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Existence of solutions for fractional difference equations via topological degree methods
Advances in Difference Equations volume 2018, Article number: 153 (2018)
Abstract
In this paper, a class of nonlinear fractional difference equations with Caputolike difference operator is considered. Some existence results for the given equation are obtained by means of topological degree methods. Three examples are constructed for the illustration of the obtained theory.
Introduction
In this paper, we consider the following nonlinear fractional difference equations:
where \({}^{C} \Delta^{\alpha}\) is a Caputolike fractional difference operator, \(f:[0,\infty)\times\mathbb{R}\to\mathbb{R}\) is a given continuous function and \(\mathbb{N}_{1\alpha}=\{1\alpha, 2\alpha,\ldots\}\).
Fractional differential operators appear naturally in modeling many phenomena in various fields of engineering, physics and economics, for example, nonlinear oscillations of earthquakes, seepage flow in porous media and dynamic traffic flow model. For details, we refer the reader to the monographs by Kilbas et al. [1], Diethelm [2], Zhou [3, 4], and to [5–12]. The theory of fractional difference equations has been rapidly developed in recent years (see [13–17]). It can elegantly describe certain behaviors in discrete fractional calculus or generalized difference equations. There are several definitions of fractional sum/difference operators proposed by many mathematicians such as Gray and Zhang [18], Atici and Eloe [19], Abdeljawad [20]. Goodrich [21] studied existence of positive solutions for fractional difference equations with initialboundary data, Chen et al. [22–24] obtained some asymptotic stability results for some nonlinear fractional difference equations. However, to the best of our knowledge, fractional difference equations have not yet been investigated with the aid of topological degree methods.
In this paper, we show existence of solutions for nonlinear fractional difference equations by applying a fixed point theorem due to Isaia [25], which was obtained via coincidence degree theory for condensing maps. The rest of the article is organized as follows. In Sect. 2, we introduce some important notions about fractional difference operators and topological degree theory, while Sect. 3 contains our main existence results for Eq. (1.1).
Preliminaries
For real numbers a and for \(\alpha\in(0,1)\), we denote \(\mathbb {N}_{a}=\{a,a+1,\ldots\}\), \(\mathbb{N}_{a+\alpha}=\{a+\alpha, a+\alpha +1,\ldots\}\) and \(\mathbb{N}_{1}=\mathbb{N}\). The forward Euler difference operator Δ is defined by \(\Delta u(t):=u(t+1)u(t)\), \(t\in \mathbb{N}_{a}\).
Definition 2.1
([19])
Let \(\alpha> 0\) be given. The fractional sum of order α is defined as
where u is given for \(s=a\) mod (1), \(\Delta^{\alpha}u(t)\) is defined for \(t=(a+\alpha)\) mod (1), and the falling factorial function is
The fractional sum \(\Delta^{\alpha}\) maps functions defined on \(\mathbb {N}_{a}\) to the functions defined on \(\mathbb{N}_{a+\alpha}\).
Definition 2.2
Let \(\mu>0\) be such that \(m1<\mu<m\), where m denotes a positive integer and \(m=\lceil\mu\rceil\), \(\lceil\cdot\rceil\) ceiling of number. Set \(\nu=m\mu\). The Caputolike fractional difference operator of order \(\alpha> 0\) is defined by
where \(\Delta^{m}\) is the mth order forward difference operator. The fractional Caputolike difference operator \({}^{C} \Delta^{\mu}\) maps functions defined on \(\mathbb{N}_{a}\) to the functions defined on \(\mathbb {N}_{a+m\mu}\).
Theorem 2.1
Let u be a real value function defined on \(\mathbb{N}_{a}\) and \(\mu,\nu >0\). Then
Lemma 2.1
([24])
Let \(0<\mu<1\). Then
where u is defined on \(\mathbb{N}_{0}\) and \({}^{C} \Delta^{\mu}\) is defined on \(\mathbb{N}_{1\mu}\).
Lemma 2.2
([13])
Assume that \(\mu+\nu+1\) is not a nonpositive integer with \(\mu\neq1\). Then
Definition 2.3
Let \(\Omega\subset X\) and \(F : \Omega\to X\) be a continuous bounded map. We say that F is σLipschitz if there exists \(\kappa\geq 0\) such that
In case \(\kappa< 1\), we call F a strict σcontraction. We say that F is σcondensing if \(\sigma(F (B)) < \sigma(B)\) for any bounded \(B \subset\Omega\) with \(\sigma(B)>0\). In other words, \(\sigma(F (B))\geq\sigma(B)\) implies \(\sigma(B) = 0\). The aforementioned σ is the Kuratowski measure of noncompactness.
Proposition 2.1
If \(F :\Omega\to X\) is compact, then F is σLipschitz with constant \(\kappa= 0\).
Let
be the family of the admissible triplets, where \(C_{\sigma}(\overline {\Omega})\) is defined by the class of all σcondensing maps \(F :\overline{\Omega}\to X\).
Theorem 2.2
([25])
Let \(F : X\to X\) be σcondensing and
If \(\mathfrak{T}\) is a bounded set in X, then there exists \(r > 0\) such that \(\mathfrak{T}\subset B_{r}(0)\), and for a degree function \(D : \mathfrak{T}\to\mathbb{Z}\),
Then F has at least one fixed point and the set of the fixed points of F lies in \(B_{r}(0)\).
The space \(l^{\infty}_{n_{0}}\) is the set of real sequences defined on the set of positive integers where any individual sequence is bounded with respect to the usual supremum norm. It is well known that \(l^{\infty}_{n_{0}}\) is a Banach space under the supremum norm.
Definition 2.4
A set Ω of sequences in \(l^{\infty}_{n_{0}}\) is uniformly Cauchy (or equiCauchy) if, for every \(\varepsilon> 0\), there exists an integer N such that \(x(i)x(j) < \varepsilon\), whenever \(i, j > N\) for any \(x =\{x(n)\}\) in Ω.
Theorem 2.3
(Discrete Arzela–Ascoli’s theorem)
A bounded uniformly Cauchy subset Ω of \(l^{\infty}_{n_{0}}\) is relatively compact.
Lemma 2.3
([23])
Assume that \(\beta>1\) and \(\gamma> 0\). Then
Lemma 2.4
([19])
Assume that the falling factorial functions are well defined, then

(i)
If \(0<\beta<1\) and \(\gamma> 0\), then \((t^{(\gamma )} )^{\beta}\leq t^{(\beta\gamma)}\);

(ii)
\(t^{(\alpha+\beta)}=(t\beta)^{(\alpha)}t^{(\beta)}\).
Remark 2.1
Obviously, If \(\beta=1\) in Lemma 2.3 and \(\beta=1\) in Lemma 2.4(i), then the resulting expressions still hold true.
Lemma 2.5
Assume that the falling factorial functions are well defined. Then
Proof
From the definition of falling factorial function \(t^{(\cdot)}\), we deduce that \(t^{(\alpha)}\) is nonincreasing for any \(\alpha\geq0\), \(t>1\), that is, \(t^{(\alpha)}\leq(t\beta)^{(\alpha)}\) for \(\alpha \geq0\), \(\beta\geq0\) and \(t>\beta1\). Indeed, it holds true obviously for \(\alpha=0\) or \(\beta=0\). Then, for \(\alpha>0\), \(\beta>0\) and \(t>\beta 1\), we find that
where \(\lambda=\frac{\alpha}{\beta+\alpha}\in(0,1)\). Then, following the logconvexity property of the gamma function, we have
Therefore, \(t^{(\alpha)}\leq(t\beta)^{(\alpha)}\) for \(\alpha>0\), \(\beta>0\) and \(t>\beta1\). By Lemma 2.4(ii), we obtain
□
Main results
In this section, we study the existence and uniqueness of solutions for nonlinear fractional difference equations.
Let \(X:=l^{\infty}_{1}\) be the set of all real sequences \(\{x(t)\} _{t=1}^{\infty}\) with norm
Then X is a Banach space.
Lemma 3.1
Let f be a realvalued function. Then the problem (1.1) has one solution if and only if u is a solution of the following fractional Taylor difference equations:
where \(0<\alpha<1\).
Proof
The proof is similar to that of [24, Lemma 2.4]. So we omit it. □
Now, for \(\gamma>0\), we define
Clearly the set S consists of nonempty bounded and closed subsets of X.
For any \(u\in S\), define an operator \(P:X\to X\) as follows:
Observe that the existence of a solution u for (1.1) is equivalent to that of a fixed point u in S such that \(u=Pu\) holds.
Lemma 3.2
Assume that

(H1)
there exist \(C_{f}> 0\), \(\beta\in(\alpha,1)\), and \(\gamma=\frac{1}{2}(\beta\alpha)\) such that
$$ \frac{C_{f}\Gamma(1\beta)}{\Gamma(1+\alpha\beta)}\leq1. $$(3.4)If f satisfies
$$\bigf (t , u)\big \leq C_{f} t^{(\beta)}, $$then the operator P is continuous and P maps S into S.
Proof
Let \(u\in S\). Since \(0<\alpha<\beta\), by the nonincreasing characteristic of \(t^{(\alpha)}\) for any \(t\in\mathbb{N}\), it follows by Lemma 2.5 together with a given \(\varepsilon>0\) that there exists \(n_{1}\in\mathbb{N}\), such that, for \(n\in\mathbb{N}_{n_{1}}\), we have
By the definition of P, Lemma 2.2, Lemma 2.4(ii), (H1), and inequality (3.4), for \(t\in\mathbb{N}\), we find that
where we applied the inequality \(1\leq\Gamma(\theta)\) for \(\theta\geq 2\). Hence \(PS\subset S\).
Let \(\{u_{m}\}_{m=1}^{\infty}\) be a sequence of S such that \(u_{m}\to u\) as \(m\to\infty\). Then, for \(t\in\mathbb{N}_{n_{1}}\), by (H1) and (3.5), we obtain
For \(t\in\{1,\ldots,n_{1}1\}\), in view of the continuity of f, we get
Thus, for \(t\in\mathbb{N}\), it is clear that
Hence P is continuous on S. □
Lemma 3.3
Assume that (H1) holds. Then PS is a compact subset of X.
Proof
From Lemma 3.2, we know that PS is a bounded subset of X. Next, we will show that P is compact.
Let \(t_{1},t_{2}\in\mathbb{N}_{n_{1}}\) with \(t_{2}>t_{1}\). From (3.5), we have
Hence, for an arbitrary choice of ε, \(\{Pu: u\in S\}\) is a uniformly Cauchy subset of X by Definition 2.4. From Lemma 3.2, we know that \(\{Pu: u\in S\}\) is bounded. Thus a direct application Theorem 2.3 implies that PS is relatively compact. □
Theorem 3.1
Assume that (H1) holds. Then the problem (1.1) has at least one solution \(u\in S\) and the set of solutions of (1.1) is bounded in S.
Proof
Let \(P: S\to S\) be the operator defined by (3.3). We know that P is continuous and bounded by Lemma 3.2. Moreover, by Lemma 3.3, P is compact. Hence, it follows by Proposition 2.1 that P is a strict σcontraction with constant zero.
Let us set
and show that \(S_{0}\) is bounded in S. Consider \(u\in S_{0}\) and \(\lambda \in[0, 1]\) such that \(u = \lambda Pu\). Using (3.6), we find that
which implies that \(S_{0}\) is bounded in S. If not, we suppose by contradiction, \(\rho:=\u\\to\infty\). Dividing both sides of (3.7) by ρ, and taking the limit \(\rho\to\infty \), we have
which is an obvious contradiction. Consequently, by Theorem 2.2, we deduce that P has at least one solution \(u^{*}\) in S. □
Theorem 3.2
Assume that (H1) holds. Furthermore, suppose that

(H2)
there exist \(L>0\) and \(\xi>\alpha\) such that
$$ \bigf(t,y)f(t,x)\big\leq Lt^{(\xi)}xy,\quad\textit{for any } x,y \in S, \textit{for } t\in\mathbb{N}_{0}. $$(3.8)
Then the problem (1.1) has a unique solution provided that
Proof
Let \(u,v\in S\) be two solutions of (1.1). Then, for \(t\in\mathbb {N}\), applying assumption (H2), we have
which implies that \(\uv\=0\) by virtue of (3.9). Hence, there exists a unique solution of (1.1). □
Example 3.1
Let us consider the following fractional difference equations:
where \(f(t,u)=\frac{0.5t^{(0.5)}\sin(u(t))}{(1+9e^{t})(1+\sin (u(t)))}\) for \(t\in\mathbb{N}_{0}\).
Since
and, from the above given data, we find that
therefore, condition (H1) holds. Thus, by Theorem 3.1, there exists at least one solution in S. Furthermore,
which implies that (H2) holds. With the given data, we also find that the inequality (3.9) holds. Thus the problem (3.10) admits a unique solution.
Theorem 3.3
Let \(\frac{1}{1\alpha}< q\). Assume that

(H3)
there exist \(C_{f}'> 0\), \(\eta\in(0,q(1\alpha)1)\) and \(\gamma\in(\frac{\eta+\alpha}{q1},\frac{1+\eta}{q})\) such that
$$ \frac{C_{f}'\Gamma(1+\gamma q)\Gamma(1+\eta\gamma q)}{\Gamma^{q}(1+\gamma )\Gamma(1+\alpha+\eta\gamma q)}\leq1. $$(3.11)If f satisfies
$$\bigf (t , u)\big \leq C_{f}'t^{(\eta)}\bigu(t)\big^{q}, $$then the problem (1.1) has at least one solution \(u\in S\) and the set of solutions of (1.1) is bounded in S.
Proof
From the definition of P, by the nonincreasing characteristic of \(t^{(\alpha)}\) for any \(t\in\mathbb{N}\), together with (H3), Lemma 2.3, Lemma 2.4(ii) and Lemma 2.5, for any \(u\in S\), we have
This shows that \(PS\subset S\).
The remaining proof concerning continuity of P is similar to that of Lemma 3.2. Consequently, PS is compact by Lemma 3.3 and hence there exists at least one solution for the problem (1.1) by Theorem 3.1. □
Corollary 3.1
Assume that \(\eta=0\) in (H3) and that
 (H3)′:

there exist \(C_{f}'> 0\), \(q>1\), and \(\gamma\in(\frac {\alpha}{q1},\frac{1}{q})\) such that
$$\bigf (t , u)\big \leq C_{f}'\bigu(t)\big^{q}. $$
Example 3.2
Consider the following fractional difference equations:
where \(f(t,u)=0.5u^{2}(t)\) for \(t\in\mathbb{N}_{0}\).
Since \(f(t,u)\leq0.5u(t)^{2}\), the condition (H3)′ holds for \(\gamma =0.25\). In consequence, there exists at least one solution for the given problem in S by Corollary 3.1.
Remark 3.1
Replacing the condition \(q>1\) of (H3) by \(0< q\leq1\), we observe that there does not exist any solution for the problem (1.1). However, with suitable conditions on f, we may have existence results for the problem (1.1).
Theorem 3.4
Let \(0< q\leq1\). Assume that

(H4)
there exist \(C_{f}''> 0\), \(\sigma\in[1+(\alpha 1)q,1)\) and \(\gamma\in(0,\frac{1\sigma}{q})\), such that
$$ \frac{C_{f}''\Gamma(1\sigma\gamma q)}{\Gamma(1+\alpha\sigma\gamma q)}\leq1. $$(3.12)If f satisfies
$$\bigf (t , u)\big \leq C_{f}''(t+1)^{(\sigma)}\bigu(t)\big^{q}, $$then the problem (1.1) has at least one solution \(u\in S\) and the set of solutions of (1.1) is bounded in S.
Proof
By definition of P, the nonincreasing characteristic of \(t^{(\alpha )}\) for any \(t\in\mathbb{N}\), together with (H4), Lemma 2.4 and Lemma 2.5, for any \(u\in S\), we have
then \(PS\subset S\). The remaining proof (concerning the continuity of P) is similar to that of Lemma 3.2. Therefore PS is compact by Lemma 3.3 and hence there exists at least one solution for the problem (1.1) by Theorem 3.1. □
Similarly, we can have another existence result for the problem (1.1) by changing the condition on σ in Theorem 3.4.
Theorem 3.5
Let \(0< q<1\). Assume that
 (H4)′:

there exist \(C_{f}''> 0\), \(\sigma\in(\alpha,1+(\alpha 1)q)\) and \(\gamma\in(0,\frac{\sigma\alpha}{1q})\), such that
$$\bigf (t , u)\big \leq C_{f}''(t+1)^{(\sigma)}\bigu(t)\big^{q}. $$
Example 3.3
Consider the following fractional difference equations:
where \(f(t,u)=0.15(t+1)^{(0.8)}\sin(u(t))^{0.5}\) for \(t\in\mathbb{N}_{0}\).
Since \(f(t,u)=0.15(t+1)^{(0.8)}\sin(u(t))^{0.5}\leq 0.15(t+1)^{(0.8)}u(t)^{0.5}\), the condition (H4) holds for \(\gamma =0.2\). Thus there exists at least one solution for the given problem in S by Theorem 3.4.
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Acknowledgements
The work was supported by the National Natural Science Foundation of China (No. 11671339) and Hunan Provincial Innovation Foundation For Postgraduate (CX2018B072).
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Each of the authors, JWH, LZ, YZ and BA, contributed equally to each part of this work. All authors read and approved the final manuscript.
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Correspondence to Yong Zhou.
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MSC
 34D20
 39A06
 39A13
Keywords
 Fractional differences equations
 Caputolike difference operator
 Existence
 Topological degree methods