Now, we are concerned with the existence and uniqueness results of the coupled system (1)–(2) in generalized Banach spaces.
Let C be the Banach space of all continuous functions v from I into \({\Bbb {R}}^{m}\) with the supremum (uniform) norm
$$\Vert v \Vert _{C}:=\sup _{t\in I} \bigl\Vert v(t) \bigr\Vert . $$
By \(L^{\infty}(I,{\Bbb {R}}_{+})\) we denote the Banach space of measurable functions from I into \({\Bbb {R}}_{+}\) which are essentially bounded.
Let \(x,y\in{\Bbb {R}}^{m}\) with \(x=(x_{1},x_{2},\dots,x_{m})\), \(y=(y_{1},y_{2},\dots,y_{m})\).
By \(x\leq y\) we mean \(x_{i}\leq y_{i}\), \(i=1,\dots,m\). Also,
$$\begin{gathered} \vert x \vert = \bigl( \vert x_{1} \vert , \vert x_{2} \vert ,\dots, \vert x_{m} \vert \bigr), \\ \max (x,y)= \bigl(\max (x_{1},y_{1}),\max (x_{2},y_{2}), \dots, \max (x_{m},y_{m}) \bigr),\end{gathered} $$
and
$${\Bbb {R}}^{m}_{+}= \bigl\{ x\in{\Bbb {R}}^{m}:x_{i}\in{ \Bbb {R}}_{+}, i=1,\dots,m \bigr\} . $$
If \(c\in{\Bbb {R}}\), then \(x\leq c\) means \(x_{i}\leq c\), \(i=1,\dots,m\).
Definition 4.1
Let X be a nonempty set. By a vector-valued metric on X we mean a map \(d:X\times X\to{\Bbb {R}}^{m}\) with the following properties:
- (i)
\(d(x,y)\geq0\) for all \(x,y\in X\), and if \(d(x,y)=0\), then \(x=y\);
- (ii)
\(d(x,y)=d(y,x)\) for all \(x,y\in X\);
- (iii)
\(d(x,z)\leq d(x,y)+d(y,z)\) for all \(x,y,z\in X\).
We call the pair \((X,d)\) a generalized metric space with
$$d(x,y):=\left ( \textstyle\begin{array}{c} d_{1}(x,y)\\ d_{2}(x,y)\\ \vdots\\ d_{m}(x,y) \end{array}\displaystyle \right ). $$
Notice that d is a generalized metric space on X if and only if \(d_{i}\), \(i=1,\dots,m\), are metrics on X.
Definition 4.2
([45])
A square matrix of real numbers is said to be convergent to zero if and only if its spectral radius \(\rho(M)\) is strictly less than 1. In other words, this means that all the eigenvalues of M are in the open unit disc, i.e., \(|\lambda|<1\) for every \(\lambda\in{\Bbb {C}}\) with \(\operatorname{det}(M-\lambda I)=0\), where I denotes the unit matrix of \(M_{m\times m}({\Bbb {R}})\).
Example 4.3
The matrix \(A\in M_{2\times2}({\Bbb {R}})\) defined by
$$A=\left ( \textstyle\begin{array}{c@{\quad}c} a & b \\ c & d \end{array}\displaystyle \right ) $$
converges to zero in the following cases:
- (1)
\(b=c=0\), \(a,d>0\), and \(\max\{a,d\}<1\).
- (2)
\(c=0\), \(a,d>0\), \(a+d<1\), and \(-1< b<0\).
- (3)
\(a+b=c+d=0\), \(a>1\), \(c>0\), and \(|a-c|<1\).
Definition 4.4
Let \((X,d)\) be a generalized metric space. An operator \(N:X\to X\) is said to be contractive if there exists a matrix M convergent to zero such that
$$d \bigl(N(x),N(y) \bigr)\leq Md(x,y)\quad \text{for all }x,y\in X. $$
In the sequel we will make use of the following fixed point theorems in generalized Banach spaces.
Theorem 4.5
[36] Let
\((X,d)\)be a complete generalized metric space and
\(N:X\to X\)be a contractive operator with Lipschitz matrixM. ThenNhas a unique fixed point
\(x_{0}\), and for each
\(x\in X\), we have
$$d \bigl(N^{k}(x),x_{0} \bigr)\leq M^{k}( M)^{-1}d \bigl(x,N(x) \bigr) \quad\textit{for all } k\in{\Bbb {N}}. $$
For \(n = 1\), we recover the classical Banach contraction fixed point result.
Theorem 4.6
([36])
LetXbe a generalized Banach space, \(D\subset E\)be a nonempty closed convex subset ofE, and
\(N:D\to D\)be a continuous operator with relatively compact range. ThenNhas at least a fixed point inD.
The following hypotheses will be used in the sequel.
- \((H_{01})\):
There exist continuous functions \(p_{i},d_{i},l_{i}:I\to {\Bbb {R}}_{+}\), \(i=1,2\), such that \(l_{i}<1\) and
$$\begin{gathered}\bigl\| f_{i}(t,u_{1},v_{1},w_{1})-f_{i}(t,u_{2},v_{2},w_{2})\bigr\| \\\quad\leq p_{i}(t) \Vert u_{1}-u_{2} \Vert +d_{i}(t) \Vert v_{1}-v_{2} \Vert +l_{i}(t) \Vert w_{1}-w_{2} \Vert \end{gathered} $$
for a.e. \(t\in I\) and each \(u_{i},v_{i},w_{i}\in{\Bbb {R}}^{m}\), \(i=1,2\).
- \((H_{02})\):
There exist continuous functions \(K_{i},P_{i},D_{i},L_{i}:I\to{\Bbb {R}}_{+}\), \(i=1,2\), such that
$$\bigl\Vert f_{i}(t,u,v,w) \bigr\Vert \leq K_{i}(t)+P_{i}(t) \Vert u \Vert +D_{i}(t) \Vert v \Vert +L_{i}(t) \Vert w \Vert $$
for a.e. \(t\in I\) and each \(u,v,w\in{\Bbb {R}}^{m}\), \(i=1,2\).
Set
$$\begin{gathered} p_{i}^{\ast}:=\sup _{t\in I}p_{i}(t),\qquad d_{i}^{\ast}:=\sup _{t\in I}d_{i}(t),\qquad l_{i}^{\ast}:=\sup _{t\in I}l_{i}(t),\qquad K_{i}^{\ast}:=\sup _{t\in I}K_{i}(t),\qquad \\ P_{i}^{\ast}:=\sup _{t\in I}P_{i}(t),\qquad D_{i}^{\ast}:=\sup _{t\in I}D_{i}(t),\qquad L_{i}^{\ast}:=\sup _{t\in I}L_{i}(t),\end{gathered} $$
and
$$\ell_{i}:=\frac{T^{\alpha_{i}}}{\varGamma_{q}(1+\alpha_{i})},\quad i=1,2. $$
The space \(C^{2}:=C\times C\) is a generalized Banach space with the norm
$$\bigl\Vert (u_{1},u_{2}) \bigr\Vert _{C^{2}} := \bigl( \Vert u_{1} \Vert _{C}, \Vert u_{2} \Vert _{C} \bigr) . $$
Definition 4.7
By a solution of problem (1)–(2) we mean a coupled continuous function \((u,v)\in C^{2}\) satisfying initial condition (2) and system (1) on I.
First, we prove an existence and uniqueness result for coupled system (1)–(2) by using Banach’s fixed point theorem type in generalized Banach spaces.
Theorem 4.8
Assume that hypothesis
\((H_{01})\)holds. If the matrix
$$M:=\left ( \textstyle\begin{array}{c@{\quad}c} \frac{\ell_{1}p_{1}^{\ast}}{1-l_{1}^{\ast}} & \frac{\ell_{1}d_{1}^{\ast }}{1-l_{1}^{\ast}}\\ [3pt] \frac{\ell_{2}p_{2}^{\ast}}{1-l_{2}^{\ast}} & \frac{\ell_{2}d_{2}^{\ast }}{1-l_{2}^{\ast}} \end{array}\displaystyle \right ) $$
converges to 0, then coupled system (1)–(2) has a unique solution.
Proof
From Lemma 2.9, we can define the operators \(N_{1},N_{2}:C^{2}\rightarrow C\) by
$$ \bigl(N_{i}(u_{1},u_{2}) \bigr) (t)=u_{0i}+ \bigl(I_{q}^{\alpha_{i}}g_{i} \bigr) (t),\quad i=1,2, t\in I, $$
(10)
where \(g_{i}(\cdot)\in C(I)\), with
$$\begin{aligned} g_{i}(t)&=f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ &=f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha _{2}}g_{2} \bigr) (t),g_{i}(t) \bigr),\quad i=1,2. \end{aligned}$$
Consider the operator \(N:C^{2}\to C^{2}\) defined by
$$ \bigl(N(u_{1},u_{2}) \bigr) (t)= \bigl( \bigl(N_{1}(u_{1},u_{2}) \bigr) (t), \bigl(N_{2}(u_{1},u_{2}) \bigr) (t) \bigr). $$
(11)
Clearly, the fixed points of the operator N are solutions of coupled system (1)–(2). We show that N satisfies all the conditions of Theorem 4.5.
For each \((u_{1},u_{2}), (v_{1},v_{2})\in C^{2}\) and \(t\in I\), we have
$$ \bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr) (t)- \bigl(N_{i}(v_{1},v_{2}) \bigr) (t) \bigr\Vert \leq \int_{0}^{t}\frac {(t-qs)^{(\alpha-i)}}{\varGamma_{q}(\alpha_{i})} \bigl\Vert g_{i}(s)-h_{i}(s) \bigr\Vert \,d_{q}s, $$
(12)
where \(g_{i}(\cdot),h_{i}(\cdot)\in C(I)\), \(i=1,2\), with
$$\begin{aligned} g_{i}(t) =&f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t),g_{i}(t) \bigr) \end{aligned}$$
and
$$\begin{aligned} h_{i}(t) =&f_{i} \bigl(t,v_{1}(t),v_{2}(t),h_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}h_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}h_{2} \bigr) (t),h_{i}(t) \bigr). \end{aligned}$$
From hypothesis \((H_{01})\), we have
$$\bigl\Vert g_{i}(t)-h_{i}(t) \bigr\Vert =p_{i}(t) \bigl\Vert u_{1}(t)-v_{1}(t) \bigr\Vert +d_{i}(t) \Vert u_{2}-v_{2} \Vert +l_{i}(t) \Vert g_{i}-h_{i} \Vert . $$
Then
$$\bigl\Vert g_{i}(t)-h_{i}(t) \bigr\Vert =p_{i}(t) \bigl\Vert u_{1}(t)-v_{1}(t) \bigr\Vert +d_{i}(t) \bigl\Vert u_{2}(t)-v_{2}(t) \bigr\Vert +l_{i}(t) \bigl\Vert g_{i}(t)-h_{i}(t) \bigr\Vert . $$
Thus
$$\Vert g_{i}-h_{i} \Vert _{C}=p_{i}^{\ast} \Vert u_{1}-v_{1} \Vert _{C}+i_{1}^{\ast} \Vert u_{2}-v_{2} \Vert _{C}+l_{i}^{\ast} \Vert g_{i}-h_{i} \Vert _{C}. $$
This implies that
$$\bigl(1-l_{i}^{\ast} \bigr) \Vert g_{i}-h_{i} \Vert _{C}=p_{i}^{\ast} \Vert u_{1}-v_{1} \Vert _{C}+d_{i}^{\ast} \Vert u_{2}-v_{2} \Vert _{C}. $$
Hence
$$\Vert g_{i}-h_{i} \Vert _{C}= \frac{p_{i}^{\ast}}{1-l_{i}^{\ast}} \Vert u_{1}-v_{1} \Vert _{C}+\frac {d_{i}^{\ast}}{1-l_{i}^{\ast}} \Vert u_{2}-v_{2} \Vert _{C}. $$
From (12), we get
$$\begin{aligned} \bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr)- \bigl(N_{i}(v_{1},v_{2}) \bigr) \bigr\Vert _{C} \leq& \int_{0}^{t}\frac {(t-qs)^{(\alpha_{i}-1)}}{\varGamma_{q}(\alpha_{i}-1)} \bigl\Vert g_{i}(s)-h_{i}(s) \bigr\Vert \,d_{q}s \\ \leq&\frac{\ell_{i}p_{i}^{\ast}}{1-l_{i}^{\ast}} \Vert u_{1}-v_{1} \Vert _{C}+\frac {\ell_{i}d_{i}^{\ast}}{1-l_{i}^{\ast}} \Vert u_{2}-v_{2} \Vert _{C}. \end{aligned}$$
Consequently,
$$d \bigl( \bigl(N(u_{1},u_{2}) \bigr), \bigl(N(v_{1},v_{2}) \bigr) \bigr)\leq Md \bigl((u_{1},u_{2}),(v_{1},v_{2}) \bigr), $$
where
$$d \bigl((u_{1},u_{2}),(v_{1},v_{2}) \bigr)= \left ( \textstyle\begin{array}{c} \Vert u_{1}-v_{1} \Vert _{C} \\ \Vert u_{2}-v_{2} \Vert _{C} \end{array}\displaystyle \right ). $$
Since the matrix M converges to zero, then Theorem 4.5 implies that coupled system (1)–(2) has a unique solution. □
Now, we prove an existence result for coupled system (1)–(2) by using Schauder’s fixed point theorem type in a generalized Banach space.
Theorem 4.9
Assume that hypothesis
\((H_{02})\)holds. Then coupled system (1)–(2) has at least one solution.
Proof
Let \(N:C^{2}\to C^{2}\) be the operator defined in (11). We show that N satisfies all the conditions of Theorem 4.6. The proof will be given in several steps.
Step 1.
N
is continuous.
Let \(\{(u_{1n},u_{2n})\}_{n}\) be a sequence such that \((u_{1n},u_{2n})\to (u_{1},u_{2})\in C^{2}\) as \(n\to\infty\). For any \(i=1,2\) and each \(t\in I\), we have
$$\bigl\Vert \bigl(N_{i}(u_{1n},u_{2n}) \bigr) (t)- \bigl(N_{i}(u_{1},u_{2}) \bigr) (t) \bigr\Vert \leq \int_{0}^{t}\frac {(t-qs)^{(\alpha_{i}-1)}}{\varGamma_{q}(\alpha_{i}-1)} \bigl\Vert g_{in}(s)-g_{i}(s) \bigr\Vert \,d_{q}s, $$
where \(g_{i}(\cdot),g_{in}(\cdot)\in C(I)\), \(i=1,2\), with
$$\begin{aligned} g_{i}(t) =&f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t),g_{i}(t) \bigr) \end{aligned}$$
and
$$\begin{aligned} g_{in}(t) =&f_{i} \bigl(t,u_{1n}(t),u_{2n}(t),g_{in}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1n} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha _{2}}g_{2n} \bigr) (t),g_{in}(t) \bigr). \end{aligned}$$
From \((H_{02})\), we have
$$\Vert g_{in}-g_{i} \Vert _{C}\leq \frac{P_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{1n}-u_{1} \Vert _{C}+\frac{D_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{2n}-u_{2} \Vert _{C}. $$
Thus,
$$\begin{aligned} \bigl\Vert \bigl(N_{i}(u_{1n},u_{2n}) \bigr) (t)- \bigl(N_{i}(u_{1},u_{2}) \bigr) (t) \bigr\Vert \leq&\frac{\ell_{i}P_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{1n}-u_{1} \Vert _{C}+\frac{\ell_{i}D_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{2n}-u_{2} \Vert _{C} \\ \to&0\quad \text{as }n\to\infty. \end{aligned}$$
Hence, we get
$$\bigl\Vert N_{i}(u_{1n},u_{2n})-N_{i}(u_{1},u_{2}) \bigr\Vert _{C}\to0 \quad \text{as }n\to\infty. $$
Consequently,
$$\begin{aligned}& \bigl\Vert N(u_{1n},u_{2n})-N(u_{1},u_{2}) \bigr\Vert _{C^{2}} \\& \quad:= \bigl( \bigl\Vert N_{1}(u_{1n},u_{2n})-N_{1}(u_{1},u_{2}) \bigr\Vert _{C}, \bigl\Vert N_{2}(u_{1n},u_{2n})-N_{2}(u_{1},u_{2}) \bigr\Vert _{C} \bigr) \\& \quad\to(0,0) \quad \text{as }n\to\infty. \end{aligned}$$
Finally, N is continuous.
Step 2.Nmaps bounded sets into bounded sets in
\(C^{2}\).
Set
$$h_{i}^{\ast}:=\sup _{t\in I}h_{i}(t),\qquad k_{i}^{\ast}:=\sup _{t\in I}k_{i}(t),\qquad l_{i}^{\ast}:=\sup _{t\in I}l_{i}(t). $$
Let \(R>0\) and set
$$B_{R}:= \bigl\{ (\mu,\nu)\in C^{2}: \Vert \mu \Vert _{C}\leq R, \Vert \nu \Vert _{C}\leq R \bigr\} . $$
For any \(i=1,2\) and each \((u,v)\in B_{R}\) and \(t\in I\), we have
$$\bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr) (t) \bigr\Vert \leq \int_{0}^{t}\frac{(t-qs)^{(\alpha _{i}-1)}}{\varGamma_{q}(\alpha_{i}-1)} \bigl\Vert g_{i}(s) \bigr\Vert \,d_{q}s, $$
where \(g_{i}(\cdot),\in C(I)\), \(i=1,2\), with
$$\begin{aligned} g_{i}(t) =&f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t),g_{i}(t) \bigr). \end{aligned}$$
Since
$$\Vert g_{i} \Vert _{C}\leq\frac{P_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{1} \Vert _{C}+\frac {D_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{2} \Vert _{C}, $$
we get
$$\bigl\Vert N_{i}(u_{1},u_{2}) \bigr\Vert _{C}\leq\frac{\ell_{i}P_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{1} \Vert _{C}+\frac{\ell_{i}D_{i}^{\ast}}{1- L_{i}^{\ast}} \Vert u_{2} \Vert _{C}. $$
Thus,
$$\bigl\Vert N_{i}(u_{1},u_{2}) \bigr\Vert _{C}\leq\frac{R\ell_{i}P_{i}^{\ast}}{1- L_{i}^{\ast }}+\frac{R\ell_{i}D_{i}^{\ast}}{1- L_{i}^{\ast}}:=M_{i}. $$
Hence,
$$\bigl\Vert \bigl(N(u,v) \bigr) \bigr\Vert _{C^{2}} \leq(M_{1},M_{2}):=M. $$
Step 3.Nmaps bounded sets into equicontinuous sets in
\(C^{2}\).
Let \(B_{R}\) be the ball defined in Step 2. For each \(t_{1},t_{2}\in I\) with \(t_{1}\leq t_{2}\) and any \((u,v)\in B_{R}\) and \(i=1,2\), we have
$$\begin{aligned}& \bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr) (t_{1})- \bigl(N_{i}(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert \\& \quad\leq \int_{0}^{t_{1}}\frac{(t_{1}-qs)^{(\alpha_{i}-1)}}{\varGamma _{q}(\alpha_{i}-1)} \bigl\Vert g_{i}(s) \bigr\Vert \,d_{q}s - \int_{0}^{t_{2}}\frac{(t_{2}-qs)^{(\alpha_{i}-1)}}{\varGamma_{q}(\alpha _{i}-1)} \bigl\Vert g_{i}(s) \bigr\Vert \,d_{q}s, \end{aligned}$$
where \(g_{i}(\cdot),\in C(I)\), \(i=1,2\), with
$$\begin{aligned} g_{i}(t) =&f_{i} \bigl(t,u_{1}(t),u_{2}(t),g_{i}(t) \bigr) \\ =&f_{i} \bigl(t,u_{01}+ \bigl(I_{q}^{\alpha_{1}}g_{1} \bigr) (t),u_{02}+ \bigl(I_{q}^{\alpha_{2}}g_{2} \bigr) (t),g_{i}(t) \bigr). \end{aligned}$$
Thus,
$$\begin{gathered} \bigl\Vert \bigl(N_{i}(u_{1},u_{2}) \bigr) (t_{1})-\bigl(N_{i}(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert \\ \quad\leq \biggl( \frac{R P_{i}^{\ast}}{1- L_{i}^{\ast}}+\frac{R D_{i}^{\ast}}{1- L_{i}^{\ast}} \biggr) \int_{0}^{t_{1}}\frac{ \vert (t_{2}-qs)^{(\alpha_{i}-1)}-(t_{1}-qs)^{(\alpha _{i}-1)} \vert }{\varGamma_{q}(\alpha_{1})}\,d_{q}s \\ \qquad{}+ \biggl(\frac{R P_{i}^{\ast}}{1- L_{i}^{\ast}}+\frac{R D_{i}^{\ast }}{1- L_{i}^{\ast}} \biggr) \int_{t_{1}}^{t_{2}}\frac{ \vert (t_{2}-qs)^{(\alpha_{i}-1)} \vert }{\varGamma_{q}(\alpha _{i})}\,d_{q}s \\ \quad\to0 \quad \text{as }t_{1}\to t_{2}. \end{gathered} $$
Hence,
$$\begin{aligned}& \bigl\Vert \bigl(N(u_{1},u_{2}) \bigr) (t_{1})- \bigl(N(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert \\& \quad= \bigl( \bigl\Vert \bigl(N_{1}(u_{1},u_{2}) \bigr) (t_{1})- \bigl(N_{1}(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert , \bigl\Vert \bigl(N_{2}(u_{1},u_{2}) \bigr) (t_{1})- \bigl(N_{2}(u_{1},u_{2}) \bigr) (t_{2}) \bigr\Vert \bigr) \\& \quad\to(0,0) \quad \text{as }t_{1}\to t_{2}. \end{aligned}$$
As a consequence of steps 1 to 3 together with Theorem 4.6, we can conclude that N has at least one fixed point in \(B_{R}\) which is a solution of our coupled system (1)–(2). □