Abstract
In this paper, we consider a discrete fractional boundary value problem with p-Laplacian operator of the form
where \(f:[\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function, and \(p>1\), \(1<\alpha ,\beta \le 2\). We study the existence and uniqueness of solution to this problem by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem and Brouwer fixed point theorem.
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1 Introduction
In recent years, fractional differential equations have seen tremendous growth. For some recent results and applications, we can see Wu et al. [1], which shows that a discrete fractional difference equation with Caputo derivative sense has an explicit solution in form of the discrete Mittag–Leffler function. Tarasov [2] formulated discrete models for dislocations in fractional nonlocal continua.
The non-locality of fractional differential equations can describe mathematical model better. However, it is difficult for us to calculate and analyze mathematical problems. With the development of computer, it is well known that discrete analogues of differential equations can be very useful, especially for using computer to simulate the behavior of solutions for certain dynamic equations.
A recent paper by Goodrich [3] explored a discrete fractional boundary value problem of the form
where \(f\in [\nu -2,\nu +b-1]_{{\mathbb {N}}_{\nu -2}}\times \mathbb {R}\rightarrow \mathbb {R}\) is a continuous function, \(g:C([\nu -2,\nu +b]_{{\mathbb {N}}_{\nu -2}},\mathbb {R})\) is given function, and \(1<\nu \le 2\). This problem was solved by the contraction mapping theorem, Brouwer fixed point theorem, and Guo–Krasnosel’skii fixed point theorem.
Pan and Han [4] studied the existence and nonexistence of positive solutions to a discrete fractional boundary value problem with a parameter
where \(1<\nu \le 2\) is a real number, \(f:[\nu -1,\nu +b]_{{\mathbb {N}}_{\nu -1}}\times {\mathbb {R}}\rightarrow (0,+\infty )\) is a continuous function, \(b\ge 2\) is an integer, \(\lambda \) is a parameter. The eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered by the properties of the Green function and Guo–Krasnosel’skii fixed point theorem in cones, some sufficient conditions of the nonexistence of positive solutions for the boundary value problem are established.
Differential equations with p-Laplacian operator are increasingly applied in real life, especially in physics and engineering [5]. Some theories of fractional differential equations with p-Laplacian operator are just beginning to be investigated. Lu and Han [6] investigated the existence of positive solutions for the eigenvalue problem of nonlinear fractional differential equation with generalized p-Laplacian operator
where \(2<\alpha \le 3,1<\beta \le 2,D_{0^{+}}^{\alpha }\) and \(D_{0^{+}}^{\beta }\) are the standard Riemann–Liouville fractional differential, \(\phi \) is the generalized p-Laplacian operator, \(\lambda >0\) is a parameter, and \(f:(0,+\infty )\rightarrow (0,+\infty )\) is a continuous function. By using the Green function’s properties and Guo–Krasnosel’skii fixed point theorem in cones, several new existence results of at least one or two positive solutions in terms of different eigenvalue interval are obtained.
Motivated by all the works above, we consider a discrete fractional boundary value problem with p-Laplacian operator
where \(p>1,1<\alpha ,\beta \le 2\), \(\Delta ^{\alpha }\) and \(\Delta ^{\beta }\) denote the Riemann–Liouville fractional differences of order \(\alpha \) and \(\beta \) respectively, \(f:[\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}\times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is a continuous function. \(\phi _{p}\) is the p-Laplacian operator, that is, \(\phi _{p}(u)=|u|^{p-2}u\), \(p>1\). Obviously, \(\phi _{p}\) is invertible and its inverse operator is \(\phi _{q}\), where \(q>1\) is a constant with \(\frac{1}{p}+\frac{1}{q}=1\).
Our work presented in this article has the following features which are worth emphasizing.
-
(i)
As far as we know, there is less literature available concerned with four-point boundary value problems of fractional difference equation which \(\Delta ^{\alpha }\) and \(\Delta ^{\beta }\) are the standard Riemann–Liouville fractional differences.
-
(ii)
We consider the boundary value problem with p-Laplacian which arises in the modeling of different physical and natural phenomena.
-
(iii)
When \(p=2\), the fractional difference equation (1.1) reduce to \( \Delta ^{\beta }(\Delta ^{\alpha }y)(t)+f(\alpha +\beta +t-1,y(\alpha +\beta +t-1))=0\) which involves mixed fractional difference equations.
-
(iv)
When \(p=2\) and \(\beta =0\), the fractional difference equation (1.1) reduce to \( \Delta ^{\alpha }y(t)+f(\alpha +t-1,y(\alpha +t-1))=0\) which is the form studied in [3] and [7].
The plan of the paper is as follows. In Sect. 2, we give some definitions and lemmas which are needed in this paper. In Sect. 3, we study the existence and uniqueness of solution to problem (1.1)–(1.3) by using Banach contraction mapping theorem and Brouwer fixed point theorem. In Sect. 4, we give examples to illustrate the theorems.
2 Preliminaries
For the convenience of the reader, we give some necessary basic definitions and lemmas in discrete fractional calculus theory.
Definition 2.1
([3]) We define \(t^{\underline{\nu }}:=\frac{\Gamma (t+1)}{\Gamma (t+1-\nu )}\) for any t and \(\nu \), for which the right-hand side is defined. We also appeal to the convention that if \(t+1-\nu \) is a pole of the Gamma function and \(t+1\) is not a pole, then \(t^{\underline{\nu }}=0\).
Definition 2.2
([3]) The \(\nu \)th fractional sum of a function f, for \(\nu >0\), is defined by
for \(t\in \{a+\nu ,a+\nu +1,\cdot \cdot \cdot \}:={{\mathbb {N}}}_{a+\nu }\). We also define the \(\nu \)th fractional difference for \(\nu >0\) by \(\Delta ^\nu f(t):=\Delta ^N\Delta ^{\nu -N}f(t)\), where \(t\in \mathbb {N}_{a+\nu }\) and \(\nu \in {{\mathbb {N}}}\) is chosen so that \(0\le N-1<\nu \le N\).
Lemma 2.1
([3]) Let \(0\le N-1<\nu \le N\), where \(N\in {{\mathbb {N}}}\) and \(N-1\ge 0\). Then
for some \(C_i\in {{\mathbb {R}}}\), with \(1\le i\le N\).
Lemma 2.2
([3]) Let t and \(\nu \) be any numbers for which \(t^{\underline{\nu }}\) and \(t^{\underline{\nu -1}}\) are defined. Then
Lemma 2.3
([3]) For t and s, for which both \( (t-s-1)^{\underline{\nu }}\) and \((t-s-2)^{\underline{\nu }}\) are defined, we find that
Lemma 2.4
([7]) et \(0\le N-1<\nu \le N\), where positive integer N greater than or equal to \(\nu \) and \(\nu >0\). Defined
Replace \(\nu \) by \(\nu -m\) to obtain
when \(\Delta ^{N}t^{N-m}=0, m=1\ldots N\), we have \(\Delta ^{\nu }t^{\underline{\nu -m}}=0, m=1\ldots N\).
We now state and prove an important lemma. This lemma will give a representation for the solution to (1.1)–(1.3), provided that the solution exists. This representation will be crucial in Sect. 3 of this paper when we prove our existence and uniqueness theorems.
Lemma 2.5
Let \(f:[\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) be given. A function y is a solution of the (1.1)–(1.3), if and only if it has the form
where H(t, s), h(s) and G(s, l) are given by
Proof
If y(t) is a solution to (1.1)–(1.3), by using Lemma 2.1, we find that
where \(t\in [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}\). Consequently, boundary condition (1.2) implies that \(C_{2}=0\) and
then we deduce that
for \(t\in [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}\). Hence, we have
Then taking p-Laplacian inverse of operators on both sides of (2.4), we find that
Next let
By using Lemma 2.1, we deduce that
Consequently, boundary condition (1.3) implies that \(D_{2}=0\) and
thus we get function (2.1) that
On the other hand, the function y(t) is satisfied to (2.1), then \(y(\alpha +\beta -4)=0\), if \(t=\alpha +\beta -4\); and \(y(\alpha +\beta +b)=0\), if \(t=\alpha +\beta +b\). It is to say function (2.1) meet the boundary condition (1.3).
What’s more, function y(t) defined by (2.1) can transform to
Then we find that
By using Lemma 2.1, we know that
By Eq. (2.5), function \(\Delta ^{\alpha }y(t)\) has the form
Then \(\Delta ^{\alpha }y(\beta -2)=0\), if \(t=\beta -2\); and then \(\Delta ^{\alpha }y(\beta +b)=0\), if \(t=\beta +b\). It is to say function (2.1) meet the boundary condition (1.2). Taking p-Laplacian operators on both sides of (2.6), we find that
or the form
By Eq. (2.7), function \(\Delta ^{\beta }[\phi _{p}(\Delta ^{\alpha }y)](t)\) has the form
By using Lemma 2.4, we know that
By Eq. (2.8), we find that
which shows that if (1.1)–(1.3) has a solution, then it can be represented by (2.1) and that every function of the form (2.1) is a solution of (1.1)–(1.3), which completes the proof. \(\square \)
3 Existence of solutions
In this section, we will show the existence of solutions for boundary value problem (1.1)–(1.3).
Lemma 3.1
The Green’s function G(s, l) and function H(t, s) satisfy the following conditions:
-
(i)
\(G(s,l)\ge 0\), for \((s,l)\in [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}\times [0,b]_{{\mathbb {N}}_{0}}\), \(H(t,s)\ge 0\), for \((t,s)\in [\alpha +\beta -4,\alpha +\beta +b]_{\mathbb {N}_{\alpha +\beta -4}}\times [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}\);
-
(ii)
\(\max _{s\in [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}}G(s,l)=G(l+\beta +1,l),\;\) for \(\;l\in [0,b]_{{\mathbb {N}}_{0}}\), \(\max _{t\in [\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}}H(t,s)=H(s+\alpha +1,s),\) for \(s\in [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}\);
-
(iii)
there exists a number\(\gamma \in (0,1)\) such that
$$\begin{aligned} \min _{s\in \left[ \frac{b+\nu }{4},\frac{3(b+\nu )}{4}\right] }G(s,l)\ge & {} \gamma \max _{s\in [\nu -2,\nu +b+1]_{\mathbb {N}_{\nu -2}}}G(s,l)\\= & {} \gamma G(l+\nu -1,l),\quad \text {for} \; l\in [0,b]_{{\mathbb {N}}_{0}}. \end{aligned}$$
Remark 3.1
We omit the proof here, which is similar to Theorem 3.2 in [7].
Lemma 3.2
([8])
-
(1)
If \(1<p<2\), \(uv>0\) and \(|u|,|v|\ge m>0\), then
$$\begin{aligned} |\phi _{p}(v)-\phi _{p}(u)|\le (p-1)m^{p-2}|v-u|. \end{aligned}$$ -
(2)
If \(p\ge 2\) and \(|u|,|v|\le M\), then
$$\begin{aligned} |\phi _{p}(v)-\phi _{p}(u)|\le (p-1)M^{p-2}|v-u|. \end{aligned}$$
Define Banach spaces
and
with maximum norm.
We now show that problem (1.1)–(1.3) has at least one solution under certain conditions. From Lemma 2.5, we observe that problem (1.1)–(1.3) may be recast as an equivalent summation equation. In order to get the main results, we introduce a operator \(T: B\rightarrow B\) by
From Lemma 2.5, we know that y is a solution of (1.1)–(1.3) if and only if y is a fixed point of the operator
which is defined by
where H(t, s) and Ty(s) are defined as (2.2) and (3.1).
We shall appeal to the contraction mapping theorem to get a unique solution of boundary value problem (1.1)–(1.3) when \(p\ge 2\).
Theorem 3.1
Suppose that f(t, y) is Lipschitz in y, that is, there exists constant \(L>0\) such that \(|f(t,y_{1})-f(t,y_{2})|\le L|y_{1}-y_{2}|\) whenever \(y_{1},y_{2}\in {\mathbb {R}},\) \(t\in [\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}\), and there exists a function A(t) such that \(|f(t,y)|\le A(t)\), for any \(y\in C\), \(t\in [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}\). If \(p\ge 2\) and \(M<1\), then the boundary value problem (1.1)–(1.3) has a unique solution.
Proof
By Lemma 3.1, for all \(y\in B\), we can get that
Let \(M_{1}=\sum _{l=0}^{b}G(l+\beta -1,l)A(l+\alpha +\beta -1)\). What’s more, for any \(y_{1},y_{2}\in B\),
We analyze the right-hand side of (3.3), by an application of Lemma 2.3, that
for \(s\in [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}\). Similarly, we have
for \(s\in [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}\). So, putting (3.3)–(3.5) together, we conclude that
and let \(M_{2}=2L\prod _{j=1}^{{b}}\left( \frac{\beta +j}{j}\right) \).
Next, we will show that S defined as (3.2) is a contraction map. To this end, we note that for given \(y_{1}, y_{2}\in C\),
for \(t\in [\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}\). For convenience, let us put
By inequality (3.7) and Eq. (3.8), we conclude that
whence by \(M<1\), we find that (1.1)–(1.3) has a unique solution. This completes the proof. \(\square \)
We shall next appeal to the contraction mapping theorem to get a unique solution of boundary value problem (1.1)–(1.3) when \(1<p<2\).
Theorem 3.2
Suppose that f(t, y) is Lipschitz in y, that is, there exists \(L>0\) such that \(|f(t,y_{1})-f(t,y_{2})|\le L|y_{1}-y_{2}|\) whenever \(y_{1},y_{2}\in {\mathbb {R}}, t\in [\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}\), and there exist a nonnegative function B(t) satisfying \(|f(t,y)|\ge B(t)\), for any \(y\in C\), \(t\in [\beta -2,\beta +b]_{{\mathbb {N}}_{\beta -2}}\). If \(1<p<2\) and \(K<1\), the boundary value problem (1.1)–(1.3) has a unique solution.
Proof
By Lemma 3.1, for all \(y\in B\), we can get that
Set \(K_{1}=\sum _{l=0}^{b}G(l+\beta -1,l)B(l+\alpha +\beta -1)\). We will show that S defined as (3.2) is a contraction map. By inequality (3.6), we note that for any \(y_{1},y_{2}\in C\), we can get that
for \(t\in [\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}\), here \(M_{2}=2L\prod _{j=1}^{b}\left( \frac{\beta +j}{j}\right) \). For convenience, we denote
By inequality (3.10) and Eq. (3.11), we conclude that
hence (1.1)–(1.3) has a unique solution. This completes the proof. \(\square \)
By weakening the condition imposed on f(t, y), we can still deduce the existence of solutions to (1.1)–(1.3). We shall appeal to Brouwer fixed point theorem to accomplish this.
Theorem 3.3
Suppose that there exists a constant \(G>0\) such that f(t, y) satisfies the inequality
Then (1.1)–(1.3) has at least one solution, say \(y_{0}\), satisfying \(|y_{0}(t)|\le G\), for all \(t\in [\alpha +\beta -4,\alpha +\beta +4]_{{\mathbb {N}}_{\alpha +\beta -4}}\).
Proof
Denote \(D=\{y\in C:\Vert y\Vert \le G\}\). Let S be the operator defined in (3.2). It is easy to prove that S is a continuous operator. By using Brouwer fixed point theorem, we will to show that \(S:D\rightarrow D\), that is, whenever \(\Vert y\Vert \le G\), it follows that \(\Vert Sy\Vert \le G\). Once this is established, we deduce the conclusion. To this end, assume that inequality (3.13) hold for f(t, y). For notational convenience in what follows, let us put
which is a positive constant. For any a \(y\in D\), from (2.3), we observe that
Note that
here we use the fact that \(t^{\underline{\beta -1}}\) is increasing in t since \(\beta -1>0\). Furthermore,
If we now put (3.15)–(3.17) together, by the definition of \(\Psi \) given in (3.14), then we find that
By inequality (3.18), we conclude that
Since
where to get inequality (3.20) we have used the fact that \(t^{\underline{\alpha -1}}\) is increasing in t since \(\alpha -1>0\), furthermore,
If we now put (3.19)–(3.21) together, then we find that
Thus, from (3.22) we deduce that \(S:D\rightarrow D\), as desired. Consequently, it follow at once by Brouwer fixed point theorem that there exists a fixed point of the map S, say \(Sy_{0}=y_{0}\) with \(y_{0}\in C\). But this function \(y_{0}\) is a solution of (1.1)–(1.3). Moreover, \(y_{0}\) satisfies the bound \(|y_{0}(t)|\le G\), for each \(t\in [\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}\). This completes the proof. \(\square \)
4 Examples
In this section, we will present some examples to illustrate main results.
Example 4.1
Consider boundary value problem of discrete fractional equation
where \(f(t,y):=t^{2}+\frac{1}{700}\sin y(t)\), for \(t\in [0,2]_{{\mathbb {N}}_{0}}, y\in {{\mathbb {R}}}\), is Lipschitz with Lipschitz constants \(L=\frac{1}{700}\). When \(p=3\), for this choice of L, inequality (3.9) is satisfied with \(M\approx 0.89<1.\) Therefore, we deduce from Theorem 3.1 that problem (4.1)–(4.3) has a unique solution.
Example 4.2
Consider boundary value problem of discrete fractional equation
where \(f(t,y):=\frac{1}{300}+\frac{1}{300}\sin y(t)\), for \(t\in [0,2]_{{\mathbb {N}}_{0}}, y\in {{\mathbb {R}}}\), is Lipschitz with Lipschitz constants \(L=\frac{1}{300}\). When \(p=\frac{3}{2}\), for this choice of L, inequality (3.12) is satisfied with \(K\approx 0.835<1.\) Therefore, we deduce from Theorem 3.2 that problem (4.4)–(4.6) has a unique solution.
Example 4.3
We suppose that \(f(t,y):=t^{2}+\sin y(t)\), for \(t\in [0,2]_{{\mathbb {N}}_{0}}, y\in {{\mathbb {R}}}\). Consider boundary value problem of discrete fractional equation
and the Banach space C in this case is \(C=\{y:[-1,5]_{{\mathbb {N}}}\rightarrow {{\mathbb {R}}}\}\).
We claim that (4.7)–(4.9) has at least one solution.Suppose that \(G=400\) and \(p=2\). To check the hypotheses of Theorem 3.3 hold, we note that
Now, it is clear that \(|f(t,y)|\le 5<26.3,\) whenever \(|y(t)|\le 400\). By Theorem 3.3 we deduce that this solution, say \(y_{0}(t)\), satisfies \(|y_{0}(t)|\le G\) for \(t\in [-1,5]_{{\mathbb {N}}}\).
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Acknowledgments
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11571202, 11572205), and supported by Shandong Provincial Natural Science Foundation (ZR2013AL003).
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Zhao, Y., Sun, S. & Zhang, Y. Existence and uniqueness of solutions to a fractional difference equation with p-Laplacian operator. J. Appl. Math. Comput. 54, 183–197 (2017). https://doi.org/10.1007/s12190-016-1003-1
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DOI: https://doi.org/10.1007/s12190-016-1003-1
Keywords
- Boundary value problem
- Fractional difference equation
- p-Laplacian operator
- Contraction mapping theorem
- Brouwer fixed point theorem