Existence of solutions for Caputo fractional iterative equations under several boundary value conditions

1.   Introduction

Boundary value and periodic problems for iterative differential equations have attracted significant interest in recent years. The investigation of iterative differential equations dates back to 1965, when Petuhov ^[1] first proposed the existence and uniqueness of solutions to the following equation:

with $z(0) = z(b) = k$ under the different ranges of parameters $\lambda$ and $k$. In general, compared with ordinary differential equations, it is more difficult to deal with such iterative differential equations since the existence of iterative terms leads to considerable difficulties of analysis and the failure of some methods, such as the monotone method ^[2], the measure of noncompactness ^[3], the upper and lower solutions method ^[4], the coincidence degree theory ^[5,6], etc. However, the fixed point method is one of the most powerful and fruitful tools studying the boundary value or periodic problems of this kind of iterative differential equation. It is worth mentioning that Ke ^[7] considered the existence of a solution for the first order iterative differential equation

where $a$ is the end point of an interval. Recently, applying Schauder's fixed point theorem, Kaufmann ^[8] further investigated the boundary value problems of the second-order iterative differential equation

where $z^{[2]}(t) = z(z(t))$, with $z(a) = a, z(b) = b$ or $z(a) = b, z(b) = a$. In ^[9], the authors studied a third-order differential equations with linear iterative source for which the existence and uniqueness of a periodic solution are established by using Banach's and Krasnoselskii's fixed point theorems. The same method is used to deal with a first-order iterative differential equations together with some properties of Green's functions in ^[10]. Periodic problems were investigated in ^[11,12,13] by Schauder's or Banach's fixed point theorem. In recent years, fractional calculus has attracted the attention of many mathematicians, as well as researchers in a number of other fields, such as engineering, chemistry and physics. It is recognized that the usage of fractional calculus in various modeling applications is quite outstanding in the process. The main reason for the broad application of fractional operators is the fact that, distinct from "integer" operators, these operators have a non-local behavior which enables us to trace the past impact of the involved phenomena ^{[14,15,16,17,18]}. Along with the recent advances in fractional differential equations, researchers have contributed many investigations that discuss the behaviors of solutions of different types of fractional differential equations ^{[19,20,21,22,23,24,25]}. In addition, since fractional derivatives and integrals have different forms, other types of fractional differential systems are also investigated in ^{[26,27,28,29,30,31,32]}, to name a few. To the best of our knowledge, there are few results on boundary value problems for fractional iterative equations. It should be noted that, the rotational periodic problem of some Caputo-fractional iterative systems is investigated in ^[33]. The existence and uniqueness of solutions for fractional iterative equations was studied in ^[34], in which the nonlinear function satisfies the Lipschitz condition.

In this paper, however, under weaker conditions of one-sided Lipschitz, we consider the existence and uniqueness for a fractional quadratic iterative differential equation with arbitrary order in the sense of Caputo. It is worth emphasizing that the solution we obtained the case of fractional order greater than 1 is monotonically continuous for the first time. Then, the existence and uniqueness for a fractional vector iterative differential equation in $R^N$ are established by the Leray-Schauder fixed point theorem and topological degree theory under periodic and integral boundary value conditions, respectively. Furthermore, applying set-value theory, we prove the existence of solutions for a fractional nonlinear control system with a disturbance. Eventually, motivated by ^[35], the existence of a solution for a fractional neural networks iterative system where neuron activations are continuous or discontinuous.

The framework of the present paper is arranged as follows. Some basic definitions and auxiliary results on fractional calculus are provided in section 2. In section 3, the existence and uniqueness of solutions of Caputo-fractional quadratic iterative equations with arbitrary order are addressed by applying the Leray-Schauder fixed point theorem and topological degree theory. In section 4, the existence and uniqueness of periodic solutions of fractional vector iterative equations are established in $R^N$. Then, the existence of an integral boundary value solution for this vector iterative equation follows in section 5. In section 6, the well-posed result for a nonlinear control system is studied by applying set-valued theory, followed the existence of solutions for a neural network iterative system in section 7 and the presentation of an example in section 8.

2.   Preliminaries

In this section, let $R^N$ be $N$ dimensional Euclid space, $\langle{\cdot}, {\cdot}\rangle$ be the inner product in $R^N$, and ${\|\cdot\|}$ be the norm of $R^N$ space. Let $T = [0, b]$ and $C(T; R^N)$ denote the space composed of all continuous functions from $T$ to $R^N$, with norm $\|\cdot\|_C = \max_{t\in T}\|\cdot\|$. Some basic definitions and fundamental facts of fractional calculus which will be used later need to be provided in the sequel. For details, we refer interest readers to ^[16,17,18].

Definition 2.1. Assume that $g$ is a function defined on the interval $[a, b]$. The Caputo fractional integral of order $\alpha > 0$ of $g$ is defined by

where $\Gamma(\cdot)$ is the Gamma function.

Definition 2.2. Let $g$ be a function defined on the interval $[a, b]$. Then, the Caputo fractional derivative of order $\alpha > 0$ of $g$ is defined by

where $g^{(n)}(t): = \frac{d^n g}{dt^n}$, $n = [\alpha]+1$, and $[\alpha]$ is defined as the maximum positive integer not exceeding the number $\alpha$.

Definition 2.3. If $g\in C^n[a, b]$, then

where $\alpha\in[n-1, n), \forall n\in \mathbb{N}$. In particular, if $g\in C^1[0, b]$ and $\alpha\in(0, 1]$, then for any $t\in [0, b]$,

Next, the following lemmas given here are of significant completing the proof of the main results.

Lemma 2.1. (^[36])Let $\alpha > 0$ and $\beta(t)$ be locally integrable on $[a, b]$, where $\beta(t)$ is a nondecreasing and nonnegative function. Assume that $h$ is a nonnegative, nondecreasing and continuous function, and $U$ is a nonnegative continuous function satisfying

Then, it holds that

where $E_\alpha$ is the single parameter Mittag-Leffler function given by $E_\alpha(x) = \sum^{\infty}_{k = 0}\frac{x^k}{\Gamma(k\alpha+1)}$.

Now, we introduce several inequalities on the fractional derivative which will be used later.

Lemma 2.2. (^[37])Let $U:T\rightarrow {R}^N$ be a continuous differentiable function, and $A\in{R}^{N\times N}$ is a positive definite matrix. Then, it holds that

for any $\alpha\in(0, 1].$

Lemma 2.3. (^[38])Assume that $U:\mathbb{R}\rightarrow \mathbb{R}_+$ is a continuous function such that

where $\omega > 0$ is a constant. Then, the following estimate holds:

Lemma 2.4. (^[39]) Suppose that ${\mathcal W}$ is a Banach space, the set ${\mathcal C}\subseteq {\mathcal W}$ is nonempty and convex with $0\in {\mathcal C}$, and $G:{\mathcal C}\rightarrow {\mathcal C}$ is an upper semicontinuous multifunction with compact convex value, which maps bounded sets to relatively compact sets. Then, one of the following statements is true:

$(i)$ $\Xi = \{{z\in {\mathcal C}:z\in\epsilon G(z), \epsilon \in (0, 1)\}}$ is an unbounded set;

$(ii)$ The multifunction $G(\cdot)$ has a fixed point, i.e., there exists $z\in {\mathcal C}$ such that $z\in G(z).$

Let $E_{\alpha, \gamma}(x): = \sum_{k = 0}^{\infty}\frac{x^{k}}{\Gamma(\alpha k +\gamma)}$ be the two parameter $(\alpha, \gamma)$-Mittag-Leffler function for $x\in \mathbb{R}$. For notational convenience, set $M_A = \max\limits_{t\in T}\|E_\alpha(-At^\alpha)\|$, $\hat{M}_A = \max\limits_{t\in T}\|E_{\alpha, \alpha}(-At^\alpha)\|$.

3.   Boundary value solution of nonlinear fractional quadratic iterative equations

This subsection investigates the existence and uniqueness of solutions of the Caputo-fractional quadratic iterative equation represented as follows:

where $z^{[2]}(t) = z(z(t))$ for any $\alpha > 0.$ In order to obtain our desired results, the following hypothesis on the nonlinear function $f$ is needed:

$(H_1)$ $f\in [a, b]\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that

(ⅰ) there exists a function $\theta_1\in L^\infty_+[a, b]$ such that $|f(t, \mu, \eta)|\leq \theta_1(t)$, for all $t\in[a, b]$, $\mu, \eta\in \mathbb{R}$;

(ⅱ) there exists a constant $W > 0$, for any $(t_1, u_1, v_1), (t_2, u_2, v_2)\in [a, b]\times \mathbb{R}\times \mathbb{R}$, such that

Theorem 3.1. Let $0 < \alpha < 1$, $z(a) = a$ and $b > \left(\frac{\|\theta_1\|_\infty}{\Gamma(\alpha)\alpha}\right)^{\frac{1}{1-\alpha}}$. If hypothesis $(H_1)$ holds, then the Caputo-fractional quadratic iterative equation (3.1) admits a unique solution.

Proof. Following Proposition 2.1, Eq (3.1) is equivalent to the following integral equation:

Define the operator $T_1:C[a, b]\rightarrow C[a, b]$ by

Now, we divide the proof into two steps.

First step. We claim the existence of a solution for Eq (3.1).

From the definition of the operator $T_1$ and the assumption $(H_1)$(ⅰ), we can deduce

which means that $T_1$ is uniformly bounded in $C[a, b]$. Now, we show the equicontinuity of $T_1$. For any $t, t+\delta\in[a, b]$ and $\delta > 0$, it follows that

Let $\delta\rightarrow0$, and then $|T_1(z(t+\delta))-T_1(z(t))|\rightarrow0$. Applying the Arzela-Ascoli theorem, it is easy to find that the operator $T_1:\Omega\rightarrow \Omega$ is completely continuous where

Then, the existence of a solution for the system (3.1) is transformed into the fixed point problem of $T_1$, i.e., $z = T_1(z)$. For this, let ${\mathbb{Q}}_\lambda(z): = z-\lambda T_{1}(z)$ with $\lambda\in[0, 1]$, and then the definition of $\Omega$ implies that $p\notin {\mathbb{Q}}(\partial \Omega)$. Thus, for any $\lambda\in[0, 1]$, we have

where $I$ is the identity map. Consequently, $T_1$ has a fixed point in $\Omega$, i.e. $z = T_1(z)$, which leads to the existence of a solution for the system (3.1).

Second step. We show the uniqueness of a solution for the system (3.1).

Assume $z_1(t), z_2(t)$ are two solutions of system (3.1) with the same initial value. Then, for all $t\in[a, b]$, we can deduce from (3.1) that

which, with multiplying by $z_1(t)-z_2(t)$, leads to

Integrate on both sides with Proposition 2.1 and apply $(H_1)$(ⅱ) to get

which, by using Lemma 2.1, yields

for any $t\in[a, b]$. This gives $z_1\equiv z_2$, and therefore, a unique solution for system (3.1) is guaranteed.

Now, we consider the case $1 < \alpha < 2$ for system (3.1), and then the hypothesis on the nonlinear function $f$ needs to be improved as follows:

$(H_2)$ $f\in [a, b]\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that

(ⅰ) there exist two constants ${\mathbb{N}}, Q > 0$ such that $-{\mathbb{N}} < f(t, \mu, \eta) < Q$ for all $(t, \mu, \eta)\in[a, b]\times\mathbb{R}^2$;

(ⅱ) there exist two constants ${M}, F > 0$ such that

for any $(t, u_1, v_1), (t, u_2, v_2)\in[a, b]\times\mathbb{R}^2$.

Theorem 3.2. Let $1 < \alpha < 2$, $z(a) = a$ and $z(b) = b$. Suppose that $b\leq a+\left(\frac{\Gamma(\alpha)\alpha}{\alpha {N}+Q}\right)^{\frac{1}{\alpha-1}}$ is satisfied, and $(H_2)$ holds. Then Eq (3.1) admits a unique solution which is monotonically increasing and continuous.

Proof. Similar to (3.2), by Proposition 2.1, the solution of (3.1) is equivalent to

Due to $z(a) = a, \; z(b) = b$, it follows from (3.3) that

which implies that

Now, substitute $z'(a)$ into (3.3), and use Green's function to turn problem (3.3) into the following integral equation:

where

Similar to the proof of Theorem 3.1, define the operator $T_2:C[a, b]\rightarrow C[a, b]$, by

First step. The existence of a solution to the problem (3.3).

In light of the definition of $T_2$, we can derive from $(H_2)$(ⅰ) that

and

Notice that $b\leq a+\left(\frac{\Gamma(\alpha)\alpha}{\alpha {\mathbb N}+Q}\right)^{\frac{1}{\alpha-1}}$, and then $({\mathbb N}+\alpha Q)(b-a)^{\alpha-1}\leq \alpha\Gamma(\alpha)$, which gives $\frac{d(T_2(z(t)))}{dt} > 0$, $\frac{d(T_2(z(t)))}{dt} < 2$ for any $t\in[a, b]$. Consequently, $T_2$ is bounded and monotone increasing. Since $T_2(z(a)) = a, \; T_2(z(b)) = b$, then we have $a\leq T_2(z(t))\leq b$, for all $t\in[a, b]$. The application of Schauder's theorem yields that there exists at least one fixed point of $T_2$ such that $z = T_2(z)$, i.e., $z$ is the solution of system (3.1).

Second step. The uniqueness of the solution of Eq (3.1).

Let $z_1, z_2$ be two solutions to Eq (3.1) with same initial data, and then it follows from $(T_2(z_i(t)))' < 2$($i = 1, 2$), (3.4) and $(H_2)$(ⅱ) that

which together with Lemma 2.1 yields

for any $t\in[a, b]$. Hence, $z_1\equiv z_2$, i.e., the uniqueness of the solution is proved, which completes the proof of Theorem 3.2.

Theorem 3.3. Let $n-1 < \alpha < n$ ($n\geq 3$), $z(a) = a,$ $z(b) = b,$ and $z^{(i)}(a) = \xi_i > 0\; (i = 1, 2, \cdots, n-1)$ with $\xi_1 > \frac{{\mathbb N}(b-a)^{\alpha-1}}{\Gamma(\alpha)}$. Assume that the hypothesis $(H_2)$ is fulfilled, and then there is a unique solution of system (3.1) which is monotone.

Proof. Integrating on both sides of system (3.1) and applying Proposition 2.1 gives

Define the operator $T_3:C[a, b]\rightarrow C[a, b]$, by

First step. We establish the existence of a solution for Eq (3.1).

The derivation rule, $(H_2)$(ⅰ) and the boundary value condition in (3.1) show that

for any $t\in [a, b]$. Also, we can find

Therefore, $T_3(z(t))$ is a bounded and monotone increasing function. By taking $T_3(z(a)) = a,$ $T_3(z(b)) = b$ into account, one has $a\leq T_3(z(t))\leq b$ for all $t\in[a, b]$. Thanks to Schauder's theorem, we can conclude that $z$ is a fixed point of $T_3$. Therefore, $z$ is the solution of Eq (3.1).

Second step. We claim the uniqueness of a solution of Eq (3.1).

Assume that $z_1, z_2\in C^{n-1}[a, b]$ are two solutions of (3.1) with the same initial data. Similar to (3.2), and from assumption ($H_2$)(ⅱ), (3.5) and (3.7), one obtains that

where

Lemma 2.1 and $z_1(0) = z_2(0)$ show $z_1(t) = z_2(t)$ for any $t\in[a, b]$, and then $z_1\equiv z_2$. The proof of Theorem 3.3 is completed.

4.   Boundary value periodic solution of Caputo-fractional quadratic iterative equations in ${R}^N$

Consider the following boundary value periodic problem of the fractional iterative vector differential equation

where $\alpha\in(0, 1),$ $z^{[2]}(t) = (z_1(\|z\|), z_2(\|z\|), \cdots, z_n(\|z\|))\in R^N$ for any $t\in T$, the linear operator $B:R^N\to R^N$ is positive definite, $f:T\times R^N\times R^N\to R^N$ is a continuous function, and $g\in L^\infty(T; R^N)$. Throughout this section, we assume that $b$ is greater than some constant $M^{\frac{1}{1-\alpha}}$ to be determined later.

We need the following assumptions:

$(H_3)$ The linear operator $B:R^N\to R^N$ is bounded and positive definite, that is, for any $z\in R^N$, there exists a constant $\xi\in \mathbb{R}_+$, such that $\langle Az, z\rangle\ge \xi\|z\|^2$.

$(H_4)$ $f:T\times R^N\times R^N\to R^N$ is a continuous function satisfying

(ⅰ) for any $x, y\in R^N$, there exists a function $\theta\in L^{\infty}_+(T)$, such that $\|f(t, x, y)\| \le \theta(t)$ for any $t\in T$;

(ⅱ) for any $t\in T,$ $(u_1, v_1),$ $(u_2, v_2)\in R^{N}\times R^{N}$, there exists a function $\mu \in L^{\infty}_+(T)$, such that

where $\|\mu\|_\infty < \xi$, and $\xi$ is a constant in $(H_3)$.

Theorem 4.1. If assumptions $(H_3), (H_4)$ hold, then the fractional iterative differential system (4.1) has a unique solution.

Proof. In view of Corollary 7.1 in ^[40], problem (4.1) is equivalent to the following integral iterative equation:

Then, it suffices to show the existence of a solution for problem (4.2). For this, the operator $T_4$: $C(T; R^N)\to C(T; R^N)$ is defined by

First, we show a priori boundedness of the solution. From the definition of operator $T_4$ and the assumption $(H_4)$(ⅰ), it can be deduced that

where

Now, let us estimate the initial value $\|z(0)\|$. Let $t = b$ in Eq (4.2), and then it follows that

Invoking $z(0) = z(b)$ and the assumption $(H_3)$, it is easy to see the determinant $\arrowvert E-E_\alpha(-Ab^\alpha)\arrowvert \not = 0$, where $E$ represents the identity matrix. Thus, we have from (4.4)

By means of hypothesis $H_4$(ⅰ) and similar to estimate (4.4), one can show

where

Substitute estimate (4.6) into estimate (4.4) to obtain

Let $M: = \frac{(M_EM_A+1)\hat{M}_A(\|\theta\|_\infty+\|g\|_\infty) }{\alpha\Gamma(\alpha)}$. From $b > M^{\frac{1}{1-\alpha}}$, we therefore have

Second, it will be shown that the existence of the solution is obtained. For this, we shall prove that the nonlinear operator $T_4$ is completely continuous. For any $z\in C(T; R^N),$ we claim $T_4(z(t))\in C(T; R^N)$. For all $t, t+\delta\in[0, b]$ and $\delta > 0$, from (4.2) one can show

As $\delta\to0$, it is easy to get that

Therefore, $T_4(z(t))\in C(T; R^N)$. Taking $z_n\to z$ in $C(T; R^N)$, we can deduce $\arrowvert T_4(z_n)-T_4(z)\arrowvert\to 0$, so $T_4$: $C(T; R^N)\to C(T; R^N)$ is continuous. According to the above prior estimate, by applying the Arzela-Ascoli theorem, it is easy to find that the operator $T_4$: $\Omega\to\Omega$ is completely continuous, where

Thus, to get the existence of the solution, it suffices to show the fixed point problem of $T_4$. Define the mapping $H_\varepsilon(z) = z-\varepsilon T_4(z)$, where $\varepsilon\in[0, 1]$. Take $p\notin H(\partial\Omega)$, and then for any $\varepsilon\in[0, 1]$, we get

So, $T_4$ has a fixed point on $\Omega$, i.e., $z = T_4(z)$, which leads to the existence of the solution.

Finally, we will establish the uniqueness of the solution of differential iterative system (4.1). If $z_1, z_2\in C(T; R^N)$ are the two solutions to problem (4.1) with the same initial data, and take the difference between the two solutions and take the inner product with $z_1-z_2$ to get

Based on the assumptions $(H_3)$ and $(H_4)$(ⅱ), and using Lemma 2.2, we can derive

Let $S(t) = \|z_1(t)-z_2(t)\|^2$, and inequality (4.9) can be simplified as

which, by applying Lemma 2.3, presents

Take $t = b$ in (4.10) to obtain

Boundary condition $z(b) = z(0)$ shows $S(b) = S(0)$. Hence, we find from (4.11) that

From the monotonicity of Mittag-Leffler function $E_\alpha(t)$ ($\alpha\in(0, 1)$) and $\|\mu\|_\infty < \xi$, it holds that

Due to

one gets $S(0) = 0$, which with (4.10) implies $S(t)\equiv0$. This means $z_1\equiv z_2$, so the uniqueness of the solution follows.

5.   Integral boundary value solution of Caputo-fractional quadratic iterative equations in ${R}^N$

Consider the following integral boundary value problem of Caputo-fractional quadratic iterative differential equations:

where $z^{[2]}(t) = (z_1(\|z\|), z_2(\|z\|), \cdots, z_n(\|z\|))$, $\lambda > 1,$ linear operator $A:R^N\rightarrow R^N$ is positive definite satisfying ($H_3$), $f:T\times R^N\times R^N\rightarrow R^N$ is a Carathéodory function satisfying ($H_4$), and $g\in L^\infty(T)$. Throughout this section, we assume that $b > M^{\frac{1}{1-\alpha}}$ with

Theorem 5.1. If the assumptions $(H_3)$ and $(H_4)$ are fulfilled, then the fractional quadratic iterative differential system (5.1) admits a unique solution.

Proof. Similar to Theorem 4.1, it is sufficient to investigate the following integral iterative equation:

Due to $z(0) = \frac{1}{\lambda b}\int_0^bz(s)ds$, define an operator ${\mathcal O}_1:C(T; R^N)\rightarrow C(T; R^N)$ by

Then, the integral boundary value problem of Eq (5.1) can be transformed into a fixed point problem:

The proof process is divided into three steps.

Step 1. The a priori boundedness of the solutions for problem (5.1).

Let $z$ be the solution of operator equation $z = {\mathcal O}_1z$. Apply the hypothesis $(H_4)$(ⅰ), to deduce

where

for any $t\in T$. Due to the monotonicity of Mittag-Leffler function $E_\alpha(-t)(t\geq0)$ and because $A$ is positive definite, it is easy to get $M_A < 1$. It follows from (5.3) that

Notice

and $b > {M}^{\frac{1}{1-\alpha}}$, and then (5.6) yields

Step 2. The existence of the solution for problem (5.1).

To begin with, we claim that ${\mathcal O}_1z\in C(T; R^N)$ for any $z\in C(T; R^N)$. For any $t, t+\delta\in T$, and $\delta > 0$, it follows from (5.2) that

As $\delta\rightarrow0$, one has $|{\mathcal O}_1z(t+\delta)-{\mathcal O}_1z(t)|\rightarrow0$, and therefore ${\mathcal O}_1z\in C(T; R^N)$. Taking $z_n\rightarrow z$ in $C(T; R^N)$, we have $z_{in}(\|z\|)\rightarrow z_{i}(\|z\|)$ for each $i = 1, 2, \cdot\cdot\cdot, n$, which together with the continuity of $(s, v)\rightarrow f(t, s, v)$, yields $|{\mathcal O}_1z_n-{\mathcal O}_1z|\rightarrow 0$. Thus, ${\mathcal O}_1:C(T; R^N)\rightarrow C(T; R^N)$ is continuous. By taking the prior estimation (Step 1) into account and applying the Arzela-Ascoli theorem, it holds that the operator ${\mathcal O}_1:\Omega\rightarrow \Omega$ is completely continuous, where

Hence, the existence of solutions for the differential iterative system (5.1) can be transformed into a fixed point problem of ${\mathcal O}_1$. Define the mapping $H_\varepsilon(z) = z-\varepsilon {\mathcal O}_1(z)$ for $z\in C(T; R^N)$, with $\varepsilon\in[0, 1]$. Let $p\not\in H_\varepsilon(\partial\Omega)$, for any $\varepsilon\in[0, 1]$, and this leads to

Therefore, the operator ${\mathcal O}_1$ has a fixed point on $\Omega$, i.e., $z = {\mathcal O}_1z$, so the existence of the solution $z$ for differential system (5.1) follows.

Step 3. The uniqueness of the solution for problem (5.1).

Let $z_1, z_2\in C(T; R^N)$ be two solutions of problem (5.1). Substitute $z_1$ and $z_2$ into (5.1), respectively, and then take a difference and the inner product with $z_1-z_2$ to get

By means of the hypotheses $(H_3)$ and $(H_4)$(ⅱ), applying Lemma 2.2, it results that

Let ${\mathcal Q}(t): = \|z_1(t)-z_2(t)\|^2$ for brevity, and the above inequality can be simplified as

Invoke Lemma 2.3 to show

In light of $z_1(0)-z_2(0) = \frac{1}{\lambda b}\int_0^b[z_1(t)-z_2(t)]dt$ in (5.7), one obtains

which leads to

with $\lambda > 1$. Since the Mittag-Leffler function $E_\alpha(-t)(t\geq0)$ is monotonically decreasing, and $\|\mu\|_\infty < {\xi}$, we can conclude that $z_1\equiv z_2$, so the iterative differential equation (5.1) has a unique solution, which yields our desired result.

6.   Control problem for a fractional iterative differential system with a disturbance

The aim of this section is to study the following Control problem for an iterative differential system with a disturbance:

where $A,$ $f$ are shown as in (5.1), $u:T\rightarrow R^N$ is a control input, $U:T\times R^{N}\rightarrow 2^{R^{N}}\setminus \{\emptyset\}$ is a multifunction of observation value, and $d\in L^\infty(T)$ is a disturbance function. The hypothesis on $U$ is presented as follows.

$H(U):\; U:T\times R^{N}\rightarrow 2^{R^{N}}\setminus \{\emptyset\}$ is a multivalued function with closed, convex value such that

(ⅰ) $(t, y)\rightarrow U(t, y)$ is graph measurable for every $(t, y)\in T\times R^N$;

(ⅱ) for almost all $t \in T,$ $y\rightarrow U(t, y)$ has a closed graph;

(ⅲ) for every $y\in R^{N}$ and all $t\in T$, there exists a function $\Phi\in L_+^\infty(T)$ such that

In this section, we assume that $b > M^{\frac{1}{1-\alpha}}$ with

Theorem 6.1. If the assumptions $(H_3), (H_4)$ and $H(U)$ hold, then the problem (6.1) admits at least one solution $z\in C(T; R^{N})$.

Proof. We first construct a closed convex subset ${\mathcal K}$ in $L^\infty(T; R^{N})$ given by

Due to Theorem 5.1, it is straightforward to derive that the iterative equation

admits a unique solution $z_h\in C(T; R^{N})$ for every $h\in {\mathcal K}$. An operator

is defined by

where

Since ${\mathcal O}_2: D({\mathcal O}_2)\rightarrow {\mathcal K}(\subset L^{\infty}(T; R^{N}))$ is a one-to-one mapping, it results that ${\mathcal O}_2^{-1}: {\mathcal K}\rightarrow D({\mathcal O}_2)$ exists. Now, we claim that the operator

is completely continuous. For this, we will claim that ${\mathcal O}_2^{-1}: {\mathcal K}\rightarrow D({\mathcal O}_2)$ is continuous. Let $h_m\rightarrow h$ in ${\mathcal K}$ as $m\rightarrow \infty$, and it remains to prove that $z_m = {\mathcal O}_2^{-1}(h_m)\rightarrow z = {\mathcal O}_2^{-1}(h)$ in $D({\mathcal O}_2)(\subset C(T; R^{N}))$. Replacing $z$ with $z_m$ in (6.2) and subtracting (6.2) implies

Then, take the inner product with $z_m-z$ on the above equation and apply Lemma 2.2 to obtain

Now, integrating in time and invoking Proposition 2.1 yields

Similar to the a priori estimate of the solution of Theorem 5.1, it is easy to verify that $\|z_m\|_C\leq b,$ which together with $z_m\in C(T; R^N)$ and the Arzela-Ascoli theorem reveals that there exists a subsequence $z_m$ (still denoted by itself) such that $z_m\rightarrow \hat{z}$ in $D({\mathcal O}_2)$ as $m\rightarrow \infty.$ By passing the limit in (6.5), this enables us to obtain

Analogous to the analysis of (5.6), set ${ {Y}} = \|\hat{z}-z\|^2$, and then we can conclude that ${ {Y}}(t)\equiv 0,$ i.e., $\hat{z}\equiv z$ in $D({\mathcal O}_2)$. This implies the continuity of operator ${\mathcal O}_2^{-1}$. In view of the a priori estimate of the solution, it is easy to deduce that ${\mathcal O}_2^{-1}(\mathcal{K})$ is a bounded set in $C(T; R^N)$. Invoking the Arzela-Ascoli theorem, ${\mathcal O}_2^{-1}({\mathcal K})\subset L^{\infty}(T; R^{N})$ is relatively compact. As a result, ${\mathcal O}_2^{-1}: {\mathcal K}\rightarrow L^{\infty}(T; R^{N})$ is completely continuous.

Now, a multivalued Nemitsky operator ${\mathcal {N}}: L^{\infty}(T; R^{N})\rightarrow 2^{L^{\infty}(T; R^{N})}$ corresponding to $U(t, z)$ is defined by

From hypothesis $H(U)$, it holds that the multivalued Nemitsky operator ${\mathcal {N}}(\cdot)$ is nonempty, closed convex value and upper hemicontinuous (Theorem 3.2, ^[41]). Thus, we can find that ${\mathcal {O}}_2^{-1}\circ{\mathcal {N}}:{\mathcal K}\rightarrow L^{\infty}(T; R^{N})$ is an upper hemicontinuous multifunction with closed, convex value, which maps a bounded set into a relatively compact set. Then, the control problem for a iterative differential system (6.1) is turned into the following fixed points problem:

For this aim, via Lemma 2.4, it remains to show that the set

is bounded. Let $z\in\Xi$, and then ${\mathcal {O}}_2(\frac{z}{\epsilon})\in {\mathcal {N}}(z)$, which gives

where $h(t)\in U(z, t)$ for all $t\in T$. Similar to (5.2), Eq (6.8) can be rewritten as

Likewise as in (5.3) and by taking $(H_4)$(ⅱ) into account, it results from (6.9) that

Similar to the estimate (5.3), it holds that $\|z(t)\|$ is uniformly bounded for any $t\in T$. Thanks to Lemma 2.4, there exists $z\in D({\mathcal O}_2)$, such that $z\in {\mathcal O}_2^{-1}\circ{\mathcal N}(z)$. It is obvious that $z$ is the solution of problem (5.1). This proof is thus complete.

7.   Integral boundary value problem of a fractional iterative neural network system

Consider the fractional iterative neural network system described as follows:

where $B = {\rm diag}(d_{1}, d_{2}, \cdots, d_{N})$ is a diagonal matrix with $d_{i} > 0\; (i = 1, 2, \cdots, N)$, $z:T\rightarrow R^{N}$ denotes the vector element of the neuron system, ${I}\in L^\infty(T; R^N)$ is the mapping of neuron inputs, and ${\mathcal F}:R^N\times R^N\rightarrow R^N$ stands for the neuron input-output continuous activation function such that

(ⅰ) For any $x, y\in R^N$, there exists a function $\widehat{\omega}\in L_+^\infty(T)$ such that $\|{\mathcal F}(x, y)\|\leq\widehat{\omega}(t), \; \forall t\in T$;

(ⅱ) For any $x_1, x_2, y_1, y_2\in R^N$, there exists a function $\widehat{\mu}\in L_+^\infty(T)$ such that

where $\|\widehat{\mu}\|_\infty < \min\{d_i:i = 1, 2, \cdot\cdot\cdot, N\}$, for almost all $t\in T.$

The initial value problem of the differential system (7.1) without iteration was discussed in ^[35], where the existence and uniqueness of the solution were demonstrated. Here, however, considering the iterative term and the integral boundary value condition, the existence of a unique integral boundary value solution to system (7.1) is guaranteed by using our results. Let $b > M^{\frac{1}{1-\alpha}}$ with

It is easy to show that all assumptions of Theorem 5.1 hold, and then the existence result for system (7.1) is provided as follows.

Theorem 7.1. If the above assumptions are fulfilled, problem (7.1) admits a unique integral boundary value solution.

It should be noted that the fractional system (7.1) without iteration was considered in Song et al. ^[35], where input function ${ I}(t)$ is assumed to be continuous. Automatically, a question is whether the fractional iteration system (7.1) has a solution, provided that ${ I}(t) = ({ I}_1(t), \cdot\cdot\cdot, {I}_N(t))$ is discontinuous. The following work is to deal with this question. For this, it is assumed that ${I}_{i}\in \Lambda\; (i = 1, 2, \cdots, N)$ are nondecreasing monotone, bounded mappings, where $\Lambda:\mathbb{R}\rightarrow \mathbb{R}$ denotes the class of functions which have a finite number of jumping discontinuities in the closed interval. If only isolated jump discontinuities for any ${ I}_{i}\; (i = 1, 2, \cdots, N)$ appear, then we can deduce

with

Hence, in this case, the iteration problem (7.1) can be rewritten as the following iteration differential inclusion:

Here, ${\mathcal Y}({ I}(t))$ can be handled as a multivalued control item of problem (6.1). Then, the argument is similar to that of Theorem 6.1, and we can conclude the following result.

Theorem 7.2. If the given assumptions are satisfied, then the solution set of iteration differential system (7.2) is nonempty.

8.   Example

As an application of the previous results, we introduce an example. Let us consider the boundary value problem of the following iterative differential equation:

From (3.1), it is easy to see that $\alpha = \frac{3}{2}$ and

Then we can directly calculate $-2\leq f(t, s, w)\leq 2$ for any $(t, s, w)\in T\times \mathbb{R}\times \mathbb{R},$ so the condition ($H_2$)(ⅰ) holds. For any $(s_1, w_1), (s_2, w_2)\in \mathbb{R}\times \mathbb{R}$ and $t\in T$, it is easy to deduce that

which yields

This implies that the condition ($H_2$)(ⅱ) holds. Assume $b < (\frac{3\Gamma(\frac{3}{2})}{10})^2$, and then apply Theorem 3.2 to conclude that Eq (8.1) admits a unique solution which is monotonically increasing continuous.

9.   Conclusions

In this article, we established the existence and uniqueness of solutions for nonlinear quadratic iterative equations in the sense of the Caputo fractional derivative with several boundary conditions by using several fixed point theorems, and we finally applied it to a neural network iterative system and a control problem of a nonlinear iteration system with a disturbance. Furthermore, when the fractional order of the differential system is greater than 1, it was shown for the first time that the solution for a nonlinear quadratic iterative equation is monotonic. Since different definitions of the fractional integral lead to different fractional order differential systems, the method applied in this article can also be applied to systems of many other fractional order iterative differential equations.

Acknowledgments

The authors are indebted to Professor Yi Cheng for his encouragement and helpful discussion. This work were partially supported by the Natural Science Foundation of Liaoning Province and of Jilin Province (No. 2020-MS-290, 20200201274JC), Jilin Provincial Department of Education Science and Technology Research Project (No. JJKH20221261KJ) and School Research Project of Bohai University. The authors are grateful to referees for their constructive comments on the first version of our paper.

Conflict of interest

The authors declare that there are no conflicts of interest.