On a class of non-local discrete boundary value problepm

Abstract

Purpose

In this article, the authors discuss the existence and multiplicity of solutions for an anisotropic discrete boundary value problem in T-dimensional Hilbert space. The approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.

Design/methodology/approach

The approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.

Findings

The authors study the existence of results for a discrete problem, with two boundary conditions type. Accurately, the authors have proved the existence of at least three solutions.

Originality/value

An other feature is that problem is with non-local term, which makes some difficulties in the proof of our results.

Citation

Ourraoui, A. and Ayoujil, A. (2022), "On a class of non-local discrete boundary value problepm", Arab Journal of Mathematical Sciences, Vol. 28 No. 2, pp. 130-141. https://doi.org/10.1108/AJMS-06-2020-0003

1. Introduction

The non-linear difference equations have been of great interest because of their important applications appearing in various fields of research, such as numerical analysis, non-linear differential equations, computer science, mechanical engineering, control systems, artificial or biological neural networks and social sciences, such as economics. To deal with these kind of problems, a various methods such as fixed points theorems, lower and upper solutions, Browder degree, variational approach and critical point theory have been applied by many different authors. For the recent progress in discrete problems, we refer the readers to valuable monograph by Agarwal [1] and the papers [2,3]. Let T≥2 be a positive integer, [a, b] be the discrete interval {a, a+1, …, b} with a and b are integers such that a<b.

In the present paper, we deal with the existence of solutions for the Neumann problem

(1){−K(I(u))(Δ(ϕp(k−1)(Δu(k−1)))+ϕp(k)(u(k)))=λf(k, u(k))+μg(k, u(k)), k∈[1,T], Δu(0)=Δu(T)=0,

as well as for the Dirichlet problem,

(2){−K(∑k=1T+1(1p(k−1)|Δu(k−1)|p(k−1)))(Δ(ϕp(k−1)(Δu(k−1))))=λf(k, u(k))+μg(k, u(k)), k∈[1, T],u(0)=u(T)=0,

where

I(u)=∑k=1T+1(1p(k−1)|Δu(k−1)|p(k−1))+∑k=1T(1p(k)|u(k)|p(k)),

Δu(k)=u(k+1)−u(k) is the forward difference operator, ϕ will stand for the homeomorphism defined by ϕs(x)=|x|s−2x, K:(0, ∞)→(0, ∞) be a non-decreasing continuous function, p:[0, T]→[2, ∞) is a bounded function while λ,μ are positive real numbers and f,g belong to ℭ which is the class of all continuous functions h which satisfy

|h(k, t)|≤C1(1+|t|r(k)−1), for all(k, t)∈[1, T]×ℝ

with r:[1, T]→[2, ∞) being a bounded function.

Equations of this type were suggested by Kirchhoff in 1883. More precisely the following model, which is called Kirchhoff equation, was introduced (see [4])

ρ∂2u∂t2−(P0h+E2L∫0L|∂u∂x|2dx)∂2u∂x2=0,

where ρ0, ρ, L and h are constants associated to the effects of the changes in the length of strings during the vibrations. It is an extension of the classical D'Alembert's wave equation. A distinguish feature of the above equation is that it contains a non-local coefficient

P0h+E2L∫0L|∂u∂x|2dx

which depends on the average

E2L∫0L|∂u∂x|2dx

of the kinetic energy 12|∂u∂x|2 on [0, L], and hence the equation is no longer a pointwise identity.

The study of these problems has received more attention. In [2,5–15], a variety of different methods were applied to obtain the existence results to the discrete boundary value problem of the following type

(3){−(Δ(ϕp(Δu(k−1))))=λf(k, u(k)), k∈[1, T],u(0)=u(T)=0,

where ϕp(x)=|x|p−2.

For example, Jiang and Zhou in [16] employing a three critical point theorem, due to Ricceri, established the existence of at least three solutions for perturbed non-linear difference equations with discrete boundary conditions. Bonanno and Candito [11], employing critical point theorems in the setting of finite dimensional Banach spaces, investigated the multiplicity of solutions for non-linear difference equations involving the p_-Laplacian. Cabada et al. in [2], based on three critical points theorems, investigated different sets of assumptions which guarantee the existence and multiplicity of solutions for difference equations involving the discrete p_-Laplacian operator. Candito and Giovannelli [12], using variational methods, established the existence of at least three solutions for the problem above. Far from being exhaustive, further details can be found in [13,17–24].

By taking into account the previous papers and inspired by [25], we study problems (1) and (2) and obtain the existence of three weak solutions by employing a kind of Ricceri's theorem [26]. As for the author's best knowledge, the present papers results are not covered in the related literature, and hence, it is original in its own right.

The structure of this paper is outlined as follows. In Section 2, some preliminary results and statement of main results are presented. In Section 3, the proof of the main results is given.

2. Preliminaries

Firstly, we recall some basic properties which will be used in the proof of the precise result.

Through the sequel, we say that the functional L∈;W if L:W→ℝ possesses the following property: (un) is a sequence in W converging weakly to u∈W and lim inf  L(un)≤L(u), then un has a subsequence converging strongly to u. When W is finite dimensional, the weak convergence coincides with the strong one.

In order to prove our main results, we will use the following Ricceri's theorem.

Theorem 2.1.

[26] Let W be a finite dimensional real Banach space, φ∈C1(W, ℝ) is coercive and belongs to ;W. The derivative of φ admits a continuous inverse on W∗; J:W→ℝ a C1 functional. Assume that φ has a strict local minimum u0 with φ(u0)=J(u0)=0.

Finally, setting assume that

α=max{0, limsup‖u‖→∞J(u)φ(u), limsup‖u‖→u0J(u)φ(u)},

β=supu∈φ−1(0,∞)J(u)φ(u),

assume that α<β.

Then, for each compact interval [a, b]⊂(1β, 1α) (with the conventions 10=∞, 1∞=0), there exists R>0 with the following property: for every λ∈[a, b] and every C1 functional ψ:W→ℝ with compact derivative, there exists δ>0 such that, for each μ∈[0, δ],

φ′(u)=λJ′(u)+μψ′(u)

has at least three solutions in W whose norms are less than R.

Denoting by F, G:[1, T]→ℝ the primitives of f and g, i.e.,

F(k, T)=∫0Tf(k, s)ds,

G(k, T)=∫0Tg(k, s)ds, k∈[0, T], t∈ℝ

and

p+≔maxk∈[0, T] p(k),

p−≔mink∈[0, T] p(k).

Solutions to (1) will be investigated in a space

W={u:[0, T+1]→ℝ:Δu(0)=Δu(T)=0},

which is a T-dimensional Hilbert space, see [5], associated with the norm

‖u‖=(∑k=1T+1|Δu(k−1)|2+∑k=1T|u(k)|2)12.

It can be verified that for all u∈W, one has

(4)‖u‖≥1⇒1p+‖u‖p−≤I(u)≤1p−‖u‖p+,

and

(5)‖u‖≤1⇒1p+‖u‖p+≤I(u)≤1p−‖u‖p−.

We list also some inequalities that will be are used later.

Lemma 2.2.

([8]) For every u∈W, we have

1. ∑k=1T|u(k)|m≤T(T+1)m−1∑k=1T+1|Δu(k−1)|m, ∀m≥2.

2. ∑k=1T+1|Δu(k−1)|p(k−1)≥T2−p−2‖u‖p−−(T+1), with‖u‖>1.

3. ∑k=1T+1|Δu(k−1)|p(k−1)≥Tp+−22‖u‖p+, with‖u‖<1.

4. ∑k=1T+1|Δu(k−1)|p(k−1)≤2m∑k=1T|u(k)|m, ∀m≥2.

We say that u∈W is a weak solution of problem (1) if

K(I(u))×(∑k=1T+1(|Δu(k−1)|p(k−1)−2Δu(k−1)Δv(k−1))+∑k=1T(1p(k)|u(k)|p(k)−2u(k)v(k)))=λ∑k=1Tf(k, u(k))v(k)+μ∑k=1Tg(k, u(k))v(k),

for any v∈W.

Define the functionals

ϕ(u)=K^(∑k=1T+1(1p(k−1)|Δu(k−1)|p(k−1))+∑k=1T(1p(k)|u(k)|p(k)))

and

J(u)=∑k=1tF(k, u(k)), ψ(u)=∑k=1TG(k, u(k))

where

K^(t)=∫0tK(s)ds.

Let C1 and C2 be positive constants such that

∑k=1T|u(k)|p+≤C2‖u‖p+ and ∑k=1T|u(k)|p−≤C1‖u‖p−, ∀u∈W.

We make the following assumptions.

K0≔inft≥0 K(t)>0.

Put

A≔max{0, limsupη→0supk∈[1,T]F(x, η)|η|p+}

and

B≔max{0, limsupη→∞supk∈[1,T]F(k, η)|η|p−}.

(F1) The following inequality holds

C0K0p+max{0, A, B} <p−supη≠0∑k=1TF(k,η)∑k=1T+1|η|p(k−1),

with C0=C1+C2.

Now, we provide an example of non-linear term which satisfies (F1).

Example:

Set f(k, u)=|u|α(k)−1−|u|β(k)−2 for all (k, u)∈[1, T]×ℝ where α and β are bounded functions such that 1≤α(k)<β(k), k∈[1, T]. Hence,

F(k, u)=(|u|α(k)α(k)−|u|β(k)β(k)).

There exists ξu∈[1, T] which depends on u such that

supk∈[1, T]F(k, u)|u|p−=(|u|α(ξu)−p−α(ξu)−|u|β(ξu)−p−β(ξu)) ≤(|u|α(ξu)−p−α−−|u|β(ξu)−p−β+) =|u|α(ξu)−p−(1α−−1β+|u|β(ξu)−α(ξu)).

For |u| large enough, we have,

|u|α(ξu)−p−(1α−−1β+|u|β(ξu)−α(ξu))≤0

Similarly,

supk∈[1,T]F(k, u)|u|p+=(|u|α(ξu)−p+α(ξu)−|u|β(ξu)−p+β(ξu)).

Since α(k)<β(k) and p+<α− then

lim supu→0supk∈[1,T]F(k, u)|u|p−=0.

Therefore,

max{0, lim sup|u|→0supk∈[1,T]F(k,u)|u|p+, lim sup|u|→∞supk∈[1,T]F(k, u)|u|p−}=0.

Besides, for u small enough we have

∑k=1T|u|(α−k)α(k)−|u|β(k)β(k)>0.

which means that (F1) is verified.

Now, we can state the first main result of this article.

Theorem 2.3.

Let f, g∈ℭ . Under the hypotheses (K1) and (F1), if we put

α∗=1p−inf{∑k=1T+1|u(k−1)|p(k−1)∑1TF(k,u(k));∑1T+1F(k, u(k)) >0},

β∗=1K0p+C0 max{A, B} .

then for each compact interval [a, b]⊂(α∗, β∗) there exists a number δ>0 with the following property: for every λ∈[a, b] and every g∈ℭ  there exists μ∗>0 such that, for each μ∈[0, μ∗] , problem (1) has at least three weak solutions whose norms are less than δ.

Now, suppose that we have:

(F1′) supu∈W∑k=1TF(k, u(k)) >0 .

(F2′) lim sup|ξ|→0supk∈[1, T]F(k, ξ)|ξ|p+≤0 .

(F3′) lim sup|ξ|→∞supk∈[1, T]F(k, ξ)|ξ|p−≤0 .

Solutions to (2) will be investigated in a space

W={u:[0, T+1]→ℝ s.t u(0)=u(T+1)=0},

which is a T-dimensional Hilbert space, see [1], with the inner product

(u, v)=∑k=1T+1Δu(k−1)Δv(k−1), for all u, v ∈W.

Therefore, the associated norm is defined by

‖u‖=(∑k=1T+1|Δu(k−1)|2)12.

Also, it is useful to introduce other norms on W, namely

|u|m=(∑k=1T|u(k)|m)1m, ∀u ∈ W and m≥2 .

It can be verified (see [15]) that

(6)T2−m2m|u|2≤|u|m≤T1m|u|2, ∀u ∈ W and m≥2 .

We report our second main result.

Theorem 2.4.

Let f, g∈ℭ and (K1) holds. Under the hypotheses (F1′)−(F3′) , if we put

θ∗=inf{∑k=1T+11p(k−1)|Δu(k−1)|p(k−1)∑1T+1F(k, u(k)); ∑1T+1F(k,u(k)) >0},

then for each compact interval Λ⊂(θ∗, ∞) there exists a number δ>0 with the following property: for every λ∈Λ and every g∈ℭ there exists μ∗>0 such that, for each μ∈[0, μ∗] , problem (2) has at least three weak solutions whose norms are less than δ.

Example:

Let consider the above example chosen for the function f, then we have

lim sup|u|→∞supk∈[1, T]F(k, u)|t|p−≤0 ,

and

lim sup|u|→0supk∈[1, T]F(k, u)|t|p+=0 .

In addition, for u small enough we have

∑k=1T(|u|(α−k)α(k)−|u|β(k)β(k)) >0 ,

which means that (F1′) is satisfied.

For function g such that g(k, u)=|u|γ(k)u if |u|≤1 and g(k, u)=|u|δ(k) if |u|≥1 where p+<γ(k)<δ(k) for all k∈[1, T], and then f and g verify the hypothesis in Theorem 2.4.

3. Proof

Proof of Theorem 2.3. It is clear that since the functional ϕ is continuously differentiable on a finite dimensional space its Gâteaux derivative is compact with

φ′(u)=K(I(u))(∑k=1T+1|Δu(k−1)|(k−1)−2Δu(k−1)Δv(k−1)+∑k=1T(|u(k)|p(k)−2u(k)v(k))

for all u, v∈W . Since W is a Hilbert space and K is continuous and strictly increasing, it follows that φ belongs to the class ;W .

(7)Let ‖u‖>1, we have ϕ(u)=K∧(I(u))≥K0‖u‖p−p+,

which means that φ is coercive. It is evident that u0=0 is the only global minimum of ϕ and that ϕ(u0)=J(u0)=0.

In view of (F1), there exist 0<ρ<1 and R>1 such that

F(x, ξ)≤(A+ε)|ξ|p+

for all k∈[1, T] and |ξ|≤ρ ,

(8)F(k, ξ)≤(B+ε)|ξ|p−

for all k∈[1, T] and |ξ|≥R .

From the fact that F is bounded on each subset of [1, T]×ℝ, we may choose r−>p+ and a suitable constant c1>0 such that

F(x, ξ)≤(A+ε)|ξ|p++c1|ξ|r−

for all (x, ξ)∈[1, T]×ℝ.

Consequently, for ‖u‖<1 with r−>p+,

J(u)≤(A+ε)∑k=1T|u(k)|p++c1∑k=1T+1|u(k)|r−

≤C2(A+ε)‖u‖p++C3‖u‖r−,

then, using the inequality (5) and the above estimation, we can write

lim sup‖u‖→0J(u)ϕ(u)≤C2(A+ε)K0p+.

On the other hand, for each ‖u‖>1, from (8) it yields

(9)J(u)ϕ(u)≤C1(B+ε)ϕ(u)‖u‖p−.

Thus we have

lim sup‖u‖→∞J(u)ϕ(u)≤p+C1K0(B+ε).

Since ε is arbitrary, so we obtain

(10)max{lim sup‖u‖→∞J(u)ϕ(u), lim sup‖u‖→0J(u)ϕ(u)}≤p+C0K0max{A, B}.

In addition, it is well known that

(11)p−supη∈ℝ/{0}∑k=1TF(k, η)∑k=1T+1|η|p(k−1)≤supu∈W/{0}J(u)ϕ(u).

From (F1) and the above inequalities (3.4), (3.4) we infer that

{lim sup‖u‖→∞J(u)ϕ(u), lim sup‖u‖→u0J(u)ϕ(u)} <supu∈ϕ−1(0,∞)J(u)ϕ(u).

Then the assumptions of Theorem 2.3 are satisfied with u0=0 and the conclusion is valid for all g∈ℭ and each interval include in (α∗, β∗)

Example

Taking K(t)=a+bt for all t>0, with a, b>0 and set

f(k, u)=h(k)(|u|r1(k)−2u−|u|r2(k)−2u)

for all (k, u)∈[1, T]×ℝ, where h:[1, T]→(0, ∞) is an arbitrary function, p+<r1(k)<r2(k), k∈[1, T] where r1, r2:[1, T]→[2, ∞) are bounded functions. It is clear that f∈ℭ. Easily we can prove that

limt→∞F(k, u)|u|p−≤0, limt→0F(k, u)|u|p+=0

So f satisfies the assumptions of Theorem 2.3 with

α∗=1p−{∑k=1T+1|u(k−1)|p(k−1)∑1TF(k, u(k)); ∑1T+1h(k)(|u|r1(k)r1(k)−|u|r2(k)r2(k)) >0}

and

β∗=∞.

Proof of Theorem 2.4. Let start by defining φ as follows:

ϕ(u)=K∧(∑k=1T+1(1p(k−1)|Δu(k−1)|p(k−1)))

Let ‖u‖>1, we have

(12)ϕ(u)≥K0‖u‖p−p+,

which means that φ is coercive. It is evident that u0=0 is the only global minimum of φ and that φ(u0)=J(u0)=0 In view of (F2′), for ε>0 there exist 0<ρ<1 such that

F(x, ξ)≤ε|ξ|p+

for all x∈[1, T] and |ξ|≤ρ . From (F3′) there exists R>1 such that

F(x, ξ)≤ε|ξ|p−

for all x∈[1, T] and |ξ|≥R . In view of the fact that F is bounded on each subset of [1, T]×ℝ, so we may choose that r−>p+ and for a suitable constant c1>0 such that

F(x, ξ)≤ε|ξ|p++c1|ξ|r−

for all (x, ξ)∈[1, T]×ℝ.

Consequently, for ‖u‖<1 with r−>p+,

J(u)≤ε∑k=1T|u(k)|p++c1∑k=1T+1|u(k)|r−

≤C2ε‖u‖p++C3‖u‖r−,

then, using the inequality (c) in Lemma 2.2 and the above estimation, we can write

lim sup∥u∥→0J(u)ϕ(u)≤C2εp+.

On the other hand, for each ∥u∥>1, it is well known that there exists h∈l1([1, T]) such that

F(x, t)≤ε|t|p−+h(x),

for all (x, t)∈[1, T]×ℝ. It yields there is C1>0 such that

(13)J(u)≤C1(ε∥u∥p−+1).

Thus, combining (12) and (13), we get

lim sup∥u∥→∞J(u)ϕ(u)≤p+C2ε.

Since ε is arbitrary and β>0, α=0, hence, all the assumptions of Theorem 2.3 are satisfied with θ∗=1β and the proof is complete.

The following corollary is a direct application of Theorem 2.4.

Corollary 3.1.

let f:ℝ→ℝ be a continuous function such that

supt F(t)>0,

where F(t)=∫0tf(s)ds,

lim sup|t|→∞f(t)|t|r1(k)<∞, r1−>p+,

lim sup|t|→0F(t)|t|p−≤0

Set

η=1p−inf{∑k=1T|Δu|p(k)∑k=1T+1F(u(k)), u∈W, ∑k=1T+1F(u(k))>0}.

Then for each compact interval [c, d]⊂[η, ∞) there exists a number ρ>0 with the following property: for every λ∈[c, d] and g in ℭ there exists σ>0 such that for μ∈[0, σ] the problem

−Δ(|Δu(k−1)|p(k−1)−2Δu(k−1))=λf(u(k))+μg(k, u(k)), k∈[1, T].u(0)=u(T+1)=0,

has at least three weak solutions whose norms are less than ρ