Homoclinic solutions in periodic difference equations with mixed nonlinearities

Lin, Guojian; Zhou, Zhan

doi:10.1002/mma.3474

Cited by 15 publications

(17 citation statements)

References 39 publications

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“…More precisely, comparing to the linking theorem, periodic approximation, generalized Nahari manifold approach, mountain pass lemma, Nehari manifold approach, and the mountain pass argument used in the known literature (for more details, please see other works) to study the open problem proposed by Pankov, in this paper, we attempt to use a new method, ie, ingeniously use a corollary with ( C ) c condition for Ekeland variational principle in the work of Motreanu et al to obtain the new existence of gap solitons of discrete periodic DNLS equation , which gives a positive answer to the aforementioned question, and the present result is very sharp.…”

Section: Introduction and Main Resultsmentioning

confidence: 62%

“…In fact, α = − ∞ implies β ≠ + ∞ , then, in order to solve the aforementioned question for the case of β ≠ + ∞ and

F_{n} false(t false) ⩾ 0

, we only need to study the case of β = + ∞ and

F_{n} false(t false) ⩽ 0

. Motivated by these ideas, in this paper, for some types of the nonlinearities f n such as

f_{n} false(t false) = O false({false| t false|}^{s_{1}} false)

as t →0 ( s 1 ∈ (0,1)), we obtain the existence of gap solitons for the case of β = + ∞ and

F_{n} false(t false) ⩽ 0

, which complements the existing ones . Moreover, coupling with the existing results, we give a complete positive answer to the open problem proposed by Pankov for the case of σ = 1 and

F_{n} false(t false) ⩾ 0

.…”

Section: Introduction and Main Resultsmentioning

confidence: 64%

“…( i i ) For the case of f n ( t )= O ( t ) as t →0, Pankov and Rothos and Lin and Zhou gave the existence of nontrivial solutions for the case of β ≠+ ∞ and

F_{n} false(t false) ⩾ 0

, and for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

, Lin and Zhou proved the nonexistence of nontrivial solution. (iii) For the case of

f_{n} false(t false) = O ((), {false| t false|}^{s_{1}})

as t →0

((), s_{1} \in false(0, 1 false))

, Lin and Zhou also obtained the nonexistence of nontrivial solution for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Let F n ( t ) be the primitive function of f n given by

F_{n} false(t false) = \int_{0}^{t} f_{n} false(x false) normald x,

and here, we mention the known results to this open problem for the case of σ =1 (the case of σ =−1 reduces to the previous one if we replace L by − L and ω by − ω ). ( i ) For the case of f n ( t )= o ( t ) as t →0, the authors in other works proved that there exist nontrivial solutions for the case of β ≠+ ∞ and

F_{n} false(t false) ⩾ 0

, and for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

, the authors in related works obtained the nonexistence of nontrivial solution. ( i i ) For the case of f n ( t )= O ( t ) as t →0, Pankov and Rothos and Lin and Zhou gave the existence of nontrivial solutions for the case of β ≠+ ∞ and

F_{n} false(t false) ⩾ 0

, and for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

, Lin and Zhou proved the nonexistence of nontrivial solution.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…( i ) For the case of f n ( t )= o ( t ) as t →0, the authors in other works proved that there exist nontrivial solutions for the case of β ≠+ ∞ and

F_{n} false(t false) ⩾ 0

, and for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

F_{n} false(t false) ⩾ 0

, and for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

, Lin and Zhou proved the nonexistence of nontrivial solution. (iii) For the case of

f_{n} false(t false) = O ((), {false| t false|}^{s_{1}})

as t →0

((), s_{1} \in false(0, 1 false))

, Lin and Zhou also obtained the nonexistence of nontrivial solution for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 4 more Smart Citations

Notes on gap solitons for periodic discrete nonlinear Schrödinger equations

Ding

Wei

2018

Math Methods in App Sciences

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Under the condition of n (u n ) = n |u n | 2 u n , by applying the linking theorem, Pankov obtained the existence of gap solitons. Pankov's result was then generalized to various type of nonlinear function f n , but most of studying works were concentrated on the case of n (t) = O (|t| s 1 ) as t → 0(s 1 ⩾ 1). In this paper, by ingeniously using Ekeland variational principle, for the nonlinearity which meets n (t) = O(|t| s 1 ) as t → 0 (s 1 ∈ (0, 1)), we obtain the existence of gap solitons, which supplements the existing ones and gives a positive answer in part to the open problem proposed by Pankov. KEYWORDS discrete nonlinear Schrödinger equations, Ekeland variational principle, gap solitons, periodic approximation Lu n − u n = n (u n ) , n ∈ Z,where {u n } n∈Z is a real valued sequence, = ±1, is the temporal frequency, f n (t) is continuous in t, f n + T (t) = f n (t) for each n ∈ Z, and f n is supposed to be gauge invariant, ie, n (e i t) = e i n (t), ∈ R, L is a Jacobi operator 9 given by Lu n = a n u n+1 + a n−1 u n−1 + b n u n , where {a n } , {b n } are real valued T-periodic sequences.Since L is a bounded and self-adjoint operator, the spectrum (L) of L has a band structure, ie, (L) is a union of a finite number of closed intervals. 9 Thus, the complement R ⧵ (L) consists of a finite number of open intervals called spectral Math Meth Appl Sci. 2018;41:6673-6682.wileyonlinelibrary.com/journal/mma

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Section: Introduction and Main Resultsmentioning

confidence: 62%

“…In fact, α = − ∞ implies β ≠ + ∞ , then, in order to solve the aforementioned question for the case of β ≠ + ∞ and

F_{n} false(t false) ⩾ 0

, we only need to study the case of β = + ∞ and

F_{n} false(t false) ⩽ 0

. Motivated by these ideas, in this paper, for some types of the nonlinearities f n such as

f_{n} false(t false) = O false({false| t false|}^{s_{1}} false)

as t →0 ( s 1 ∈ (0,1)), we obtain the existence of gap solitons for the case of β = + ∞ and

F_{n} false(t false) ⩽ 0

, which complements the existing ones . Moreover, coupling with the existing results, we give a complete positive answer to the open problem proposed by Pankov for the case of σ = 1 and

F_{n} false(t false) ⩾ 0

.…”

Section: Introduction and Main Resultsmentioning

confidence: 64%

“…( i i ) For the case of f n ( t )= O ( t ) as t →0, Pankov and Rothos and Lin and Zhou gave the existence of nontrivial solutions for the case of β ≠+ ∞ and

F_{n} false(t false) ⩾ 0

, and for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

, Lin and Zhou proved the nonexistence of nontrivial solution. (iii) For the case of

f_{n} false(t false) = O ((), {false| t false|}^{s_{1}})

as t →0

((), s_{1} \in false(0, 1 false))

, Lin and Zhou also obtained the nonexistence of nontrivial solution for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Let F n ( t ) be the primitive function of f n given by

F_{n} false(t false) = \int_{0}^{t} f_{n} false(x false) normald x,

F_{n} false(t false) ⩾ 0

, and for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

F_{n} false(t false) ⩾ 0

, and for the case of β =+ ∞ and

F_{n} false(t false) ⩾ 0

, Lin and Zhou proved the nonexistence of nontrivial solution.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…( i ) For the case of f n ( t )= o ( t ) as t →0, the authors in other works proved that there exist nontrivial solutions for the case of β ≠+ ∞ and