On the generalized Gronwall inequalities involving $\psi$-fractional integral operator with applications

1.   Introduction

The fractional Gronwall inequalities are effective tools to study the qualitative and quantitative properties of solution for fractional differential and integral equations ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]} by giving the explicit bounds of solutions. Further detail on fractional Gronwall inequalities mainly involving the Riemann-Liouville fractional integrals ^{[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]}, the Caputo fractional integrals ^[17], the Hadamard fractional integrals ^[18], the Katugampola fractional integrals ^[19,20], and the generalized proportional fractional integrals ^[21].

In ^[22], the authors produced the $\psi$-Hilfer fractional derivative as the Riemann-Liouville fractional derivative and the Caputo fractional derivative. In ^[23], considering the continuous dependence of the solution on the order and the initial condition of $\psi$-fractional differential equations, the authors presented the following theorem involving the $\psi$-fractional integral operator.

Theorem 1.1. ^[23] 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 $\psi'(t)\neq 0, t\in [a,b]$. Assume that $(1)$ $u$ and $v$ are nonnegative; $(2)$ $g$ is nonnegative and nondecreasing. If

Then

More inequalities related to $\psi$-fractional integral operator, see ^[24,25,26] for details.

As the generalizations of the classical fractional calculus operators, the $\psi$-fractional operator (i.e. the fractional derivative and integral of a function $f$ with respect to another function $\psi$) has wide applications and properties ^{[27,28,29,30,31,32,33,34]} according to the choice of the $\psi$-function, which makes the Riemann-Liouville, Hadamard, Katugampola, etc fractional integral operators and the properties of above operators can be unified and considered as a whole.

Motivated by ^[23], in order to release the limitation of the number of nonlinear terms, new generalized forms of Theorem 1.1 are presented in this article, which is effective in dealing with neutral fractional differential equations involving $\psi$-fractional integral operator.

The organization of this paper is: In Section 2, we give some preliminaries. In Section 3, main results are obtained. In Section 4, the applications of (1.1) are given. In Section 5, an example is given to illustrate our result. In Section 6, the paper is concluded.

2.   Preliminaries

We introduce some basic definitions and properties of the calculus theory, please see the details in ^[27,34].

Definition 2.1. ^[27,34] Let $f$ be an integrable function defined on $[a, b]$ and $\psi\in C^1([a, b])$ be an increasing function with $\psi'(t) \neq 0, t\in [a, b]$. The left $\psi$-Riemann-Liouville fractional integral operator of order $\gamma$ of a function $f$ is defined by

Definition 2.2. ^[27,34] Let $\gamma\in(n-1,n), f\in C^n([a, b])$ and $\psi\in C^n([a, b])$ be an increasing function with $\psi'(t) \neq 0, t\in [a, b]$. The left $\psi$-Caputo fractional derivative of order $\gamma$ of a function $f$ is defined by

where $n=[\alpha]+1, f^{[n]}(s):=(\frac{1}{\psi'(t)}\frac{d}{dt})^nf(t)$ on $[a, b]$.

Theorem 2.1. ^[35] Let $X$ be a Banach space$,$ $F:X \rightarrow X$ be a completely continuous operator. If the set $E(F)=\{y\in X: y= lFy\ \hbox{for some}\ l\in [0,1]\}$ is bounded, then $F$ has at least a fixed point.

3.   Results

Theorem 3.1. Assume that $x, a$ are integrable and nonnegative functions and $b_{j},j=1,2,\cdots,m$ are continuous integrable and nonnegative functions with $t\in [a,b]$. Let $\psi\in C^1([a,b])$ be an increasing function with $\psi'(t)\neq 0, t\in [a,b]$. If

then

provided that there exist a constant $M > 0$ such that

where $\alpha_{j_n}\in \{\alpha_1,\alpha_2,\cdots,\alpha_m\}, n\in N, b(t)=\max\{b_j(t)\}\leq M,j=1,2,\cdots,m$.

Proof. Let

Then by (3.1), we get

By the monotonicity of the operators $A$ and (3.1) and mathematical induction, for $t\in [a,b]$, we have

i.e.

where $A^0a(t)=a(t)$.

For $t\in [a,b]$, by mathematical induction, we will show that

$n\in N,$ and $\lim\limits_{n\rightarrow \infty}A^nx(t)=0$.

For $n=1$, the conclusion in (3.7) holds naturally. Using the change of variables $\theta=\frac{\psi(s)-\psi(\tau)}{\psi(t)-\psi(\tau)}$ and the Beta function $\frac{\Gamma(\alpha_j)\Gamma(\beta_j)}{\Gamma(\alpha_j+\beta_j)}=B(\alpha_j,\beta_j)$, we have

For $t\in [a,b]$, we can suppose

For $n=k+1$, using the non-increasing properties of $b_{j}(t), j=1,2,\cdots,m, t\in [a,b]$, we have

Since $b_{j}(t), j=1,2,\cdots,m$ are all continuous functions on $[a,b]$, then there exist a constant $M > 0$ such that $b(t)=\max\{b_{j}(t)\}\leq M, j=1,2,\cdots,m$. So we have

Consider the infinite series of number $\sum\limits_{n=1}^{\infty}M^{n}\sum\limits_{j_1=1}^m\sum\limits_{j_2=1}^m \cdots \sum\limits_{j_n=1}^m \frac{\Gamma(\alpha_{j_1})\Gamma(\alpha_{j_2})\cdots \Gamma(\alpha_{j_n})}{\Gamma\Big(\sum\limits_{\nu=1}^{n}\alpha_{j_\nu}\Big)}$, by virtue of the ratio test

to the infinite series of number and the asymptotic approximation in ^[36], we get

which implies that $A^nx(t)$ is convergent. Hence the conclusion in (3.2) holds.

Theorem 3.2. Under the hypotheses of Theorem $3.1$ and let $a(t)$ be a nondecreasing function for $t\in [a,b]$. Then

Proof. Since $a(t)$ ia a nondecreasing function, for $\alpha_j, j=1,2,\cdots,m$, then we get

So from (3.2) and (3.14), we have

4.   Applications

Consider the following neutral fractional equations involving $\psi$-fractional integral operator

where $\gamma>0, \gamma_i>0, i=1,2,\cdots,l$.

$(H_1)$ For the functions $f,g_i\in C(J\times R, R)$, there are some constants $c_i,c\geq 0$ such that

$(H'_1)$ For the functions $f,g_i\in C(J\times R, R)$, there are some constants $c_i,c\geq 0$ such that

$(H_2)$ $H=\sum\limits_{i=1}^l\frac{c_i(\psi(b)-\psi(a))^{\gamma_i}}{\Gamma(\gamma_i+1)}+\frac{c(\psi(b)-\psi(a))^{\gamma}}{\Gamma(\gamma+1)}<1$.

By using Definitions 2.1 and 2.2, we get the following result.

Lemma 4.1. Under the hypotheses $(H_1), (H_2)$. $x(t)$ satisfies $(4.1)$ if and only if $x(t)$ satisfies the equality

where

Theorem 4.1. Under the hypotheses $(H_1), (H_2)$. Then $(4.1)$ has a unique solution on $J$.

Proof. For $x\in C(J,R)$, denote by

with

On $B_r$, we define the operator $\Gamma x$ as

By $(H_1),(H_2)$, we have

Then for $x,y\in C(J,R)$, by $(H_2)$, we get

i.e. the operator $\Gamma$ has a unique solution on $J$.

Theorem 4.2. Under the hypotheses $(H'_1), (H_2)$. Then $(4.1)$ has at least one solution on $J$.

Proof. Consider the Cauchy problem (4.1). Define the operator $\Gamma$ as in (4.4).

Claim 1: $\Gamma$ is continuous. Let $x_n$ be a sequence such that $x_n\rightarrow x\in C^{1}(J, R)$. Then since $g_i,f$ are continuous and $(H'_1)$, then we have

Thus $(\Gamma x_n)\rightarrow (\Gamma x)$ in $C^{1}(J,R)$ and $\Gamma$ is continuous.

Claim 2: $\Gamma$ maps bounded sets into bounded sets in $C^{1}(J, R)$. Denote by $B_r$ as in Theorem 4.1. Then as (4.5), we get that $\|(\Gamma x)\|\leq r,\ t\in J,$ which implies that $\|\Gamma x\|\leq r$ and the operator $\Gamma$ is uniformly bounded.

Claim 3: $\Gamma$ maps bounded sets into equi-continuous sets of $C^{1}(J,R)$. For any $x\in B_r$, where $B_r$ is defined as in Claim 2. As $t_1\rightarrow t_2$ for $t_1,t_2\in J$, we have

Thus $\|(\Gamma x)(\hat{t}_2)-(\Gamma x)(\hat{t}_1)\| \rightarrow 0, \ \hbox{as}\ \hat{t}_1\rightarrow \hat{t}_2.$ As a consequence of Claims 1–3, it follows that $\Gamma: C^{1}(J,R) \rightarrow C^{1}(J,R)$ is continuous and completely continuous.

Claim 4: We show that the set $K=\{x\in C^{1}(J,R): x=\lambda \Gamma x\ \hbox{for some}\ 0<\lambda<1\}$ is bounded. Let $x\in K$, then $x=\lambda \Gamma x$ for some $0 < \lambda < 1$. Thus we have

By $(H'_1)$, we have

and Theorem 3.2 implies that

where $C=\max\{c_1,\cdots,c_l,c\}, \alpha_{j_{k}}\in \{\gamma_1,\cdots,\gamma_l,\gamma\},k\in N$ and which shows that the set $K$ is bounded.

By Theorem 2.1, the operator $\Gamma$ has a fixed point, which is a solution of problem (4.1).

5.   An example

Consider the following neutral $\psi$-fractional differential equation

where $\gamma=\frac{2}{3}, \gamma_1=\frac{3}{4}, i=1, g_1(t,x(t))=\frac{\sqrt{t}}{5}sin x(t), f(t,x(t))=\frac{lnt}{4}arctan x(t)$. Then $g_1,f$ are continuous and satisfy the assumptions $(H_1), (H_2)$ with $\psi(t)=\sqrt[3]{t}, c_1=\frac{\sqrt{6}}{5}, c=\frac{ln6}{4}$ and

Then by Theorem 4.1, (5.1) has a unique solution $x(t)$ on the interval $[1,6]$.

By Theorem 4.2, (5.1) also has at least one solution $x(t)$ on the interval $[1,6]$.

6.   Conclusions

In this paper, we obtained a new generalized Gronwall inequality involving $\psi$-ractional integral operator that include the results in ^[23]. Furthermore, the Riemann-Liouville, the Hadamard, the Katugampola fractional integrals etc can be considered uniformly. The feasibility of the main results is checked by considering the existence of solutions of a type of neutral fractional differential equation involving $\psi$-fractional derivative. In the future, we will consider the stabilities for the neutral $\psi$-fractional differential equation.

Acknowledgements

We are really thankful to the reviewers for their careful reading of our manuscript and their many insightful comments and valuable suggestions that have improved the quality of our manuscript. This work is supported by the Key Natural Science Project of Anhui Provincial Education Department (KJ2018A0027).

Conflict of interest

The authors declare that there are no conflicts of interest.