 (1 + 2pk 2 δlN ) ξ¯2 1 c2 ( , ¯ max|u|≤c H(u) ) = (0.096, 0.184), 2k 2 βkbk1 pα H(ξ)

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by applying Remark 3.2 and Theorem 3.3, we conclude that there are at least two nontrivial solutions to Hamiltonian systems (1.1). References

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[1] J. Nieto, D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal. 10 (2009) 680-690. [2] J. Nieto, Variational formulation of a damped Dirichlet impulsive problem, Appl. Math. Lett. 23 (2010) 940-942. [3] J. Sun, H. Chen, J. Nieto, M. Otero-Novoa, Multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. 72 (2010) 4575-4586. [4] G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012) 205-220. [5] G. Bonanno, G. D’Agu`ı, Two Non-Zero Solutions for Elliptic Dirichlet Problems, Z. Anal. Anwend. 35 (2016) 449-464. [6] G. Bonanno, R. Livrea, Multiple periodic solutions for Hamiltonian systems with not coercive potential, J. Math. Anal. Appl. 363 (2010) 627-638. [7] G. Bonanno, P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differ. Equations 12 (2008) 3031-3059. [8] J. Liu, Z. Zhao, An application of variational methods to second-order impulsive differential equation with derivative dependence, Electron. J. Differ. Eq. 2014 (2014) 1-13. [9] J. Liu, Z. Zhao, Multiple solutions for impulsive problems with non-autonomous perturbations, Appl. Math. Lett. 64 (2017) 143-149. [10] J. Liu, Z. Zhao, Variational approach to second-order damped Hamiltonian systems with impulsive effects, J. Nonlinear Sci. Appl. 9 (2016) 3459-3472. [11] J. Liu, Z. Zhao, T. Zhang, Multiple solutions to damped Hamiltonian systems with impulsive effects, Appl. Math. Lett. 91 (2019) 173-180. [12] J. Liu, Z. Zhao, W. Yu, T. Zhang, Triple solutions for a damped impulsive differential equation, Adv. Differ. Equ. 2019 (2019) 1-8.

*Author Contributions Section

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Author contributions All authors contributed equally to the writing of this paper.

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All authors read and approved the final manuscript.