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).