The fractional diabetic model (11) can be reduced to an integral equation of Volterra type when the integral related to the CF derivative is employed. We recall that the Caputo–Fabrizio non-integer order integral of \(l(\eta )\) is the average of \(l(\eta )\) and its Riemann–Liouville integral of fractional order.

Now using the integral operator of fractional order introduced by Losada and Nieto [21] on the system (11), we get

$$\begin{aligned} \begin{aligned}& C(\eta ) - g_{1}(\eta ) = \frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} \bigl\{ - (\lambda + \theta )C(\eta ) + \lambda N(\eta ) \bigr\} \\ & \phantom{C(\eta ) - g_{1}(\eta ) =}{}+ \frac{2\sigma}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \bigl\{ - (\lambda + \theta )C(s) + \lambda N(s) \bigr\} \,ds, \\ &N(\eta ) - g_{2}(\eta ) = \frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} \bigl\{ I - (\nu + \delta )C(\eta ) - \mu N(\eta ) \bigr\} \\ &\phantom{N(\eta ) - g_{2}(\eta ) = }{}+ \frac{2\sigma}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \bigl\{ I - (\nu + \delta )C(s) - \mu N(s) \bigr\} \,ds. \end{aligned} \end{aligned}$$

(20)

A possibility of reducing Eq. (20) to the iterative approach is presented as follows:

$$ C_{0}(\eta ) = g_{1}(\eta ),\qquad D_{0}(\eta ) = g_{2}(\eta ). $$

(21)

Now, we get the subsequent iterative algorithm

$$\begin{aligned} \begin{aligned}& C_{(n + 1)}(\eta ) = \frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} \bigl\{ - (\lambda + \theta )C_{n}(\eta ) + \lambda N_{n}(\eta ) \bigr\} \\ &\phantom{C_{(n + 1)}(\eta ) =}{}+ \frac{2\sigma}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \bigl\{ - (\lambda + \theta )C_{n}(s) + \lambda N_{n}(s) \bigr\} \,ds, \\ &N{}_{(n + 1)}(\eta ) = \frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} \bigl\{ I - (\nu + \delta )C_{n}(\eta ) - \mu N_{n}(\eta ) \bigr\} \\ &\phantom{N{}_{(n + 1)}(\eta ) =} {}+ \frac{2\sigma}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \bigl\{ I - (\nu + \delta )C_{n}(s) - \mu N_{n}(s) \bigr\} \,ds. \end{aligned} \end{aligned}$$

(22)

Here we assume that we can get the exact solution by taking the limit as *n* tends to infinity.

### 4.1 Existence of the solution by Picard–Lindelof approach

### Theorem 1

*We discuss the existence of the solution by applying the Picard–Lindelof approach*.

### Proof

We express the operators as

$$\begin{aligned} \begin{aligned} &k_{1}(\eta,C) = - (\lambda + \theta )C + \lambda N, \\ &k_{2}(\eta,N) = I - (\nu + \delta )C - \mu N, \end{aligned} \end{aligned}$$

(23)

where \(k_{1}(\eta,C)\) and \(k_{2}(\eta,N)\) are contractions with respect to *C* and *N* for the first and second functions, respectively.

Let

$$ A_{1} = \sup_{C_{a,b_{1}}} \bigl\Vert K_{1}( \eta,C) \bigr\Vert ,\qquad A_{2} = \sup_{C_{a,b_{2}}} \bigl\Vert K_{2}(\eta,N) \bigr\Vert , $$

(24)

where

$$\begin{aligned} \begin{aligned} &C_{a,b_{1}} = \vert \eta - a,\eta + a \vert \times [ C - b_{1,}C + b_{1} ] = A_{1} \times B_{1}, \\ &C_{a,b_{2}} = \vert \eta - a,\eta + a \vert \times [ N - b_{2,}N + b_{2} ] = A_{1} \times B_{2}. \end{aligned} \end{aligned}$$

(25)

On consideration of the Picard operator, we have

$$ \phi:C(A_{1},B_{1},B_{2}) \to C(A_{1},B_{1},B_{2}), $$

(26)

given as follows:

$$ \phi \xi (\eta ) = \xi_{0}(\eta ) + \Delta \bigl( \eta,\xi (\eta ) \bigr)\frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} + \frac{2\sigma}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \Delta \bigl( s,\xi (s) \bigr) \,ds, $$

(27)

where \(\xi (\eta ) = \{ C(\eta ),N(\eta ) \},\xi_{0}(\eta ) = \{ g_{1},g_{2} \}\text{ and }\Delta ( \eta,\xi (\eta ) ) = \{ k_{1}(\eta,C(\eta ),k_{2}(\eta,N(\eta ) \}\).

Next we suppose the solution of the fractional diabetic model are bounded within a time period,

$$\begin{aligned} &\bigl\Vert \xi (\eta ) \bigr\Vert _{\infty} \le \max \{ B_{1},B_{2} \}, \end{aligned}$$

(28)

$$\begin{aligned} &\bigl\Vert \xi (\eta ) - \xi_{0}(\eta ) \bigr\Vert = \biggl\Vert \Delta \bigl( \eta,\xi (\eta ) \bigr)\frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} + \frac{2\sigma}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \Delta \bigl( s,\xi (s) \bigr) \,ds \biggr\Vert \\ &\phantom{\bigl\Vert \xi (\eta ) - \xi_{0}(\eta ) \bigr\Vert }\le \frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} \bigl\Vert \Delta \bigl( \eta,\xi (\eta ) \bigr) \bigr\Vert + \frac{2\sigma}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \bigl\Vert \Delta \bigl( s,\xi (s) \bigr) \bigr\Vert \,ds \\ &\phantom{\bigl\Vert \xi (\eta ) - \xi_{0}(\eta ) \bigr\Vert }\le \biggl( \frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} + \frac{2\sigma \eta_{0}}{(2 - \sigma )M(\sigma )} \biggr)\max \{ B_{1},B_{2} \} \le \rho B \le \alpha, \end{aligned}$$

(29)

where we demand that

$$ \rho < \frac{\alpha}{B}. $$

(30)

By employing the fixed point theorem pertaining to Banach space along with the metric, we obtain

$$ \Vert \phi \xi_{1} - \phi \xi_{2} \Vert _{\infty} = \sup_{\eta \to A} \vert \xi_{1} - \xi_{2} \vert . $$

(31)

Now we have

$$\begin{aligned} \Vert \phi \xi_{1} - \phi \xi_{2} \Vert = {}&\biggl\Vert \bigl\{ \Delta \bigl( \eta,\xi_{1}(\eta ) \bigr) - \Delta \bigl( \eta,\xi_{2}(\eta ) \bigr) \bigr\} \frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} \\ &{} + \frac{2\sigma}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \bigl\{ \Delta \bigl( s, \xi_{1}(s) \bigr) - \Delta \bigl( s,\xi_{2}(s) \bigr) \bigr\} \,ds \biggr\Vert \\ \le{}& \frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )} \bigl\Vert \Delta \bigl( \eta,\xi_{1}(\eta ) \bigr) - \Delta \bigl( \eta,\xi_{2}(\eta ) \bigr) \bigr\Vert \\ &{}+ \frac{2\sigma}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \bigl\Vert \Delta \bigl( s, \xi_{1}(s) \bigr) - \Delta \bigl( s,\xi_{2}(s) \bigr) \bigr\Vert \,ds \\ \le{}& \frac{2(1 - \sigma )}{(2 - \sigma )M(\sigma )}\beta \bigl\Vert \xi_{1}(\eta ) - \xi_{2}(\eta ) \bigr\Vert + \frac{2\sigma \beta}{(2 - \sigma )M(\sigma )} \int_{0}^{\eta} \bigl\Vert \xi_{1}(s) - \xi_{2}(s) \bigr\Vert \,ds \\ \le {}&\biggl\{ \frac{2(1 - \sigma )\beta}{(2 - \sigma )M(\sigma )} + \frac{2\sigma \beta \eta_{0}}{(2 - \sigma )M(\sigma )} \biggr\} \bigl\Vert \xi_{1}(\eta ) - \xi_{2}(\eta ) \bigr\Vert \\ \le{}& \rho \beta \bigl\Vert \xi_{1}(\eta ) - \xi_{2}(\eta ) \bigr\Vert , \end{aligned}$$

(32)

with \(\beta < 1\). Since *ξ* is a contraction, we have \(\rho \beta < 1\), hence the defined operator *ϕ* is also a contraction. Therefore, the diabetic model involving CF derivative given in Eq. (11) has a unique set of solutions. □