This section deals with preliminaries and some concepts.

### Definition 2.1

[12]

For a continuous function \(g: \mathbb{R} \rightarrow\mathbb{R}\) and a constant \(\ell>0\), the forward operator \(FW(\ell)\) is defined by the equality

$$FW(\ell)g(x):= g(x+\ell). $$

The fractional difference on the right and of the order ℘, \(0< \wp< 1\) of \(g(x)\) is defined by the formula

$$\bigtriangleup^{\wp} g(x):= \bigl(FW(\ell) -1 \bigr) ^{\wp}g(x) $$

with its fractional derivative on the right

$$g^{(\wp)}_{+}(x)= \lim_{\ell\rightarrow0} \frac{\bigtriangleup^{\wp} [g(x)-g(0)]}{\ell^{\wp}}. $$

### Definition 2.2

[13]

Let \((\Xi, \|\cdot\|)\) be a real Banach space with the dual space \(\Xi^{*}\). A function \(\varphi: \Xi\rightarrow\mathbb{R}\) is called Gâteaux differentiable at \(x \in\Xi\) if \(\chi:= \varphi'(x) \in\Xi^{*}\) satisfies

$$\lim_{\ell\rightarrow0^{+} } \frac{\varphi(x+ \ell y)-\varphi(x)}{\ell }= \varphi'(x) (y),\quad \forall y \in\Xi. $$

### Definition 2.3

[13]

A function \(\varphi: \Xi\rightarrow\mathbb{R}\) is called Gâteaux differentiable and verifies the Palais-Smale condition ((PS)-condition) if any bounded sequence \(\{x_{n}\}\) with \(\lim_{n\rightarrow \infty} \|\varphi'(x_{n})\|_{\Xi^{*}}=0\) has a convergent subsequence.

Combining Definitions 2.1 and 2.2, we obtain a fractional Gâteaux derivative.

### Definition 2.4

Let \((\Xi, \|\cdot\|)\) be a real Banach space with the dual space \(\Xi^{*}\). A function \(\varphi: \Xi\rightarrow\mathbb{R}\) has a fractional Gâteaux derivative, of the order \(0< \wp<1\) at \(x \in\Xi\) if \(\varphi^{(\wp)}(x) \in\Xi^{*}\) exists such that for a constant \(\ell>0\), the forward operator \(FW_{y}(\ell)\), \(y \in\Xi\) is defined by the equality

$$FW_{y}(\ell)\varphi(x):= \varphi(x+\ell y), $$

with the fractional difference on the right

$$\bigtriangleup^{\wp}_{y} \varphi(x):= \bigl(FW_{y}( \ell) -1 \bigr) ^{\wp}\varphi(x), $$

and its fractional derivative on the right

$$\lim_{\ell\rightarrow0^{+} } \frac{\bigtriangleup^{\wp}_{y}[\varphi (x)-\varphi(0)]}{\ell^{\wp}}= \varphi^{(\wp)}(x) (y), \quad \forall y \in\Xi. $$

The next results show some properties of the fractional Gâteaux derivative, which basically are generalizations of some results given in [14]. Therefore, we skip the proofs.

### Theorem 2.1

*Suppose that* Ξ *is a real Banach space and*
\(F, G: \Xi\rightarrow\mathbb{R}\)
*are two continuously fractional Gâteaux differentiable functions*. *Assume that*

*and that*
\(x_{0} \in\Xi\)
*and*
\(\rho_{1}, \rho_{2} \in \mathbb{R}\)
*exist*, *with*
\(\rho_{1} < F(x_{0}) < \rho_{2}\), *thereby obtaining*

$$\begin{aligned}& \sup_{y \in F^{-1}\,]\rho_{1}, \rho_{2}[} G(y) < G(x_{0})+ \rho_{2} - F(x_{0}), \end{aligned}$$

(1)

$$\begin{aligned}& \sup_{y \in F^{-1}\,]{-}\infty, \rho_{1} ]} G(y) < G(x_{0})+ \rho_{1} - F(x_{0}). \end{aligned}$$

(2)

*Presume that*
*T*
*satisfies the* (*PS*)-*condition*. *Then*
\(y_{0} \in F^{-1}\,]\rho_{1}, \rho_{2}[\)
*such that*
\(T(y_{0}) < T(y)\), \(y \in F^{-1}\,]\rho_{1}, \rho_{2}[\), *and*
\(T'(y_{0})=0\).

### Theorem 2.2

*Let* Ξ *be a real Banach space*, *and let*
\(F, G: \Xi\rightarrow\mathbb{R}\)
*be two continuous fractional Gâteaux differentiable functions*. *Assume that*

*and that*
\(x_{0} \in\Xi\)
*and*
\(\rho\in \mathbb{R}\)
*such that*
\(\rho> F(x_{0})\)

$$ \sup_{y \in F^{-1}\,]{-}\infty, \rho[} G(y) < G(x_{0})+ \rho- F(x_{0}). $$

(3)

*We infer that*
*T*
*satisfies the* (*PS*)-*condition*. *Then*
\(y_{0} \in F^{-1}\,]{-}\infty, \rho[\)
*such that*
\(T(y_{0}) < T(y)\), \(y \in F^{-1}\,]{-}\infty, \rho[\), *and*
\(T'(y_{0})=0\).

### Theorem 2.3

*Let* Ξ *be a real Banach space and let*
\(F, G: \Xi\rightarrow\mathbb{R}\)
*be two continuous fractional Gâteaux differentiable functions*. *Assume that*

*where*
*T*
*is bounded from below*, *and*
\(x_{1} \in\Xi\)
*and*
\(\rho\in \mathbb{R}\)
*with*
\(\rho< F(x_{1})\)
*such that*

$$ \sup_{y \in F^{-1}\,]{-}\infty, \rho]} G(y) < G(x_{1})+ \rho- F(x_{1}). $$

(4)

*We deduce that*
*T*
*satisfies the* (*PS*)-*condition*. *Then*
\(y_{1} \in F^{-1}\,]\rho,+\infty[\)
*such that*
\(T(y_{1}) \leq T(y)\), \(y \in F^{-1}\,]\rho,+\infty[\), *and*
\(T'(y_{1})=0\).

We apply the above critical point theorems to find solutions to the fractional difference equation, taking the type

$$\begin{aligned} &{-}\bigtriangleup^{\wp}_{y} \bigl( \psi_{p}\bigl(\bigtriangleup^{\wp}_{y} \mu_{k-1}\bigr) \bigr) +\sigma_{k} \psi_{p}( \mu_{k})= h(k,\mu_{k}), \quad k \in[1, N], \\ &\mu_{0}= \mu_{N+1}=0, \end{aligned}$$

(5)

where \(h: [1, N] \times\mathbb{R} \rightarrow\mathbb{R}\) is a continuous function, \(\sigma_{k}\geq0\), and \(\psi_{p}(\varsigma):= |\varsigma| ^{p-1} \varsigma\), \(1< p< +\infty\).

Define the Banach space as

$$\mathcal{B}:= \bigl\{ \mu:[0,N+1] \rightarrow\mathbb{R} : \mu_{0}=\mu _{n+1}=0\bigr\} $$

endowed with a discrete norm

$$\|\mu\| := \Biggl( \sum^{N+1}_{k=1} \bigl| \bigtriangleup^{\wp}_{y,k-1}\mu \bigr|^{p}+ \sigma_{k}|\mu|^{p} \Biggr)^{1/p} $$

such that (see [7, Lemma 2.2])

$$\max_{k \in[1,N]} |\mu_{k}| \leq\frac{(N+1)^{(p-1)/p} }{2}\|\mu\|, \quad \forall \mu\in\mathcal{B}. $$

Let

$$F(\mu) := \frac{\|\mu\|^{p}}{p}, \qquad G(\mu):= \sum^{N}_{k=1} H(k,\mu_{k}),\qquad T(\mu):= F(\mu) -G(\mu),\quad \forall \mu\in \mathcal{B}, $$

where

$$H(k,t):= \int^{t}_{0}h(k,u)\,du, \quad\forall (k,t) \in[1, N]\times \mathbb{R}. $$

Note that \(T \in C^{1}(\mathcal{B}, \mathbb{R})\), \(T(0)=0\), and that all the critical points of *T* are the solutions of (5).