On coupled Gronwall inequalities involving a $\psi$-fractional integral operator with its applications

1.   Introduction

The fractional calculus is an important branch of mathematics and it has wide applications to many fields of science and engineering. We know that the fractional calculus is a wonderful technique to understand of memory and hereditary properties of materials and processes. Some contributions to fractional calculus have been carried out, see the monographs ^[1,2,3], and the references cited therein.

The theory of generalized fractional calculus was proposed by Kiryakova in ^[4]. One of the proposed generalizations of the fractional calculus operators is the $\psi$-fractional operator which has wide applications, some properties of this operator could be found in ^{[5,6,7,8,9,10]}. The Gronwall inequality plays an important role in the study of qualitative and qualitative properties of solution of fractional differential and integral equations ^{[11,12,13,14,15,16,17,18]}. In order to work with continuous dependence of differential equations via $\psi$-Hilfer fractional derivative, the generalized Gronwall inequality by means of the fractional integral with respect to another function $\psi$ was first given and proved by Vanterler et al. in ^[19]. Indeed, they obtained the theorem given below.

Theorem 1.1 ^[19]. Let $u, v$ be two integrable functions and $g$ continuous, with domain $[a, b]$. Let $\psi \in C^1[a, b]$ be an increasing function such that $\psi^{\prime}(t) \not = 0$, $\forall t \in [a, b]$. Assume that

(1) $u$ and $v$ are nonnegative;

(2) $g$ is nonnegative and nondecreasing.

If

then

The Hilfer version of the fractional derivative with another function called $\psi$-Hilfer FDO has been presented by Sousa et al ^[20]. Recently, the existence and uniqueness of the solution of a nonlinear $\psi$-Hilfer fractional differential equations with different kinds of initial and boundary conditions and the Ulam-Hyers stabilities of its solutions have been investigated ^{[21,22,23,24,25]}.

The Ulam stability, which can be considered as a special type of date dependence was initiated by Ulam ^[26,27]. Since then, there are many development of this field, we refer the reader to ^{[28,29,30,31,32]} and the references therein.

The main objective of this paper is to extend Theorem 1.1 to the generalized coupled Gronwall inequality by the implementation of $\psi$-fractional operator. As applications, we prove the existence and uniqueness of solutions for the following nonlinear delay coupled $\psi$-Caputo fractional differential system

where $F, G \in C([t_0, t_1] \times \mathbb{R}^2, \mathbb{R})$, $\phi, \theta \in C[t_0- \tau, t_0]$, and $\mbox{}_{t_0}^C D_{\psi}^{\alpha} x(t)$ is the left Caputo fractional derivative of $x$ of order $\alpha$, $0 < \alpha \le 1$ with respect to the continuous function $\psi$ with $\psi^{\prime}(t) > 0$, $t \in [t_0, t_1]$. The meaning of $\mbox{}_{t_0}^C D_{\psi}^{\beta} y(t)$ ($0 < \beta \le 1$) is the same as $\mbox{}_{t_0}^C D_{\psi}^{\alpha} x(t)$. Moreover, we investigate the Ulam-Hyers stability of solutions for (1.1). Our results extend the main results of ^[33].

This paper is organized as follows: In Section 2, we give some notations, definitions and preliminaries. Section 3 is devoted to proving a new generalized coupled Gronwall inequality. In Section 4, the existence and uniqueness of the solution of system (1.1) are given and proved, and the Ulam-Hyers stability theorem of (1.1) is obtained. In Section 5, an example is given to illustrate our theoretical result. Finally, the paper is concluded in Section 6.

2.   Preliminaries

In this section, we provided some basic definitions and lemmas which are used in the sequel.

Definition 2.1 ^[10,34]. Let $\alpha > 0$, $f$ be an integrable function defined on $[a, b]$ and $\psi \in C^1([a, b])$ be an increasing function with $\psi^{\prime}(t) \not = 0$ for all $t \in [a, b]$. The left $\psi$-Riemann-Liouville fractional integral operator of order $\alpha$ of a function $f$ is defined by

Definition 2.2 ^[10,34]. Let $n-1 < \alpha < n$, $f \in C^n([a, b])$ and $\psi \in C^n([a, b])$ be an increasing function with $\psi^{\prime}(t) \not = 0$ for all $t \in [a, b]$. The left $\psi$-Caputo fractional derivative of order $\alpha$ of a function $f$ is defined by

where $n = [\alpha]+1$ and $f^{[n]}(t) : = \left(\frac{1}{\psi^{\prime}(t)} \frac{d}{dt}\right)^n f(t)$ on $[a, b]$.

Lemma 2.1 ^[34]. Let $\alpha > 0$ and $\beta > 0$, then

(i) ${ \mbox{}_{t_0} I_{\psi}^{\alpha} (\psi(s) - \psi(t_0))^{\beta-1}(t) = \frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)} (\psi(t) - \psi(t_0))^{\beta+\alpha-1}, }$

(ii) ${ \mbox{}_{t_0}^C D_{\psi}^{\alpha} (\psi(s) - \psi(t_0))^{\beta-1}(t) = \frac{\Gamma(\beta)}{\Gamma(\beta-\alpha)} (\psi(t) - \psi(t_0))^{\beta-\alpha-1}, }$

(iii) ${ \mbox{}_{t_0}^C D_{\psi}^{\alpha} (\psi(s) - \psi(t_0))^k(t) = 0, \quad n-1 < \alpha < n, \quad k = 0, 1, ..., n-1. }$

In the following, we will give the combinations of the fractional integral and the fractional derivatives of a function with respect to another function.

Lemma 2.2 ^[34]. Let $f \in C^n([a, b])$ and $n-1 < \alpha < n$. Then we have

(1) ${ \mbox{}_{t_0}^C D_{\psi}^{\alpha} \mbox{}_{t_0} I_{\psi}^{\alpha} f(t) = f(t) }$;

(2) ${ \mbox{}_{t_0} I_{\psi}^{\alpha} \mbox{}_{t_0}^C D_{\psi}^{\alpha} f(t) = f(t) - \sum \limits_{k = 0}^{n-1} \frac{f^{[k]}(t_0^+)}{k!} (\psi(t) - \psi(t_0))^k }$.

In particular, given $\alpha \in (0, 1)$, one has

Let $X = C([t_0- \tau, t_1], \mathbb{R}) \cap C^1([t_0, t_1], \mathbb{R})$, then the space $X$ is a Banach space with respect to the norm defined by $\|u\| = \max_{t \in [t_0- \tau, t_1]} |u(t)|$.

The following definition of Ulam stability of (1.1) is similar to the definition stated in ^[35].

Definition 2.3. System (1.1) is said to be Ulam-Hyers stable if there exists a real number $c$ such that for all $\epsilon > 0$ and for each $(u, v) \in X \times X$ with $(u(t), v(t)) = (\phi(t), \theta(t))$ for $t \in [t_0- \tau, t_0]$ satisfying the inequalities

there exists a solution $(x, y) \in X \times X$ of (1.1) satisfying

3.   A generalized coupled Gronwall inequality

Now we state and prove a new generalized coupled Gronwall inequality as follows.

Theorem 3.1. Assume that $x, y$, and $a_i$ ($i = 1, 2$) are integrable and nonnegative functions, and $b_i$ ($i = 1, 2$) are continuous, nonnegative and nondecreasing functions, with domain $[t_0, t_1]$. Let $\psi \in C^1[t_0, t_1]$ be an increasing function such that $\psi^{\prime}(t) \not = 0$, $\forall t \in [t_0, t_1]$.

If

then

and

Proof. Let

and

Then from system (3.1), one has

By (3.4) and the monotonicity of the operators $A$ and $B$, we obtain

Thus, through iteration, for $n \in \mathbb{N}$, one has

Similarly, we have

where $(AB)^0 a_1(t) = a_1(t)$ and $(BA)^0 a_2(t) = a_2(t)$.

In the following, we will prove that

where $t \in [t_0, t_1]$, and

We know that (3.7) is true for $n = 1$. In fact, one has

Introducing a change of variables $v = \frac{\psi(s) - \psi(\tau)}{\psi(t) - \psi(\tau)}$ and using the definition of beta function, we obtain

Thus

Similarly, one has

Now, by using mathematical induction, for $n = k$ and $t \in [t_0, t_1]$, we obtain

For $n = k+1$ and $t \in [t_0, t_1]$, we get by using the nondecreasing of functions $b_1(t)$ and $b_2(t)$, and the induction hypothesis that

Similar to the proof of (3.10), we can obtain

That is, (3.7) is proved. Now we prove that (3.8) holds. Since $b_1$ and $b_2$ are two continuous functions on $[t_0, t_1]$, there exists a constant $M > 0$ such that $b_1(t) \le M$ and $b_2(t) \le M$ for $t \in [t_0, t_1]$. Thus, we have

Consider the series

Using the ratio test to the series and the asymptotic approximation ^[36], we obtain

Hence, the series converges and we conclude that

Similarly, we can obtain that (3.3) holds.

Corollary 3.2. Under the hypotheses of Theorem 3.1, assume that $a_1(t)$ and $a_2(t)$ are two nondecreasing functions for $t \in [t_0, t_1]$. Then

and

Proof. Since $a_2$ is nondecreasing, one has

Thus, from (3.2) and (3.14), we get

Similarly, we obtain (3.13) holds.

4.   Main results

By Lemma 2.2, we can easily show that the following lemma holds.

Lemma 4.1. $(x(t), y(t))$ satisfies (1.1) if and only if $(x(t), y(t))$ satisfies the coupled integral system

The product space $X \times X$ is a Banach space with norm $\|(u, v)\| = \|u\| + \|v\|$. Now we give and prove the existence uniqueness theorem.

Theorem 4.2. Let

(H1) $F, G \in C([t_0, t_1] \times \mathbb{R}^2, \mathbb{R})$ and $\phi, \theta \in C[t_0- \tau, t_0]$;

(H2) there exist two positive constants $L_1$ and $L_2$ such that

(H3) ${M = \max \left\{\frac{2 L_1 (\psi(t_1) - \psi(t_0))^{\alpha}}{\Gamma(\alpha+1)}, \frac{2 L_2 (\psi(t_1) - \psi(t_0))^{\beta}}{\Gamma(\beta+1)}\right\} < 1. }$

Then the system (1.1) has a unique solution in $X \times X$.

Proof. Define the operator $T(x, y)(t) : = (T_1 x(t), T_2 y(t))$ as follows :

For $t \in [t_0- \tau, t_0]$, we have $|T_1 x(t) - T_1 u(t)| = 0$ and $|T_2 y(t) - T_2 v(t)| = 0$ if $(x, y), (u, v) \in C([t_0- \tau, t_1], \mathbb{R}) \times C([t_0- \tau, t_1], \mathbb{R})$. For $t \in [t_0, t_1]$, one has

Similarly, we can obtain

Thus, by (4.1) and (4.2), we have

which implies that the operator $T$ is a contraction by (H3). Thus $T$ has a unique fixed point by Banach fixed point theorem.

Theorem 4.3. Under the hypotheses of Theorem 4.2, system (1.1) is Ulam-Hyers stable.

Proof. Let $(u(t), v(t)) \in X \times X$ be a solution of the inequalities (2.3) and (2.4), and let $(x(t), y(t))$ be the unique solution of system (1.1) satisfying the conditions

Thus we have

Which is guaranteed by Theorem 4.2. Obviously, $(u(t), v(t))$ satisfies (2.3)-(2.4) if and only if there exist two functions $h_1(t), h_2(t) \in C[t_0, t_1]$ such that $|h_i(t)| \le \epsilon$ ($i = 1, 2$) and

Applying the $\psi$-fractional integral (2.1) to both sides of (4.3) and using Lemma 2.2 we obtain

Similarly, we get

For $t \in [t_0- \tau, t_0]$, we have $|x(t) - u(t)| = 0$ and $|y(t) - v(t)| = 0$. For $t \in [t_0, t_0 + \tau]$, one has

Similarly, for $t \in [t_0, t_0 + \tau]$, we get

Using (4.5) and (4.7), (4.6) and (4.8), respectively, we obtain

By using Corollary 3.2, we get from (4.9) and (4.10) that

Therefore, for any $t \in [t_0, t_0 + \tau]$, one has

Similarly, we have

For $t \in [t_0 + \tau, t_1]$, we adopt the similar steps as above, we may have

and

Let $z(t) = \max \limits_{r \in [-\tau, 0]} |u(t+r) - x(t+r)|$ and $w(t) = \max \limits_{r \in [-\tau, 0]} |v(t+r) - y(t+r)|$, then we obtain by (4.11) and (4.12) that

and

By utilizing Corollary 3.2, for any $t \in [t_0 + \tau, t_1]$, we get by (4.13) and (4.14) that

and

Since $|u(t) - x(t)| \le z(t)$ and $|v(t) - y(t)| \le w(t)$, for each $t \in [t_0 + \tau, t_1]$, we have

and

5.   Example

Example 5.1. Consider the following coupled delay $\psi$-Caputo fractional differential system

Here

It is easy to know that $F$ is continuous with the Lipschitz constant $L_1 = \frac{\ln6}{4}$, and $G$ is continuous with the Lipschitz constant $L_2 = \frac{\sqrt{6}}{5}$. Since $\psi(t) = \sqrt[3]{t}$, $\alpha = \frac{2}{3}$ and $\beta = \frac{3}{4}$, we have

and

Thus, all the conditions of Theorem 4.3 are satisfied. Hence (5.1) is Ulam-Hyers stable.

6.   Conclusions

In this paper, we introduced and proved a new generalized coupled Gronwall inequality. We examined the validity and applicability of our results by considered the existence and uniqueness of solutions of nonlinear delay coupled $\psi$-Caputo fractional differential system. Moreover, some result to verify sufficient conditions has been provided in this paper to determine the Ulam-Hyers stability of solutions for the considered system. Finally, a example is given to illustrate the effectiveness and feasibility of our criterion.

In the future, we will consider the nonlinear delay coupled $\psi$-Hilfer fractional differential systems, and we will study the existence and multiplicity of solutions, and the Ulam-Hyers and Ulam-Hyers-Rassias stabilities for there systems.

Acknowledgments

We are really thankful to the reviewers for their careful reading of our manuscript and their many insightful comments and suggestions that have improved the quality of our manuscript. This work is supported by Natural Science Foundation of China (11571136).

Conflict of interest

The authors declare that there are no conflicts of interest.