On sequences of large homoclinic solutions for a difference equation on the integers involving

On sequences of large homoclinic solutions for a difference equation on the integers involving

oscillatory nonlinearities

Robert Stegli ´nski

Institute of Mathematics, Technical University of Lodz, Wolczanska 215, 90-924 Lodz, Poland

Received 9 March 2016, appeared 2 June 2016 Communicated by Stevo Stevi´c

Abstract. In this paper, we determine a concrete interval of positive parameters λ, for which we prove the existence of infinitely many homoclinic solutions for a discrete problem

−_∆ a(k)φp(_∆u(k−1))+b(k)φp(u(k)) =λf(k,u(k)), k∈_Z,

where the nonlinear term f : Z×_R → _Rhas an appropriate oscillatory behavior at infinity, without any symmetry assumptions. The approach is based on critical point theory.

Keywords: difference equations, discrete p-Laplacian, variational methods, infinitely many solutions.

2010 Mathematics Subject Classification: 39A10, 47J30, 35B38.

1 Introduction

In the present paper we deal with the following nonlinear second-order difference equation:

(−_∆ a(k)φp(_∆u(k−1))+b(k)φp(u(k)) =λf(k,u(k)) for allk∈_Z

u(k)→0 as|k| →_∞. (1.1)

Here p > 1 is a real number, λ is a positive real parameter, φ_p(t) = |t|^p−2t for all t ∈ _R, a,b : Z → (0,+_∞), while f : Z×_R → _R is a continuous function. Moreover, the forward difference operator is defined as ∆u(k−1) = u(k)−u(k−1). We say that a solution u = {u(k)}of (1.1) is homoclinic if lim_|k|→∞u(k) =0.

The problem (1.1) is in a class of partial difference equations which usually describe the evolution of certain phenomena over the course of time. The theory of nonlinear discrete dynamical systems has been used to examine discrete models appearing in many fields such as computing, economics, biology and physics.

BEmail: robert.steglinski@p.lodz.pl

Boundary value problems for difference equations can be studied in several ways. It is well known that variational method in such problems is a powerful tool. Many authors have applied different results of critical point theory to prove existence and multiplicity results for the solutions of discrete nonlinear problems. Studying such problems on bounded discrete intervals allows for the search for solutions in a finite-dimensional Banach space (see [1,2,5, 6,14]). The issue of finding solutions on unbounded intervals is more delicate. To study such problems directly by variational methods, [13] and [8] introduced coercive weight functions which allow for preservation of certain compactness properties onl^p-type spaces.

The goal of the present paper is to establish the existence of a sequence of homoclinic solu- tions for the problem (1.1), which has been studied recently in several papers. Infinitely many solutions were obtained in [20] by employing Nehari manifold methods, in [9] by applying a variant of the fountain theorem (but see Section 5), and in [18] by use of the Ricceri’s theorem (see [3,17]). In this present paper, the result will be achieved by providing the nonlinearity with a suitable oscillatory behavior. For this kind of nonlinearity see [10–12]. We refer to [7,15,16,19] for related results that involve differential operators with variable exponents.

A special case of our contributions reads as follows. For b : Z → _R and the continuous mapping f :Z×_R→_Rdefine the following conditions:

(_B) _b(_k)≥b0>_{0 for allk} ∈_Z, b(_k)→+_∞as |k| →+_∞;

(F₁) lim

t→0

|f(k,t)|

|t|^p−1 =0 uniformly for allk∈ _Z;

(F₂) there are sequences {c_n},{d_n} such that 0 < c_n < d_n < c_n+1, lim_n→_∞c_n = +_∞ and f(k,t)≤0 for everyk ∈_Zandt∈ [cn,dn],n ∈_N

(F₃) there isr<0 such that sup_t∈[r,dn]|F(·.t)| ∈l₁for all n∈_N;

(F₄⁺) lim sup

(k,t)→(+_∞,+_∞)

F(k,t)

[a(k+1) +a(k) +b(k)]t^p = +_∞; (F₄⁻) lim sup

(k,t)→(−_∞,+_∞)

F(k,t)

[a(k+1) +a(k) +b(k)]t^p = +_∞;

(_F5) _sup

k∈_Z

lim sup

t→+_∞

F(k,t)

[a(k+1) +a(k) +b(k)]t^p

!

= +_∞,

where F(k,t)is the primitive function of f(k,t), that is F(k,t) = Rt

0 f(k,s)ds for everyt ∈ _R andk ∈Z. The solutions are found in the normed space(X,k·k), where

X= (

u:Z→_{R :}

∑

k∈_Z

a(k)|_∆u(k−₁)|^p+b(k)|u(k)|^p< _∞ )

and

kuk=

∑

k∈_Z

a(k)|_∆u(k−1)|^p+b(k)|u(k)|^p

!¹_p .

Theorem 1.1. Assume that(A), (F₁),(F₂)and(F₃)are satisfied. Moreover, assume that at least one of the conditions(F₄⁺),(F₄⁻),(F5)is satisfied. Then, for anyλ>0,the problem(1.1)admits a sequence of non-negative solutions in X whose norms tend to infinity.

The plan of the paper is as follows: Section 2 is devoted to our abstract framework, while Section 3 is dedicated to the main result. In Section 4 we give two examples of the indepen- dence of conditions (F₄⁺)and(F₅). Finally, we compare our result with other known results.

2 Abstract framework

We begin by defining some Banach spaces. For all 1 ≤ p < +∞, we denote `^p the set of all functionsu:Z→_Rsuch that

kuk^p_p =

∑

k∈_Z

|u(k)|^p <+_∞.

Moreover, we denote`^∞ the set of all functionsu:Z→_{Rsuch that} kuk_∞ =sup

k∈_Z

|u(k)|<+_∞

We set

X= (

u:Z→_R :

∑

k∈_Z

a(k)|_∆u(k−1)|^p+b(k)|u(k)|^p< _∞ )

and

kuk=

∑

k∈_Z

a(k)|_∆u(k−1)|^p+b(k)|u(k)|^p

!¹_p . Clearly we have

kuk_∞ ≤ kuk_p ≤b⁻

1 p

0 kuk for allu∈ X. (2.1)

As is shown in [8, Proposition 3], (X,k · k)is a reflexive Banach space and the embedding X,→l^p is compact.

Let

Φ(u):= ¹ p

∑

k∈_Z

a(k)|_∆u(k−1)|^p+b(k)|u(k)|^p for all u∈ X and

Ψ(u):=

∑

k∈_Z

F(k,u(k)) for all u∈l^p where F(k,s) =Rs

0 f(k,t)dtfors∈_Randk ∈_{Z. Let} J :X→_Rbe the functional associated to problem (1.1) defined by

J_λ(u) =_Φ(u)−λΨ(u)_.

Proposition 2.1. Assume that(A)and(F₁)are satisfied. Then (a) _Φ∈C¹(X)_;

(b) _Ψ∈C¹(l^p) and Ψ ∈C¹(X);

(c) J_λ ∈C¹(X)and every critical point u∈X of J_λ is a homoclinic solution of problem(1.1);

(d) J_λ is sequentially weakly lower semicontinuous functional on X.

This version of the proposition, parts (a),(b) and (c), can be proved essentially by the same way as Propositions 5, 6 and 7 in [8], wherea(k)≡1 onZand the norm onXis slightly different. See also Lemma 2.3 in [9]. The proof of part(d)is standard.

3 Main theorem

Now we will formulate and prove a stronger form of Theorem1.1. Let B±:= lim sup

(k,t)→(±_∞,+_∞)

F(k,t)

[a(k+1) +a(k) +b(k)]t^p and

B₀:=sup

k∈_Z

lim sup

t→+_∞

F(k,t)

[a(k+1) +a(k) +b(k)]t^p

! . SetB=max{B±,B0}. For conveniece we put ₊¹_∞ =0.

Theorem 3.1. Assume that(A), (F₁),(F₂)and(F₃)are satisfied and assume that B > 0. Then, for anyλ > _Bp¹ ,the problem(1.1)admits a sequence of non-negative solutions in X whose norms tend to infinity.

Proof. Putλ> _Bp¹ and putΦ,Ψand J_λ as in the previous section. By Proposition2.1we need to find a sequence{u_n}of critical points of J_λ with non-negative terms whose norms tend to infinity.

Let {cn},{dn}be sequences andr < 0 a number satisfying conditions (F₂)and(F₃). For everyn∈ _Ndefine the set

W_n={u∈ X:r≤ u(k)≤ d_nfor every k∈_Z}.

Claim 3.2. For every n∈ _N,the functional Jλ is bounded from below on Wnand its infimum on Wn

is attained.

Clearly, the setW_nis weakly closed inX. By condition(F₃)we have J(u) = ¹

p

∑

k∈_Z

a(k)|_∆u(k−1)|^p+b(k)|u(k)|^p−λ

∑

k∈_Z

F(k,u(k))

≥ −λ

∑

k∈_Z

max

t∈[r,dn]F(k,t)>−_∞

foru ∈ W_n. Thus, J_λ is bounded from below on W_n. Let η_n = inf_WnJ_λ and{u˜_l}be sequence inXsuch thatηn≤ J_λ(u˜_l)≤ηn+¹_l for alll∈_{N. Then}

1

pku˜_lk^p= ¹ p

∑

k∈_Z

a(k)|_∆u˜_l(k−1)|^p+b(k)|u˜_l(k)|^p= J(u˜_l) +λ

∑

k∈_Z

F(k, ˜u_l(k))

≤ η_n+1+λ

∑

k∈_Z

tmax∈[r,dn]F(k,t)

for all l ∈ _{N, i.e.} {u˜_l} is bounded in X. So, up to subsequence, {u˜_l} weakly converges in Xto someun ∈Wn. By the sequentially weakly lower semicontinuity of Jλ we conclude that J_λ(u_n) =η_n=inf_WnJ_λ. This proves Claim3.2.

Claim 3.3. For every n∈_N,let un ∈Wnbe such that J_λ(un) =inf_Wn J_λ. Then,0≤ un(k)≤cnfor all k ∈_Z.

LetK={k∈ _Z:u_n(k)∈/[0,c_n]}and suppose thatK6=∅. We then introduce the sets K− ={k∈K: un(k)<0} and K+={k ∈K: un(k)> cn}.

Thus, K=K−∪K+.

Define the truncation function γ : R → _R by γ(s) = min(s+,cn), where s+ = max(s, 0). Now, set w_n = γ◦u_n. Clearly w_n ∈ X. Moreover, w_n(k) ∈ [0,c_n] for every k ∈ _{Z; thus} w_n∈W_n.

We also have thatw_n(k) =u_n(k)for allk∈_Z\K,w_n(k) =0 for allk∈K−, andw_n(k) =c_n for all k∈K+. Furthermore, we have

J_λ(wn)−J_λ(un) = ¹ p

∑

k∈_Z

a(k) (|_∆wn(k−1)|^p− |_∆un(k−1)|^p) + +¹

p

∑

k∈_Z

b(k) |w_n(k)|^p− |u_n(k)|^p−λ

∑

k∈_Z

[F(k,w_n(k))−F(k,u_n(k))]

=: 1 pI₁+ ¹

pI₂−λI₃.

(3.1)

Sinceγis a Lipschitz function with Lipschitz-constant 1, andw=γ◦u, we have˜ I₁=

∑

k∈_Z

a(k) (|_∆wn(k−1)|^p− |_∆un(k−1)|^p)

=

∑

k∈_Z

a(k) (|w_n(k)−w_n(k−₁)|^p− |u_n(k)−u_n(k−₁)|^p)

≤0.

(3.2)

Moreover, we have I₂ =

∑

k∈_Z

b(k) |w_n(k)|^p− |u_n(k)|^p =

∑

k∈K

b(k) |w_n(k)|^p−(u_n(k))^p

=

∑

k∈K−

−b(k)|u_n(k)|^p+

∑

k∈K+

b(k)[c^p_n− |u_n(k)|^p]

≤0.

(3.3)

Next, we estimateI3. First,F(k,s) =0 fors≤0,k∈Z, and consequently∑k∈K−[F(k,wn(k))− F(k,u_n(k))] =0. By the mean value theorem, for everyk ∈ K+, there existsξ_k ∈ [c_n,u_n(k)]⊂ [c_n,d_n]such thatF(k,w_n(k))−F(k,u_n(k)) =F(k,c_n)−F(k,u_n(k)) = f(k,ξ_k)(c_n−u_n(k)). Tak- ing into account hypothesis (F2), we have thatF(k,wn(k))−F(k,un(k))≥ 0 for everyk∈ K+. Consequently,

I₃ =

∑

k∈_Z

[F(k,w_n(k))−F(k,u_n(k))] =

∑

k∈K

[F(k,w_n(k))−F(k,u_n(k))]

=

∑

k∈K+

[F(k,wn(k))−F(k,un(k))]≥0. (3.4)

Combining relations (3.2)–(3.4) with (3.1), we have that J_λ(w_n)−J_λ(u_n)≤0.

But J_λ(w_n) ≥ J_λ(u_n) = _infWnJ_λ since w_n ∈ W_n. So, every term in J_λ(w_n)−J_λ(u_n) _{should be} zero. In particular, fromI₂, we have

k∈

∑

|u_n(k)|^p=

∑

k∈K+

[c^p_n− |u_n(k)|^p] =0,

which imply thatu_n(k) = 0 for every k∈ K−andu_n(k) = c_n for everyk ∈ K+. By definition of the setsK− andK+, we must have K− = K+ = _∅, which contradictsK−∪K+ = K 6= _∅; thereforeK=∅. This proves Claim3.3.

Claim 3.4. For every n∈_N,let u_n∈W_nbe such that J_λ(u_n) =inf_WnJ_λ. Then, u_nis a critical point of J_λ.

It is sufficient to show that un is local minimum point of Jλ in X. Assuming the contrary, consider a sequence {v_i} ⊂ X which converges to u_n and J_λ(v_i) < J_λ(u_n) = inf_WnJ_λ for all i∈ _N. From this inequality it follows thatv_i ∈/ W_n for anyi ∈ _N. Since v_i → u_n in X, then due to (2.1),v_i →uninl_∞as well. Choose a positiveδsuch thatδ < ¹₂min{−r,dn−cn}. Then, there existsi_δ ∈ _N such thatkv_i−u_nk_∞ < δ for every i≥ i_δ. By using Claim 3.3 and taking into account the choice of the numberδ, we conclude that r < v_i(k) < d_n for all k ∈ _Zand i≥i_δ, which contradicts the factv_i ∈/Wn. This proves Claim3.4.

Claim 3.5. For every n∈_N,letη_n=inf_WnJ_λ. Thenlim_n→+_∞η_n =−_∞.

Firstly, we assume that B = B±. Without loss of generality we can assume that B = B+. We begin withB = +∞. Then there exists a numberσ > _λ¹_p, a sequence of positive integers {kn}and a sequence of real numbers{tn}which tends to +∞, such that

F(k_n,t_n)>σ(a(k_n+1) +a(k_n) +b(k_n))t^p_n

for alln∈N. Up to extracting a subsequence, we may assume thatd_n ≥t_n≥1 for alln∈_N.

Define inXa sequence{w_n}such that, for everyn∈_N,w_n(k_n) =t_nandw_n(k) =0 for every k∈_Z\{kn}. It is clear thatwn∈Wn. One then has

J_λ(wn) = ¹ p

∑

k∈_Z

a(k)|_∆wn(k−1)|^p+b(k)|wn(k)|^p−λ

∑

k∈_Z

F(k,wn(k))

< ¹

p(a(k_n+1) +a(k_n))t_n^p+ ¹

pb(k_n)t_n^p−λσ(a(k_n+1) +a(k_n) +b(k_n))t_n^p

= 1

p −λσ

(a(k_n+₁) +a(k_n) +b(k_n))t^p_n

which gives lim_n→+_∞J(w_n) = −_∞. Next, assume that B < +_∞. Since λ> _Bp¹ , we can fix ε<B−_λp¹ . Therefore, also taking{kn}a sequence of positive integers and{tn}a sequence of real numbers with lim_n→+_∞t_n= +_∞andd_n ≥t_n≥1 for alln∈_Nsuch that

F(k_n,t_n)>(B−ε)(a(k_n+1) +a(k_n) +b(k_n))t^p_n for alln∈_N, choosing{w_n}inW_nas above, one has

J_λ(w_n)<

1

p −λ(B−ε)

(a(k_n+1) +a(k_n) +b(k_n))t_n^p.

So, also in this case, lim_n→+_∞J(w_n) =−_∞.

Now, assume that B = B0. We begin with B = +∞. Then there exists a number σ > _λp¹ and an index k₀ ∈_Zsuch that

lim sup

t→+_∞

F(k₀,t)

(a(k₀+1) +a(k₀) +b(k₀))|t|^p >σ.

Then, there exists a sequence of real numbers{t_n}such that lim_n→+_∞t_n= +_∞and F(k0,tn)>σ(a(k0+1) +a(k0) +b(k0))t^pn

for all n ∈ N. Up to considering a subsequence, we may assume that d_n ≥ t_n ≥ 1 for all n ∈ _N. Thus, take in X a sequence {w_n} such that, for every n ∈ _N, w_n(k₀) = t_n and wn(k) =0 for everyk∈_Z\{k0}. Then, one has wn∈Wn and

J_λ(w_n) = ¹ p

∑

k∈_Z

a(k)|_∆w_n(k−1)|^p+b(k)|w_n(k)|^p−λ

∑

k∈_Z

F(k,w_n(k))

< ¹

p(a(k0+1) +a(k0))t^p_n+ ¹

pb(k0)t_n^p−λσ(a(k0+1) +a(k0) +b(k0))t_n^p

= 1

p−λσ

(a(k₀+1) +a(k₀) +b(k₀))t_n^p,

which gives lim_n→+_∞J(w_n) = −∞. Next, assume that B < +_{∞. Since} λ> _Bp¹ , we can fix ε>0 such thatε< B− _λp¹ . Therefore, there exists an indexk₀∈_{Zsuch that}

lim sup

t→+_∞

F(k₀,t)

(a(k₀+1) +a(k₀) +b(k₀))t^p >B−ε.

and taking {t_n}a sequence of real numbers with lim_n→+_∞t_n = +_∞ andd_n ≥ t_n ≥ 1 for all n∈_Nand

F(k0,tn)> (B−ε) (a(k0+1) +a(k0) +b(k0))t^p_n for all n∈N, choosing{w_n}_inW_nas above, one has

J_λ(w_n)<

1

p −λ(B−ε)

(a(k₀+1) +a(k₀) +b(k₀))t^p_n. So, also in this case, limn→+_∞J_λ(w_n) =−∞. This proves Claim3.5.

Now we are ready to end the proof of Theorem3.1. With Proposition2.1, Claims3.3–3.5, up to a subsequence, we have infinitely many pairwise distinct non-negative homoclinic solu- tionsunof (1.1) withun∈Wn. To finish the proof, we will prove thatkunk →+_∞asn→+_∞.

Let us assume the contrary. Therefore, there is a subsequence{un_i}of{un}which is bounded in X. Thus, it is also bounded inl_∞. Consequently, we can findm₀ ∈ _Nsuch that u_ni ∈W_m0 for alli∈N. Then, for everyn_i ≥m0one has

η_m0 = inf

Wm0

J ≤ J(u_ni) =inf

W_niJ =η_ni ≤η_m0,

which proves thatη_ni =η_m0 for alln_i ≥m₀, contradicting Claim3.5. This concludes our proof.

Remark 3.6. Theorem1.1follows now from Theorem3.1.

4 Examples

Now, we will show the example of a function for which we can apply Theorem1.1. First we give an example of a function f for which(F₄⁺)arise, but (F₅)is not satisfied.

Example 4.1.Let{a(k)},{b(k)}be two sequences of positive numbers such that lim_k→+_∞b(k) = +_{∞. Let} {c_n},{d_n} be sequences such that 0 < c_n < d_n < c_n+1 and lim_n→_∞c_n = +_{∞. Let} {h_n}be a sequence such that

h_n >n (a(n+₁) +a(n) +b(n))c_n^p₊₁

for every n ∈ N. For every nonpositive integer k let f(k,·) _{: R} → _R be identically zero function. For every positive integer k let f(k,·) : R → _R be any nonnegative continuous function such that f(k,t) = 0 for t ∈ _R\(d_k,c_k+1) and Rc_k+1

d_k f(k,t)dt = h_k. The conditions (F₁)_and(F₂)are now obviously satisfied.

Set F(_k,t) _:= Rt

0 f(_k,s)_{ds for every t} ∈ _R and k ∈ Z. Since for every n ∈ _N and all r< 0 only finitely many max_t∈[r,dn]F(k,t)is nonzero, (F₃)is satisfied. By our choosing of the sequence{h_n}we have

lim sup

(k,t)→(+_∞,+_∞)

F(k,t)

(a(k+1)a(k) +b(k))|t|^p ≥ lim

n→+_∞

F(n,c_n+1)

(a(n+1) +a(n) +b(n))c_n^p₊₁

= lim

n→+_∞

h_n

(a(n+1) +a(n) +b(n))c_n^p₊₁ = +_∞ and

sup

k∈_Z

lim sup

t→+_∞

F(k,t)

(a(k+1) +a(k) +b(k))|t|^p

!

=0.

Now we give an example of a function f for which (F₅)arises, but(F₄⁺)is not satisfied.

Example 4.2.Let{a(k)},{b(k)}be two sequences of positive numbers such that lim_k→+_∞b(k) = +_{∞. Let} {c_n},{d_n} be sequences such that 0 < c_n < d_n < c_n+1 and lim_n→_∞c_n = +_{∞. Let} {hn}be a sequence of nonnegative numbers satisfying

∑ⁿ_k=1h_k

(a(1) +a(0) +b(0))c^p_n+1 >n

for everyn∈_{N. Let ˜}f :R→_Rbe the continuous nonnegative function given by f˜(s):=

∑

n∈_N

2hn

c_n+1−dn−2

s− ¹

2(dn+c_n+1)

·1_[dn,cn+1]

where1_[d,c] is the indicator of the interval [d,c]. We check at once that, for everyn ∈_N,

Z _cn+1

dn

f˜(s)ds=hn.

Set f(0,s):= f^˜(s)fors∈_Rand f(k,s) =0 fork∈_Z\{0}ands∈_{R. Set}F(k,t):= Rt

0 f(k,s)ds for everyt ∈ _{R and}k ∈ _{Z. Then} F(_0,c_n+1) = _∑ⁿ_k=1h_k. The conditions (F₁)_,(F₂)_and(F₃) _are

satisfied and sup

k∈_Z

lim sup

t→+_∞

F(k,t)

(a(k+1) +a(k) +b(k))|t|^p

!

=_{lim sup}

t→+_∞

F(0,t)

(a(1) +a(0) +b(0))|t|^p

≥ lim

n→+_∞

F(0,c_n+1)

(a(1) +a(0) +b(0))c_n^p₊₁

= lim

n→+_∞

∑ⁿk=1h_k

(a(1) +a(0) +b(0))c_n^p₊₁ = +_∞.

Moreover,

lim sup

(k,t)→(+_∞,+_∞)

F(k,t)

(a(k+1) +a(k) +b(k))t^p =0.

5 Comparison with other known results

In the paper [9], the following theorem is presented.

Theorem 5.1. Assume that a function b:Z→_Rand a continuous function f :Z×_R→_Rsatisfy conditions:

(B) b(k)≥b₀ >0for all k∈_{Z, b}(k)→+_∞as|k| →+_∞;

(H₁) sup

|t|≤T

|F(·.t)| ∈l₁for all T >0;

(H2) f(k,−t) =−f(k,t)for all k ∈_Zand t∈ _R;

(H₃) there exist d>0and q> p such that |F(k,t)| ≤d|t|^qfor all k∈_Zand t∈_R;

(H₄) lim

|t|→+_∞ f(k,t)t

|t|^p = +_∞uniformly for all k∈_Z;

(H5) there existsσ≥1such thatσF(k,t)≥ F(k,st)for k ∈_Z,t ∈_R,and s∈ [0, 1], where F(k,t)is the primitive function of f(k,t), that is F(k,t) = Rt

0 f(k,s)ds for every t ∈ _R and k∈_Z,andF(k,t) =t f(k,t)−pF(k,t). Then, for anyλ>0, problem(1.1)has a sequence{u_n(k)}

of nontrivial solutions such that J_λ(un)→+_∞as n→+_∞.

As an example of function, which satisfied conditions(H₁)–(H₅)is given the function f(k,t) = ¹

k^µ|t|^p−2tln 1+|t|^ν, (k,t)∈_Z×_R

withµ>1 andν≥1. But the theorem cannot be applied to this function, because it does not satisfy the condition(H₄). Moreover, the conditions(H₁)and(H₄)are contradictory. Indeed, since p>1 the hypothesis(H₄)does give usT₁ >0 such that|f(k,t)| ≥1 for all|t| ≥T₁ and k ∈ _{Z. Put}α_k = F(k,T₁)for allk ∈_{Z. Then} {α_k} ∈ l₁, by (H₁). As f is continuous we have forT >T₁ andk ∈_Z

|F(_k,T)|=

Z _T

0 f(_k,t)_dt

=

Z _T1

0 f(_k,t)_dt+

Z _T

T1

f(_k,t)_dt

=

α_k+

Z _T

T1

f(_k,t)_dt

≥

Z _T

T₁ f(k,t)dt

− |α_k|=

Z _T

T₁

|f(k,t)|dt− |α_k| ≥(T−T₁)− |α_k|,

and so|F(·,T)|∈/l₁, contrary to(H₁).

In the paper [20], the problem (1.1) witha(k)≡1 andλ=1 was considered. The authors obtained infinitely many pairs of homoclinic solutions assuming, among other things, that f(k,t)is odd in t for each k ∈ _{Z, i.e.}(H₂). Our Theorem 3.1 has no symmetry assumptions and, for instance, the function in our Example 1 is not odd. On the other hand, Example 7 in [20] shows the function f : Z×_R → _R satisfying assumptions of the main theorem in [20]

with f(k,t)>0 for allt >1 andk∈Z. Such a function does not satisfy (F₂)and Theorem3.1 does not apply to it.

In the paper [18], the problem (1.1) with a(k) ≡ 1 was considered and the following theorem was obtained.

Theorem 5.2. Assume that a function b:Z→_Rand a continuous function f :Z×_R→_Rsatisfy conditions:

(B) b(k)≥b0>0for all k∈_{Z, b}(k)→+_∞as|k| →+_∞;

(F₁) lim_t→0|f(k,t)|

|t|^p−1 =0uniformly for all k∈_Z.

Put

A:=lim inf

t→+_∞

∑k∈_Zmax_|ξ|≤tF(k,ξ)

t^p ,

B±,± := lim sup

(k,t)→(±_∞,±_∞)

F(k,t) (2+b(k))|t|^p, B± :=sup

k∈_Z

lim sup

t→±_∞

F(k,t) (2+b(k))|t|^p

!

and B := max{B±,±,B±},where F(k,t)is the primitive function of f(k,t). If A< b₀·B, then for eachλ∈ I := _Bp¹ ,_Ap^b0

problem(1.1)admits a sequence of solutions.

As the example 3 in [18] shows, for any two strictly positive real numbers α,β there is a continuous function f :Z×_R→_Rsuch that A=αandB= β. So, if we choose^˙ α,β>0 with α≥ b₀·β, we will not be able to apply the above theorem. Since this example is similar to our Example 1, the function f satisfies the condition(F2)and(F3), and we can apply Theorem3.1 to obtain a sequence of solutions. On the other hand, as f in example 3 in [18] is non-negative, it is easy to see, that we can modify it in the way, that for some (or even infinitely many)kwe have f(k,t) > 0 for allt ≥ 1 and the interval I differ by as little as we wish. Therefore, such an f does not satisfy(F₂)and cannot be used in Theorem3.1.

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