In this section, our aim is to give existence and uniqueness theorems of solutions for a general form of the interval-valued delay fractional integral equation by using some recent results of fixed point of weakly contractive mappings on partially ordered sets, and in the next section, we use these results in order to investigate the existence and uniqueness results of solutions for an interval-valued delay fractional differential equation. For a positive number σ, we denote by \(C_{\sigma}\) the space \(C([-\sigma, 0], K_{C}(\mathbb {R}))\) equipped with the metric defined by
$${H_{\sigma}} [ {X,Y} ] = \mathop{\sup} _{t \in [ { - \sigma,0} ]} {H} \bigl[ {X(t),Y(t)} \bigr]. $$
Define \(I = [a, b], J=[a - \sigma, a] \cup I = [a - \sigma, b]\). Then, for each \(t \in I\), we denote by \(X_{t}\) the element of \(C_{\sigma}\) defined by \(X_{t}(s)=X(t+s)\), \(s \in[-\sigma,0]\).
Interval-valued delay fractional integral equation: Consider the following interval-valued delay fractional integral equation:
$$\begin{aligned} \textstyle\begin{cases} X(t) \ominus_{g}X(a)= \frac{1}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1} F(s,{X(s),X_{s}})\,ds},& t \in[a,b],\\ X(t) = \varphi(t-a), & t \in[a-\sigma, a], \end{cases}\displaystyle \end{aligned}$$
(3.1)
where \(F: [a,b] \times K_{C}(\mathbb {R}) \times C_{\sigma} \to K_{C}(\mathbb {R}), \varphi(t-a) \in K_{C}(\mathbb {R})\). We say that a continuous interval function \(X: [a,b] \to K_{C}(\mathbb {R})\) is a solution to the interval fractional integral equation (3.1) if it satisfies equation (3.1). Let us suppose that \(X \in C([a,b],K_{C}(\mathbb {R}))\) is w-monotone on \([a,b]\) and satisfies (3.1). As X is w-monotone on \([a,b]\), then it follows that \(X(t)\ominus_{g} X(a)\) is w-increasing on \([a,b]\). Hence, from (3.1) it follows that the right-hand side of (3.1) must be w-increasing on \([a,b]\) (see [38]). Furthermore, we observe that if a continuous interval function X is a unique w-monotone solution of (3.1) on \([a,b]\), then the function \(Y(t):=X(t) \ominus_{g}X(a)\) is w-increasing on \([a,b]\). In addition, the function Y may create two solutions of (3.1): a unique w-increasing solution of (3.1) and a unique w-decreasing solution of (3.1) on \([a,b]\).
Remark 3.1
If \(X \in C([a,b], K_{C}(\mathbb {R}))\) is such that \(w(X(t))\geq w(X(a))\) for all \(t\in{}[ a,b]\), then (3.1) can be rewritten as
$$ \textstyle\begin{cases} X(t) = \varphi(0) + \frac{1}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1} F(s,{X(s),X_{s}}) \,ds},& t \in[a,b], \\ X(t) = \varphi(t-a), &t \in[a-\sigma, a]. \end{cases} $$
If \(X \in C([a,b], K_{C}(\mathbb {R}))\) is such that \(w(X(t))\le w(X(a))\) for all \(t\in{}[ a,b]\), then (3.1) can be rewritten as
$$ \textstyle\begin{cases} X(t) = \varphi(0) \ominus \frac{(-1)}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1} F(s,{X(s),X_{s}}) \,ds},& t \in[a,b], \\ X(t) = \varphi(t-a),& t \in[a-\sigma, a]. \end{cases} $$
Definition 3.1
A w-monotone interval function \(X^{L} \in C([a,b], K_{C}(\mathbb {R}))\) is a lower solution for (3.1) if
$$\begin{aligned} \textstyle\begin{cases} X^{L}(t) \ominus_{g}X^{L}(a)= \frac{1}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1} F(s,{X^{L}(s),X^{L}_{s}})\,ds},& t \in[a,b], \\ X^{L}(t) = \varphi^{L}(t-a) \preceq\varphi(t-a), & t \in[a-\sigma, a]. \end{cases}\displaystyle \end{aligned}$$
(3.2)
A w-monotone interval-valued function \(X^{U}\in C([a,b],K_{C}(\mathbb {R}))\) is an upper solution for (3.1) if it satisfies the reverse inequalities of (3.2).
In the following, for given \(k>0\), we consider the set \(\mathbb {B}_{k}\) of all continuous interval functions \(X \in C([a-\sigma, b], K_{C}(\mathbb {R}))\) such that \(\sup_{t \in[a-\sigma, b]}\{H[X(t),\mathbf{0}] \exp (-k(t+\sigma)) \}< \infty\). On \(\mathbb {B}_{k}\) we can define the following metric:
$$\begin{aligned} & H_{k}[X,Y]=\sup_{t \in[a-\sigma, b]} \bigl\{ H \bigl[X(t),Y(t) \bigr] \exp \bigl(-k(t + \sigma) \bigr) \bigr\} , \\ &\quad X, Y \in C \bigl([a- \sigma, b], K_{C}(\mathbb {R}) \bigr), \end{aligned}$$
(3.3)
where \(k>0\) is large enough such that \(k> 2^{1/\alpha}\). It is well known that \((C([a, b], K_{C}(\mathbb {R})), H_{k})\) is a complete metric space.
Theorem 3.1
Let
\(F \in C([a,b] \times K_{C}(\mathbb {R}) \times C_{\sigma}, K_{C}(\mathbb {R}))\)
and suppose that
\(F(t,A,B)\)
is nondecreasing in
A
and
B
for each
\(t \in[a,b]\), that is, if
\(A \succeq C\)
and
\(B \succeq D\), then
\(F(t,A,B)\succeq F(t,C,D)\). Moreover, assume that the following conditions are satisfied:
-
(A1)
there exists a
w-monotone lower solution
\(X^{L} \in C([a,b], K_{C}(\mathbb {R}))\)
for problem (3.1);
-
(A2)
\(F (t,A,B)\)
is weakly contractive for comparable elements, that is, for some altering distance functions
\(\mathbb {T}_{1}\)
and
\(\mathbb {T}_{2}\), it holds
$$\begin{aligned} &\mathbb {T}_{1} \bigl( H \bigl[ F (t,A,B), F (t,C,D) \bigr] \bigr)\\ &\quad \le \bigl[ \mathbb {T}_{1} \bigl(H[ A, B] \bigr) - \mathbb {T}_{2} \bigl(H[ A, B] \bigr) \bigr] + \bigl[ \mathbb {T}_{1} \bigl(H_{\sigma}[ A, B] \bigr) - \mathbb {T}_{2} \bigl(H_{\sigma}[ A, B] \bigr) \bigr], \end{aligned}$$
if
\(A \succeq C, B \succeq D\)
and
\(t \in[a,b]\). Then there exists a unique
w-monotone solution
X
for problem (3.1) in some intervals
\([a, \mathbb {T}]\), with
\(\mathbb {T} \le b\).
Proof
Let \(\mathcal {X}(t):= X(t) \ominus_{g} X(a), t \in[a-\sigma ,b]\). We define the operator \(\mathbb {Q}: C([a-\sigma,b], K_{C}(\mathbb {R})) \to C([a-\sigma,b], K_{C}(\mathbb {R}))\) by
$$\begin{aligned} (\mathbb {Q} \mathcal {X}) (t) = \textstyle\begin{cases} \varphi(t-a) \ominus_{g} \varphi(0), &t \in[a -\sigma, a], \\ \frac{1}{\Gamma(\alpha)} \int_{a}^{t}{(t-s)^{\alpha-1}F (s, X(s),X_{s} ) \,ds},& t \in[a, b].\end{cases}\displaystyle \end{aligned}$$
We check that the conditions in Theorem 2.1 are satisfied. Indeed, let \(X\succeq Y\) and \(X_{t} \succeq Y_{t}\) on \([a,b]\), then we have \((\mathbb {Q} \mathcal {X}) (a) = (\mathbb {Q} \mathcal {Y}) (a), t \in[a-\sigma,a]\), and for \(t \in[a,b]\)
$$\begin{aligned} F(t,X,X_{t}) \succeq F(t,Y,Y_{t}). \end{aligned}$$
From the result of Remark 2.1-(i), we obtain
$$\begin{aligned} \mathbb {Q} \mathcal {X} (t) &= \frac{1}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1} F \bigl(s,{X(s)},X_{s} \bigr)\,ds} \succeq\frac{1}{\Gamma (\alpha)} \int_{a}^{t}{(t-s)^{\alpha-1} F \bigl(s,{Y(s)},Y_{s} \bigr) \,ds}\\ &= (\mathbb {Q} \mathcal {Y}) (t). \end{aligned}$$
Then \(\mathbb {Q} \mathcal {X} \succeq\mathbb {Q} \mathcal {Y}\) whenever \(X\succeq Y, X_{t} \succeq Y_{t}\) on \([a,b]\), and consequently, the operator \(\mathbb {Q}\) is nondecreasing. Now, condition (A2) shows that
$$\begin{aligned} H \bigl[F \bigl(t,X(t),X_{t} \bigr), F \bigl(t,Y(t),Y_{t} \bigr) \bigr] \le H \bigl[X(t),Y(t) \bigr] + H_{\sigma}[X_{t},Y_{t}] \end{aligned}$$
(3.4)
for all \(X \succeq Y, X_{t} \succeq Y_{t}\) for \(t \in[a,b]\). Indeed, from (A2) we get
$$\begin{aligned} \mathbb {T}_{1} \bigl( H \bigl[F \bigl(t,X(t),X_{t} \bigr), F \bigl(t,Y(t),Y_{t} \bigr) \bigr] \bigr) \le\mathbb {T}_{1} \bigl(H \bigl[ X(t), Y(t) \bigr] \bigr) + \mathbb {T}_{1} \bigl(H_{\sigma}[ X_{t}, Y_{t}] \bigr) \end{aligned}$$
(3.5)
for all \(X \succeq Y, X_{t} \succeq Y_{t}\). If inequality (3.4) is not true, then for all \(X \succeq Y, X_{t} \succeq Y_{t}\) we have
$$\begin{aligned} H \bigl[X(t),Y(t) \bigr] + H_{\sigma}[X_{t},Y_{t}] < \max H \bigl[F \bigl(t,X(t),X_{t} \bigr), F \bigl(t,Y(t),Y_{t} \bigr) \bigr]. \end{aligned}$$
Then, since \(\mathbb {T}_{1}\) is nondecreasing, for all \(X \succeq Y, X_{t} \succeq Y_{t}\) it holds
$$\begin{aligned} \mathbb {T}_{1} \bigl(H \bigl[X(t),Y(t) \bigr] \bigr) +\mathbb {T}_{1} \bigl(H_{\sigma}[X_{t},Y_{t}] \bigr) \le \mathbb {T}_{1} \bigl( H \bigl[F \bigl(t,X(t),X_{t} \bigr), F \bigl(t,Y(t),Y_{t} \bigr) \bigr] \bigr). \end{aligned}$$
Therefore, from (3.5),
$$ \mathbb {T}_{1} \bigl(H \bigl[X(t),Y(t) \bigr] \bigr) +\mathbb {T}_{1} \bigl(H_{\sigma}[X_{t},Y_{t}] \bigr) = \mathbb {T}_{1} \bigl( H \bigl[F \bigl(t,X(t),X_{t} \bigr), F \bigl(t,Y(t),Y_{t} \bigr) \bigr] \bigr) $$
for all \(X \succeq Y, X_{t} \succeq Y_{t}\). From (A2) it follows that \(0 \le- ( \mathbb {T}_{2}(H[ X(t), Y(t)]) +\mathbb {T}_{2}(H_{\sigma }[ X_{t}, Y_{t}]) )\), and therefore,
$$\begin{aligned} \mathbb {T}_{2} \bigl(H \bigl[ X(t), Y(t) \bigr] \bigr) = \mathbb {T}_{2} \bigl(H_{\sigma}[ X_{t}, Y_{t}] \bigr) = 0. \end{aligned}$$
As \(\mathbb {T}_{2}\) is an altering distance function, we have that \(H[ X(t), Y(t)] =0, H_{\sigma}[ X_{t}, Y_{t}] =0\) for all \(X \succeq Y, X_{t} \succeq Y_{t}\). This infers a contradiction, that is, \(H[F(t,X(t),X_{t}), F(t,Y(t),Y_{t})] =0\). Thus, inequality (3.4) is true. Next, for \(X \succeq Y, X_{t} \succeq Y_{t}\) and \(t \in[a,b]\), we get
$$\begin{aligned} &H \bigl[ (\mathbb {Q} \mathcal {X}) (t), (\mathbb {Q} \mathcal {Y}) (t) \bigr]\\ &\quad \le \frac {1}{\Gamma(\alpha)} \int_{a}^{t}{(t-s)^{\alpha-1} \Bigl( H \bigl[X(s),Y(s) \bigr] + \sup_{\theta\in[s-\sigma,s]}H \bigl[X(\theta), Y(\theta) \bigr] \Bigr) \,ds}. \end{aligned}$$
By the definition of metric (3.3), it follows that \(H[X(s),Y(s)] \le H_{k}[X,Y] e^{k(s+\sigma)}\) for all \(t \ge a -\sigma\) and \(\mathop{\sup} _{\theta\in[s-\sigma,s]} H[X(\theta), Y(\theta )]\)
\(\le H_{k}[X,Y] e^{k(s+\sigma)}\) for all \(s \ge a\). Then, for all \(t \ge a\), we obtain
$$\begin{aligned} &H \bigl[ (\mathbb {Q} \mathcal {X}) (t), (\mathbb {Q} \mathcal {Y}) (t) \bigr] \\ &\quad\le \frac {2}{\Gamma(\alpha)} \int_{a}^{t}{(t-s)^{\alpha-1} H_{k}[X,Y]e^{k(s+\sigma)} \,ds}, \end{aligned}$$
and so
$$\begin{aligned} H_{k}[ \mathbb {Q} \mathcal {X}, \mathbb {Q} \mathcal {Y}] &\le \frac {2H_{k}[X,Y]}{\Gamma(\alpha)}\mathop{\sup} _{t \in[a,b]} \int _{a}^{t}{(t-s)^{\alpha-1} e^{k(s-t)} \,ds} \\ &\le \frac{2H_{k}[X,Y]}{k^{\alpha}\Gamma(\alpha)}\mathop{\sup} _{t \in[a,b]} \int_{0}^{k(t-a)}{u^{\alpha-1}e^{-u} \,du} \\ &\le \frac{2 H_{k}[X,Y]}{k^{\alpha}}. \end{aligned}$$
Therefore, it holds that
$$\begin{aligned} \mathbb {T}_{1} \bigl(H_{k}[ \mathbb {Q} \mathcal {X}, \mathbb {Q} \mathcal {Y}] \bigr)& \le \mathbb {T}_{1} \biggl( \frac{ 2 H_{k}[X,Y]}{k^{\alpha}} \biggr) \\ & = \mathbb {T}_{1} \bigl(H_{k}[X,Y] \bigr) - \biggl[ \mathbb {T}_{1} \bigl(H_{k}[X,Y] \bigr) - \mathbb {T}_{1} \biggl( \frac{2H_{k}[X,Y]}{k^{\alpha}} \biggr) \biggr]. \end{aligned}$$
Then, if \(\mathbb {T}_{2} (t) = \mathbb {T}_{1}(t) - \mathbb {T}_{1} ( 2t /k^{\alpha } )\), it follows that
$$\begin{aligned} \mathbb {T}_{1} \bigl(H_{k}[ \mathbb {Q} \mathcal {X}, \mathbb {Q} \mathcal {Y}] \bigr) \le \mathbb {T}_{1} \bigl(H_{k}[X,Y] \bigr) - \mathbb {T}_{2} \bigl(H_{k}[X,Y] \bigr) \end{aligned}$$
for all \(X \succeq Y\). Finally, using the existence of the lower solution, we check that \(\mathcal {X}\) is such that \(\mathcal {X}^{L} \preceq \mathbb {Q} \mathcal {X}^{L}\). Indeed, since \(X^{L}(t) =\xi(t-a) \preceq\varphi(t-a)\) for \(t \in[a-\sigma,a]\) and for \(t \in[a,a+p]\),
$$\begin{aligned} X^{L} (t) \ominus_{g} X(a) \preceq\frac{1}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1} F \bigl(s,{X^{L}(s),X_{s}^{L}} \bigr)\,ds}, \end{aligned}$$
it follows that
$$\begin{aligned} \textstyle\begin{cases} \mathcal {X}^{L}(t):= X^{L}(t) \ominus_{g} \varphi(0) \preceq \varphi(t - {a}) \ominus_{g} \varphi(0) = \mathbb {Q} \mathcal {X}^{L}(t), & t \in[ {a} - \sigma, a], \\ \mathcal {X}^{L}(t) \preceq\frac{1}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1}F (s, X^{L}(s),X^{L}_{s} ) \,ds}= \mathbb {Q} \mathcal {X}^{L}(t), & t \in[a, a+p]. \end{cases}\displaystyle \end{aligned}$$
As the operator \(\mathbb {Q}\) satisfies all hypotheses of Theorem 2.1, \(\mathbb {Q}\) has a fixed point in \(C([a-\sigma,b], K_{C}(\mathbb {R}))\). Moreover, since every pair of interval-valued functions in \(C([a-\sigma,b], K_{C}(\mathbb {R}))\) has an upper bound (see Lemma 2.2), the operator \(\mathbb {Q}\) has a unique fixed point \(\mathcal {X}\) and \(\mathcal {X}\) is the unique solution to (3.1). □
Remark 3.2
The conclusion of Theorem 3.1 is still valid if the existence of a w-monotone lower solution for problem (3.1) is replaced by the existence of a w-monotone upper solution for problem (3.1).
Interval-valued delay fractional differential equation: Let us consider again the interval-valued delay fractional differential equation with Caputo generalized Hukuhara fractional differentiability under the form
$$\begin{aligned} \textstyle\begin{cases} ({^{C}} \mathcal{D}_{a^{+}}^{\alpha}X ) (t) = F (t,{X(t)},X_{t} ),& t \in [a,b] \\ X(t) = \varphi(t-a),& t \in[a-\sigma,a]. \end{cases}\displaystyle \end{aligned}$$
(3.6)
Denote by \(C^{1,F}([a,b], K_{C}(\mathbb {R}))\) the space of interval-valued functions which are continuous Caputo gH-fractional differentiable on \([a,b]\). A solution \(X\in C([a-\sigma ,b],K_{C}(\mathbb {R}) ) \cap C^{1,F}([a,b], K_{C}(\mathbb {R})) \) of (3.6) is said to be w-monotone if it is w-increasing or w-decreasing on \([a,b]\).
Lemma 3.1
Let
F
be interval functions such that
\(F \in C([a,b],K_{C}(\mathbb {R}))\)
for any
\(X \in K_{C}(\mathbb {R})\). Then a
w-monotone interval function
\(X\in C([a-\sigma,b],K_{C}(\mathbb {R}))\)
is a solution of initial value problem (3.6) if and only if
X
satisfies the interval fractional integral equation (3.1) and the interval-valued function
\(t\mapsto\Im^{\alpha}_{a^{+}}{\mathbb {F}(t) }\)
is
w-increasing on
\([a,b]\), where
$$\begin{aligned} \mathbb {F}(t): = F \bigl(t,{X(t)},X_{t} \bigr),\quad t \in[a,b]. \end{aligned}$$
(3.7)
Proof
The proof of this lemma is similar to the proof of Lemma 3.1 in [38]. □
Corollary 3.1
If a
w-monotone interval function
X
is a solution of (3.1) such that the function
\(t\mapsto\Im^{\alpha}_{a^{+}}{\mathbb {F}(t) }\)
is
w-increasing on
\([a,b]\), then
X
is a
w-monotone solution of (3.6).
Definition 3.2
A w-monotone interval function \(X^{U} \in X\in C([a-\sigma ,b],K_{C}(\mathbb {R}))\cap C^{1,F}([a,b], K_{C}(\mathbb {R}))\) is an upper solution for (3.6) if
$$\begin{aligned} \begin{aligned}&\bigl({^{C}}\mathcal{D}_{a^{+}}^{\alpha}X^{U} \bigr) (t) \succeq F \bigl(t, X^{U}(t),X_{t}^{U} \bigr), \quad t \in[a,b],\\ & X^{U}(t) = \xi(t-a) \succeq\varphi(t-a), \quad t \in[a- \sigma,a]. \end{aligned} \end{aligned}$$
(3.8)
A w-monotone interval-valued function \(X^{L}\in C([a-\sigma ,b],K_{C}(\mathbb {R}))\cap C^{1,F}([a,b], K_{C}(\mathbb {R}))\) is a lower solution for (3.6) if it satisfies the reverse inequalities of (3.8).
Corollary 3.2
Let
\(F \in C([a,b], K_{C}(\mathbb {R}))\)
and suppose that
\(F(t,A,B)\)
is nondecreasing in
\(A, B\)
for each
\(t \in[a,b]\), that is, if
\(A \succeq C, B \succeq D\), then
\(F(t,A,B)\succeq F(t,C,D)\). Moreover, assume that the following conditions are satisfied:
-
(A3)
there exists a
w-monotone upper solution
\(X^{U} \in C([a-\sigma,b], K_{C}(\mathbb {R}))\cap C^{1,F}([a,b], K_{C}(\mathbb {R}))\)
for problem (3.6);
-
(A4)
for an altering distance function
\(\mathbb {T}_{3}\), it holds
$$\begin{aligned} &H \bigl[ F (t,A,B), F (t,C,D) \bigr] \\ &\quad\le \bigl( H[ A, B] + H_{\sigma}[B,D] \bigr) - \bigl(\mathbb {T}_{3} \bigl(H[ A, B] \bigr)+\mathbb {T}_{3} \bigl(H_{\sigma}[ B, D] \bigr) \bigr) \end{aligned}$$
if
\(A \succeq C, B \succeq D\)
and
\(t \in[a,b]\). Then there exists a unique
w-monotone solution
X
for problem (3.6) in some intervals
\([a, \mathbb {T}]\), with
\(\mathbb {T} \le b\).
Proof
In the same way as the proof of Theorem 3.1, let \(\mathcal {X}: = X(t) \ominus_{g} X(a), t \in[a-\sigma,b]\), and we define the operator \(\mathbb {P}: C([a-\sigma,b], K_{C}(\mathbb {R})) \to C([a-\sigma,b], K_{C}(\mathbb {R})) \) by \((\mathbb {P} \mathcal {X})(t) = \varphi(t-a) \ominus_{g} \varphi(0), t \in[a-\sigma,a]\) and
$$\begin{aligned} (\mathbb {P} \mathcal {X}) (t) = \frac{1}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1} F \bigl(s,{X(s),X_{s}} \bigr)\,ds}, \quad t \in[a, b]. \end{aligned}$$
From the proof of Theorem 3.1 it is easy to see that from hypothesis of the nondecreasing property of F with respect to the second and third variables, the operator \(\mathbb {P}\) is nondecreasing, that is, \(\mathbb {P} \mathcal {X} \succeq\mathbb {P} \mathcal {Y}\) whenever \(X\succeq Y\). On the other hand, hypothesis (A4) is implied from hypothesis (A2) with considering \(\mathbb {T}_{1} (u) = u\). Therefore, we can easily infer that the operator \(\mathbb {P}\) is contractive-like. Finally, hypothesis (A3) infers that \(\mathcal {X}^{U} \succeq\mathbb {P} \mathcal {X}^{U}\). Indeed, since \(\mathcal {X}^{U}\) is an upper solution and \(X^{U}(t) \succeq X(t), t \in[a-\sigma,a]\), from Lemma 3.1 for \(t \in[a,b]\) we get
$$\begin{aligned} X^{U}(t) \ominus_{g} \varphi(0) & \succeq \bigl( \Im^{\alpha}_{a^{+}} {^{C}}\mathcal{D}_{a^{+}}^{\alpha}X \bigr) (t) \\ & \succeq\frac{1}{\Gamma(\alpha)} \int_{a}^{t}{(t-s)^{\alpha -1}F \bigl(s,{X^{U}(s)},X^{U}_{s} \bigr) \,ds} \\ &= \bigl(\mathbb {P} \mathcal {X}^{U} \bigr) (t),\quad t \in[a,b]. \end{aligned}$$
Thus \(\mathcal {X}^{U} \succeq\mathbb {P} \mathcal {X}^{U}\). We see that the operator \(\mathbb {P}\) verifies all the hypotheses of Theorem 2.1. In consequence, \(\mathbb {P}\) has a fixed point in \(C([a-\sigma,b], K_{C}(\mathbb {R}))\). Furthermore, the space \(C([a-\sigma ,b], K_{C}(\mathbb {R}))\) satisfies that every pair of elements of \(C([a-\sigma,b], K_{C}(\mathbb {R}))\) has an upper bound (see Lemma 2.2). It follows that \(\mathbb {P}\) has a unique fixed point. The proof is complete. □
In the following corollary, we analyze the dependence of the solution on the order and the initial condition for problem (3.6).
Theorem 3.2
Let
F
satisfy the assumptions of Corollary
3.2, and let
\(\alpha\in(0,1)\), \(\delta>0\)
such that
\(0 < \alpha- \delta< \alpha <1\). For
\(t \in[a,\mathbb {T}]\), assume that
X
and
Z
are the solutions of the initial value problem (3.6) and
$$\begin{aligned} \begin{aligned} &\bigl({^{C}}\mathcal{D}_{a^{+}}^{\alpha-\delta}Z \bigr) (t) = F \bigl(t,{Z(t)},Z_{t} \bigr), \quad t \in[a,b],\\ & Z(t) = \psi(t-a),\quad t \in[a-\sigma,a], \end{aligned} \end{aligned}$$
(3.9)
respectively. Then the following holds:
$$\begin{aligned} H \bigl[X(t), Z(t) \bigr] \le B(t) + \int_{a}^{t} { \sum_{i = 1}^{\infty}{ \biggl( \frac{2}{\Gamma(\alpha)}\Gamma(\alpha-\delta) \biggr)^{i} \frac {(t-s)^{i(\alpha- \delta) -1}}{\Gamma(i(\alpha- \delta))}} B(s) \,ds}, \end{aligned}$$
where
$$\begin{aligned} B(t): = {}&H \bigl[X(a), Z(a) \bigr] \\ &{}+ \biggl\vert \frac{(t-a)^{\alpha- \delta}}{\alpha- \delta} \biggl( \frac{1}{\Gamma(\alpha- \delta)} - \frac{1}{\Gamma (\alpha)} \biggr) \biggr\vert \sup _{t \in[0,\mathbb {T}] } F \bigl(t,Y(t),Y_{t} \bigr) \\ &{} + \biggl\vert \frac{(t-a)^{\alpha- \delta}}{(\alpha- \delta)\Gamma(\alpha )} - \frac{(t-a)^{\alpha}}{\Gamma(\alpha+1)} \biggr\vert \sup _{t \in [0,\mathbb {T}] } F \bigl(t,X(t),X_{t} \bigr). \end{aligned}$$
Proof
From Lemma 3.1, the solutions of the initial value problems (3.6) and (3.9) are given by
$$ X(t) \ominus_{g}\varphi(0)= \frac{1}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1} F \bigl(s,{X(s)},X_{s} \bigr)\,ds},\quad t \in[a,\mathbb {T}] $$
and
$$ Z(t) \ominus_{g}\psi(0)= \frac{1}{\Gamma(\alpha-\delta)} \int _{a}^{t}{(t-s)^{\alpha- \delta-1} F \bigl(s,{Z(s)},Z_{s} \bigr)\,ds}, \quad t \in [a,\mathbb {T}], $$
respectively. Observe that for \(t \in \mathbb {T}\),
$$\begin{aligned} &H \bigl[X(t), Z(t) \bigr]\\ &\quad \le H \bigl[\varphi(0), \psi(0) \bigr] \\ &\qquad{}+ H \biggl[\frac{1}{\Gamma( \alpha- \delta)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} F \bigl(s, Z(s),Z_{s} \bigr) \,ds},\\ &\qquad \frac{1}{\Gamma( \alpha )} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} F \bigl(s, Z(s),Z_{s} \bigr) \,ds} \biggr] \\ &\qquad{}+H \biggl[ \frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} F \bigl(s, Z(s),Z_{s} \bigr) \,ds}, \frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} F \bigl(s, X(s),X_{s} \bigr) \,ds} \biggr] \\ &\qquad{}+H \biggl[ \frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} F \bigl(s, X(s),X_{s} \bigr) \,ds}, \frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha -1} F \bigl(s, X(s),X_{s} \bigr) \,ds} \biggr] \\ &\quad\le H \bigl[\varphi(0), \psi(0) \bigr] + \biggl\vert \frac{(t-a)^{\alpha- \delta }}{\alpha- \delta} \biggl( \frac{1}{\Gamma(\alpha- \delta)} - \frac {1}{\Gamma(\alpha)} \biggr) \biggr\vert \sup _{t \in[a,\mathbb {T}] } F \bigl(t,Z(t),Z_{t} \bigr) \\ &\qquad{}+\frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} \Bigl(H \bigl[ X(s), Z(s) \bigr] + \sup_{\theta\in[s-\sigma, s] } H \bigl[ X(\theta), Z(\theta) \bigr] \Bigr)\,ds} \\ &\qquad {}+ \biggl\vert \frac{(t-a)^{\alpha- \delta}}{(\alpha- \delta)\Gamma(\alpha )} - \frac{(t-a)^{\alpha}}{\Gamma(\alpha+1)} \biggr\vert \sup _{t \in [a,\mathbb {T}] } F \bigl(t,X(t),X_{t} \bigr). \end{aligned}$$
Putting \(k(s) = \sup_{\theta\in[s-\sigma, s] } H[ X(\theta), Z(\theta )] \) for any \(s \in[a,\mathbb {T}]\), we have, by generalized Gronwall’s inequality (see Theorem 1 in [44]), that
$$\begin{aligned} H \bigl[X(t), Z(t) \bigr] \le B(t) + \int_{a}^{t} { \sum_{i = 1}^{\infty}{ \biggl( \frac{2}{\Gamma(\alpha)}\Gamma(\alpha-\delta) \biggr)^{i} \frac {(t-s)^{i(\alpha- \delta) -1}}{\Gamma(i(\alpha- \delta))}} B(s) \,ds}. \end{aligned}$$
□
Remark 3.3
Under the hypothesis of Corollary 3.2, if \(\delta=0\), then we get the following estimate:
$$\begin{aligned} H \bigl[X(t), Z(t) \bigr] \le H \bigl[\varphi(0), \psi(0) \bigr] \sum _{i = 0}^{\infty}{\frac {2^{i} (t-a)^{i \alpha}}{\Gamma(i\alpha+ i)} }. \end{aligned}$$
In the sequel, we show that the solutions of initial value problem (3.6) depend continuously on the initial condition, the order and the right-hand side of equation.
Theorem 3.3
Let
\(F, G\)
satisfy the assumptions of Corollary
3.2, and let
\(\alpha\in(0,1)\), \(\delta>0\)
such that
\(0 < \alpha- \delta< \alpha<1\). For
\(t \in[a,\mathbb {T}]\), assume that
X
and
Z
are the solutions of initial value problem (3.6) and
$$\begin{aligned} \bigl({^{C}}\mathcal{D}_{a^{+}}^{\alpha-\delta}Z \bigr) (t) = G \bigl(t,{Z(t)},Z_{t} \bigr),\quad t \in[a,b],\qquad Z(t) = \psi(t-a),\quad t \in[a-\sigma,a], \end{aligned}$$
(3.10)
respectively. Assume also that there exists a positive constant
ε
such that
$$H \bigl[ F(t,A,B), G(t,A,B) \bigr] \le\varepsilon,\quad t \in[a,\mathbb {T}]. $$
Then the following holds:
$$\begin{aligned} H \bigl[X(t), Z(t) \bigr] \le C(t) + \int_{a}^{t} { \sum_{i = 1}^{\infty}{ \biggl( \frac{2}{\Gamma(\alpha)}\Gamma(\alpha-\delta) \biggr)^{i} \frac {(t-s)^{i(\alpha- \delta) -1}}{\Gamma(i(\alpha- \delta))}} C(s) \,ds}, \end{aligned}$$
where
$$\begin{aligned} C(t): ={}& H \bigl[\varphi(0), \psi(0) \bigr] + \biggl\vert \frac{(t-a)^{\alpha- \delta }}{\alpha- \delta} \biggl( \frac{1}{\Gamma(\alpha- \delta)} - \frac {1}{\Gamma(\alpha)} \biggr) \biggr\vert \sup _{t \in[0,\mathbb {T}] } F \bigl(t,Y(t),Y_{t} \bigr) \\ & {}+ \biggl\vert \frac{(t-a)^{\alpha- \delta}}{(\alpha- \delta)\Gamma(\alpha )} - \frac{(t-a)^{\alpha}}{\Gamma(\alpha+1)} \biggr\vert \sup _{t \in [0,\mathbb {T}] } F \bigl(t,X(t),X_{t} \bigr) + \frac{\varepsilon(t-a)^{\alpha-\sigma }}{\Gamma(\alpha) (\alpha-\sigma)}. \end{aligned}$$
Proof
Let \(X, Z\) denote the solutions of problems (3.6) and (3.10), respectively. For \(t \in[a,\mathbb {T}]\), we get
$$\begin{aligned} H \bigl[ F(t,X,X_{t}), G(t,Z,Z_{t}) \bigr] &\le H \bigl[ F(t,X,X_{t}), G(t,X,X_{t}) \bigr] + H \bigl[ G(t,X,X_{t}), G(t,Z,Z_{t}) \bigr] \\ & \le\varepsilon+ H[X,Z] + H_{\sigma}[X_{t},Z_{t}]. \end{aligned}$$
Therefore, we obtain the following estimate:
$$\begin{aligned} &H \bigl[X(t), Z(t) \bigr] \\ &\quad \le H \bigl[\varphi(0), \psi(0) \bigr] \\ &\qquad{}+ H \biggl[\frac{1}{\Gamma( \alpha- \delta)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} G \bigl(s, Z(s),Z_{s} \bigr) \,ds}, \\ &\qquad\frac{1}{\Gamma( \alpha )} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} G \bigl(s, Z(s),Z_{s} \bigr) \,ds} \biggr] \\ &\qquad{}+H \biggl[ \frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} G \bigl(s, Z(s),Z_{s} \bigr) \,ds}, \frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} F \bigl(s, X(s),X_{s} \bigr) \,ds} \biggr] \\ &\qquad{}+H \biggl[ \frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} F \bigl(s, X(s),X_{s} \bigr) \,ds}, \frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha -1} F \bigl(s, X(s),X_{s} \bigr) \,ds} \biggr] \\ &\quad \le C(t) +\frac{1}{\Gamma( \alpha)} \int_{a}^{t} {(t-s)^{\alpha- \delta-1} \Bigl(H \bigl[ X(s), Z(s) \bigr] + \sup_{\theta\in[s-\sigma, s] } H \bigl[ X(\theta), Z(\theta) \bigr] \Bigr)\,ds}. \end{aligned}$$
Putting \(k(s) = \sup_{\theta\in[s-\sigma, s] } H[ X(\theta), Z(\theta )] \) for any \(s \in[a,\mathbb {T}]\) we obtain, by generalized Gronwall’s inequality (see Theorem 1 in [44]), that
$$\begin{aligned} H \bigl[X(t), Z(t) \bigr] \le C(t) + \int_{a}^{t} { \sum_{i = 1}^{\infty}{ \biggl( \frac{2}{\Gamma(\alpha)}\Gamma(\alpha-\delta) \biggr)^{i} \frac {(t-s)^{i(\alpha- \delta) -1}}{\Gamma(i(\alpha- \delta))}} C(s) \,ds}. \end{aligned}$$
□
The following corollary shows a new technique to find the exact solutions of interval-valued delay fractional differential equation by using the solutions of interval-valued delay integer order differential equation.
Corollary 3.3
Assume that the conditions of Corollary
3.2
hold. Then a solution of (3.6), \(X_{FO}\), is given by
$$\begin{aligned} \textstyle\begin{cases} X_{FO}(t)= X_{IO} ( \frac{(t-a)^{\alpha}}{\Gamma(\alpha+1)} ),& t \in[a,b]\\ X_{FO}(t) = \varphi(t-a),& t \in[a-\sigma,a], \end{cases}\displaystyle \end{aligned}$$
where
\(X_{IO}( v )\)
is a solution of IVP of the interval-valued delay integer order differential equation
$$\begin{aligned} \textstyle\begin{cases} X'_{IO}(v) = F^{*}(v, X_{IO}(v),X_{IO,v}),& v \in[0, (b-a)^{\alpha }/\Gamma(\alpha+1)] \\ X_{IO}(v) = \varphi (t-a - [(t-a)^{\alpha} -v \Gamma(\alpha +1) ]^{1/\alpha} ), & v \in[-\sigma,0], \end{cases}\displaystyle \end{aligned}$$
(3.11)
where
\(F^{*}(v, X_{IO}(v),X_{IO,v}) = F( k(t,v),X_{FO}(k(t,v)), X_{FO,k(t,v)})\), and
\(k(t,v): =t - ( [t-a]^{\alpha}-v \Gamma(\alpha +1))^{1/\alpha}\).
Proof
From Corollary 3.2, we infer that the solution of problem (3.6), \(X_{FO}\), exists and is given by
$$\begin{aligned} \textstyle\begin{cases} X_{FO}(t) \ominus_{g} X_{FO}(a) = \frac{1}{\Gamma(\alpha)} \int _{a}^{t}{(t-s)^{\alpha-1} F(s, X_{FO}(s),X_{FO,s})\,ds},& t \in[a,b] \\ X_{FO}(t) = \varphi(t-a ), &t \in[a-\sigma,a]. \end{cases}\displaystyle \end{aligned}$$
(3.12)
Let \(s = t - [ (t-a)^{\alpha} - v \Gamma(\alpha+1)]^{1/\alpha}, t \in [a,b]\). Then the interval delay fractional integral equation (3.12) can be written as
$$\begin{aligned} X_{FO}(t) \ominus_{g} X_{FO}(a) &= \int_{{0}}^{(t-a)^{\alpha}/\Gamma (\alpha+1)} { F \bigl( k(s,v),X_{FO} \bigl(k(s,v) \bigr), X_{FO,k(s,v)} \bigr)\,dv } \\ & = \int_{{0}}^{(t-a)^{\alpha}/\Gamma(\alpha+1)} { F^{*} \bigl(v, X_{IO}(v),X_{IO,v} \bigr) \,dv }. \end{aligned}$$
(3.13)
On the other hand, from the interval differential equation (3.11) we obtain
$$\begin{aligned} X_{IO}(v) \ominus_{g} X_{IO}(0) = \int_{0}^{v} { F^{*} \bigl(v, X_{IO}(v),X_{IO,v} \bigr) \,dv}, \end{aligned}$$
(3.14)
where \(v \in[0, (b-a)^{\alpha}/\Gamma(\alpha+1)]\). From (3.13), (3.14), \(X_{FO}(a) = X_{IO}(0)=\varphi(0)\) and as \(0 \le(t-a)^{\alpha}/\Gamma(\alpha+1) \le (b-a)^{\alpha }/\Gamma(\alpha+1)\), we get
$$\begin{aligned} X_{FO}(t) \ominus_{g} \varphi(0) &= X_{IO}(v) \ominus_{g} \varphi(0) \\ &= X_{IO} \biggl(\frac{(t-a)^{\alpha}}{\Gamma(\alpha+1)} \biggr) \ominus _{g} \varphi(0). \end{aligned}$$
The proof is completed. □
Example 3.4
Consider the fractional order initial value problem for interval-valued delay differential equation given by
$$\begin{aligned} \textstyle\begin{cases} ({^{C}} \mathcal{D}_{0^{+}}^{0.5 }X ) (t) = \lambda_{1} X(t-1) - \lambda _{2}t, &t \in[0,1] \\ X(t) = [t-1,t],& t \in[-1,0], \end{cases}\displaystyle \end{aligned}$$
(3.15)
where \(\lambda_{1}, \lambda_{2} \in\mathbb {R} \backslash\{0\}\). Using the result of Corollary 3.3 for this example, we get
$$\begin{aligned} F^{*} \bigl(v, X_{IO}(v),X_{IO,v} \bigr) = \lambda_{1} X \bigl(k(t,v)-1 \bigr) - \lambda_{2} \bigl( 2 \sqrt{t}\Gamma(3/2) v - v^{2} \Gamma^{2}(3/2) \bigr). \end{aligned}$$
The corresponding differential equation of this fractional initial value problem (3.15) is
$$\begin{aligned} \textstyle\begin{cases} X_{IO}'(v) = \lambda_{1} X_{IO} (k(t,v)-1 ) - \lambda_{2} ( 2 \sqrt {t}\Gamma(3/2) v - v^{2} \Gamma^{2}(3/2) ), &v \in [0,1/\Gamma(3/2) ] \\ X_{IO}(v) = [k(t,v) -1,k(t,v) ],& v \in[-1,0], \end{cases}\displaystyle \end{aligned}$$
where \(k(t,v) = ( 2 \sqrt{t}\Gamma(3/2) v - v^{2} \Gamma^{2}(3/2) )\). Then, by using the method of steps, we get the following problem:
$$\begin{aligned} \textstyle\begin{cases} X_{IO}'(v) = \lambda_{1} [k(t,v) -2, k(t,v) -1 ] - \lambda_{2} k(t,v), \quad v \in [0,1/\Gamma(3/2) ] \\ X_{IO}(0) = [-1,0]. \end{cases}\displaystyle \end{aligned}$$
Case 1. \(\lambda_{1}, \lambda_{2} \in\mathbb {R}^{+}\) and \(X_{IO}\) is w-increasing. Then we obtain the solution \(X_{IO}(v) = [ \underline{X}_{IO}(v), \overline{X}_{IO}(v)]\), where
$$\begin{aligned} \underline{X}_{IO}(v)&= (\lambda_{1} - \lambda_{2}) \biggl( \sqrt{t} \Gamma (3/2)v^{2} - \frac{\Gamma(3/2)v^{3}}{3} \biggr) - 2 \lambda_{1} v-1, \\ \overline{X}_{IO}(v)&= (\lambda_{1} - \lambda_{2}) \biggl( \sqrt{t} \Gamma (3/2)v^{2} - \frac{\Gamma(3/2)v^{3}}{3} \biggr) - \lambda_{1} v. \end{aligned}$$
From the result of Corollary 3.3, the solution of the given fractional order (3.17) is \(X_{FO}(t) = X_{IO} ( \frac {t^{0.5}}{\Gamma(1.5)} ) = [\underline{X}_{FO}(t), \overline{X}_{FO}(t)]\), where
$$\begin{aligned} \underline{X}_{FO}(t) = (\lambda_{1} -\lambda_{2}) \frac{2t^{3/2}}{3\Gamma (3/2)} - \frac{2\lambda_{1} \sqrt{t}}{\Gamma(3/2)}-1, \qquad \overline{X}_{FO}(t)=( \lambda_{1} -\lambda_{2}) \frac{2t^{3/2}}{3\Gamma(3/2)} - \frac {\lambda_{1} \sqrt{t}}{\Gamma(3/2)}. \end{aligned}$$
Case 2. \(\lambda_{1}, \lambda_{2} \in\mathbb {R}^{-}\) and \(X_{IO}\) is w-decreasing. Similar to Case 1, we obtain
$$\begin{aligned} \bigl[\underline{X}_{FO}(t),\overline{X}_{FO}(t) \bigr]= ( \lambda_{1} -\lambda_{2}) \frac {2t^{3/2}}{3\Gamma(3/2)} + \biggl[ - \frac{2\lambda_{1} \sqrt{t}}{\Gamma (3/2)}-1, - \frac{\lambda_{1} \sqrt{t}}{\Gamma(3/2)} \biggr]. \end{aligned}$$
In the sequel we consider an example which corresponds to the interval version of the problem of the fish population size over time, and the propositional harvesting model in uncertain environment is presented to show the efficiency of the approach. First of all, we recall the framework of the fish population growth model in the situation where quantity of fish is precisely described. Under simplified conditions such as a constant environment (and with no migration), it can be shown that the change in population size p through time t (the time horizon is from zero to \(b>0\)) will depend on three factors including birth rate, death rate and harvest rate, and given by
$$\begin{aligned} \frac{dp(t)}{dt} = \beta p(t) - \bigl(m +c p(t) \bigr)p(t) - h(t),\quad p(0)=z_{0}, \end{aligned}$$
(3.16)
where \(\beta p(t)\) is the birth rate, \((m +c p(t))p(t)\) is the death rate (here, the natural mortality coefficient m is augmented by the term \(cp(t)\) which accounts for overcrowding), \(h(t)\) is the harvest rate which depends on time t, and \(\beta, m, c\) are negative proportionality constants. The symbol \(p_{0}\) denotes the initial population size and p denotes the current population size. If the partial information in the classical population model (3.16) may be known or in the parameters used in the above model may be uncertainty, then model (3.16) can be appropriate by interval or fuzzy theory. Therefore, the corresponding to (3.16) model incorporating uncertainty could be the interval fish population growth model by using the concept of interval Caputo fractional derivative. Furthermore, in the classical population model (3.16), it is considered that the birth rate changes immediately as soon as a change in the number of individuals is produced. However, the members of the population must reach a certain degree of development to give birth to new individuals and this suggests an introduction of a delay term into the problem.
Example 3.5
From the crisp problem (3.16), in this example we choose a form (representation) for the corresponding interval fractional order initial value problem with delay as follows:
$$\begin{aligned} \textstyle\begin{cases} ({^{C}} \mathcal{D}_{0^{+}}^{0.5 }P ) (t)+H(t) = (\beta- m) P(t) + P(t-1),& t \in[0,1] \\ P(t) = [t+1,t+2] \in K_{C}(\mathbb {R}),& t \in[-1,0], \end{cases}\displaystyle \end{aligned}$$
(3.17)
where \((\beta-m) \in\mathbb {R} \backslash\{0\}\) and \(H(t): = [0,h t^{2}] \in K_{C}(\mathbb {R})\) is the harvest rate, for \(h >0\), and there is no overcrowding, i.e., \(c=0\). Using the result of Corollary 3.3 for this example, we get
$$\begin{aligned} F^{*} \bigl(v,P_{IO}(v),P_{IO,v} \bigr) = \lambda_{1} P_{IO}(v) + P_{IO} \bigl(k(t,v) -1 \bigr) \ominus \bigl[ 0, h k^{2}(t,v) \bigr], \end{aligned}$$
where \(\lambda_{1}:=\beta- m\). The corresponding differential equation of this fractional initial value problem (3.17) is
$$\begin{aligned} \textstyle\begin{cases} P'_{IO}(v) = \lambda_{1} P_{IO}(v) + P_{IO}(k(t,v)-1) \ominus[ 0, h k^{2}(t,v)], &v \in[0,1/\Gamma(3/2)]\\ P_{IO}(v) = [k(t,v) +1, k(t,v) +2],& v \in[-1,0], \end{cases}\displaystyle \end{aligned}$$
where \(k(t,v) = ( 2 \sqrt{t}\Gamma(3/2) v - v^{2} \Gamma^{2}(3/2) )\). Then, by using the method of steps, we get the following problem:
$$\begin{aligned} \textstyle\begin{cases} P'_{IO}(v) = \lambda_{1} P_{IO}(v) + [k(t,v), k(t,v) - h k^{2}(t,v) + 1 ],& v \in[0,1/\Gamma(3/2)] \\ P_{IO}(0) = [1, 2]. \end{cases}\displaystyle \end{aligned}$$
Case 1. \(\lambda_{1} \in\mathbb {R}^{+}\) and \(P_{IO}\) is w-increasing. Then we obtain the solution \(P_{IO}(v) = [ \underline{P}_{IO}(v), \overline{P}_{IO}(v)]\), where
$$\begin{aligned} \underline{P}_{IO}(v)={}& \frac{ \Gamma(3/2)}{\lambda_{1}^{3}} \bigl[ \Gamma (3/2) \lambda_{1} \bigl( \lambda_{1} v^{2} + 2v \bigr) - 2\lambda_{1} \sqrt{t}(\lambda_{1} v +1) + 2\Gamma(3/2) \bigr], \\ &{}+ e^{\lambda_{1} v} \biggl[ 1 + \frac{ \Gamma(3/2)}{\lambda_{1}^{3}} \bigl( 2 \lambda _{1} \sqrt{t} -2\Gamma(3/2) \bigr) \biggr], \\ \overline{P}_{IO}(v) ={}& \biggl[ 2+ \frac{ \Gamma(3/2) }{\lambda_{1}^{5}} \biggl( -24 h \Gamma^{3}(3/2) - 2 \Gamma(3/2) \lambda_{1}^{2} + \frac{\lambda_{1}^{4}}{\Gamma(3/2)} \\ &{}+ 2\lambda _{1}^{3} \sqrt{t} - 8h \Gamma(3/2)\lambda_{1}^{2} t + 24 h \Gamma^{2}(3/2) \lambda _{1} \sqrt{t} \biggr) \biggr] e^{\lambda_{1} v} \\ &{} + \biggl[ \frac{ \Gamma(3/2) }{\lambda_{1}^{5}} \bigl( 24h\Gamma^{3}(3/2) +2 \Gamma(3/2) \lambda_{1}^{2} -\lambda_{1}^{4} / \Gamma(3/2)- 2\lambda_{1}^{3} \sqrt { t}\\ &{} + 8h\Gamma(3/2) \lambda_{1}^{2} t - 24h\Gamma^{2}(3/2) \lambda_{1} \sqrt {t} \bigr) \\ &{}+ \frac{ \Gamma(3/2)v }{\lambda_{1}^{4}} \bigl( -2\Gamma(3/2)\lambda_{1}^{2} + 24h\Gamma^{3}(3/2) - 2 \lambda_{1}^{3}\sqrt{t} \\ &{}-24h\Gamma^{2}(3/2) \lambda_{1} \sqrt{t} - 8h \Gamma(3/2) \lambda_{1}^{2} t \bigr) \\ &{} + \frac{ \Gamma(3/2)v^{2} }{\lambda_{1}^{3}} \bigl( \Gamma(3/2)\lambda_{1}^{2} +12h\Gamma^{3}(3/2) - 12 \Gamma^{2}(3/2) \lambda_{1}h\sqrt{t} + 4h \Gamma (3/2) \lambda_{1}^{2} t \bigr) \\ &{} + {\frac{ \Gamma(3/2)v^{3} }{\lambda_{1}^{2}} \bigl( 4 h\Gamma ^{3}(3/2) - 4 h \Gamma^{2}(3/2) \lambda_{1} \sqrt{t} \bigr) + \frac{ \Gamma ^{4}(3/2)hv^{4} }{\lambda_{1}}} \biggr]. \end{aligned}$$
From the result of Corollary 3.3, the solution of the given fractional order (3.17) is \(P_{FO}(t) = P_{IO} ( \frac {t^{0.5}}{\Gamma(1.5)} ) = [\underline{P}_{FO}(t), \overline{P}_{FO}(t)]\), where
$$\begin{aligned} \underline{P}_{FO}(t) ={}& \frac{1}{\lambda_{1}^{3}} \biggl( - \lambda_{1}^{2} t + \frac{\pi}{2} \biggr) + \exp \biggl( \frac{2\lambda_{1} \sqrt{t}}{\sqrt{\pi }} \biggr)\biggl( 1 + \frac{1}{\lambda_{1}^{3}}( \lambda_{1} \sqrt{\pi t} - \pi/2) \biggr), \\ \overline{P}_{FO}(t)={}& \exp \biggl( \frac{2\lambda_{1} \sqrt{t}}{\sqrt{\pi}} \biggr) \\ &{}\times \biggl( 2 + \frac{1}{ \lambda_{1}^{5}} \biggl[ -\frac{3h}{2}\pi^{2} - \pi \frac{\lambda_{1}^{2} }{2}+ \lambda_{1}^{4}+ \lambda_{1}^{3} \sqrt{\pi t} - 2h \pi\lambda_{1}^{2} t + 3h \pi \lambda_{1} \sqrt{\pi t} \biggr] \biggr) \\ &{}+ \biggl[ {\frac{1}{\lambda_{1}^{5}} \biggl( \frac{3h}{2}\pi^{2} + \pi \frac {\lambda_{1}^{2} }{2}- \lambda_{1}^{4}- \lambda_{1}^{3} \sqrt{\pi t} + 2h \pi \lambda_{1}^{2} t - 3h \pi \lambda_{1} \sqrt{\pi t} \biggr)} \\ &{}+\frac{1}{\lambda_{1}^{4}} \bigl( 3h \pi\sqrt{\pi t} - \lambda_{1}^{2} \sqrt {\pi t} - 2\lambda_{1}^{3} t - 6h \pi \lambda_{1} t - 4h\sqrt{\pi}\lambda _{1}^{2}t^{3/2} \bigr) \\ &{}+ {\frac{1}{\lambda_{1}^{3}} \bigl( \lambda_{1}^{2} t + 3 h\pi t - 6h \sqrt{\pi} \lambda_{1} t^{3/2} + 4 h \lambda_{1}^{2} t^{2} \bigr) +\frac {4h}{\lambda_{1}^{2}} \bigl( 2 \sqrt{\pi}t^{3/2} - \lambda_{1} t^{2} \bigr) + \frac{h}{\lambda_{1}}t^{2}} \biggr]. \end{aligned}$$
In our numerical simulations, we use the value of parameters \(\lambda_{1} = 1\) and \(h=0.1\). The w-increasing solution of (3.17) is shown in Figure 1.
Case 2. \(\lambda_{1} \in\mathbb {R}^{-}\) and \(P_{IO}\) is w-decreasing. Similar to Case 1, from the result of Corollary 3.3, the solution of the given fractional order (3.17) is \(P_{FO}(t) = P_{IO} ( \frac{t^{0.5}}{\Gamma (1.5)} ) = [\underline{P}_{FO}(t), \overline{P}_{FO}(t)]\), where
$$\begin{aligned} \underline{P}_{FO}(t)={}& \exp \biggl( \frac{2\lambda_{1} \sqrt{t}}{\sqrt{\pi }} \biggr)\\ &{}\times \biggl( 1 + \frac{1}{ \lambda_{1}^{5}} \biggl[ -\frac{3h}{2}\pi^{2} - \pi \frac{\lambda_{1}^{2} }{2}+ \lambda_{1}^{4}+ \lambda_{1}^{3} \sqrt{\pi t} - 2h \pi\lambda_{1}^{2} t + 3h \pi \lambda_{1} \sqrt{\pi t} \biggr] \biggr) \\ &{}+ \biggl[ {\frac{1}{\lambda_{1}^{5}} \biggl( \frac{3h}{2}\pi^{2} + \pi \frac {\lambda_{1}^{2} }{2}- \lambda_{1}^{4}- \lambda_{1}^{3} \sqrt{\pi t} + 2h \pi \lambda_{1}^{2} t - 3h \pi \lambda_{1} \sqrt{\pi t} \biggr)} \\ &{}+\frac{1}{\lambda_{1}^{4}} \bigl( 3h \pi\sqrt{\pi t} - \lambda_{1}^{2} \sqrt {\pi t} - 2\lambda_{1}^{3} t - 6h \pi \lambda_{1} t - 4h\sqrt{\pi}\lambda _{1}^{2}t^{3/2} \bigr) \\ &{}+ {\frac{1}{\lambda_{1}^{3}} \bigl( \lambda_{1}^{2} t + 3 h\pi t - 6h \sqrt{\pi} \lambda_{1} t^{3/2} + 4 h \lambda_{1}^{2} t^{2} \bigr) +\frac {4h}{\lambda_{1}^{2}} \bigl( 2 \sqrt{\pi}t^{3/2} - \lambda_{1} t^{2} \bigr) + \frac{h}{\lambda_{1}}t^{2}} \biggr], \\ \overline{P}_{FO}(t) = {}&\frac{1}{\lambda_{1}^{3}} \biggl( - \lambda_{1}^{2} t + \frac{\pi}{2} \biggr) + \exp \biggl( \frac{2\lambda_{1} \sqrt{t}}{\sqrt{\pi }} \biggr) \biggl( 2 + \frac{1}{\lambda_{1}^{3}}( \lambda_{1} \sqrt{\pi t} - \pi/2) \biggr). \end{aligned}$$
In our numerical simulations, we use the value of parameters \(\lambda_{1} =-2\) and \(h = 0.4\). The w-decreasing solution of (3.17) is shown in Figure 2.