Abstract
This article studies the existence and uniqueness of solutions for coupled systems of nonlinear impulsive quantum difference equations with coupled and uncoupled boundary conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. Examples illustrating our results are also presented.
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1 Introduction and preliminaries
In this paper, we concentrate on the study of the existence and uniqueness of solutions for a coupled system of nonlinear impulsive quantum difference equations,
where \(0=t_{0}< t_{1}< t_{2}<\cdots<t_{k}<\cdots<t_{m}<t_{m+1}=T\), \(f, g:J\times \mathbb{R}^{2}\rightarrow\mathbb{R}\) are continuous functions, \(I_{k}, I_{k}^{*}\in C(\mathbb{R},\mathbb{R})\), \(\Delta u(t_{k})=u(t_{k}^{+})-u(t_{k})\), \(u(t_{k}^{+})=\lim_{h\to0^{+}}u(t_{k}+h)\), \(u\in\{ x,y\}\), for \(k=1,2,\ldots,m\), and \(0< p_{k},q_{k}<1\) for \(k=0, 1, 2, \ldots ,m\) are given quantum numbers, \(a_{i}\), \(b_{i}\), \(\lambda_{i}\), \(i=1,2\) are real constants with \(a_{1}a_{2}\ne b_{1}b_{2}\).
The notions of quantum calculus on finite intervals, \(q_{k}\)-derivatives, and \(q_{k}\)-integrals were introduced in [1]. For a fixed \(k\in {\mathbb{N}}\cup\{0\}\) let \(J_{k}:=[t_{k}, t_{k+1}]\subset\mathbb{R}\) be an interval and \(0< q_{k}<1\), \(k=1,2,\ldots,m\) be a constant. We define the \(q_{k}\)-derivative of a function \(f:J_{k}\rightarrow\mathbb{R}\) at a point \(t\in J_{k}\) as follows.
Definition 1.1
Assume \(f:J_{k}\rightarrow\mathbb{R}\) is a continuous function and let \(t\in J_{k}\). Then the expression
is called the \(q_{k}\)-derivative of function f at t.
We say that f is \(q_{k}\)-differentiable on \(J_{k}\) provided \(D_{q_{k}}f(t)\) exists for all \(t\in J_{k}\). Note that if \(t_{k}= 0\) and \(q_{k}=q\) in (1.2), then \(D_{q_{k}}f=D_{q}f\), where \(D_{q}\) is the well-known q-derivative of the function \(f(t)\), defined by
The \(q_{k}\)-integral is defined as follows.
Definition 1.2
Assume \(f:J_{k}\rightarrow\mathbb{R}\) is a continuous function. Then the \(q_{k}\)-integral is defined by
for \(t\in J_{k}\). Moreover, if \(a\in(t_{k},t)\), then the definite \(q_{k}\)-integral is defined by
Note that if \(t_{k}=0\) and \(q_{k}=q\), then (1.4) reduces to q-integral of a function \(f(t)\), defined by \(\int_{0}^{t}f(s)\,d_{q}s=(1-q)t\sum_{n=0}^{\infty}q^{n}f(q^{n}t)\) for \(t\in[0,\infty)\).
For the basic properties of the \(q_{k}\)-derivative and the \(q_{k}\)-integral we refer to [1].
The book by Kac and Cheung [2] covers many of the fundamental aspects of the quantum calculus. In recent years, the topic of q-calculus has attracted the attention of several researchers and a variety of new results can be found in [3–15] and the references cited therein.
Impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. The recent development in this field has been motivated by many applied problems, such as control theory, population dynamics, and medicine. For some recent works on the theory of impulsive differential equations, we refer the interested reader to the monographs [16–18]. Moreover, the interested reader is referred to [19–24] for some recent results on impulsive \(q_{k}\)-difference equations.
In this paper we prove existence and uniqueness results for the impulsive boundary value problem (1.1) by using Banach’s contraction mapping principle and Leray-Schauder’s nonlinear alternative. The rest of this paper is organized as follows: In Section 2 we present an auxiliary lemma which is used to convert the impulsive boundary value problem (1.1) into an equivalent integral equation. In Section 3, we establish an existence and uniqueness result via Banach’s contraction principle, and an existence result by applying Leray-Schauder’s alternative. Results on uncoupled integral boundary conditions case are in Section 4. Examples illustrating our results are also presented.
2 An auxiliary lemma
Let \(J=[0, T]\), \(J_{0}=[t_{0}, t_{1}]\), \(J_{k}=(t_{k}, t_{k+1}]\) for \(k=1,2,\ldots ,m\). To define the solutions of problem (1.1) we need the following lemma, which deals with a linear variant of problem (1.1) and gives a representation of the solutions.
Lemma 2.1
Given \(\phi, \psi\in C(J, {\mathbb{R}})\), the unique solution of the problem
is
and
where
Proof
For \(t\in J_{0}\), \(q_{0}\)-integrating (2.1), it follows that
which leads to
For \(t\in J_{1}\), taking the \(q_{1}\)-integral for (2.1), we get
Since \(x(t_{1}^{+})=x(t_{1})+I_{1}(x(t_{1}))\), we have
Again \(q_{2}\)-integrating (2.1) from \(t_{2}\) to t, where \(t\in J_{2}\), then
Repeating the above process, for \(t\in J\), we obtain
In the same way, we can obtain
In particular, for \(t=T\), we have
Applying the boundary conditions of (2.1), we get the system
from which we have
and
Substituting the values of \(x(0)\) and \(y(0)\) in (2.5) and (2.6), we obtain the solutions (2.2) and (2.3). □
3 Main results
Let \(PC(J, \mathbb{R})\) = {\(x: J\rightarrow\mathbb{R}\); \(x(t)\) is continuous everywhere except for some \(t_{k}\) at which \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) exist and \(x(t_{k}^{-})=x(t_{k})\), \(k=1,2,\ldots,m\)}. \(PC(J, \mathbb{R})\) is a Banach space with the norm \(\|x\|_{PC}=\sup\{ |x(t)|, t\in J\}\). Let us introduce the space \(X=\{x(t) ; x(t)\in PC([0,T])\}\) endowed with the norm \(\|x\|=\sup\{|x(t)|, t\in[0,T]\}\). Obviously \((X,\| \cdot\|)\) is a Banach space. Also let \(Y=\{y(t) ; y(t)\in PC([0,T])\} \) 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 2.1, we define an operator \(\mathcal{T}: X\times Y\to X\times Y\) by
where
and
For the sake of convenience, we set
The first result is concerned with the existence and uniqueness of solutions for the problem (1.1) and is based on Banach’s contraction mapping principle.
Theorem 3.1
Assume that:
- (H1):
-
The functions \(f, g: [0,T]\times{\mathbb{R}}^{2}\to {\mathbb{R}}\) are continuous and there exist constants \(K_{i}, L_{i}>0\), \(i=1,2\) such that for all \(t\in[0,T]\) and \(u_{i}, v_{i}\in{\mathbb{R}}\), \(i=1,2\),
$$\bigl|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\bigr|\le K_{1}|u_{1}-v_{1}|+K_{2}|u_{2}-v_{2}| $$and
$$\bigl|g(t,u_{1},u_{2})-g(t,v_{1},v_{2})\bigr|\le L_{1}|u_{1}-v_{1}|+L_{2}|u_{2}-v_{2}|. $$ - (H2):
-
The functions \(I_{k},I_{k}^{*}:{\mathbb{R}}\rightarrow {\mathbb{R}}\) are continuous and there exist constants \(K_{3}, L_{3}>0\) such that for all \(t\in[0,T]\) and \(u_{3}, v_{3}\in{\mathbb{R}}\), \(k=1,2,\ldots,m\),
$$\bigl|I_{k}(u_{3})-I_{k}(v_{3})\bigr|\le K_{3}|u_{3}-v_{3}| $$and
$$\bigl|I_{k}^{*}(u_{3})-I_{k}^{*}(v_{3})\bigr|\le L_{3}|u_{3}-v_{3}|. $$
In addition, assume that
where \(M_{i}\), \(i=1,2,4,5\), are given by (3.1)-(3.2) and (3.4)-(3.5). Then the boundary value problem (1.1) has a unique solution.
Proof
Define \(\sup_{t\in[0,T]}f(t,0,0)=N_{1}<\infty\), \(\sup_{t\in [0,T]}g(t,0,0)=N_{2}<\infty\), \(\sup\{|I_{k}(0)|:k=1,2,\ldots,m\} =N_{3}<\infty \) and \(\sup\{|I_{k}^{*}(0)|:k=1,2,\ldots,m\}=N_{4}<\infty\) such that
where \(M_{3}\) and \(M_{6}\) are defined by (3.3) and (3.6), respectively.
We show that \(\mathcal{T}B_{r}\subset B_{r}\), where \(B_{r}=\{(x,y)\in X\times Y: \|(x,y)\|\le r\}\).
For \((x,y)\in B_{r}\), we have
In the same way, we can obtain
Consequently, \(\|\mathcal{T}(x,y)(t)\|\le r\).
Now for \((x_{2},y_{2}), (x_{1},y_{1})\in X\times Y\) and for any \(t\in[0,T]\), we get
and consequently we obtain
Similarly,
It follows from (3.7) and (3.8) that
Since \(M_{1}+M_{2}+M_{4}+M_{5}<1\), therefore, \(\mathcal{T}\) is a contraction operator. So, by Banach’s fixed point theorem, the operator \(\mathcal {T}\) has a unique fixed point, which is the unique solution of problem (1.1). This completes the proof. □
In the next result, we prove the existence of solutions for problem (1.1) by applying the Leray-Schauder alternative.
For the sake of convenience, we set
and
Lemma 3.1
(Leray-Schauder alternative) ([25], p.4)
Let \(F: E\to E\) be a completely continuous operator (i.e., a map that is 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.
Theorem 3.2
Assume that:
- (H3):
-
The functions \(f, g: [0,T]\times{\mathbb{R}}^{2}\to {\mathbb{R}}\) are continuous and there exist constants \(A_{i}, B_{i} \ge0\) (\(i=1, 2\)) and \(A_{0}, B_{0}>0\) such that \(\forall x_{i}\in{\mathbb{R}}\) (\(i=1, 2\))
$$\bigl|f(t,x_{1},x_{2})\bigr|\le A_{0}+A_{1}|x_{1}|+A_{2}|x_{2}| $$and
$$\bigl|g(t,x_{1},x_{2})\bigr|\le B_{0}+B_{1}|x_{1}|+B_{2}|x_{2}|. $$ - (H4):
-
The functions \(I_{k},I_{k}^{*}:{\mathbb{R}}\rightarrow {\mathbb{R}}\) are continuous and there exist constants \(A_{4}, B_{4}\geq0\) and \(A_{3}, B_{3}>0\) such that \(\forall x_{3}\in{\mathbb{R}}\), \(k=1,2,\ldots,m\)
$$\bigl|I_{k}(x_{3})\bigr|\le A_{3}+A_{4}|x_{3}| $$and
$$\bigl|I_{k}^{*}(x_{3})\bigr|\le B_{3}+B_{4}|x_{3}|. $$
In addition it is assumed that
where \(M_{7}\), \(M_{8}\), \(M_{10}\), \(M_{11}\) are given by (3.9)-(3.10) and (3.12)-(3.13). Then there exists at least one solution for the boundary value problem (1.1).
To prove the theorem we use the following lemma.
Lemma 3.2
Assume that (H3) and (H4) hold. Then the operator \(\mathcal {T}:X\times Y\to X\times Y\) is completely continuous.
Proof
By continuity of functions f and g, the operator \(\mathcal{T}\) is continuous.
Let \(\Theta\subset X\times Y\) be bounded. Then there exist positive constants \(P_{1}\), \(P_{2}\), \(P_{3}\), and \(P_{4}\) such that
Then for any \((x,y)\in\Theta\), we have
Similarly, we get
Thus, it follows from the above inequalities that the operator \(\mathcal {T}\) is uniformly bounded.
Next, we show that \(\mathcal{T}\) is equicontinuous. Let \(\nu_{1}, \nu_{2} \in(t_{l},t_{l+1})\) for some \(l=0,1,\ldots,m\) with \(\nu_{1}< \nu_{2}\). Then we have
Analogously, we can obtain
Therefore, the operator \(\mathcal{T}(x,y)\) is equicontinuous, and thus the operator \(\mathcal{T}(x,y)\) is completely continuous. □
Proof of Theorem 3.2
By Lemma 3.2 the operator \(\mathcal{T}(x,y)\) is completely continuous.
Now, it will be verified that the set \({\mathcal{E}}=\{(x,y)\in X\times Y| (x,y)=\lambda\mathcal{T}(x,y), 0\le\lambda\le1\}\) is bounded. Let \((x,y)\in{\mathcal{E}}\), then \((x,y)=\lambda\mathcal{T}(x,y)\). For any \(t\in[0,T]\), we have
Then
and
Hence we have
and
which imply that
Consequently,
for any \(t\in[0,T]\), where \(M_{0}\) is defined by (3.15), which proves that \({\mathcal{E}}\) is bounded. Thus, by Lemma 3.1, the operator \(\mathcal{T}\) has at least one fixed point. Hence the boundary value problem (1.1) has at least one solution. The proof is complete. □
3.1 Examples
Example 3.1
Consider the following coupled system of impulsive quantum difference equations with coupled boundary conditions
Here \(q_{k}=(2k+1)/(k^{2}+k+2)\), \(p_{k}=(\sqrt{k+1})/(e^{k}+1)\), \(k=0,1,2,3\), \(m=3\), \(T=2\), \(a_{1}=2\), \(a_{2}=3\), \(b_{1}=4\), \(b_{2}=-2\), \(\lambda_{1}=5\), \(\lambda_{2}=-6\), \(f(t,x,y)=(t\cos^{2}(\pi t)|x|)/(((3e^{t}+4)^{2})(|x|+1))+((t+1)|y|)/(((2t+4)^{3})(|y|+1))+3/2\), \(g(t,x,y)=(\sin x)/(2^{t+1}+5)^{2}+(e^{-2(t+1)}\cos y)/7+(t^{2}+1)/3\), \(I_{k}(x)=|x|/(6(k+5)+|x|)\), and \(I_{k}^{*}(y)=|y|/(7(k+4)+|y|)\). We have \(|f(t,x_{1},y_{1})-f(t,x_{2},y_{2})|\leq( (2/49)|x_{1}-x_{2}|+(3/64)|y_{1}-y_{2}|)\), \(|g(t,x_{1},y_{1})-g(t,x_{2},y_{2})|\leq( (1/49)|x_{1}-x_{2}|+(1/(7e^{2}))|y_{1}-y_{2}|)\), \(|I_{k}(x)-I_{k}(y)|\leq (1/36)|x-y|\), and \(|I_{k}^{*}(x)-I_{k}^{*}(y)|\leq(1/35)|x-y|\). We can find
With the given values, it is found that \(K_{1}=2/49\), \(K_{2}=3/64\), \(K_{3}=1/36\), \(L_{1}=1/49\), \(L_{2}=1/(7e^{2})\), \(L_{3}=1/35\), \(M_{1}\simeq 0.29422\), \(M_{2}\simeq0.25393\), \(M_{4}\simeq0.11127\), \(M_{5}\simeq 0.22224\),, and
Thus all the conditions of Theorem 3.1 are satisfied. Therefore, by the conclusion of Theorem 3.1, problem (3.16) has a unique solution on \([0, 2]\).
Example 3.2
Consider the following coupled system of impulsive quantum difference equations with coupled boundary conditions:
Here \(q_{k}=(k+1)/(\sqrt{k^{2}+e^{k}+1})\), \(p_{k}=(\sin(((k+1)\pi)/10))/3\), \(k=0,1,2,\ldots,8\), \(m=8\), \(T=3\), \(a_{1}=-1\), \(a_{2}=2\), \(b_{1}=5\), \(b_{2}=3\), \(\lambda_{1}=-2\), \(\lambda_{2}=5\), \(f(t,x,y)=(1/4)+(\sin x)/(2(t+5)^{2})+(\tan ^{-1} y)/(7\pi^{2})\), \(g(t,x,y)=((t+2)/e)+(x\cos y)/40+(y)/(2^{t}+45)\), \(I_{k}(x)=(\tan^{-1}(x/8))/4+2\), and \(I_{k}^{*}(y)=(\sin(y/6))/5+3\). We get
Since \(|f(t,x,y)|\leq A_{0}+A_{1}|x|+A_{2}|y|\), \(|g(t,x,y)|\leq B_{0}+B_{1}|x|+B_{2}|y|\), where \(A_{0}=1/4\), \(A_{1}=1/50\), \(A_{2}=1/(7\pi^{2})\), \(B_{0}=5/e\), \(B_{1}=1/40\), \(B_{2}=1/46\), it is found that \(M_{7}\simeq 0.62765\), \(M_{8}\simeq0.27696\), \(M_{10}\simeq0.19588\), \(M_{11}\simeq 0.63239\). Furthermore,
and
Thus all the conditions of Theorem 3.2 holds true and consequently the conclusion of Theorem 3.2; problem (3.17) has at least one solution on \([0, 3]\).
4 Uncoupled boundary conditions case
In this section, we consider again the system
Lemma 4.1
(Auxiliary lemma)
For \(h\in C([0,T], {\mathbb{R}})\), the unique solution of the problem
is given by
where
In view of Lemma 4.1, we define an operator \(\mathfrak{T}: X\times Y\to X\times Y\) by
where
and
where
We remark that \(\mathfrak{T}_{1}\) depends only on f and \(\mathfrak {T}_{2}\) only on g. We call the above system, for convenience, a ‘coupled system with uncoupled boundary conditions’.
In the sequel, we set the constants
Now we present the existence and uniqueness result for problem (4.1). We do not provide the proof of this result as it is similar to the one for Theorem 3.1.
Theorem 4.1
Assume that:
- (H5):
-
The functions \(f, g: [0,T]\times{\mathbb{R}}^{2}\to {\mathbb{R}}\) are continuous and there exist constants \(\overline{K}_{i}, \overline{L}_{i}>0\), \(i=1,2\) such that for all \(t\in[0,T]\) and \(u_{i}, v_{i}\in{\mathbb{R}}\), \(i=1,2\),
$$\bigl|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\bigr|\le \overline{K}_{1}|u_{1}-v_{1}|+ \overline{K}_{2}|u_{2}-v_{2}| $$and
$$\bigl|g(t,u_{1},u_{2})-g(t,v_{1},v_{2})\bigr|\le \overline{L}_{1}|u_{1}-v_{1}|+ \overline{L}_{2}|u_{2}-v_{2}|. $$ - (H6):
-
The functions \(I_{k},I_{k}^{*}:{\mathbb{R}}\rightarrow {\mathbb{R}}\) are continuous and there exist constants \(\overline{K}_{3}, \overline {L}_{3}>0\) such that for all \(t\in[0,T]\) and \(u_{3}, v_{3}\in{\mathbb{R}}\), \(k=1,2,\ldots,m\)
$$\bigl|I_{k}(u_{3})-I_{k}(v_{3})\bigr|\le \overline{K}_{3}|u_{3}-v_{3}| $$and
$$\bigl|I_{k}^{*}(u_{3})-I_{k}^{*}(v_{3})\bigr|\le \overline{L}_{3}|u_{3}-v_{3}|. $$
In addition, assume that
where \(\overline{M}_{1}\), \(\overline{M}_{2}\), \(\overline{M}_{4}\), \(\overline {M}_{5}\) are given by (4.5)-(4.6) and (4.8)-(4.9), respectively. Then the boundary value problem (4.1) has a unique solution.
Example 4.1
Consider the following coupled system of impulsive quantum difference equations with uncoupled boundary conditions
Here \(q_{k}=(2/7)^{k}\), \(p_{k}=((3+k)/(4+2k))^{k}\), \(k=0,1,2,3,4\), \(m=4\), \(T=1\), \(a_{1}=3\), \(a_{2}=4\), \(b_{1}=-8\), \(b_{2}=5\), \(\lambda_{1}=7\), \(\lambda _{2}=2\), \(f(t,x,y)=(\sin(\pi t)|x|)/(((e^{t}+5)^{2})(|x|+1))+(\pi ^{t}|y|)/(((t+4)^{3})(|y|+1))+3\), \(g(t,x,y)=(\cos x)/(10(2^{t}+4))+(|y|)/(6\pi(t+3))+2\), \(I_{k}(x)=|x|/(3(k+9)+|x|)\), and \(I_{k}^{*}(y)=|y|/(5(k+6)+|y|)\). Since \(|f(t,x_{1},y_{1})-f(t,x_{2},y_{2})|\leq( (1/36)|x_{1}-x_{2}|+(\pi/64)|y_{1}-y_{2}|)\), \(|g(t,x_{1},y_{1})-g(t,x_{2},y_{2})|\leq( (1/50)|x_{1}-x_{2}|+(1/(18\pi))|y_{1}-y_{2}|)\), \(|I_{k}(x)-I_{k}(y)|\leq (1/30)|x-y|\), and \(|I_{k}^{*}(x)-I_{k}^{*}(y)|\leq(1/35)|x-y|\). We can find
With the given values, it is found that \(\overline{K}_{1}=1/36\), \(\overline{K}_{2}=\pi/64\), \(\overline{K}_{3}=1/30\), \(\overline{L}_{1}=1/50\), \(\overline{L}_{2}=1/(18\pi)\), \(\overline{L}_{3}=1/35\), \(\overline {M}_{1}\simeq0.41889\), \(\overline{M}_{2}\simeq0.12763\), \(\overline {M}_{4}\simeq0.03111\), \(\overline{M}_{5}\simeq0.20529\), and
Thus all the conditions of Theorem 4.1 are satisfied. Therefore, by the conclusion of Theorem 4.1, problem (4.11) has a unique solution on \([0, 1]\).
The second result dealing with the existence of solutions for the problem (4.1) is analogous to Theorem 3.2 and is given below.
In the sequel, we set constants
Theorem 4.2
Assume that:
- (H7):
-
The functions \(f, g: [0,T]\times{\mathbb{R}}^{2}\to {\mathbb{R}}\) are continuous and there exist constants \(\overline{A}_{i}, \overline{B}_{i} \ge0 \) (\(i=1, 2\)) and \(\overline{A}_{0}, \overline{B}_{0}>0\) such that \(\forall x_{i}\in{\mathbb{R}} \) (\(i=1, 2\))
$$\bigl|f(t,x_{1},x_{2})\bigr|\le\overline{A}_{0}+ \overline{A}_{1}|x_{1}|+\overline{A}_{2}|x_{2}| $$and
$$\bigl|g(t,x_{1},x_{2})\bigr|\le\overline{B}_{0}+ \overline{B}_{1}|x_{1}|+\overline{B}_{2}|x_{2}|. $$ - (H8):
-
The functions \(I_{k},I_{k}^{*}:{\mathbb{R}}\rightarrow {\mathbb{R}}\) are continuous and there exist constants \(\overline{A}_{4}, \overline {B}_{4}\geq0\) and \(\overline{A}_{3}, \overline{B}_{3}>0\) such that \(\forall x_{3}\in{\mathbb{R}}\), \(k=1,2,\ldots,m\),
$$\bigl|I_{k}(x_{3})\bigr|\le\overline{A}_{3}+ \overline{A}_{4}|x_{3}| $$and
$$\bigl|I_{k}^{*}(x_{3})\bigr|\le\overline{B}_{3}+ \overline{B}_{4}|x_{3}|. $$
In addition it is assumed that
where \(\overline{M}_{7}\), \(\overline{M}_{8}\), \(\overline{M}_{10}\), \(\overline{M}_{11}\) are given by (4.12)-(4.13) and (4.15)-(4.16), respectively. Then the boundary value problem (4.1) has at least one solution.
Proof
Setting
the proof is similar to that of Theorem 3.2. So we omit it. □
Example 4.2
Consider the following coupled system of impulsive quantum difference equations with uncoupled boundary conditions:
Here \(q_{k}=2/(3+k)\), \(p_{k}=(3+k)/(5+2k+k^{2})\), \(k=0,1,2,\ldots,9\), \(m=9\), \(T=1\), \(a_{1}=2\), \(a_{2}=3\), \(b_{1}=-7\), \(b_{2}=-5\), \(\lambda_{1}=-3\), \(\lambda _{2}=-10\), \(f(t,x,y)=3+(\sin(x/2))/20+(\tan^{-1} (y/4))/(t+3)^{2}\), \(g(t,x,y)=4+(tx)/(10(2^{t}+1))+(\sin(\pi t)y)/(2(2t+5)^{2})\), \(I_{k}(x)=(\sin (\pi x/2))/(10\pi^{2})+1/2\), and \(I_{k}^{*}(y)=y/(20e+y^{2})+\pi/4\). We get
Since \(|f(t,x,y)|\leq\overline{A}_{0}+\overline{A}_{1}|x|+\overline {A}_{2}|y|\), \(|g(t,x,y)|\leq\overline{B}_{0}+\overline{B}_{1}|x|+\overline {B}_{2}|y|\), where \(\overline{A}_{0}=3\), \(\overline{A}_{1}=1/40\), \(\overline {A}_{2}=1/36\), \(\overline{B}_{0}=4\), \(\overline{B}_{1}=1/20\), \(\overline {B}_{2}=1/50\), it is found that \(\overline{M}_{7}\simeq0.40377\), \(\overline {M}_{8}\simeq0.06667\), \(\overline{M}_{10}\simeq0.175\), \(\overline {M}_{11}\simeq0.64941\). Furthermore,
and
Thus all the conditions of Theorem 4.2 holds true and consequently the conclusion of Theorem 4.2; problem (4.18) has at least one solution on \([0, 1]\).
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Acknowledgements
We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-GOV-58-10.
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Tariboon, J., Ntouyas, S.K. & Thiramanus, P. Impulsive quantum difference systems with boundary conditions. Adv Differ Equ 2015, 163 (2015). https://doi.org/10.1186/s13662-015-0506-7
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DOI: https://doi.org/10.1186/s13662-015-0506-7