Abstract
In this paper, we investigate the existence and the uniqueness of solutions for coupled and uncoupled systems of fractional q-integro-difference equations with nonlocal fractional q-integral boundary conditions. The existence and the uniqueness of the solutions are established by using the Banach contraction principle, while the existence of solutions is derived by applying Leray-Schauder’s alternative. Examples illustrating our results are also presented.
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1 Introduction
In this paper, we investigate a coupled system of fractional q-integro-difference equations with nonlocal fractional q-integral boundary conditions given by
where \(0< p,q,r,z,m,n,h,k<1\) are quantum numbers, \(\eta,\xi,\theta ,\tau\in(0,T)\) are fixed points, \(\delta,\varepsilon,\gamma,\kappa ,\mu,\nu>0\), and \(\lambda_{1},\lambda_{2}\in\mathbb{R}\) are given constants, \(D_{\omega}^{\rho}\) is the fractional ω-derivative of Riemann-Liouville type of order ρ, when \(\rho\in\{\alpha ,\beta\}\) and \(\omega\in\{p,q\}\), \(I_{\phi}^{\psi}\) is the fractional ϕ-integral of order ψ with \(\phi\in\{ r,z,m,n,h,k\}\) and \(\psi\in\{\delta,\varepsilon,\gamma,\kappa,\mu ,\nu\}\) and \(f,g:[0,T]\times\mathbb{R}\times\mathbb{R}\to\mathbb {R}\) are continuous functions.
The early work on q-difference calculus or quantum calculus dates back to Jackson’s paper [1]. Basic definitions and properties of quantum calculus can be found in the book [2]. The fractional q-difference calculus had its origin in the works by Al-Salam [3] and Agarwal [4]. Motivated by recent interest in the study of fractional-order differential equations, the topic of q-fractional equations has attracted the attention of many researchers. The details of some recent development of the subject can be found in [5–18], and the references cited therein, whereas the background material on q-fractional calculus can be found in a recent book [19].
Recently in [20], we have studied the existence and the uniqueness of solutions of a class of boundary value problems for fractional q-integro-difference equations with nonlocal fractional q-integral conditions which have different quantum numbers. Here we extend the results of [20] to a coupled system of fractional q-integro-difference equations with nonlocal fractional q-integral boundary conditions.
The paper is organized as follows: In Section 2 we will present some useful preliminaries and lemmas. Some auxiliary lemmas are presented in Section 3. In Section 4, we establish an existence and a uniqueness result via the Banach contraction principle, and an existence result by applying Leray-Schauder’s alternative. Results on the uncoupled integral boundary conditions case are contained in Section 5. Examples illustrating our results are also presented.
2 Preliminaries
To make this paper self-contained, below we recall some well-known facts on fractional q-calculus. The presentation here can be found in, for example, [6, 19].
For \(q\in(0,1)\), define
The q-analog of the power function \((a-b)^{k}\) with \(k\in\mathbb {N}_{0}:=\{0,1,2,\ldots\}\) is
More generally, if \(\gamma\in{\mathbb{R}}\), then
Note if \(b=0\), then \(a^{(\gamma)}=a^{\gamma}\). We also use the notation \(0^{(\gamma)}=0\) for \(\gamma>0\). The q-gamma function is defined by
Obviously, \(\Gamma_{q}(t+1)=[t]_{q}\Gamma_{q}(t)\).
The q-derivative of a function h is defined by
and q-derivatives of higher order are given by
The q-integral of a function h defined on the interval \([0, b]\) is given by
If \(a\in[0, b]\) and h is defined in the interval \([0, b]\), then its integral from a to b is defined by
Similar to derivatives, an operator \(I_{q}^{k}\) is given by
The fundamental theorem of calculus applies to these operators \(D_{q}\) and \(I_{q}\), i.e.,
and if h is continuous at \(t=0\), then
Definition 2.1
Let \(\nu\geq0\) and h be a function defined on \([0, T]\). The fractional q-integral of Riemann-Liouville type is given by \((I_{q}^{0} h)(t)=h(t)\) and
Definition 2.2
The fractional q-derivative of Riemann-Liouville type of order \(\nu\ge 0\) is defined by \((D_{q}^{0}h)(t)=h(t)\) and
where l is the smallest integer greater than or equal to ν.
Definition 2.3
For any \(t, s>0\),
is called the q-beta function.
The expression of q-beta function in terms of the q-gamma function can be written as
Lemma 2.4
[4]
Let \(\alpha, \beta\ge0\) and f be a function defined in \([0,T]\). Then the following formulas hold:
-
(1)
\((I_{q}^{\beta}I_{q}^{\alpha} f)(t)=(I_{q}^{\alpha+\beta}f)(t)\),
-
(2)
\((D_{q}^{\alpha}I_{q}^{\alpha} f)(t)=f(t)\).
Lemma 2.5
[6]
Let \(\alpha>0\) and n be a positive integer. Then the following equality holds:
3 Some auxiliary lemmas
The following formulas have been modified from Lemmas 3.2 and 7 in [21] and [20], respectively.
Lemma 3.1
Let \(x,y,z>0\) and \(0< u,v,w<1\). Then, for \(\phi\in\mathbb{R}_{+}\), we have
-
(i)
\(I_{u}^{x}I_{v}^{y}(1)(\phi)=\frac{\Gamma _{u}(y+1)}{\Gamma_{u}(x+y+1)\Gamma_{v}(y+1)}\phi^{x+y}\);
-
(ii)
\(I_{u}^{x}I_{v}^{y}I_{w}^{z}(1)(\phi)=\frac{\Gamma _{u}(y+z+1)\Gamma_{v}(z+1)}{\Gamma_{u}(x+y+z+1)\Gamma_{v}(y+z+1)\Gamma _{w}(z+1)}\phi^{x+y+z}\).
Lemma 3.2
Given \(u,v\in C([0, T], \mathbb{R})\), the unique solution of the problem
is
and
where
Proof
From \(1<\alpha\leq2\), we let \(n=2\). Applying Lemma 2.5, the equations in (3.1) can be expressed as equivalent integral equations
for \(c_{1},c_{2},d_{1},d_{2}\in\mathbb{R}\). The conditions \(x(0)=0\) and \(y(0)=0\) imply that \(c_{2}=0\) and \(d_{2}=0\), respectively. Taking the Riemann-Liouville fractional ϕ-integral of order \(\psi>0\) for (3.4) and (3.5), we have the system
Substituting \((\psi,\phi,t)\) by \((\gamma,m,\eta)\), \((\nu,k,\tau)\) in (3.6), and \((\kappa,n,\xi)\), \((\mu,h,\theta)\) in (3.7) and using Lemma 2.4 with nonlocal conditions in (3.1), we have
and
Substituting the values of \(c_{1}\), \(c_{2}\), \(d_{1}\), and \(d_{2}\) in (3.4) and (3.5), we obtain the solutions (3.2) and (3.3) as required. □
4 Main results
Let \(\mathcal{C} = C([0,T],\mathbb{R})\) denotes the Banach space of all continuous functions from \([0,T]\) to \(\mathbb{R}\). Let us introduce the space \(X=\{x(t)|x(t)\in C([0,T],\mathbb{R})\}\) endowed with the norm \(\|x\|=\sup\{|x(t)|, t\in[0,T]\}\). Obviously \((X,\|\cdot\|)\) is a Banach space. Also let \(Y=\{y(t)|y(t)\in C([0,T],\mathbb{R})\}\) be endowed with the norm \(\|y\|=\sup\{|y(t)|, t\in[0,T]\}\). Obviously the product space \((X\times Y, \|(x,y)\|)\) is a Banach space with norm \(\|(x,y)\|=\|x\|+\|y\|\).
In view of Lemma 3.2, we define an operator \(\mathcal {K}: X\times Y\to X\times Y\) by
where
and
For the sake of convenience, we set
Theorem 4.1
Assume that \(f,g:[0,T]\times\mathbb{R}^{2}\to\mathbb{R}\) are continuous functions and there exist positive constants \(M_{i}\), \(N_{i}\), \(i=1,2\), such that for all \(t\in[0, T]\) and \(u_{i},v_{i}\in\mathbb{R}\), \(i=1,2\),
and
In addition, we suppose that
where
Then the system (1.1) has a unique solution on \([0, T]\).
Proof
Firstly, we define \(\sup_{t\in[0, T]}|f(t,0,0)|=G_{1}<\infty\) and \(\sup_{t\in[0, T]}|g(t,0,0)|=G_{2}<\infty\) such that
where
We will show that \(\mathcal{K}B_{r}\subset B_{r}\), where \(B_{r}=\{(x,y)\in X\times Y:\|(x,y)\|\leq r\}\).
For \((x,y)\in B_{r}\), taking into account Lemma 3.1, we have
In a similar way, we get
Consequently, \(\|\mathcal{K}(x,y)(t)\|\leq r\).
Next, for \((x_{2},y_{2}), (x_{1},y_{1})\in X\times Y\), and for any \(t\in[0, T]\), we have
Therefore, we have
In the same way, we have
which implies
It follows from (4.3) and (4.4) that
Since \(B_{1}+B_{2}+C_{1}+C_{2}<1\), therefore, \(\mathcal{K}\) is a contraction operator. So, by Banach’s fixed point theorem, the operator \(\mathcal {K}\) has a unique fixed point, which is the unique solution of problem (1.1). The proof is completed. □
In the next result, we prove the existence of solutions for the problem (1.1) by applying the Leray-Schauder alternative.
Lemma 4.2
(Leray-Schauder alternative, see [22], p.4)
Let \(F: E\to E\) be a completely continuous operator (i.e., a map that restricted to any bounded set in E is compact). Let
Then either the set \({\mathcal{E}}(F)\) is unbounded, or F has at least one fixed point.
For convenience, we set constants
and
Theorem 4.3
Assume that there exist real constants \(P_{i}, Q_{i} \ge0\) (\(i=1, 2\)), and \(P_{0}>0\), \(Q_{0}>0\) such that for all \(u_{i},v_{i}\in{\mathbb{R}}\) (\(i=1, 2\)) we have
In addition it is assumed that
Then there exists at least one solution for the system (1.1).
Proof
We first prove that the operator \(\mathcal{K}:X\times Y\to X\times Y\) is completely continuous. The continuity of functions f and g imply that the operator \(\mathcal{K}\) is continuous. Let \(\Phi\subset X\times Y\) be a bounded set. Then there exist positive constants \(D_{1}\) and \(D_{2}\) such that
Then for any \((u_{1},u_{2}), (v_{1},v_{2})\in\Phi\), and using Lemma 3.1, we have
In the same way, we deduce that
Thus, it follows from the above inequalities that the operator \(\mathcal{K}\) is uniformly bounded.
Next, we show that \(\mathcal{K}\) is equicontinuous. Let \(t_{1}, t_{2} \in [0,T]\) with \(t_{1}< t_{2}\). Then we have
Analogously, we can get
Therefore, the operator \(\mathcal{K}(x,y)\) is equicontinuous, and thus the operator \(\mathcal{K}(x,y)\) is completely continuous.
Finally, it will be verified that the set \({\mathcal{E}}=\{(x,y)\in X\times Y : (x,y)=\lambda\mathcal{K}(x,y), 0\le\lambda\le1\}\) is bounded. Let \((x,y)\in{\mathcal{E}}\), then \((x,y)=\lambda\mathcal {K}(x,y)\). For any \(t\in[0,T]\), we have
Then we have
and
which yields
and
Therefore, we have
and, consequently,
for any \(t\in[0,T]\), which proves that \({\mathcal{E}}\) is bounded. Thus, by Lemma 4.2, the operator \(\mathcal{K}\) has at least one fixed point. Hence the system (1.1) has at least one solution. The proof is complete. □
4.1 Examples
In this subsection, we present some examples to illustrate our results.
Example 4.4
Consider the following coupled system of fractional q-integro-difference equations:
Here \(\alpha=3/2\), \(\delta=\pi\), \(\beta=4/3\), \(\varepsilon=\pi/2\), \(\gamma=7/6\), \(\kappa=\sqrt{2}\), \(\mu=e\), \(\nu=\sqrt{3}\), \(q=1/2\), \(r=1/4\), \(p=1/3\), \(z=1/5\), \(m=1/8\), \(n=1/6\), \(h=1/9\), \(k=1/7\), \(\eta=3/2\), \(\xi=1/2\), \(\theta=1/3\), \(\tau=5/3\), \(\lambda_{1}=\sqrt{2}\), \(\lambda_{2}=\sqrt{3}/2\), \(T = 2\), \(f(t,x,I_{r}^{\delta}y) = (|x|\cos^{2}\pi t)/((e^{t}+4)^{2}(4+|x|))+(e^{-t^{2}}/((t+8)^{2}))I_{1/4}^{\pi}y+\sqrt{2}/2\), and \(g(t,y,I_{r}^{\delta}x) = (|y|\sin^{2}\pi t)/((11+t)^{2}(1+|y|))+(1/(e^{t}+8)^{2}) I_{1/5}^{\pi/2}x+\sqrt{3}\). Since
and
then the assumptions of Theorem 4.1 are satisfied with \(M_{1} = 1/100\), \(M_{2} = 1/64\), \(N_{1} = 1/121\), and \(N_{2} = 1/81\). By using the Maple program, we can find that
and
Therefore, we get
Hence, by Theorem 4.1, the problem (4.5) has a unique solution on \([0,2]\).
Example 4.5
Consider the following coupled system of fractional q-integro-difference equations with fractional q-integral conditions:
Here \(\alpha=4/3\), \(\delta=\sqrt{3}\), \(\beta=7/5\), \(\varepsilon=\sqrt {2}\), \(\gamma=\sqrt{5}\), \(\kappa=1/3\), \(\mu=\sqrt{\pi}\), \(\nu =\sqrt{2}/2\), \(q=1/3\), \(r=1/5\), \(p=1/4\), \(z=1/6\), \(m=1/2\), \(n=1/3\), \(h=1/7\), \(k=1/8\), \(\eta=\pi/5\), \(\xi=2\pi/5\), \(\theta=3\pi/5\), \(\tau=4\pi/5\), \(\lambda_{1}=-\sqrt{5}\), \(\lambda_{2}=-\sqrt{2}\), \(T = \pi\), \(f(t,x,I_{r}^{\delta}y) = (4e^{-t}/(t+11)^{2})(|x|/(2+|x|))+(1/(e^{-t^{2}}+8)^{2})I_{1/5}^{\sqrt {3}}y+\sqrt{2}\), and \(g(t,y,I_{r}^{\delta}x) = (\cos^{2} 2\pi t/(10+t)^{2})(|y|/(1+|y|))+(1/(e^{t}+7)^{2}) I_{1/6}^{\sqrt{2}}x+(1/2)\). Since
and
then the assumptions of Theorem 4.3 are satisfied with \(P_{0} =\sqrt{2}\), \(P_{1} = 1/72\), \(P_{2} = 1/81\), \(Q_{0} = 1/2\), \(Q_{1} = 1/100\), and \(Q_{2} = 1/81\). By using the Maple program, we can find that
and
and
Therefore, we get
Hence, by Theorem 4.3, the problem (4.6) has at least one solution on \([0,\pi]\).
5 Uncoupled integral boundary conditions case
In this section we consider the following system:
Lemma 5.1
(Auxiliary lemma, see [20])
For \(h\in C([0,T], {\mathbb{R}})\), the unique solution of the problem
is given by
where
5.1 Existence results for uncoupled case
In view of Lemma 5.1, we define an operator \(\mathcal{T}: X\times Y\to X\times Y\) by
where
and
where
In the sequel, we set constants
Now we present the existence and the uniqueness result for the problem (5.1). We do not provide the proof of this result as it is similar to the one for Theorem 4.1.
Theorem 5.2
Assume that \(f, g: [0,T]\times{\mathbb{R}}^{2}\to{\mathbb{R}}\) are continuous functions and there exist constants \(\overline{K}_{i}\), \(\overline{L}_{i}\), \(i=1,2\) such that for all \(t\in[0,T]\) and \(u_{i}, v_{i}\in {\mathbb{R}}\), \(i=1,2\),
and
In addition, assume that
Then the boundary value problem (5.1) has a unique solution on \([0, T]\).
The second result deals with the existence of solutions for the problem (5.1), is analogous to Theorem 4.3 and is given below.
Theorem 5.3
Assume that there exist real constants \(\bar{m}_{i}, \bar{n}_{i} \ge0\) (\(i=1, 2\)), and \(\bar{m}_{0}>0\), \(\bar{n}_{0}>0\) such that \(\forall x_{i} \in {\mathbb{R}}\) (\(i=1, 2\)) we have
In addition it is assumed that
where \(U_{i}\), \(V_{i}\), \(i=1,2\), are given by
Then the boundary value problem (5.1) has at least one solution on \([0, T]\).
Proof
By setting
the proof is similar to that of Theorem 4.3. So we omit it. □
5.2 Examples
In this subsection, we present two examples of uncoupled case of nonlocal conditions.
Example 5.4
Consider the following system of fractional q-integro-difference equations with q-integral conditions:
Here \(\alpha=3/2\), \(\delta=\sqrt{2}\), \(\beta=5/4\), \(\varepsilon =\sqrt{3}\), \(\gamma=\sqrt{2}/2\), \(\kappa=\sqrt{3}/2\), \(\mu=\pi\), \(\nu=\sqrt{\pi}\), \(q=1/9\), \(r=1/8\), \(p=1/7\), \(z=1/6\), \(m=1/3\), \(n=1/4\), \(h=1/5\), \(k=1/6\), \(\eta=3/4\), \(\xi=9/4\), \(\theta=3/2\), \(\tau=3\), \(\lambda_{1}=1/2\), \(\lambda_{2}=-1/3\), \(T = 3\), \(f(t,x,I_{r}^{\delta}y) =( e^{-t\sin\pi t} /(t+3)^{2})(|x|/(9+|x|))+(\cos ^{2}\pi t/\pi(t+7)^{2}) I_{1/8}^{\sqrt{2}}y-(1/3)\), and \(g(t,y,I_{r}^{\delta}x) = (2\pi e^{-2t}/ (7\pi+t)^{2})(|y|/(2+|y|))+(\sin 2\pi t/(3e^{t}+5)^{2}) I_{1/6}^{\sqrt{3}}x+(1/3)\). Since
and
then the assumptions of Theorem 5.2 are satisfied with \(\overline{M}_{1} = 1/81\), \(\overline{M}_{2} = 1/49\pi\), \(\overline {N}_{1} = 1/36\pi\), and \(\overline{N}_{2} = 1/64\). By using the Maple program, we can find that
and
Therefore, we get
Hence, by Theorem 5.2, the problem (5.4) has a unique solution on \([0,3]\).
Example 5.5
Consider the following system of fractional q-integro-difference equations:
Here \(\alpha=\sqrt{\pi}\), \(\delta=1/2\), \(\beta=\pi/2\), \(\varepsilon=3/2\), \(\gamma=4/5\), \(\kappa=2/3\), \(\mu=\sqrt{3}\), \(\nu=1/3\), \(q=\sqrt{2}/2\), \(r=\sqrt{3}/2\), \(p=\pi/4\), \(z=\pi/5\), \(m=\pi/6\), \(n=\pi/7\), \(h=\pi/8\), \(k=\pi/9\), \(\eta=1\), \(\xi=3/4\), \(\theta=1/2\), \(\tau=1/4\), \(\lambda_{1}=-\sqrt{3}/2\), \(\lambda_{2}=5\), \(T = 1\), \(f(t,x,I_{r}^{\delta}y) = (25e^{-t}/(e^{-t}+4)^{2})(|x|/(5+|x|))+(3\pi ^{2}/(t+3\pi)^{2})I_{\sqrt{3}/2}^{1/2}y+(1/\sqrt{5})\), and \(g(t,y,I_{r}^{\delta}x) = (9e^{-t\cos^{2} \pi t}/(t+6)^{2})(|y|/(1+|y|))+(6/(t+6)^{2}) I_{\pi/5}^{3/2}x+(\sqrt{2}/3)\). Since
and
then the assumptions of Theorem 5.3 are satisfied with \(\bar{m}_{0} = 1/\sqrt{5}\), \(\bar{m}_{1} = 1/5\), \(\bar{m}_{2} = 1/3\), \(\bar{n}_{0} =\sqrt{2}/3\), \(\bar{n}_{1} = 1/4\), and \(\bar{n}_{2} = 1/6\). By using the Maple program, we can find that
and
Therefore, we get
Hence, by Theorem 5.3, the problem (5.5) has at least one solution on \([0,1]\).
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This paper was supported by the Thailand Research Fund under the project RTA5780007.
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Suantai, S., Ntouyas, S.K., Asawasamrit, S. et al. A coupled system of fractional q-integro-difference equations with nonlocal fractional q-integral boundary conditions. Adv Differ Equ 2015, 124 (2015). https://doi.org/10.1186/s13662-015-0462-2
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DOI: https://doi.org/10.1186/s13662-015-0462-2