On a class of non-local discrete boundary value problepm

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

as well as for the Dirichlet problem,

where

$Δu(k)=u(k+1)−u(k)$ 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

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])

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

which depends on the average

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

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.

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 $liminfL(un)≤L(u),$ 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.

Finally, setting assume that

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 $μ∈[0,δ],$

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.,

and

Solutions to (1) will be investigated in a space

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

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

and

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

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

for any $v∈W$⁠.

Define the functionals

and

where

Let $C1$ and $C2$ be positive constants such that

We make the following assumptions.

Put

and

$(F1)$ The following inequality holds

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,

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

For $|u|$ large enough, we have,

Similarly,

Since $α(k)<β(k)$ and $p+<α−$ then

Therefore,

Besides, for $u$ small enough we have

which means that $(F1)$ is verified.

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

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

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 $;W.$

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

for all $k∈[1,T]$ and $|ξ|≤ρ,$

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

for all $(x,ξ)∈[1,T]×ℝ.$

Consequently, for $‖u‖<1$ with $r−>p+,$

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

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

Thus we have

Since $ε$ is arbitrary, so we obtain

In addition, it is well known that

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

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

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

So $f$ satisfies the assumptions of Theorem 2.3 with

and

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

Let $‖u‖>1,$ we have

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

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

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

for all $(x,ξ)∈[1,T]×ℝ.$

Consequently, for $‖u‖<1$ with $r−>p+,$

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

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

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

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

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.