Abstract
This article aims to organize positive solutions to discrete fractional boundary value problems for continuous Gâteaux differentiable functions. A generalized Gâteaux derivative is introduced using a fractional discrete operator for a Jumarie fractional operator. The method of finding solutions is based on critical point theorems of finite dimensional Banach spaces.
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1 Introduction
In 1974, Diaz and Osler [1] imposed a discrete fractional difference operator based on an infinite series. In 1988, Gray and Zhang [2] presented a class of fractional difference operators and introduced the Leibniz formula. In 2009, Atici and Eloe [3] introduced methods for composing fractional sums and differences. Mathematicians, physicists, and engineers have recently employed fractional calculus to solve varieties of models of applied problems in different fields. In a previous work, we developed various classes of discrete fractional operators with one or two parameters for image and signal processing [4–6].
Researchers in the fields of computer science, neural networks, control systems, food processing, and economics rely on mathematical modeling because it naturally entails nonlinear difference equations. Therefore, many authors have widely developed various procedures and patterns, such as upper and lower solutions, fixed point theorems, and the Brouwer degree, to study discrete models [7–11]. Critical point theory has recently attracted the attention of many researchers. The theory guarantees the outcomes of both ordinary and partial differential problems. Hence, critical point theory is the main tool for finding solutions to fractional discrete nonlinear equations.
This study aims to establish positive solutions to discrete fractional boundary value problems for continuous Gâteaux differentiable functions. A generalized Gâteaux derivative is introduced using a fractional discrete operator. Some results are refined in this direction. The method for finding solutions is based on critical point theorems of finite dimensional Banach spaces. The applications are illustrated to study the abstract outcomes.
2 Main methods
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
The fractional difference on the right and of the order ℘, \(0< \wp< 1\) of \(g(x)\) is defined by the formula
with its fractional derivative on the right
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
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
with the fractional difference on the right
and its fractional derivative on the right
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
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 \(\rho> F(x_{0})\)
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
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
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
endowed with a discrete norm
such that (see [7, Lemma 2.2])
Let
where
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).
3 Main findings
In this section, we introduce at least one solution to problem (5) with the following results.
Theorem 3.1
Consider three positive constants C, \(C_{1}\), and \(C_{2}\) such that
where \(\overline{\sigma}:= \sum^{N}_{k=1} \sigma_{k}\). If
and
where \((2C_{i})^{p}\neq(2+\overline{\sigma} )(N+1) ^{p-1}C^{p}\), \(i=1,2\), then (5) has at least one non-trivial solution \(\mu^{*}\) such that
Proof
We aim to apply Theorem 2.1 on T, because the critical points of T are solutions to (5). Choose \(\omega\in \mathcal{B}\) and define
Thus, we obtain
According to condition (6), a computation implies that
We proceed to consider conditions (1) and (2):
Similarly, we have
Thus, in virtue of Theorem 2.1, we conclude that T has at least one critical point \(\mu^{*}\) such that
and we obtain
□
By letting \(C_{1}=0\), \(C_{2}=C\), we obtain the following result.
Corollary 3.1
Consider two positive constants \(C_{1}\) and C with
If
and
where \(2^{p}\neq(2+\overline{\sigma} )(N+1) ^{p-1}\neq0\), then (5) has at least one non-trivial solution \(\mu^{*}\) such that
Theorem 3.2
Consider two positive constants C and ϵ such that
If
where \((2\epsilon)^{p}\neq(2+\overline{\sigma} )(N+1) ^{p-1}C^{p}\), then (5) has at least one non-trivial solution \(\mu^{*}\) such that
Proof
We aim to employ Theorem 2.2 on T, where all critical points of T are solutions to (5). Choose \(\omega\in\mathcal{B}\) as in Theorem 2.1. We have
According to condition (8), a calculation yields
We proceed to consider condition (3):
Thus, in view of Theorem 2.2, we observe that T has at least one critical point \(\mu^{*}\) such that
and we obtain
This completes the proof. □
Theorem 3.3
Consider two positive constants C and γ such that
If
where \((2\gamma)^{p}\neq(2+\overline{\sigma} )(N+1) ^{p-1}C^{p}\), then (5) has at least one non-trivial solution.
Proof
Our aim is to employ Theorem 2.3 on T, because the critical points of T are solutions to (5). We put \(\omega\in\mathcal{B}\) as in Theorem 3.1 such that
According to condition (9), a manipulation leads to
Thus, we consider condition (4):
Therefore, in virtue of Theorem 2.3, we conclude that T has at least one critical point. The boundedness of the solution is due to the (SP)-condition. The above completes the proof. □
4 Multiplicity findings
We impose multiplicity solutions in this section.
Theorem 4.1
Let the hypothesis of Theorem 3.1 hold. Assume that \(h(k,0) \neq0\), \(k \in [1,N]\). Consider two constants \(\kappa_{1}>p\) and \(\kappa_{2}>0\) such that for \(|\chi| \geq\kappa_{2}\), one has
If \(T'\) is a Lipschitz continuous functional in ℬ, then (5) admits at least two solutions.
Proof
In view of Theorem 3.1, (5) has at least one solution, say \(\mu_{1}\). We use the mountain pass theorem to identify the second solution. The functional \(T= F-G\) is obviously in class \(C^{1}(\mathcal{B},\mathbb{R})\) with \(T(0)=0\). Assume that \(\mu_{1}\) is a strict local minimum for T in ℬ. Thus a constant \(\varrho>0\) satisfies
After integrating (10), we observe that there are two positive constants \(b_{1}\) and \(b_{2}\) such that
We then proceed to identify the second solution of (5). Let \(\nu\in \mathcal{B}\backslash \{0\}\). A calculation then yields
Given that \(\kappa_{1}>p\), we have
This formula implies that the functional T has a critical point \(\mu_{2}:=\inf_{\nu\in\mathcal{B}} \max_{t \in\mathbb{R}} T(\nu(t))\), which is the second solution for (5). This condition completes the proof. □
In the same manner as in Theorem 4.1, we have the following results.
Theorem 4.2
Let the hypothesis of Theorem 3.2 be satisfied. Assume that \(h(k,0) \neq0\), \(k \in [1,N]\). Consider the assertion of (10). If \(T'\) is a Lipschitz continuous functional in ℬ, then (5) admits at least two solutions.
Theorem 4.3
Let the hypothesis of Theorem 3.3 hold. Assume that \(h(k,0) \neq0\), \(k \in [1,N]\). Consider the inequality (10). If \(T'\) is a Lipschitz continuous functional in ℬ, then (5) admits at least two solutions.
5 Positive findings
We focus on the positive results of problem (5). For this purpose, we assume
where \(\lambda:[1,N] \rightarrow\mathbb{R}\) is a non-negative, non-zero function and \(\hbar: [0, +\infty) \rightarrow\mathbb{R}\) is a continuous function with \(\hbar(0)=0\). We have the following result.
Theorem 5.1
Let \(h(k,t)\) be defined in (11) and (8) let be satisfied. Then the problem
has at least one positive solution \(\mu_{k}\).
Proof
By Theorem 3.2, we conclude that (12) has at least one solution \(\mu^{*}\). If \(\sigma_{k} =0\) and \(\mu_{j}:=\min_{k\in[1,N]} \mu_{k}\), then we obtain
However, \(\psi_{p}\) is strictly monotone, and \(\mu_{j}-\mu_{j-1}=0\) and \(\mu_{j+1}-\mu_{j}=0\). Consequently, we have a positive solution. Let \(\sigma_{k}\neq0\), we attain
and
so \(\mu_{j} \geq0\) and \(\mu\geq0\). This completes the proof. □
In the same manner as Theorem 5.1, we have the following.
Theorem 5.2
Let \(h(k,t)\) be defined in (11) and let (9) hold. Then problem (13) has at least one positive solution \(\mu_{k}\).
6 Example
Assume the problem
Let \(h (\zeta):= \frac{\lambda}{8}+ |\zeta|^{2} \zeta\), \(\zeta\in \mathbb{R}\). It is clear that \(h(0)\neq0\), when \(\lambda\neq0\) and \(\lim_{\zeta\rightarrow0^{+}} \frac{h(\zeta)}{\zeta}= +\infty\). Moreover, we let \(\kappa_{1}=4 >p=3\) and \(\kappa_{2}=1 \Rightarrow|\zeta| \geq1\). Thus, we conclude that
Obviously, \(T'(\mu_{k})= ( \frac{\|\mu\|^{p}}{p}- \sum^{N}_{k=1} H(k,\mu_{k}) )'\) is a Lipschitz continuous functional. Hence in view of Theorem 4.1, problem (13) has at least two solutions.
Now let \(\sigma_{k}=0\) and \(\lambda=0\). We then have the following problem:
such that \(\hbar(t):=t|t|^{2}\), \(t \in[0,\infty)\). Hence, in view of Theorem 5.1, where inequality (8) is satisfied for \(\overline{\sigma}=0\), problem (14) has at least one positive solution.
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Acknowledgements
The authors would like to thank the referees for giving useful suggestions for improving the work. This research is supported by Project No.: RG312-14AFR from the University of Malaya.
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Ibrahim, R.W., Jalab, H.A. Discrete boundary value problem based on the fractional Gâteaux derivative. Bound Value Probl 2015, 23 (2015). https://doi.org/10.1186/s13661-015-0287-7
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DOI: https://doi.org/10.1186/s13661-015-0287-7