Abstract
In this paper, by means of Darbo’s fixed point theorem, we establish the existence of solutions to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space. Additionally, as an application, we give an example to demonstrate the main result.
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1 Introduction
Throughout this paper, we denote \(\mathbb{N}_{a}=\{a, a+1, a+2, \ldots\}\) and \(\mathbb{N}_{a}^{b}=\{a, a+1, \ldots, b\}\) for \(a,b\in\mathbb{R}\) with \(b-a\in\mathbb{N}_{1}\). Moreover, for any Banach-valued function u on \(\mathbb{N}_{a}\), we make the convention that \(\sum_{s=k_{1}}^{k_{2}}u(s)=\theta\) if \(k_{1},k_{2}\in\mathbb{N}_{a}\) with \(k_{1}>k_{2}\), where θ is the zero element of a given Banach space.
In this paper, we consider the following discrete fractional mixed type sum-difference equation boundary value problem in Banach space E:
where \(\beta\in(1,2]\), \(\Delta^{\beta}\) denotes the discrete Riemann-Liouville fractional difference of order β, \(f:\mathbb{N}_{\beta-1}\times E\times E\times E\rightarrow E\) is a continuous function, θ is the zero element of E, \(\Delta^{\beta-1}u(\infty) =\lim_{t\rightarrow+\infty}\Delta^{\beta-1}u(t) =u_{\infty}\in E\), and
where \(k:D\rightarrow\mathbb{R}\), \(D=\{(t,s)\in\mathbb{N}_{0}\times\mathbb{N}_{0}:s\leq t\}\), \(h:\mathbb{N}_{0}\times\mathbb{N}_{0}\rightarrow\mathbb{R}\).
In the last years, discrete fractional calculus and fractional difference equations with various boundary conditions have been studied more intensively. For details, see [1–16]; we particularly should note that the recent monograph of Goodrich and Peterson [1] is an extremely useful textbook for readers to obtain the fundamental background in discrete fractional calculus. However, as far as we know, most of the recent papers, mainly based on the Krasnosel’skii fixed point theorem, are concerning about the existence of positive solutions for discrete fractional boundary value problems on a finite interval, and few papers can be found in the literature for discrete boundary value problems on an infinite interval [16].
Lv and Feng [16] initially introduced some basic conceptions and fundamental results on discrete fractional calculus for any Banach-valued function and also, using of the contraction mapping principle, investigated the existence and uniqueness of solutions for a class of fractional mixed type sum-difference equation boundary value problems on discrete infinite intervals in Banach spaces. This is the first attempt to study the discrete fractional difference equation boundary value problems in abstract spaces.
It is well known that the measure of noncompactness is a very powerful tool to deal with differential equations [17–20], difference equations [21–23], integration equations [24], and differential inclusions [25, 26]. So, in this paper, we l employ noncompact measures and Darbo’s fixed point theorem to establish some conditions for the existence of solutions to problem (1.1). We point out that the main result is even new and efficient for integer order case of \(\beta=2\).
The outline for the remainder of this paper is as follows. In Section 2, we recall some useful preliminaries. In Section 3, we establish the existence result of problem (1.1), and finally we present in Section 4 an example illustrating the abstract theory.
2 Preliminaries
In this section, we begin by presenting here some necessary definitions for discrete fractional calculus, and more preliminary facts can be found, for example, in [1, 16, 27, 28].
Definition 2.1
For any t and ν, the falling factorial function is defined as
provided that the right-hand side is well defined. We make 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
The νth discrete fractional sum of a function \(f:\mathbb{N}_{a}\rightarrow E\) for \(\nu>0\) is defined by
Also, we define the trivial sum \(\Delta_{a}^{-0}f(t)=f(t)\), \(t\in \mathbb{N}_{a}\).
Definition 2.3
The νth discrete Riemann-Liouville fractional difference of a function \(f:\mathbb{N}_{a}\rightarrow E\) for \(\nu>0\) is defined by
where n is the smallest integer greater than or equal to ν, and \(\Delta^{n}\) is the nth-order forward difference operator. If \(\nu =n\in\mathbb{N}_{1}\), then \(\Delta_{a}^{n}f(t)=\Delta^{n}f(t)\).
We denote by \(C(\mathbb{N}_{a}^{b}, E)\) the Banach space of all functions \(\varpi: \mathbb{N}_{a}^{b}\rightarrow E\) with the usual supremum norm \(\|\varpi\|_{0}=\sup\{\|\varpi(t)\|: t\in\mathbb {N}_{a}^{b}\}\). Define the space
equipped with the norm
Furthermore, by means of the linear functional analysis theory we can easily verify that \((X,\|\cdot\|_{X} )\) is a Banach space. It is worth reminding that here we use α, \(\alpha_{C}\), and \(\alpha_{X}\) to denote the Kuratowski noncompactness measure of bounded sets in Banach spaces \(E, C(\mathbb{N}_{a}^{b},E)\), and X, respectively. For more details on the Kuratowski noncompactness measure, we refer the reader to [29, 30]. We state the following properties of the Kuratowski measure of noncompactness and the Darbo’s fixed point theorem, which are needed for the sequel discussion.
Lemma 2.1
([22])
Let \(A\subseteq C(\mathbb {N}_{a}^{b}, E)\) be bounded. Then
-
(i)
\(\alpha_{C}(A)=\alpha (A (\mathbb{N}_{a}^{b} ) )\);
-
(ii)
\(\alpha (A (\mathbb{N}_{a}^{b} ) ) =\sup_{t\in\mathbb{N}_{a}^{b}}\alpha (A (t ) )\),
where \(A(t)=\{\varpi(t) : \varpi\in A\}\) and \(A (\mathbb{N}_{a}^{b} )=\bigcup_{t\in\mathbb{N}_{a}^{b}}A(t)\).
Lemma 2.2
([30])
Let D be a bounded, closed, and convex subset of a Banach space E. If an operator \(\mathcal {A}:D\rightarrow D\) is a strict set contraction, then \(\mathcal{A}\) has a fixed point in D.
Remark 2.1
A bounded and continuous operator \(\mathcal{A}:D\rightarrow E\) is called a strict set contraction if there is a constant \(\lambda\in[0,1)\) such that \(\alpha (\mathcal{A}S)\leq\lambda\alpha(S)\) for any bounded set \(S\subset D\).
3 Main result
In this section, we establish the existence of solutions for problem (1.1) by using Darbo’s fixed point theorem. For convenience and shortness of our presentation, for any \(u\in X\), we denote
for further discussion and list the following conditions:
(C1)
(C2) There exist nonnegative numbers \(q_{i}\), \(i\in\mathbb{N}_{1}^{3}\), and functions \(p_{1}, p_{2}:\mathbb{N}_{\beta -1}\rightarrow[0,\infty)\) with \(p_{1}^{*}=\sum_{t=\beta-1}^{\infty} p_{1}(t)(1+t^{\underline{\beta-1}}) <\Gamma(\beta)\) and \(p_{2}^{*}=\sum_{t=\beta-1}^{\infty} p_{2}(t)<+\infty\) such that
for \((t,u,v,w)\in\mathbb{N}_{\beta-1}\times E\times E\times E\);
(C3) For any positive number \(r\in\mathbb {N}_{\beta-1}\), \(f(t,u,v,w)\) is uniformly continuous on \(\mathbb{N}_{\beta-1}^{r}\times B_{E}(\theta,r)\times B_{E}(\theta,r)\times B_{E}(\theta,r)\), where \(B_{E}(\theta,r)=\{x\in E : \|x\|\leq r\}\);
(C4) There exist functions \(l_{i}:\mathbb {N}_{\beta-1}\rightarrow[0,+\infty)\), \(i\in\mathbb{N}_{1}^{3}\), such that
for all bounded sets \(V_{i}\subset E\), \(i\in\mathbb{N}_{1}^{3}\), and
Moreover, we set
when (C1) holds.
Next, we state and prove the following lemmas, which are necessary for the proof of the main result.
Lemma 3.1
Assume that (C1) and (C2) hold. Then, for any \(u\in X\),
Proof
For any \(u\in X\), \(t\in\mathbb{N}_{0}\), using (C1), (C2), and the monotonicity of \(t^{\underline{\beta-1}}\) for \(t\in\mathbb{N}_{\beta-1}\) produces
By summing both sides of (3.2) we get (3.1). So, this proof is completed. □
Lemma 3.2
Let \(h:\mathbb{N}_{0}\rightarrow E\) be given, and \(\beta\in(1,2]\). The unique solution of
is
where
Remark 3.1
The proof of Lemma 3.2 is similar to that of Lemma 3.2 in [16]. Hence, we omit it here. Moreover, in view of the expression of \(G(t,s)\), we can easily verify that \(G(t,s)\geq0\) and \(\frac{G(t,s)}{1+t^{\underline{\beta-1}}}<\frac {1}{\Gamma(\beta)}\) for \((t,s)\in\mathbb{N}_{\beta-2}\times\mathbb{N}_{0}\).
For any \(u\in X\), define the operator \(\mathcal{F}\) by
and due to Lemma 3.1 and Remark 3.1, we have
which implies that \(\mathcal{F}:X\rightarrow X\) is well defined and bounded. Furthermore, from Lemma 3.2 we know that the existence of solutions for problem (1.1) is equivalent to that of fixed points of \(\mathcal{F}\) in X.
Lemma 3.3
Suppose that (C1), (C2), and (C3) are satisfied. Then the operator \(\mathcal{F}:X\rightarrow X\) is continuous.
Proof
Let \(\{u_{n}\}_{n=1}^{\infty}\subset X\) and \(u\in X\) be such that \(\|u_{n}-u\|_{X}\rightarrow0\) as \(n\rightarrow\infty\). Then, \(\{u_{n}\}_{n=1}^{\infty}\) is a bounded set of X, that is, there exists \(M>0\) such that \(\|u_{{n}}\|_{X}\leq M\) for \(n\in\mathbb {N}_{1}\). By taking limit we also have that \(\|u\|_{X}\leq M\).
In view of (C2), for any \(\epsilon>0\), there exists a positive integer L such that
On the other hand, condition (C3) yields that there exists \(N>0\) such that for any \(n>N\) and \(t\in\mathbb{N}_{0}^{L}\),
Therefore, for \(t\in\mathbb{N}_{\beta-2}^{L+\beta-1}\) and \(n>N\), by (C1), (C2), (3.3)-(3.5), and Remark 3.1 we obtain that
Meanwhile, for \(t\in\mathbb{N}_{L+\beta}\) and \(n>N\), applying (3.3)-(3.5) again, we can easily verify that
Then, we conclude that \(\|\mathcal{F}u_{n}-\mathcal{F}u\|_{X}\leq \epsilon\) for \(n>N\). So \(\mathcal{F}\) is continuous, and the proof is completed. □
Lemma 3.4
Let B be a bounded subset of X. If (C1) and (C2) hold, then for any \(\epsilon>0\), there exists a positive number \(N\in\mathbb{N}_{\beta-2}\) such that \(\|\frac{(\mathcal{F}u)(t_{2})}{1+t_{2}^{\underline{\beta-1}}} -\frac{(\mathcal{F}u)(t_{1})}{1+t_{1}^{\underline{\beta-1}}} \| <\epsilon\) for each \(u\in B\) and any \(t_{1}, t_{2}\in\mathbb{N}_{N}\).
Proof
In view of Lemma 3.1 and the boundedness of B, there exists \(M>0\) such that
Note that
Observing (3.7) together with \(\lim_{t\rightarrow+\infty}\frac {t^{\underline{\beta-1}}}{1+t^{\underline{\beta-1}}}=1\), we only need to show that for any \(\epsilon>0\), there exists sufficiently large positive number \(N\in\mathbb{N}_{\beta-2}\) such that, for any \(t_{1},t_{2}\in\mathbb{N}_{N}\),
Relation (3.6) yields that there exists a positive number \(L\in\mathbb {N}_{0}\) such that
On the other hand, from the monotonicity of \(\iota^{\underline{\beta -1}}\) we can declare that \(\lim_{t\rightarrow+\infty}\frac {(t-L-1)^{\underline{\beta-1}}}{1+t^{\underline{\beta-1}}}=1\).
In fact, for any \(t\in\mathbb{N}_{L+\beta+1}\), \(L\in\mathbb {N}_{0}\), we have
which implies that
So, there exists \(N\in\mathbb{N}_{L+\beta+1}\) such that for any \(t_{1},t_{2}\in\mathbb{N}_{N}\) and \(s\in\mathbb{N}_{0}^{L}\),
Now taking \(t_{1},t_{2}\in\mathbb{N}_{N}\), by (3.8) and (3.9) we get
Therefore, the proof is completed. □
Lemma 3.5
Let B be a bounded subset of X. If (C1) and (C2) hold, then
Proof
First, we claim that \(\alpha_{X}(\mathcal{F}B) \leq\sup_{t\in\mathbb{N}_{\beta-2}} \alpha (\frac{(\mathcal{F}B)(t)}{1+t^{\underline{\beta-1}}} )\).
In view of Lemma 3.4, we know that for any \(\epsilon>0\), there exists a positive number \(N\in\mathbb{N}_{\beta-2}\) such that, for \(t_{1}, t_{2}\in\mathbb{N}_{N}\),
Denote by \(\mathcal{F}B|_{\mathbb{N}_{\beta-2}^{N}}\) the restriction of \(\mathcal{F}B\) on \(\mathbb{N}_{\beta-2}^{N}\). By Lemma 2.1 we obtain
So, there exists a partition of B such that \(B=\bigcup_{i=1}^{n}B_{i}\), \(\mathcal{F}B|_{\mathbb{N}_{\beta-2}^{N}}=\bigcup_{i=1}^{n} \mathcal{F}B_{i}|_{\mathbb{N}_{\beta-2}^{N}}\) and
where \(\operatorname{diam}_{X}(\cdot)\) denotes the diameter of a bounded subset of X. Moreover, for all \(\mathcal{F}u_{1}, \mathcal{F}u_{2}\in \mathcal{F}B_{i}\), \(i\in\mathbb{N}_{1}^{n}\), and \(t\in\mathbb{N}_{N}\), (3.10) and (3.11) imply that
Hence, it follows from (3.11) and (3.12) that
Since \(\mathcal{F}B=\bigcup_{i=1}^{n}\mathcal{F}B_{i}\), we get that \(\alpha_{X}(\mathcal{F}B) \leq\sup_{t\in\mathbb{N}_{\beta-2}} \alpha (\frac{(\mathcal{F}B)(t)}{1+t^{\underline{\beta-1}}} ) +3\epsilon\), which by the arbitrariness of ϵ implies that
Next, we show that \(\sup_{t\in\mathbb{N}_{\beta-2}} \alpha (\frac{(\mathcal{F}B)(t)}{1+t^{\underline{\alpha-1}}} ) \leq\alpha_{X}(\mathcal{F}B)\). For any given \(\epsilon>0\), there exists a partition \(\mathcal {F}B=\bigcup_{i=1}^{n}\mathcal{F}B_{i}\) such that \(\mathrm {diam}_{X}(\mathcal{F}B_{i})<\alpha_{X}(\mathcal{F}B)+\epsilon\). Therefore, for any \(t\in\mathbb{N}_{\beta-2}\) and \(u_{1}, u_{2}\in B_{i}\), we obtain
In accordance with \((\mathcal{F}B)(t)=\bigcup_{i=1}^{n}\mathcal {F}B_{i}(t)\), we get \(\alpha (\frac{(\mathcal{F}B)(t)}{1+t^{\underline{\beta-1}}} ) \leq\alpha_{X}(\mathcal{F}B)+\epsilon\). Because ϵ is arbitrary, we have
Consequently, the proof of this lemma is complete. □
With all auxiliary results in hand, now we state the main result.
Theorem 3.1
If (C1)-(C4) hold, then problem (1.1) has at least one solution u in X.
Proof
Choose
and let
Then, for any \(u\in B\), by Lemma 3.1 and Remark 3.1 we have
which implies that \(\mathcal{F}: B\rightarrow B\).
Set \(D=\overline{\mathrm{co}}(\mathcal{F}B)\). Obviously, D is a bounded, convex, and closed subset of B. In the sequel, we show that the operator \(\mathcal{F}: D\rightarrow D\) is a strict contraction.
Observing that \(\mathcal{F}D\subset\mathcal{F}B\subset D\), together with Lemma 3.3, we know that \(\mathcal{F}: D\rightarrow D\) is bounded and continuous. Finally, we show that there exists a constant \(\lambda \in[0,1)\) such that \(\alpha_{X}(\mathcal{F}V)\leq\lambda\alpha_{X}(V)\) for any \(V\subset D\). Moreover, in view of Lemma 3.5, we only need to verify
For a positive integer \(n>t-\beta\), define
Then from (C1) and (C2), for any \(u\in V\), we have
which implies that
where \(d_{\mathrm{H}}(\cdot,\cdot)\) denotes the Hausdorff metric in space E. So, by the properties of noncompactness measure we obtain
Now we estimate
In view of (C1) and (C2), we know that \(\{g_{u}(t) : u\in V \}\) is bounded. Relying on Lemma 2.1, Lemma 3.5, and (C4), we have
By (3.14) we immediately get
So (3.13) holds with \(\lambda\in[0,1)\), and from Lemma 2.2 we immediately obtain that problem (1.1) has at least one solution in D. Hence, the proof is completed. □
4 An example
Example 4.1
Consider the following infinite system of scalar discrete fractional difference equations:
Conclusion
System (4.1) has at least one solution \(\{u_{n}(t)\}_{n=1}^{\infty}\) such that \(u_{n}(t)\rightarrow0\) as \(n\rightarrow\infty\) for \(t\in\mathbb{N}_{-1/2}\).
Proof
Let \(E=c_{0}=\{u=(u_{1},u_{2},\ldots,u_{n},\ldots):u_{n}\rightarrow0\} \). Evidently, \((E,\|\cdot\|)\) is a Banach space with the norm \(\|u\| =\sup_{n}|u_{n}|\) for any \(u\in E\). Then infinite system (4.1) can be regarded as a boundary value problem of the form (1.1) in the Banach space E. In this case, \(\beta=3/2\), \(\theta=(0,0,\ldots,0,\ldots)\in E\), \(u_{\infty}=(1,1/2!,\ldots,1/n!,\ldots)\in E\),
and \(f=(f_{1},f_{2},\ldots,f_{n},\ldots)\) with
where \(t\in\mathbb{N}_{1/2}\) and \(u=(u_{1},u_{2},\ldots,u_{n},\ldots), v=(v_{1},v_{2},\ldots,v_{n},\ldots), w=(w_{1},w_{2},\ldots ,w_{n},\ldots)\in E\). From the expression of \(f_{n}\) we easily to see that \(f:\mathbb{N}_{1/2}\times E\times E\times E \rightarrow E\) is continuous. Furthermore, we can easily verify that
So, condition (C1) is satisfied. On the other hand, using the simple inequality
we see from (4.2) that
and, therefore,
where
which implies
So condition (C2) is satisfied. We can also verify that (C3) holds by (4.2). Finally, we check condition (C4). Let \(f=f^{(1)}+f^{(2)}\), where
Then we obtain that for any bounded sets \(V_{i}\subset E\), \(i\in\mathbb {N}_{1}^{3}\), \(\alpha (f^{2}(t,V_{1},V_{2},V_{3}) )=0\), \(t\in\mathbb{N}_{1/2}\). In fact, since \(V_{i}\), \(i\in\mathbb{N}_{1}^{3}\), are bounded, there exists \(r>0\) such that \(V_{i}\subset B_{E}(\theta, r)\), \(i\in\mathbb{N}_{1}^{3}\). Let \(\{ u^{(m)}\}_{m=1}^{\infty}\in V_{1}\), \(\{v^{(m)}\}_{m=1}^{\infty}\in V_{2}\), \(\{w^{(m)}\}_{m=1}^{\infty}\in V_{3}\). Then for any fixed \(t\in\mathbb{N}_{1/2}\), we have
Therefore, \(\{f^{(2)}_{n} (t,u^{(m)},v^{(m)},w^{(m)} ) \}\) is bounded, and so, by the diagonal method we can choose a subsequence \(\{m_{k}\}\subset\{m\}\) such that
So \(z=(z_{1},z_{2},\ldots,z_{n},\ldots)\in c_{0}=E\), and it is easy to see from (4.3)-(4.5) that
Thus, we have proved that \(f^{2}(t,V_{1},V_{2},V_{3})\) is relatively compact in E and
On the other hand, for any \(t\in\mathbb{N}_{1/2}\), \(u,\overline{u}\in V_{1}\), \(v,\overline{v}\in V_{2}\), \(w,\overline{w}\in V_{3}\), we have
and, therefore,
In view of \(\sum_{t=1/2}^{\infty}\frac {3^{-t-1/2}}{(1+t^{\underline{1/2}})}<1/2<\Gamma(3/2)\), we get that condition (C4) holds with \(l_{1}(t)=\frac {3^{-t-1/2}}{(1+t^{\underline{1/2}})^{2}}\) and \(l_{2}(t)=l_{3}(t)=0\). So by Theorem 3.1 our conclusion follows. □
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Acknowledgements
The authors would like to express their thanks to the referees for their valuable comments and suggestions. This work was supported by the Longdong University Grant XYZK-1402, the Science Research Project of Gansu University 2015A-149, and the Science and Technology Plan Projects in Gansu Province 145RJYM286.
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Lv, W., Zhu, Xy. Solvability for a discrete fractional mixed type sum-difference equation boundary value problem in a Banach space. Bound Value Probl 2016, 77 (2016). https://doi.org/10.1186/s13661-016-0585-8
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DOI: https://doi.org/10.1186/s13661-016-0585-8