A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

1.   Introduction

In this paper, we discuss certain discrete fractional boundary value problems in the form

where $1 < p < \infty$, $\alpha\in(0, 1)$, $\lambda > 0$, ${_{L+1}\nabla^{\alpha}_\ell}$ and ${_\ell\nabla^{\alpha}_0}$ are the right and left discrete nabla fractional difference operators, $f:[1, L]_{ {\mathbb{N}}_0}\times {\mathbb{R}}\to {\mathbb{R}}$ is continuous, and $\varphi_p$ stands for the operator that is defined in the usual way by $\varphi_p(s) = |s|^{p-2}s$.

In this paper, we want to utilize a version of Ricceri's variational principle as given in ^[1]. We build the energy functional and we take some assumptions on the nonlinear term to get a functional that satisfies the conditions in the key theorem. In fact, by first requiring a simple algebraic inequality condition on the nonlinear term for small values of the parameter and requiring an additional asymptotical behavior of the potential at zero if $f(\ell, 0) = 0$, the existence of one nontrivial solution is achieved. Moreover, we deduce the existence of solutions for small positive values of the parameter such that the corresponding solutions have smaller and smaller energies as the parameter goes to zero. Then, under an appropriate oscillating behavior of the nonlinear term, we discuss the existence of an unbounded sequence of solutions. We give exact collections for the parameter for each results, which cannot be found in other works in the literature related to these types of discrete problems.

As is well known, fractional differential equations (FDEs) are valuable tools when modeling many phenomena in various areas of science and engineering. We refer to ^{[2,3,4,5,6,7,8]} and the references therein for a range of applications of FDEs in control, electrochemistry, viscoelasticity, electromagneticity, porous media, and other fields. Classical tools that have been employed in the study of nonlinear FDEs (see ^[9,10,11] and their references) include fixed point theory, monotone iterative methods, coincidence theory, and the upper and lower solutions method.

Thanks to their wide range of applications in many fields such as science, economics, ecology, neural networks, cybernetics, etc., nonlinear difference equations have been studied extensively for the last $50$ years. In addition, boundary value problems involving difference equations have received a lot of attention, see, e.g., ^{[12,13,14,15]} and references therein. Difference equations subjected to many different kinds of boundary conditions have also been extensively studied by using various techniques. A popular technique has been to use variational methods ^[16,17,18]. In the last few years, many researchers have investigated nonlinear problems of this type through various approaches. Moreover, such boundary value problems for ordinary differential equations, difference equations, and dynamic equations on time scales have been studied extensively, but there are only a few papers dealing with fractional boundary value problems, besides ^[19,20,21], especially for discrete fractional boundary value problems involving Caputo fractional difference operators. For example, Lv ^[22], by using the fixed point theorem of Schaefer, under certain nonlinear growth assumptions, obtained the existence of solutions to a discrete fractional boundary value problem. Furthermore, in ^[23,24], some existence and multiplicity results for $(p, q)$-Laplacian problems were considered. For example, in ^[23], by using suitable variational arguments and Ljusternik–Schnirelmann category theory, the multiplicity and concentration of positive solutions for $(p, q)$-Laplacian problems were obtained.

In view of the facts presented above, in the current study, we discuss the existence of at least one solution to the boundary value problem $(P^f_\lambda)$, as well as the existence of an infinite number of solutions of $(P^f_\lambda)$. Our primary tool is [1, Theorem 2.1], which is a more precise variant of the famous variational principle of Ricceri ^[25]. In Theorem 9 below, we show that, subject to certain assumptions, the boundary value problem $(P^f_\lambda)$ possesses at least one nontrivial solution. We also offer Example 17, where all hypotheses of our Theorem 9 are satisfied. We present a series of remarks concerning our results. Moreover, under suitable assumptions of the oscillatory behavior at infinity of the nonlinearity, we investigate the existence of an infinite number of solutions for the boundary value problem $(P^f_\lambda)$. We prove the existence of a definite interval about $\lambda$, in which the boundary value problem $(P^f_\lambda)$ admits a sequence of solutions, which is unbounded in the space ${\mathcal{V}}$, to be introduced later (Theorem 18). We present an example that illustrates Theorem 20 (see Example 21). Moreover, some corollaries of Theorem 18 are offered. Under different assumptions, we ensure the existence of a sequence of pairwise different solutions that strongly converges to zero (see Theorem 26).

The paper is set up as follows. In Section 2, we remind the reader of some basic definitions and our main tool. In Sections 3 and 4, we state and prove our main results.

2.   Preliminaries

The key argument in our results is the next version of the variational principle by Ricceri [25, Theorem 2.1], as presented in ^[1].

Theorem 1. Suppose that $X$ is a reflexive real Banach space. Assume that $\Phi, \Psi:X\to {\mathbb{R}}$ are Gâteaux-differentiable functionals such that $\Psi$ is sequentially weakly upper semicontinuous, and $\Phi$ is strongly continuous, sequentially weakly lower semicontinuous, and coercive. For every $r > \inf_X\Phi$, let us put

and

One then has the following.

$(a)$ For every $r > \inf_X\Phi$ and each $\lambda\in\left(0, \dfrac{1}{\varphi(r)}\right)$, the restriction of the functional $I_\lambda = \Phi-\lambda\Psi$ to $\Phi^{-1}(]-\infty, r[)$ possesses a global minimum, and this global minimum is a critical point, i.e., a local minimum, of $I_\lambda$ in $X$.

$(b)$ If $0 < \theta < \infty$, then, for all $\lambda\in\left(0, \dfrac{1}{\theta}\right)$, either

$(b _1)$ $I_\lambda$ possesses a global minimum, or

$(b _2)$ for all $n\in {\mathbb{N}}$, $I_\lambda$ has a critical point $u_n$, and

$(c)$ If $0 < \delta < \infty$, then, for all $\lambda\in\left(0, \dfrac{1}{\delta}\right)$, either

$(c _1)$ $\Phi$ possesses a global minimum, which is a local minimum of $I_\lambda$, or

$(c _2)$ a sequence of pairwise different critical points of $I_\lambda$ exists, and this sequence converges weakly to a global minimum of $\Phi$.

We refer to ^[9,26,27,28], in which Theorem 1 was applied successfully in order to prove the existence of at least one nontrivial solution for certain boundary value problems, and ^{[29,30,31,32,33]}, in which Theorem 1 has been successfully employed in order to prove the existence of an infinite number of solutions for certain boundary value problems.

In this section, we present several foundational definitions, notations, and results that are used in the remainder of this paper.

Definition 2 (see ^[34]). (ⅰ) For $m\in {\mathbb{N}}$, the $m$ rising factorial of $t$ is defined as

(ⅱ) For $t\in {\mathbb{R}}\setminus\{\ldots, -2, -1, 0\}$ and $\alpha\in {\mathbb{R}}$, the $\alpha$ rising function is increasing on ${\mathbb{N}}_0$ and

Definition 3 (see ^[34]). For $f$ defined on ${\mathbb{N}}_{a-1}\cap{_{b+1} {\mathbb{N}}}$, $a < b$, $\alpha\in(0, 1)$, the left Caputo discrete fractional nabla difference operator is given by

the right Caputo discrete fractional nabla difference operator is given by

the left Riemann discrete fractional nabla difference operator is given by

the right Riemann discrete fractional nabla difference operator is given by

where $\rho(\ell) = \ell-1$ is the backward jump operator.

Example 4. Let us give an example concerning Definition 3. Consider $f: {\mathbb{N}}_{a-1}\cap{_{b+1} {\mathbb{N}}}\to {\mathbb{R}}$ defined by $f(\ell)\equiv 1$. For this $f$, from (1) and (2), we have

The relations among the right and left Riemann and Caputo nabla fractional difference operators are

Thus, by (3)–(5), we have

With respect to the domains of the various fractional-type difference operators, we observe the following:

(ⅰ) The nabla left fractional operator ${_\ell\nabla^{\alpha}_{a-1}}$ maps functions defined on $_{a-1} {\mathbb{N}}$ to functions defined on $_a {\mathbb{N}}$.

(ⅱ) The nabla right fractional operator ${_{b+1}\nabla^{\alpha}_\ell}$ maps functions defined on $_{b+1} {\mathbb{N}}$ to functions defined on $_b {\mathbb{N}}$.

One can show that for $\alpha\to 0$, we have ${_\ell\nabla^{\alpha}_{a-1}} f(\ell)\to f(\ell)$ and for $\alpha\to 1$, we have ${_\ell\nabla^{\alpha}_{a-1}} f(\ell)\to\nabla f(\ell)$. We note that the Caputo and Riemann nabla fractional difference operators for $0 < \alpha < 1$ coincide when $f$ vanishes at the end points, i.e., $f(a-1) = 0 = f(b+1)$ (see (4), (5), and ^[2]). So, for convenience, for the remainder of this paper, we use the symbol ${_\ell\nabla^{\alpha}_{a-1}}$ instead of ${^{{\rm{C}}}_\ell\nabla^{\alpha}_{a-1}}$ or ${^{{\rm{R}}}_\ell\nabla^{\alpha}_{a-1}}$ and ${_{b+1}\nabla^{\alpha}_\ell}$ instead of ${^{{\rm{C}}}_{b+1}\nabla^{\alpha}_\ell}$ or ${^{{\rm{R}}}_{b+1}\nabla^{\alpha}_\ell}$.

Now we present the discrete fractional summation by parts formula.

Theorem 5. (see [35, Theorem 4.4]). For $f, g: {\mathbb{N}}_a\cap{_b {\mathbb{N}}}\to {\mathbb{R}}$, $a < b$, and $\alpha\in(0, 1)$, the formulas

and

hold.

In order to give the variational formulation of the boundary value problem $(P^f_\lambda)$, let us introduce the finite $L$-dimensional Banach space

which is equipped with the norm

According to the definition of the norm, the following lemma is obvious.

Lemma 6. For all $\alpha\in(0, 1)$ and for all $v\in {\mathcal{V}}$, we have

For every $v\in {\mathcal{V}}$, we define the functionals $\Phi$ and $\Psi$ as

and

and we put

Definition 7. A (weak) solution of $(P^f_\lambda)$ is defined to be any function $v\in {\mathcal{V}}$ such that

for every $\tilde{v}\in {\mathcal{V}}$.

Note that since ${\mathcal{V}}$ is a finite-dimensional space, every weak solution is a usual solution of the boundary value problem $(P^f_\lambda)$.

Lemma 8. Let $v\in {\mathcal{V}}$. Then $v$ is a critical point of $I_\lambda$ in ${\mathcal{V}}$ if and only if $v$ solves $(P^f_\lambda)$.

Proof. First assume that $v\in {\mathcal{V}}$ is a critical point of $I_\lambda$. Then, for any $\tilde{v}\in {\mathcal{V}}$, (9) holds. Bearing in mind that $\tilde{v}\in {\mathcal{V}}$ is arbitrary, we get

for all $\ell\in[1, L]_{ {\mathbb{N}}}$. Therefore, $v$ solves $(P^f_\lambda)$. Since $v$ was chosen arbitrarily, we deduce that all critical points of the functional $I_\lambda$ in ${\mathcal{V}}$ solve $(P^f_\lambda)$. Conversely, if $v$ solves $(P^f_\lambda)$, then, by reversing the above steps, the proof is completed. □

Put

3.   Existence of one solution

The following is our main result concerning the existence of a solution of $(P^f_\lambda)$.

Theorem 9. Assume that $f(\ell, 0) = 0$ and

and there are discrete intervals $D = [1, L_1]_{ {\mathbb{N}}_0}\subseteq[1, L]_{ {\mathbb{N}}_0}$ and $B = [1, L_2]_{ {\mathbb{N}}_0}\subset[1, L_1]_{ {\mathbb{N}}_0}$ with $L_1, L_2\geq 2$, such that

and

Then, for every

the boundary value problem $(P^f_\lambda)$ admits at least one nontrivial solution $v_\lambda\in {\mathcal{V}}$.

Proof. Our goal is to apply Theorem 1 to $(P^f_\lambda)$. We utilize the functionals $\Phi$ and $\Psi$ as introduced in (7) and (8), respectively. Let us demonstrate that $\Phi$ and $\Psi$ meet the required assumptions of Theorem 1. As ${\mathcal{V}}$ is embedded compactly in $(\text{C}^0([1, L]_{ {\mathbb{N}}_0}), {\mathbb{R}})$, we know that $\Psi$ is Gâteaux-differentiable, and its Gâteaux derivative $\Psi'(v)\in {\mathcal{V}}^*$ at $v\in {\mathcal{V}}$ is given by

for each $\tilde{v}\in {\mathcal{V}}$, and $\Psi$ is sequentially weakly upper semicontinuous. Furthermore, $\Phi$ is also Gâteaux-differentiable, and its Gâteaux derivative at $v\in {\mathcal{V}}$ is the functional $\Phi'(v)\in {\mathcal{V}}^{*}$ given by

for every $\tilde{v}\in {\mathcal{V}}$. Furthermore, by the definition of $\Phi$, we observe that it is sequentially weakly lower semicontinuous and strongly continuous. Now, in light of (7), for each $v\in {\mathcal{V}}$, we obtain

By employing the left inequality of (11), we get

In other words, $\Phi$ is coercive. Using (10), there is $\bar{\theta} > 0$ with

Put

From the way $\Phi$ is defined and considering (6), (7), and (11), since $r > 0$, we have

which implies

By considering these computations, as $0\in\Phi^{-1}(-\infty, r)$ and $\Phi(0) = \Psi(0) = 0$, we get

Hence, we put

At this point, thanks to Theorem 1, for each $\lambda\in(0, \lambda^*)\subseteq\left(0, \dfrac{1}{\varphi(r)}\right)$, Theorem 1 shows that $I_\lambda$ possesses at least one critical point (local minimum) $v_\lambda\in\Phi^{-1}(-\infty, r)$. We show that $v_\lambda$ cannot be the trivial function. Let us prove

Due to our required conditions at zero, we can find a sequence $\{\xi_k\}\subset {\mathbb{R}}^+$ that converges to zero, and $\zeta > 0$ and $\kappa$ with

and

Let $[1, L_3]_{ {\mathbb{N}}_0}\subset[1, L_2]_{ {\mathbb{N}}_0}$, where $L_3\geq 2$, and let $\tilde{v}\in {\mathcal{V}}$ be such that

(ⅰ) $\tilde{v}(\ell)\in[0, 1]$ for all $\ell\in[1, L]_{ {\mathbb{N}}_0}$,

(ⅱ) $\tilde{v}(\ell) = 1\in {\mathbb{R}}$ for all $\ell\in[1, L_3]_{ {\mathbb{N}}_0}$,

(ⅲ) $\tilde{v}(\ell) = 0$ for all $\ell\in[ L_1+1, L]_{ {\mathbb{N}}_0}$.

Therefore, we fix an arbitrary $Y > 0$ and $\eta > 0$ with

Thus, there is $n_0\in {\mathbb{N}}$ with $\varepsilon^n < \zeta$ and

for all $n > n_0$. Next, for all $n > n_0$, using $0\leq\xi_n \tilde{v}(\ell) < \zeta$ for all large enough $n$, by (11), we get

Since $Y$ can be as large as we desire, we conclude that

which implies (12). Thus, we have a sequence $\{ \upsilon_n\}\subset {\mathcal{V}}$ that converges strongly to zero, $\upsilon_n\in\Phi^{-1}(-\infty, r)$, and

Since $v_\lambda$ is a global minimum of the restriction of $I_\lambda$ to $\Phi^{-1}(-\infty, r)$, we get

and thus $v_\lambda$ is not trivial. The proof is complete. □

Some comments are given next.

Remark 10. In Theorem 9, we sought for the critical points of $I_\lambda$, a functional that is intrinsically connected to the boundary value problem $(P^f_\lambda)$. We observe that, in a general case, $I_\lambda$ may not be bounded in ${\mathcal{V}}$. To see this, consider, for instance, the situation when $f(\xi) = 1+\xi^{\gamma-1}$ for each $\xi\in {\mathbb{R}}$, where $\gamma > p$. Let $v\in {\mathcal{V}}\setminus\{0\}$ and $\mu\in {\mathbb{R}}$. We then find

as $\mu\to\infty$. Thus, [36, (I$_2$) in Theorem 2.2] is not satisfied. Therefore, we are unable to use direct minimization to find the critical points of $I_\lambda$.

Remark 11. Note that $I_\lambda$ is not coercive. To see this, let $s\in(p, \infty)$ and define $F$ by $F(\xi) = |\xi|^s$ for each $\xi\in {\mathbb{R}}$. Let $v\in {\mathcal{V}}\setminus\{0\}$ and $\mu\in {\mathbb{R}}$. We then find

as $\mu\to-\infty$.

Remark 12. If, in Theorem 9, $f$ satisfies $f(\ell, x)\geq 0$ for almost every $(\ell, x)\in[1, L]_{ {\mathbb{N}}_0}\times {\mathbb{R}}$, then (10) assumes the simpler form

Moreover, if the assumption

is satisfied, then (14) automatically holds.

Remark 13. From (13), we can directly see that

is indeed a negative map. Furthermore, we have

Indeed, by considering that $\Phi$ is coercive and that for $\lambda\in(0, \lambda^*)$, the solution $v_{\lambda}\in\Phi^{-1}(-\infty, r)$, we have the existence of $\mathcal{L} > 0$ with $\| v_\lambda\|\leq\mathcal{L}$ for every $\lambda\in(0, \lambda^*)$. Next, it is also easy to see that $\mathcal{M} > 0$ exists with

for each $\lambda\in(0, \lambda^*)$. Since $v_\lambda$ is a critical point of $I_\lambda$, $I'_\lambda(v_\lambda)(\tilde{v}) = 0$ for any $\tilde{v}\in {\mathcal{V}}$ and each $\lambda\in(0, \lambda^*)$. In particular, $I'_\lambda(v_\lambda)(v_\lambda) = 0$, i.e.,

for every $\lambda\in(0, \lambda^*)$. Then, since

by using (17), it is concluded that

for all $\lambda\in(0, \lambda^*)$. If we now let $\lambda\to 0^+$, by (16), then we get $\lim_{\lambda\to0^+}\| v_\lambda\| = 0$. One has

Finally, we prove that

decreases strictly in $(0, \lambda^*)$. We observe that for all $v\in {\mathcal{V}}$, we have

Now, let $0 < \lambda_1 < \lambda_2 < \lambda^*$ and let $v_{\lambda_1}$, $v_{\lambda_2}$ be the global minima of the functional $I_{\lambda_i}$ restricted to $\Phi(-\infty, r)$ for $i = 1, 2$. Moreover, let

for $i = 1, 2$. Obviously, (15), in connection with (19) and $\lambda > 0$, yields

In addition,

because $0 < \lambda_1 < \lambda_2$. Then, by observing (19)–(20) and as $0 < \lambda_1 < \lambda_2$, we get

so that (18) decreases strictly for $\lambda\in(0, \lambda^*)$. Since $\lambda < \lambda^*$ is arbitrary, we see that (18) indeed decreases strictly in $(0, \lambda^*).$

Remark 14. We note that Theorem 9 represents a bifurcation result in the sense that $(0, 0)$ belongs to the closure of

in ${\mathcal{V}}\times {\mathbb{R}}$. To observe this, by Remark 13, we get

Therefore, there are sequences $\{ v_j\}$ in ${\mathcal{V}}$ and $\{\lambda_i\}$ in ${\mathbb{R}}^+$ (here, $v_j = v_\lambda$) with

as $j\to\infty$. In addition, we want to emphasize that because

is a strictly decreasing map, for all $\lambda_1, \lambda_2\in(0, \lambda^*)$ such that $\lambda_1\neq\lambda_2$, the solutions $\bar{ v}_{\lambda_1}$ and $\bar{ v}_{\lambda_2}$ ensured by Remark 13 are distinct.

Remark 15. Under the assumption that $f\geq 0$, the solution that is ensured by Theorem 9 is nonnegative. To observe this, suppose that $v_0$ is a nontrivial solution of $(P^f_\lambda)$. Assume that

has positive measure. Put $\bar{ \tilde{v}}(\ell) = \min\{0, v_0(\ell)\}$ for each $\ell\in[1, L]_{ {\mathbb{N}}_0}$. We obtain $\bar{ \tilde{v}}\in {\mathcal{V}}$ and

Thus, from our imposed data sign assumptions, we have

Hence, $v_0 = 0$ in $\mathcal{A}$, which is impossible.

The next result concerns a particular case of the previously presented results, in which the function $f$ depends only on the second variable, considering the nonautonomous case of the problem.

Theorem 16. Let $f: {\mathbb{R}}\to {\mathbb{R}}$ be nonnegative with $f(0) = 0$. Let $F(\xi) = \int_0^\xi f(x){\rm d}x$ for all $x\in {\mathbb{R}}$. Assume that

Then, for every

the boundary value problem

possesses at least one nontrivial solution $v_\lambda\in {\mathcal{V}}$ satisfying

and the real function

is negative and decreases strictly in $\left(0, \dfrac{(L+1)^{\frac{p(p-2)}{4}}}{ L p} \sup_{\theta > 0}\dfrac{{\theta}^p}{F(\theta)}\right)$.

Finally, we present the following example to illustrate Theorem 16.

Example 17. Let $p = 4$ and $L = 2$. We consider the problem

where

By simple computations, we have

We see that all assumptions of Theorem 16 are satisfied, and this implies that the boundary value problem (21), for each $\lambda\in\left(0, \dfrac{9}{8}\right)$, admits at least one nontrivial solution $v_\lambda\in {\mathcal{V}}$ such that

and the real function

is negative and strictly decreasing in $\left(0, \dfrac{9}{8}\right)$.

4.   Existence of an infinite number of solutions

Put

Our main result concerning the existence of infinitely many solutions of $(P^f_\lambda)$ is as follows.

Theorem 18. Assume that two sequences $\{a_n\}$ and $\{b_n\}$ exist with

such that

In this case, for all $\lambda\in\left(\dfrac{1}{\mathcal{B}^\infty}, \dfrac{1}{\mathcal{A}^\infty}\right)$, the boundary value problem $(P^f_\lambda)$ possesses an unbounded sequence of solutions.

Proof. Our aim is to employ Theorem 1. We utilize the functionals $\Phi$ and $\Psi$ as introduced in (7) and (8), respectively. Therefore, we observe that the regularity assumptions on $\Phi$ and $\Psi$, as required in Theorem 1, are satisfied. Hence $v\in {\mathcal{V}}$ is a solution of $(P^f_\lambda)$ if and only if $v$ is a critical point of the function $I_\lambda$. Put

We see that $r_n > 0$ for all $n\in {\mathbb{N}}$. From the way $\Phi$ is defined and in light of (6), (8), and (11), for each $r_n > 0$, we have

which implies

Now, for each $n\in {\mathbb{N}}$, we define

Clearly, $\upsilon\in {\mathcal{V}}$. Since $\upsilon$ vanishes at the end points (that is, $\upsilon(0) = 0 = \upsilon(L+1)$), its Riemann and Caputo fractional differences coincide. Hence, for any $\ell\in {\mathbb{N}}_1\cap{_ L {\mathbb{N}}}$, we have

and

Thus,

We have

In addition, from $(\rm A_1)$, we have $\Phi(\upsilon_n) < r_n$. Thus, for all large enough values of $n$, we obtain

Hence, due to $(\rm A_2)$, we get

Now, we can verify that $I_\lambda$ is unbounded from below. First, assume that $\mathcal{B}^\infty = \infty$. Accordingly, fix $N$ such that

and let $c_n > 0$ for all $n\in {\mathbb{N}}$ such that $c_n\to\infty$ as $n\to\infty$ and

For each $n\in {\mathbb{N}}$, define

Thus, $y_n\in {\mathcal{V}}$ and

Therefore,

that is, $\lim_{n\to\infty}I_\lambda(y_n) = -\infty$. Next, suppose $\mathcal{B}^\infty < \infty$. As $\lambda > \dfrac{1}{\mathcal{B}^\infty}$, we can find $\varepsilon > 0$ with $\varepsilon < \mathcal{B}^\infty-\dfrac{1}{\lambda}$. Thus, again letting $c_n > 0$ for all $n\in {\mathbb{N}}$ such that $c_n\to\infty$ as $n\to\infty$ and

as argued above, and by letting $y_n\in {\mathcal{V}}$ as before, we obtain

Therefore, $\lim_{n\to\infty}I_\lambda(y_n) = -\infty$. Hence, in either case, $I_\lambda$ is not bounded from below. This completes the proof. □

Remark 19. If $\{a_n\}$ and $\{b_n\}$ are real sequences such that $\lim_{n\to\infty}b_n = \infty$ and such that $(\rm A_1)$ from Theorem 18 is satisfied, then, assuming that $\mathcal{A}_\infty = 0$ and $\mathcal{B}^\infty = \infty$, Theorem 18 ensures that for each $\lambda > 0$, the boundary value problem $(P^f_\lambda)$ admits an infinite number of solutions.

Theorem 20. Assume that

In this case, for all

the boundary value problem $(P^f_\lambda)$ possesses an unbounded sequence of solutions.

Proof. We pick $b_n > 0$ for all $n\in {\mathbb{N}}$ with $b_n\to\infty$ as $n\to\infty$ and

Now, as $\Phi(0) = \Psi(0) = 0$, we may take $a_n = 0$ for all $n\in {\mathbb{N}}$ in (22), and then the conclusion follows from Theorem 1. □

Now, we present an example that illustrates Theorem 20.

Example 21. Let $p = 4$ and $L = 3$. We consider the boundary value problem

where

Some computation yields

Since

and

we clearly see that all assumptions of Theorem 20 are satisfied, and then (23), for every $\lambda\in\left(\frac{1}{8}, \infty\right)$, has an unbounded sequence of solutions in $\left\{ v:[0, 4]_{ {\mathbb{N}}_0}\to {\mathbb{R}}: v(0) = v(4) = 0\right\}$.

Here, we point out several simple corollaries of our main results.

Corollary 22. Suppose that there are real sequences $\{a_n\}$ and $\{b_n\}$ with $b_n\to\infty$ as $n\to\infty$ and such that $(\rm A_1)$ from Theorem 18 holds, $\mathcal{A}_\infty < 1$, and $\mathcal{B}_\infty > 1$. Then, the boundary value problem

possesses an unbounded sequence of solutions.

Corollary 23. Suppose $\mathcal{B}_\infty > 1$ and

Then the boundary value problem $(P^f)$ possesses an unbounded sequence of solutions.

Corollary 24. Suppose that there are real sequences $\{a_n\}$ and $\{b_n\}$ with $b_n\to\infty$ as $n\to\infty$ and such that $(\rm A_1)$ from Theorem 18 holds, $f_1\in\text{C}([1, L]_{ {\mathbb{N}}_0}\times {\mathbb{R}}, {\mathbb{R}})$, and

Moreover, assume

and

Then, for all $f_i\in\text{C}([1, L]_{ {\mathbb{N}}_0}\times {\mathbb{R}}, {\mathbb{R}})$, by writing

for $2\leq i\leq n$, such that

and

for all

the boundary value problem

has an unbounded sequence of solutions.

Proof. Put $F(\ell, \xi) = \sum\limits_{i = 1}^nF_i(\ell, \xi)$ for $(\ell, \xi)\in[1, L]_{ {\mathbb{N}}_0}\times {\mathbb{R}}$. ($\rm A_4$), along with

ensures

Moreover, Assumption ($\rm A_4$), together with the condition

implies

Hence, an application of Theorem 18 completes the proof. □

Corollary 25. Let $f_1\in\text{C}([1, L]_{ {\mathbb{N}}_0}\times {\mathbb{R}}, {\mathbb{R}})$ and put

Assume that

and

Then, for all $f_i\in\text{C}([1, L]_{ {\mathbb{N}}_0}\times {\mathbb{R}}, {\mathbb{R}})$, by writing

for $2\leq i\leq n$, such that

and

for all

the boundary value problem (24) has an unbounded sequence of solutions.

Now put

With a proof similar to the proof of Theorem 18, but this time using Theorem 1 (c) instead of Theorem 1 (b), we can establish the next result. The proof is similar to the proof of Theorem 18, but here we have a real sequence $\{e_n\}$ which tends to zero at $\infty$ constructing $r$, because in Theorem 1 (c) for $\delta$, it requires $r\to(\inf_X\Phi)^+$ instead of in Theorem 1 (b) for $\theta$, it requires $r\to\infty$.

Theorem 26. Suppose that there are real sequences $\{d_n\}$ and $\{e_n\}$ with $\lim_{n\to\infty}e_n = 0$ such that

Then, for each

the boundary value problem $(P^f_\lambda)$ possesses a sequence of pairwise different solutions that strongly converges to $0$ in ${\mathcal{V}}$.

Theorem 27. Suppose

Then, for each

the boundary value problem $(P^f_\lambda)$ has a sequence of pairwise different solutions that converges strongly to $0$ in ${\mathcal{V}}$.

Remark 28. By employing Theorem 26, we may obtain results that are similar to Remark 19 and Corollaries 22–25.

5.   Conclusions

In this paper, we investigated the existence of one and of infinitely many solutions for a class of discrete fractional boundary value problems. As a matter of fact, by demanding an algebraic condition on the nonlinear term for small values of the parameter and requiring an additional asymptotical behavior of the potential at zero, we obtain the existence of at least one nontrivial solution for the problem. Moreover, under suitable assumptions on the oscillatory behavior at infinity of the nonlinearity, for exact collections of the parameter, we get the existence of a sequence of solutions for the problem. The main results improve and extend recent results from the literature. We also presented some examples that illustrate the applicability of the main results. The main technique of the proofs involves variational methods and critical point theorems for smooth functionals.

Conflict of interest

There is no conflict of interest.

Use of AI tools declaration

No AI tools have been used in the preparation of this study.