In this section, by using fixed point theorems, we look into the existence of solution for the boundary value problem (1). In the first step, by applying the following lemma, we break down the solution kind of hybrid fractional differential inclusion (1).
Lemma 10
Let
\(y \in C(J,\mathbb{R})\)
and
$$\frac{x(\cdot) - f(\cdot, x(\cdot), I^{\beta _{1}} h_{1}(\cdot, x(\cdot)), I^{\beta _{2}} h _{2}(\cdot, x(\cdot)), \ldots , I^{\beta _{n}} h_{n}(\cdot, x(\cdot)) )}{g(\cdot, x(\cdot), I ^{\gamma _{1}}x(\cdot), I^{\gamma _{2}}x(\cdot), \ldots , I^{\gamma _{m}} x(\cdot))} \in AC_{\delta }^{2} [1, e]. $$
Then the unique solution of the hybrid fractional differential equation
$$ \begin{aligned}[b] {}^{C}_{H} &D^{\alpha } \biggl[ \frac{x(t) - f ( t, x(t), I^{ \beta _{1}} h_{1}(t, x(t)), I^{\beta _{2}} h_{2}(t, x(t)), \ldots , I ^{\beta _{n}} h_{n}(t, x(t)) )}{g ( t, x(t), I^{\gamma _{1}} x(t), I^{\gamma _{2}} x(t), \ldots , I^{\gamma _{m}} x(t) ) } \biggr] \\ & \quad = y(t), \end{aligned} $$
(3)
via the boundary conditions
\(x(1) =\mu (x)\)
and
\(x(e) = \eta (x)\), is
$$ \begin{aligned}[b] x(t) = {}& g \bigl( t, x(t), I^{\gamma _{1}} x(t), I^{\gamma _{2}} x(t), \ldots , I^{\gamma _{m}} x(t) \bigr) \\ & {} \times \biggl[ \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{t}{s} \biggr)^{\alpha -1} \frac{y(s)}{ s} \,ds + a_{0} \\ & {} + \biggl( a_{1} - \frac{1}{\varGamma ( \alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{ \alpha -1} \frac{y(s)}{s} \,ds - a _{0} \biggr) \ln t \biggr] \\ & {} + f \bigl( t, x(t), I^{\beta _{1}} h_{1} \bigl( t, x(t) \bigr), I^{\beta _{2}} h_{2}\bigl(t, x(t)\bigr), \ldots , I^{\beta _{n}} h_{n} \bigl(t, x(t)\bigr) \bigr) \end{aligned} $$
(4)
for all
\(t\in J\), where
$$\begin{aligned} &a_{0}=\frac{ \mu (x) -f (1, \mu (x), \overbrace{0, \ldots , 0}^{n} ) }{g (1, \mu (x), \underbrace{0, \ldots , 0}_{m} )}, \\ &a_{1}=\frac{ \eta (x) - f ( t, x(t), I^{\beta _{1}} h _{1}(t, x(t)), I^{\beta _{2}} h_{2}(t, x(t) ), \ldots , I^{\beta _{n}} h _{n} (t, x(t)) ) |_{t = e}}{ g ( t, x(t), I^{\gamma _{1}} x(t), I^{\gamma _{2}} x(t), \ldots , I^{\gamma _{m}} x(t) ) |_{t = e}}. \end{aligned}$$
Proof
Let \(x(t)\) be a solution of equation (3). By using the operator \(I^{\alpha }\) on both sides of equation (3) and applying Lemma 3, we have
$$\begin{aligned} &\frac{x(t) - f ( t, x(t), I^{\beta _{1}} h_{1}(t, x(t)), I^{ \beta _{2}}h_{2}(t, x(t) ) , \ldots , I^{\beta _{n}} h_{n}(t, x(t)) )}{ g (t, x(t), I^{\gamma _{1}} x(t), I^{\gamma _{2}}x(t), \ldots , I^{\gamma _{m}} x(t) )} \\ &\quad = I^{\alpha }y(t) + c_{0} + c_{1} \ln t, \end{aligned}$$
where \(c_{0}, c_{1} \in \mathbb{R}\) denote arbitrary constants. Thus
$$\begin{aligned} x(t) = {}& g \bigl(t, x(t), I^{\gamma _{1}}x(t), I^{\gamma _{2}} x(t), \ldots , I^{\gamma _{m}} x(t) \bigr) \\ & {} \times \biggl[ \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha -1} \frac{y(s)}{s}\,ds + c_{0} + c _{1} \ln t \biggr] \\ & {} + f \bigl( t, x(t), I^{\beta _{1}}h_{1}\bigl(t, x(t)\bigr), I^{\beta _{2}}h_{2}\bigl(t, x(t)\bigr), \ldots , I^{\beta _{n}} h_{n}\bigl(t, x(t)\bigr) \bigr). \end{aligned}$$
Now, by utilizing the boundary conditions, we conclude \(c_{0}=a_{0}\) and
$$c_{1}= a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1}\frac{y(s)}{s}\,ds - a_{0}. $$
Conversely, it is easy to check that equation (4) satisfies the boundary conditions \(x(1) = \mu (x)\) and \(x(e) = \eta (x)\). On the other hand, by deformation equation (4) to
$$\begin{aligned} & \frac{x(t)-f(t, x(t), I^{\beta _{1}}h_{1}(t, x(t)), I^{\beta _{2}}h _{2}(t, x(t)), \ldots , I^{\beta _{n}}h_{n}(t, x(t)))}{g(t, x(t), I ^{\gamma _{1}}x(t), I^{\gamma _{2}}x(t), \ldots , I^{\gamma _{m}}x(t))} \\ &\quad = I^{\alpha }y(t) + a_{0} + \biggl( a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1} ^{e} \biggl(\ln \frac{ 1}{s} \biggr)^{\alpha -1} \frac{y(s)}{s} \,ds - a_{0} \biggr) \ln t, \end{aligned}$$
and by applying Lemmas 2 and 4, we have
$${}^{C}_{H}D^{\alpha } \biggl[ \frac{ x(t) -f ( t, x(t), I^{\beta _{1}} h_{1}(t, x(t)), I^{\beta _{2}} h_{2}(t, x(t)), \ldots , I^{\beta _{n}} h_{n}(t, x(t)) ) }{g (t, x(t), I^{\gamma _{1}}x(t), I ^{\gamma _{2}}x(t), \ldots , I^{\gamma _{m}}x(t) ) } \biggr] = y(t), $$
where \(a_{0}\), \(a_{1}\) are defined in Lemma 10. This finishes our proof. □
A function x belonging to \(C(J,\mathbb{R})\) is a solution of problem (1) whenever it satisfies the boundary conditions, and there exists a function \(y \in S_{K,x}\) such that
$$\begin{aligned} \begin{aligned} x(t) ={}&g \bigl(t, x(t), I^{\gamma _{1}}x(t), I^{\gamma _{2}} x(t), \ldots , I^{\gamma _{m}}x(t) \bigr) \\ & {} \times \biggl[ \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{ \alpha -1} \frac{y(s)}{s} \,ds + a_{0} \\ & {} + \biggl( a_{1} \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{ \alpha - 1} \frac{ y(s)}{s} \,ds - a_{0} \biggr) \ln t \biggr] \\ & {} + f \bigl(t, x(t), I^{\beta _{1}}h_{1}\bigl(t, x(t)\bigr), I^{\beta _{2}}h_{2}\bigl(t, x(t)\bigr), \ldots , I^{\beta _{n}}h_{n}\bigl(t, x(t)\bigr) \bigr) \end{aligned} \end{aligned}$$
for \(t\in J\), where
$$\begin{aligned} &a_{0}= \frac{\mu (x)- f(1, \mu (x), \overbrace{0, \ldots , 0}^{n})}{ g ( 1, \mu (x), \underbrace{0,\ldots , 0}_{m})}, \\ &a_{1}= \frac{\eta (x) - f ( t, x(t), I^{\beta _{1}} h _{1}(t, x(t)), I^{\beta _{2}}h_{2}(t, x(t)), \ldots , I^{\beta _{n}}h _{n}(t, x(t)) ) |_{t=e}}{ g ( t, x(t), I ^{\gamma _{1}}x(t), I^{\gamma _{2}}x(t), \ldots , I^{\gamma _{m}}x(t) ) |_{t=e}}. \end{aligned}$$
Let \(\mathcal{X}=C(J,\mathbb{R})\) denote the Banach algebra of all continuous functions from J to \(\mathbb{R}\) endowed with the norm defined by \(\|x\|=\sup \{|x(t)| : t\in J\}\).
Theorem 11
Suppose that
-
\((H_{1})\)
:
-
The multifunction
\(K: J\times \mathbb{R}\to P_{cp, cv}( \mathbb{R})\)
is an
\(L^{1}\)-Caratheodory such that
$$\bigl\Vert K(t, x) \bigr\Vert = \sup \bigl\{ \vert y \vert : y \in K(t, x) \bigr\} \leq p(t) \varphi \bigl( \vert x \vert \bigr) $$
for
\((t, x)\in J\times \mathbb{R}\), where
\(p\in C(J, \mathbb{R^{+}})\)
and
\(\varphi : [0, \infty ) \to (0, \infty )\)
is a continuous and increasing function;
-
\((H_{2})\)
:
-
Function
\(f: J\times \mathbb{R}^{n+1}\to \mathbb{R}\)
is continuous, and there exists
\(\phi \in C(J, \mathbb{R}^{+})\)
such that
$$\bigl\vert f ( t, x_{1}, x_{2}, \ldots , x_{n+1} ) - f \bigl( t, x'_{1}, x'_{2}, \ldots , x'_{n+1} \bigr) \bigr\vert \leq \phi (t) \sum_{i=1}^{n+1} \bigl\vert x_{i} - x'_{i} \bigr\vert , $$
where
\(x_{i}, x'_{i}\in \mathbb{R}\)
for
\(i=1, 2, \ldots , n+1\), \(t\in J\), and
\(n\in \mathbb{N}\);
-
\((H_{3})\)
:
-
Function
\(g: J \times \mathbb{R}^{m+1} \to \mathbb{R} - \{0\}\)
is continuous, and there exists
\(\psi \in C (J, \mathbb{R}^{+})\)
such that
$$\bigl\vert g (t , x_{1}, x_{2}, \ldots , x_{m+1} ) - f \bigl( t, x'_{1}, x'_{2}, \ldots , x'_{m+1} \bigr) \bigr\vert \leq \psi (t) \sum_{i=1}^{m+1} \bigl\vert x_{i} - x'_{i} \bigr\vert , $$
where
\(x_{i}, x'_{i}\in \mathbb{R}\)
for
\(i=1,2, \ldots , m+1\), \(t\in J\), and
\(m\in \mathbb{N}\);
-
\((H_{4})\)
:
-
Functions
\(h_{i}: J\times \mathbb{R}\to \mathbb{R}\)
are continuous for
\(i=1,2, \ldots , n\), and there exists
\(p_{i}\in C(J, \mathbb{R}^{+})\)
for
\(i=1, 2, \ldots , n\)
such that
\(\vert h_{i}(t, x) - h_{i}(t, x') \vert \leq p_{i}(t) \vert x- x' \vert \)
for
\(i=1,2, \ldots , n\), where
\(x, x'\in \mathbb{R}\), \(n\in \mathbb{N}\), \(t\in J\);
-
\((H_{5})\)
:
-
There exist constants
\(M_{0}, M_{1}>0\)
such that
$$\biggl\vert \frac{ \eta (x) - f ( t, x(t), I^{\beta _{1}}h _{1}(t, x(t) ), I^{\beta _{2}} h_{2}(t, x(t) ), \ldots , I^{\beta _{n}} h_{n} (t, x(t)) ) |_{t=e}}{ g ( t, x(t), I ^{\gamma _{1}} x(t), I^{\gamma _{2}}x(t), \ldots , I^{\gamma _{m}} x(t) ) |_{t=e}} \biggr\vert < M_{1} $$
and
$$\biggl\vert \frac{ \mu (x) - f(1, \mu (x), \overbrace{0, \ldots , 0}^{n})}{ g ( 1, \mu (x), \underbrace{0, \ldots , 0}_{m} )} \biggr\vert < M _{0} $$
for all
\(x\in C(J, \mathbb{R})\).
The fractional differential inclusion (1) admits a solution whenever, for
\(i=1,2, \ldots , n\),
$$\begin{aligned} &\biggl( \frac{2 \Vert p \Vert \varphi (r)}{ \varGamma ( \alpha + 1)} + 2M _{0} + M_{1} \biggr) \Vert \psi \Vert \Biggl( 1 + \sum_{i=1} ^{m} \frac{1}{\varGamma (\gamma _{i} + 1)} \Biggr) \\ & \quad {}+ \Vert \phi \Vert \Biggl( 1 + \sum_{i=1}^{n} \frac{ \Vert p_{i} \Vert }{\varGamma (\beta _{i} + 1)} \Biggr) < \frac{1}{2}, \end{aligned}$$
where
\(G_{0} = \sup_{t\in J} | g (t, \underbrace{ 0, \ldots , 0}_{m+1} ) |\), \(F_{0} = \sup_{t\in J} | f ( t , \underbrace{0, \ldots , 0}_{n+1} ) |\), \(H_{i, 0} = \sup_{t \in J} |h_{i}(t, 0) |\), and
$$\begin{aligned} r &> \frac{G_{0} [ \frac{2 \Vert p \Vert \varphi (r) }{\varGamma ( \alpha + 1)} + 2M_{0} + M_{1} ] + \Vert \phi \Vert \sum_{i=1}^{n} \frac{ H _{i,0}}{ \varGamma ( \beta _{i} + 1)} + F_{0}}{ 1 - \Vert \psi \Vert ( 1 + \sum_{i=1}^{m} \frac{1}{\varGamma (\gamma _{i} + 1)} ) ( \frac{2 \Vert p \Vert \varphi (r) }{ \varGamma (\alpha + 1)} + 2M_{0} + M _{1} ) - \Vert \phi \Vert ( 1 + \sum_{i=1}^{n}\frac{ \Vert p_{i} \Vert }{ \varGamma ( \beta _{i} + 1)} )}. \end{aligned}$$
Proof
Let \(\overline{B_{r}(0)} = \{ x\in \mathcal{X}: \|x\|\leq r \}\). Define operators \(A:\overline{B_{r}(0)}\to \mathcal{X}\), \(B:\overline{B_{r}(0)}\to P(\mathcal{X})\) and \(C:\overline{B_{r}(0)} \to \mathcal{X}\) by
$$\begin{aligned} &(Ax) (t) =g \bigl( t, x(t), I^{\gamma _{1}}x(t), I^{\gamma _{2}}x(t), \ldots , I^{\gamma _{m}}x(t) \bigr), \\ &(Bx) (t)= \biggl\{ v(t) : v(t) = \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha -1} \frac{y(s)}{s} \,ds + a_{0} \\ & \hphantom{(Bx) (t)= }{}+ \biggl( a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1} ^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{ \alpha - 1}\frac{y(s)}{s} \,ds - a_{0} \biggr) \ln t, y\in S_{K, x} \biggr\} , \\ &(Cx) (t)= f \bigl( t, x(t), I^{\beta _{1}}h_{1}\bigl(t, x(t) \bigr), I^{\beta _{2}}h_{2}\bigl(t, x(t)\bigr), \ldots , I^{\beta _{n}}h_{n}\bigl(t, x(t)\bigr) \bigr), \end{aligned}$$
respectively. We establish that the operators A, B, and C satisfy all the conditions of Theorem 9. Since the operator K is \(L^{1}\)-Caratheodory, \(S_{K, x}\neq \emptyset \) for all \(x\in \mathcal{X}\). First, we show that Bx is a convex subset of \(\mathcal{X}\) for all \(x\in \overline{B_{r}(0)}\). Consider \(x\in \overline{B _{r}(0)}\) and \(v_{1}, v_{2} \in Bx\). Choose \(y_{1}, y_{2}\in S_{K, x}\) such that
$$\begin{aligned} & v_{1}(t) =\frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha -1}\frac{y_{1}(s)}{s}\,ds + a_{0} \\ & \hphantom{v_{1}(t) =}{} + \biggl( a_{1} - \frac{1}{\varGamma ( \alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1}\frac{y_{1}(s)}{s} \,ds - a _{0} \biggr) \ln t, \\ &v_{2}(t) =\frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha -1}\frac{y_{2}(s)}{s} \,ds + a_{0} \\ & \hphantom{v_{2}(t) =}{} + \biggl( a_{1} \frac{1}{ \varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{1}{s} \biggr)^{ \alpha -1}\frac{y_{2}(s)}{s}\,ds -a _{0} \biggr) \ln t. \end{aligned}$$
At present, for any \(0\leq \lambda \leq 1\), we have
$$\begin{aligned} \lambda v_{1}(t) + (1-\lambda ) v_{2}(t) ={}& \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha -1}\frac{ \lambda y_{1}(s)+ (1-\lambda ) y_{2}(s)}{s} \,ds + a_{0} \\ &{} + \biggl[ a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1}\frac{\lambda y_{1}(s) + (1- \lambda ) y_{2}(s)}{s} \,ds - a_{0} \biggr] \ln t. \end{aligned}$$
Since K has convex values, hence \(S_{K, x}\) is convex and \(\lambda y_{1}+ (1-\lambda )y_{2}\in S_{K, x}\). So \(\lambda v_{1}+ (1- \lambda ) v_{2} \in Bx\), which results in that Bx is a convex subset of \(\mathcal{X}\). Now, we show that Bx is a compact subset of \(\mathcal{X}\) for all \(x\in \overline{B_{r}(0)}\). Let \(x\in \overline{B _{r}(0)}\) and \(\{v_{n}\}\) be a sequence in Bx. For each n, choose \(y_{n}\in S_{K,x}\) such that
$$\begin{aligned} v_{n}(t) ={}& \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha -1}\frac{y_{n}(s)}{s}\,ds + a_{0} \\ & {} + \biggl( a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1}\frac{y_{n}(s)}{s}\,ds - a _{0} \biggr) \ln t. \end{aligned}$$
Since K is compact-valued, \(\{y_{n}\}\) has a subsequence which pointwise converges to y. We denote this subsequence again by \(\{y_{n}\}\). Since \(y(t)\in K(t, x(t))\) and \(\|K(t, x(t)) \|\leq p(t) \varphi (|x(t)|)\), by using Lebesgue dominated convergence theorem \(y\in S_{K, x}\), we have
$$\begin{aligned} v_{n}(t)\to v(t) ={}& \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha -1} \frac{y(s)}{s}\,ds + a_{0} \\ & {} + \biggl( a_{1} - \frac{1}{ \varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1}\frac{y(s)}{s}\,ds -a_{0} \biggr) \ln t \end{aligned}$$
pointwise on J. Let \(t_{1}< t_{2}\in J\). Then we have
$$\begin{aligned} \bigl\vert v_{n}(t_{2})-v_{n}(t_{1}) \bigr\vert \leq{}& \frac{1}{\varGamma (\alpha )} \biggl\vert \int _{1}^{t_{2}} \biggl( \ln \frac{ t_{2}}{s} \biggr)^{ \alpha -1} \frac{y _{n}(s) }{s}\,ds \\ & {} - \int _{1}^{t_{1}} \biggl(\ln \frac{ t_{1}}{s} \biggr)^{ \alpha - 1} \frac{y_{n}(s)}{s}\,ds \biggr\vert \\ & {} + \biggl\vert a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl(\ln \frac{ 1}{s} \biggr)^{\alpha -1} \frac{ y_{n}(s)}{s}\,ds - a _{0} \biggr\vert \\ & {} \times (\ln t_{2}-\ln t_{1}) \\ \leq {}& \frac{1}{\varGamma (\alpha )} \int _{1}^{t_{1}} \biggl\vert \biggl(\ln \frac{ t_{2}}{s} \biggr)^{\alpha -1} - \biggl( \ln \frac{ t_{2}}{s} \biggr)^{\alpha -1} \biggr\vert \frac{ \Vert p \Vert \varphi (r)}{s} \,ds \\ & {} + \frac{1}{\varGamma (\alpha )} \int _{t_{1}}^{t_{2}} \biggl\vert \biggl( \ln \frac{ t_{2}}{s} \biggr)^{\alpha -1} \biggr\vert \frac{ \Vert p \Vert \varphi (r)}{s} \,ds \\ & {} + \biggl( \vert a_{1} \vert + \frac{1}{ \varGamma (\alpha )} \int _{1}^{e} \biggl\vert \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1} \biggr\vert \frac{ \Vert p \Vert \varphi (r)}{s} \,ds + \vert a_{0} \vert \biggr) \\ & {} \times (\ln t_{2}-\ln t_{1}). \end{aligned}$$
As \(t_{1}\to t_{2}\), the right-hand side of the above inequality tends to zero. Hence \(\{ v_{n} \}\) is an equicontinuous sequence. Therefore \(\{ v_{n} \}\) has a subsequence that uniformly converges to \(v\in Bx\), which proves that Bx is a compact subset of \(\mathcal{X}\). In this section, we shall show that the operator B is compact on \(\overline{B_{r}(0)}\). Let \(S\subset \overline{B_{r}(0)}\). We prove that \(\overline{B(S)}\) is a compact set in \(\mathcal{X}\). To do this, we apply Arzelá–Ascoli theorem and show that \(B(S)\) is a uniformly bounded and equicontinuous set. Let \(x\in S\), \(v\in Bx\), and \(t_{1}< t_{2}\in J\). Choose \(y\in S_{K,x}\) such that
$$\begin{aligned} v(t) ={}& \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl(\ln \frac{ t}{s} \biggr)^{\alpha -1}\frac{y(s)}{s}\,ds + a_{0} \\ & {} + \biggl( a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl(\ln \frac{ 1}{s} \biggr)^{\alpha -1}\frac{y(s)}{s} \,ds - a_{0} \biggr) \ln t. \end{aligned}$$
Therefore, we have
$$\begin{aligned} \bigl\vert v(t) \bigr\vert \leq{}& \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha - 1} \frac{ \vert y(s) \vert }{s} \,ds + \vert a_{0} \vert \\ & {} + \biggl( \vert a_{1} \vert + \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha - 1}\frac{ \vert y(s) \vert }{s} \,ds + \vert a_{0} \vert \biggr) \ln t \\ \leq{} & \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{ \alpha -1} \frac{ \Vert p \Vert \varphi (r)}{s} \,ds \\ & {} + M_{0} + \biggl( M_{1} + \frac{1}{\varGamma (\alpha )} \int _{1} ^{e} \biggl(\ln \frac{ 1}{s} \biggr)^{ \alpha -1}\frac{ \Vert p \Vert \varphi (r)}{s} \,ds + M_{0} \biggr) \\ \leq{} &\frac{2 \Vert p \Vert \varphi (r)}{\varGamma (\alpha +1)}+2M_{0}+M_{1}. \end{aligned}$$
Thus, \(\|v\|\leq \frac{2\|p\|\varphi (r)}{\varGamma (\alpha +1)}+2M_{0}+M _{1}\). Also, we have
$$\begin{aligned} \bigl\vert v(t_{2}) - v(t_{1}) \bigr\vert \leq {}& \frac{1}{\varGamma (\alpha )} \biggl\vert \int _{1}^{t_{2}} \biggl( \ln \frac{ t_{2}}{s} \biggr)^{\alpha -1} \frac{y(s)}{s} \,ds - \int _{1}^{t_{1}} \biggl( \ln \frac{ t_{1}}{s} \biggr) ^{\alpha -1} \frac{y(s)}{s} \,ds \biggr\vert \\ & {} + \biggl\vert a_{0} - \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl(\ln \frac{ 1}{s} \biggr)^{\alpha -1} \frac{ y(s) }{ s } \,ds - a _{0} \biggr\vert ( \ln t_{2} - \ln t_{1} ) \\ \leq{}& \frac{1}{\varGamma (\alpha )} \int _{1}^{t_{1}} \biggl\vert \biggl( \ln \frac{ t_{2}}{s} \biggr)^{\alpha -1} - \biggl( \ln \frac{ t_{2}}{s} \biggr)^{ \alpha -1} \biggr\vert \frac{ \Vert p \Vert \varphi (r)}{s} \,ds \\ & {} + \frac{1}{ \varGamma (\alpha )} \int _{t_{1}}^{t_{2}} \biggl\vert \biggl( \ln \frac{ t_{2}}{s} \biggr)^{\alpha -1} \biggr\vert \frac{ \Vert p \Vert \varphi (r)}{s} \,ds \\ & {} + \biggl( \vert a_{1} \vert + \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl\vert \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1} \biggr\vert \frac{ \Vert p \Vert \varphi (r)}{s} \,ds + \vert a_{0} \vert \biggr) \\ & {} \times (\ln t_{2}-\ln t_{1}). \end{aligned}$$
As \(t_{1}\to t_{2}\), the right-hand side of the above inequality tends to zero. Thus, \(B(S)\) is a uniformly bounded and equicontinuous set in \(\mathcal{X}\) and B is a compact operator. In the following, we shall show that the operator B is upper semi-continuous on \(\overline{B _{r}(0)}\). To do this, we apply Lemma 7 and show that B has a closed graph. Let \(x_{n}\to x_{0}\) and \(v_{n}\in Bx_{n}\) (\(n\geq 1\)) with \(v_{n}\to v_{0}\). We prove that \(v_{0}\in Bx_{0}\). For each natural number n, choose \(y_{n}\in S_{K, x_{n}}\) such that
$$\begin{aligned} v_{n}(t) ={}& \frac{1}{\varGamma ( \alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha - 1} \frac{y_{n}(s)}{s}\,ds + a_{0} \\ & {} + \biggl( a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{ \alpha -1} \frac{y_{n}(s)}{s}\,ds - a_{0} \biggr) \ln t. \end{aligned}$$
Define the continuous linear operator \(\theta :L^{1}(J,\mathbb{R}) \to \mathcal{X}\) by
$$\begin{aligned} \theta (y) (t) ={}& \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\alpha - 1} \frac{ y(s)}{s} \,ds + a_{0} \\ & {} + \biggl( a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1} \frac{y(s)}{s} \,ds -a _{0} \biggr) \ln t. \end{aligned}$$
By using Lemma 6, \(\theta o S_{K}\) is a closed graph operator. Since \(v_{n}\in \theta ( S_{K, x_{n}})\) for all n and \(x_{n}\to x_{0}\), there exists \(y_{0}\in S_{K, x_{0}}\) such that
$$\begin{aligned} v_{0}(t) ={}& \frac{1}{\varGamma (\alpha )} \int _{1}^{t} \biggl(\ln \frac{ t}{s} \biggr)^{ \alpha - 1} \frac{y_{0}(s)}{s} \,ds + a_{0} \\ & {} + \biggl( a_{1} - \frac{1}{\varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1} \frac{y_{0}(s)}{s}\,ds - a _{0} \biggr) \ln t. \end{aligned}$$
Hence, \(v_{0}\in Bx_{0}\). This implies that B has a closed graph, and as a result, B is upper semi-continuous. Now, we show that the operators A, C are Lipschitz on \(\overline{B_{r}(0)}\). Let \(x, y \in \overline{B_{r}(0)}\). Then we have
$$\begin{aligned} \bigl\vert Ax(t)-Ay(t) \bigr\vert ={}& \bigl\vert g\bigl(t, x(t), I^{\gamma _{1}}x(t), I^{\gamma _{2}}x(t), \ldots , I^{\gamma _{m}}x(t)\bigr) \\ & {} - g\bigl(t, y(t), I^{\gamma _{1}}y(t), I^{\gamma _{2}}y(t), \ldots , I^{\gamma _{m}}y(t)\bigr) \bigr\vert \\ \leq{}& \psi (t) \Biggl[ \bigl\vert x(t) - y(t) \bigr\vert + \sum _{ i=1}^{m} \bigl\vert I^{\gamma _{i}}x(t)- I^{\gamma _{i}}y(t) \bigr\vert \Biggr] \\ \leq {}& \psi (t) \Biggl[ \bigl\vert x(t) - y(t) \bigr\vert \\ & {} + \sum_{i=1}^{m}\frac{1}{ \varGamma (\gamma _{i})} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{\gamma _{i} -1} \frac{ \vert x(s) - y(s) \vert }{s} \,ds \Biggr] \\ \leq{}& \Vert \psi \Vert \Biggl[ \Vert x-y \Vert + \Vert x-y \Vert \sum_{i=1}^{m} \frac{1}{ \varGamma (\gamma _{i} + 1)} \Biggr], \end{aligned}$$
which implies that
$$\Vert Ax-Ay \Vert \leq \underbrace{ \Vert \psi \Vert \Biggl(1 +\sum _{i=1}^{m} \frac{1}{ \varGamma ( \gamma _{i}+1)} \Biggr)}_{q_{1}} \Vert x - y \Vert =q_{1} \Vert x-y \Vert . $$
Also,
$$\begin{aligned} \bigl\vert Cx(t) - Cy(t) \bigr\vert ={}& \bigl\vert f \bigl( t, x(t), I^{\beta _{1}} h_{1}\bigl(t, x(t)\bigr), I^{\beta _{2}} h_{2}\bigl(t, x(t)\bigr), \ldots , I^{\beta _{n}} h_{n} \bigl(t, x(t)\bigr) \bigr) \\ & {} - f \bigl( t, y(t), I^{\beta _{1}} h_{1}\bigl(t, y(t) \bigr), I^{\beta _{2}}h_{2}\bigl(t, y(t)\bigr), \ldots , I^{\beta _{n}} h_{n}\bigl(t, y(t)\bigr) \bigr) \bigr\vert \\ \leq {}& \phi (t) \Biggl[ \bigl\vert x(t) - y(t) \bigr\vert \\ & {} + \sum_{i=1}^{n} \bigl\vert I^{\beta _{i}} h_{i}\bigl(t, x(t)\bigr) - I ^{\beta _{i}} h_{i} \bigl(t, y(t)\bigr) \bigr\vert \Biggr] \\ \leq{} & \phi (t) \Biggl[ \bigl\vert x(t) - y(t) \bigr\vert \\ & {} + \sum_{i=1}^{n} \frac{1}{\varGamma (\beta _{i})} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{ \beta _{i} - 1} \frac{ \vert h_{i}(s, x(s)) - h_{i}(s, y(s)) \vert }{s} \,ds \Biggr] \\ \leq {}&\phi (t) \Biggl[ \bigl\vert x(t) - y(t) \bigr\vert \\ & {} + \sum_{i=1}^{n} \frac{1}{ \varGamma (\beta _{i})} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{ \beta _{i} - 1} \frac{ p_{i}(s) \vert x(s)-y(s) \vert }{s} \,ds \Biggr] \\ \leq{} &\Vert \phi \Vert \Biggl( \Vert x - y \Vert + \Vert x - y \Vert \sum_{i=1}^{n} \frac{ \Vert p_{i} \Vert }{\varGamma ( \beta _{i} +1)} \Biggr) \\ ={}& \underbrace{ \Vert \phi \Vert \Biggl( 1+\sum _{i=1}^{n} \frac{ \Vert p_{i} \Vert }{ \varGamma (\beta _{i}+1)} \Biggr)}_{q_{2}} \Vert x-y \Vert = q_{2} \Vert x-y \Vert . \end{aligned}$$
Since
$$\begin{aligned} M &= \bigl\Vert B\bigl( \overline{B_{r}(0)}\bigr) \bigr\Vert = \sup_{x\in \overline{B_{r}(0)}} \bigl\{ \sup Bx(t), t\in J\bigr\} \leq \frac{2 \Vert p \Vert \varphi (r)}{\varGamma (\alpha +1)}+2M_{0}+M_{1}, \end{aligned}$$
we have
$$\begin{aligned} Mq_{1} + q_{2} \leq{}& \biggl( \frac{2 \Vert p \Vert \varphi (r)}{ \varGamma ( \alpha +1)} + 2M_{0} + M_{1} \biggr) \Vert \psi \Vert \Biggl( 1 + \sum_{i=1}^{m} \frac{1}{ \varGamma (\gamma _{i}+1)} \Biggr) \\ & {} + \Vert \phi \Vert \Biggl( 1 + \sum_{i=1}^{n} \frac{ \Vert p_{i} \Vert }{ \varGamma ( \beta _{i}+1)} \Biggr) < \frac{1}{2}. \end{aligned}$$
Hence, all the conditions of Theorem 9 hold. Let \(u\in \mathcal{X}\), \(\|u\|=r\), \(\mu >1\), and \(\mu u\in AuBu+Cu\). Then we have
$$\begin{aligned} \bigl\vert u(t) \bigr\vert \leq{}& \bigl\vert g \bigl( t, u(t), I^{\gamma _{1}}u(t), I^{\gamma _{2}}u(t), \ldots , I^{\gamma _{m}}u(t) \bigr) \bigr\vert \\ & {} \times \biggl\vert \frac{1}{ \varGamma (\alpha )} \int _{1}^{t} \biggl( \ln \frac{ t}{s} \biggr)^{ \alpha -1}\frac{y(s)}{s} \,ds + a_{0} \\ & {} + \biggl( a_{1} - \frac{1}{ \varGamma (\alpha )} \int _{1}^{e} \biggl( \ln \frac{ 1}{s} \biggr)^{\alpha -1}\frac{y(s)}{s} \,ds - a _{0} \biggr) \ln t \biggr\vert \\ & {} + \bigl\vert f \bigl( t, u(t), I^{\beta _{1}} h_{1} \bigl(t, u(t)\bigr), I ^{\beta _{2}} h_{2}\bigl(t, u(t)\bigr), \ldots , I^{\beta _{n}} h_{n}\bigl(t, u(t)\bigr) \bigr) \bigr\vert \\ \leq{}& \bigl[ \bigl\vert g \bigl( t, u(t), I^{\gamma _{1}}u(t), I^{\gamma _{2}}u(t), \ldots , I^{\gamma _{m}}u(t) \bigr) - g (t,\underbrace{0, 0, \ldots , 0}_{m+1} ) \bigr\vert \\ & {} + \bigl\vert g ( t, \underbrace{0, 0, \ldots , 0}_{m+1} ) \bigr\vert \bigr] \biggl( \frac{ 2 \Vert p \Vert \varphi ( \Vert u \Vert )}{ \varGamma ( \alpha + 1)} + 2M_{0} + M _{1} \biggr) \\ & {} + \bigl\vert f \bigl( t, u(t), I^{\beta _{1}} h_{1} \bigl(t, u(t)\bigr), I ^{\beta _{2}} h_{2} \bigl(t, u(t)\bigr), \ldots , I^{\beta _{n}} h_{n}\bigl(t, u(t)\bigr) \bigr) \\ & {} - f ( t, \underbrace{0, 0, \ldots , 0}_{n+1}) \bigr\vert + \bigl\vert f (t,\underbrace{0, 0, \ldots , 0}_{n+1} ) \bigr\vert \\ \leq{}& \Biggl[ \psi (t) \Biggl( \bigl\vert u(t) \bigr\vert + \sum _{i=1}^{m} \bigl\vert I^{ \gamma _{i}}u(t) \bigr\vert \Biggr) + G_{0} \Biggr] \biggl[ \frac{ 2 \Vert p \Vert \varphi ( \Vert u \Vert )}{ \varGamma (\alpha +1)} + 2M_{0} + M_{1} \biggr] \\ & {} + \phi (t) \Biggl( \bigl\vert u(t) \bigr\vert + \sum _{i=1}^{n} \bigl\vert I^{\beta _{i}} h_{i}\bigl(t, u(t) \bigr) \bigr\vert \Biggr) + F_{0} \\ \leq {}& \Biggl[ \Vert \psi \Vert \Vert u \Vert \Biggl( 1 + \sum _{i=1}^{m} \frac{1}{ \varGamma (\gamma _{i}+1)} \Biggr) + G_{0} \Biggr] \biggl[ \frac{2 \Vert p \Vert \varphi ( \Vert u \Vert )}{ \varGamma (\alpha +1)} + 2M_{0}+M_{1} \biggr] \\ & {} + \Vert \phi \Vert \Biggl( \Vert u \Vert + \sum _{i=1}^{n} \frac{ \Vert p_{i} \Vert \Vert u \Vert + H_{i,0}}{ \varGamma (\beta _{i}+1)} \Biggr) + F_{0} \end{aligned}$$
for some \(y\in S_{K,u}\). Therefore
$$\begin{aligned} \Vert u \Vert \leq {}&\Biggl[ \Vert \psi \Vert \Vert u \Vert \Biggl( 1 + \sum_{i=1}^{m}\frac{1}{ \varGamma (\gamma _{i} + 1)} \Biggr) + G_{0} \Biggr] \biggl[ \frac{2 \Vert p \Vert \varphi ( \Vert u \Vert )}{\varGamma (\alpha +1) } + 2M_{0}+M_{1} \biggr] \\ & {} + \Vert \phi \Vert \Biggl( \Vert u \Vert + \sum _{i=1}^{n} \frac{ \Vert p_{i} \Vert \Vert u \Vert + H_{i,0}}{ \varGamma (\beta _{i}+1)} \Biggr) + F_{0}. \end{aligned}$$
As a result, we have
$$\Vert u \Vert \leq \frac{G_{0} [\frac{2 \Vert p \Vert \varphi ( \Vert u \Vert )}{\varGamma (\alpha +1)}+2M_{0}+M_{1} ]+ \Vert \phi \Vert \sum_{i=1}^{n}\frac{H_{i,0}}{\varGamma ( \beta _{i}+1)}+F_{0}}{1- \Vert \psi \Vert (1+\sum_{i=1}^{m}\frac{1}{\varGamma (\gamma _{i}+1)} ) ( \frac{2 \Vert p \Vert \varphi ( \Vert u \Vert )}{\varGamma (\alpha +1)}+2M_{0}+M_{1} )- \Vert \phi \Vert (1+\sum_{i=1}^{n}\frac{ \Vert p_{i} \Vert }{ \varGamma (\beta _{i}+1)} )}. $$
Since \(\|u\|=r\), we have
$$r \leq \frac{ G_{0} [\frac{2 \Vert p \Vert \varphi (r)}{ \varGamma ( \alpha + 1)} + 2M_{0} + M_{1} ] + \Vert \phi \Vert \sum_{i=1}^{n}\frac{H_{i,0}}{ \varGamma (\beta _{i} + 1)} + F_{0}}{1- \Vert \psi \Vert (1 + \sum_{i=1}^{m}\frac{1}{ \varGamma (\gamma _{i}+1)} ) ( \frac{2 \Vert p \Vert \varphi (r)}{ \varGamma ( \alpha +1)} + 2M_{0}+M_{1} )- \Vert \phi \Vert (1+\sum_{i=1}^{n}\frac{ \Vert p_{i} \Vert }{\varGamma (\beta _{i}+1)} )} < r. $$
This is an apparent contradiction, and conclusion (ii) of Theorem 9 does not hold. Hence, there exists \(x\in \overline{B_{r}(0)}\) such that \(x\in Ax Bx + Cx\), and it is a solution of hybrid fractional differential inclusion (1). □
In the following, we review the existence of solution for the hybrid fractional differential equation
$$ \textstyle\begin{cases} {}^{C}_{H} D^{\alpha } [ \frac{x(t)-f(t, x(t), I^{\beta _{1}} h _{1}(t, x(t)), I^{\beta _{2}}h_{2}(t, x(t)), \ldots , I^{\beta _{n}}h _{n}(t, x(t)))}{g(t, x(t), I^{\gamma _{1}} x(t), I^{\gamma _{2}}x(t), \ldots , I^{\gamma _{m}}x(t))} ] =k(t, x(t)), \\ x(1)=\mu (x), \qquad x(e)=\eta (x), \end{cases} $$
(5)
where \({}^{C}_{H} D^{\alpha }\) and \(I^{\alpha }\) denote the Caputo–Hadamard fractional derivative and Hadamard integral of order α, respectively, \(t \in J=[1,e]\), \(n, m \in \mathbb{N}\), \(1<\alpha \leq 2\), \(\beta _{i}>0\)
\((i=1,2, \ldots , n)\), \(\gamma _{i}>0\)
\((i=1,2, \ldots , m)\), the functions \(f: J\times \mathbb{R}^{n+1} \to \mathbb{R}\), \(g:J\times \mathbb{R}^{m+1} \to \mathbb{R}-\{0\}\), \(h_{i}: J\times \mathbb{R} \to \mathbb{R}\)\((i=1,2, \ldots , n)\), \(k: J \times \mathbb{R} \to \mathbb{R}\), and \(\mu , \eta : C(J, \mathbb{R}) \to \mathbb{R}\) satisfy certain conditions. It is clear that equation (5) is a special case of inclusion (1), where \(K (t, x(t)) = \{k(t, x(t))\}\). With argument similar to the proof of Theorem 11, and applying Theorem 8, we can prove the following theorem.
Theorem 12
Suppose that
-
\((H'_{1})\)
:
-
The function
\(k: J\times \mathbb{R}\to \mathbb{R}\)
is a continuous map such that
\(|k(t, x)|\leq p(t) \varphi (|x|)\)
for all
\((t, x)\in J \times \mathbb{R}\), where
\(p\in C(J, \mathbb{R^{+}})\)
and
\(\varphi : [0, \infty )\to (0, \infty )\)
is a continuous and increasing function;
-
\((H'_{2})\)
:
-
If
\(\frac{(x_{1}, x_{2}, \ldots , x_{m+1})}{g(t, x _{1}, x_{2}, \ldots , x_{m+1} )} = \frac{(y_{1}, y_{2}, \ldots , y _{m+1})}{g(t, y_{1}, y_{2}, \ldots , y_{m+1} )}\), then
$$(x_{1}, x_{2}, \ldots , x_{m+1})=(y_{1}, y_{2}, \ldots , y_{m+1}) $$
for all
\(t\in J\)
and
\((x_{1}, x_{2}, \ldots , x_{m+1}) \in \mathbb{R} ^{m+1}\).
Also, assume that conditions
\((H_{2})\), \((H_{3})\), \((H_{4})\), and
\((H_{5})\)
in Theorem
11
hold. If
$$\begin{aligned} &\biggl( \frac{2 \Vert p \Vert \varphi (r)}{ \varGamma (\alpha +1)}+2M_{0}+M_{1} \biggr) \Vert \psi \Vert \Biggl(1+\sum_{i=1}^{m} \frac{1}{\varGamma (\gamma _{i}+1)} \Biggr) \\ & \quad {}+ \Vert \phi \Vert \Biggl( 1 + \sum_{i=1}^{n} \frac{ \Vert p_{i} \Vert }{ \varGamma (\beta _{i}+1)} \Biggr) < \frac{1}{2}, \end{aligned}$$
where
r
is the constant that was introduced in Theorem
11, then the fractional differential equation (5) has at least one solution on
J.
Example 1
Consider the following hybrid Caputo–Hadamard fractional differential inclusion:
$$ \textstyle\begin{cases} {}^{C}_{H} D^{\frac{3}{2}} [ \frac{x(t)- 3 - \frac{t^{2}}{ 100(1+t ^{2})} ( \sin x(t)+\cos (I^{\frac{1}{3}}(\sin t \tan ^{-1} x(t)) )+\frac{ \vert I^{\frac{1}{2}}(\frac{t^{2} \vert x(t) \vert }{5(1+ \vert x(t) \vert )}) \vert }{1+ \vert I ^{\frac{1}{2}}(\frac{t^{2} \vert x(t) \vert }{ 5 ( 1+ \vert x(t) \vert )}) \vert } )}{ 1 + \frac{e ^{-\pi t}}{ 1000 (1+e^{ - \pi t})} ( \frac{ \vert x(t) \vert + \vert I^{ \frac{2}{3}} x(t) \vert + \vert I^{\frac{3}{4}} x(t) \vert }{ 1 + \vert x(t) \vert + \vert I^{ \frac{2}{3}}x(t) \vert + \vert I^{\frac{3}{4}}x(t) \vert } )} ] \in [0, e^{t} \sin t +2], \\ x(1) =\sin (x(\frac{3}{2})), \qquad x(e)=\cos (x(\frac{5}{2})). \end{cases} $$
(6)
Here \(\alpha =\frac{3}{2}\), \(m=n=2\), \(\beta _{1}=\frac{1}{3}\), \(\beta _{2}=\frac{1}{2}\), \(\gamma _{1}=\frac{2}{3}\), \(\gamma _{2}= \frac{3}{4}\),
$$\begin{aligned} &g(t, x_{1}, x_{2}, x_{3})= 1 + \frac{( \vert x_{1} \vert + \vert x_{2} \vert + \vert x_{3} \vert ) e ^{-\pi t}}{ 1000 ( 1 + ( \vert x_{1} \vert + \vert x_{2} \vert + \vert x_{3} \vert )(1+e^{-\pi t}) ) }, \\ &f(t, x_{1}, x_{2}, x_{3})=3 + \frac{t^{2}}{100 (1 +t^{2})} \biggl(\sin x _{1}+\cos x_{2}+ \frac{ \vert x_{3} \vert }{1+ \vert x_{3} \vert } \biggr), \end{aligned}$$
\(K(t,x)=[0, e^{t}\sin x +2]\), \(h_{1}(t, x)=\sin t \tan ^{-1}x\), \(h_{2}(t, x) = \frac{t^{2}|x|}{ 5(1+|x|)}\), \(\mu (y) =\sin (y( \frac{3}{2}))\), and \(\eta (y)=\cos (y(\frac{5}{2}))\) for all x, \(x_{1}\), \(x_{2}\), and \(x_{3}\in \mathbb{R}\), \(t\in J\) and \(y \in C(J, \mathbb{R})\). Obviously,
$$\begin{aligned} & \bigl\vert g (t, x_{1}, x_{2}, x_{3} )- g \bigl( t, x'_{1}, x'_{2}, x'_{3} \bigr) \bigr\vert \leq \frac{e^{-\pi t}}{ 1000 ( 1 + e ^{-\pi t})}\sum _{i=1}^{3} \bigl\vert x_{i}-x'_{i} \bigr\vert , \\ & \bigl\vert f ( t, x_{1}, x_{2}, x_{3} ) - f \bigl( t, x'_{1}, x'_{2}, x'_{3} \bigr) \bigr\vert \leq \frac{t^{2}}{ 100(1+t ^{2} )}\sum _{i=1}^{3} \bigl\vert x_{i}-x'_{i} \bigr\vert , \end{aligned}$$
\(|h_{1}(t, x_{1})-h_{1}(t, x'_{1})| \leq \sin t|x_{1}-x'_{1}|\), \(|h_{2}(t, x_{1})-h_{2}(t, x'_{1})| \leq \frac{t^{2}}{5}|x_{1}-x'_{1}|\),
$$\begin{aligned}& \biggl\vert \frac{\eta (y) - f(t, y(t), I^{\beta _{1}}h_{1}(t, y(t)), I^{ \beta _{2}}h_{2}(t, y(t))) |_{t=e}}{ g(t, x(t), I^{ \gamma _{1}}x(t), I ^{\gamma _{2}}x(t)|_{t=e}} \biggr\vert < 4.03, \\& \biggl\vert \frac{\mu (y)-f(1, \mu (y),0,0)}{g(1, \mu (y),0, 0)} \biggr\vert < 4.03, \end{aligned}$$
\(\|K(t, x)\|\leq e^{t} + 2\), \(F_{0}\leq 3.01\), \(G_{0}=1\), and \(H_{1,0}= H_{2,0}=0\) for all \(x_{1}\), \(x_{2}\), \(x_{3}\), \(x'_{1}\), \(x'_{2}\), \(x'_{3}\), and \(x \in \mathbb{R}\), \(t \in J\), and \(y\in C(J, \mathbb{R})\). Now, by setting \(p(t)=e^{t}+2\), \(\varphi (x)=1\), \(\psi (t) = \frac{e^{-\pi t}}{1000(1 +e^{-\pi t})}\), \(\phi (t) = \frac{t^{2}}{100(1 + t^{2})}\), \(p_{1}(t)=\sin t\), \(p_{2}(t)=\frac{t^{2}}{5}\), \(M_{0}=4.03\), and \(M_{1}= 4.03\) for all \(t\in J\) and \(x\in \mathbb{R}^{+}\), we have \(\|p\|= e^{e}+2\), \(\|\phi \|\leq \frac{1}{100}\), \(\|\psi \|\leq \frac{1}{1000}\), \(\|p_{1}\|=1\), and \(\|p_{2}\|=\frac{e^{2}}{5}\). Hence
$$\begin{aligned} &\biggl( \frac{2 \Vert p \Vert \varphi (r)}{ \varGamma (\alpha + 1)} + 2M_{0} + M _{1} \biggr) \Vert \psi \Vert \Biggl( 1 + \sum_{i=1}^{m} \frac{1}{ \varGamma ( \gamma _{i}+1)} \Biggr) \\ & \quad {}+ \Vert \phi \Vert \Biggl( 1 +\sum_{i=1}^{n} \frac{ \Vert p_{i} \Vert }{ \varGamma (\beta _{i}+1)} \Biggr) \leq 0.14606 < \frac{1}{2}. \end{aligned}$$
Therefore, all the conditions in Theorem 11 are satisfied, and problem (6) has at least one solution on \(\overline{B_{r}(0)}\), where
$$ r >\frac{G_{0} [\frac{2 \Vert p \Vert \varphi (r)}{\varGamma (\alpha +1)}+2M _{0}+M_{1} ]+ \Vert \phi \Vert \sum_{i=1}^{n} \frac{H_{i,0}}{ \varGamma (\beta _{i}+1)} + F_{0}}{1 - \Vert \psi \Vert (1+\sum_{i=1}^{m} \frac{1}{ \varGamma (\gamma _{i} + 1)} ) ( \frac{2 \Vert p \Vert \varphi (r)}{ \varGamma ( \alpha + 1)} + 2M_{0} + M_{1} ) - \Vert \phi \Vert ( 1 +\sum_{i=1}^{n}\frac{ \Vert p_{i} \Vert }{ \varGamma (\beta _{i} + 1)} )} \geq 47.68. $$