New generalizations for Gronwall type inequalities involving a ψ–fractional operator and their applications

1 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 12435, Saudi Arabia 2 Department of Mathematics, Faculty of Sciences, University of M’hamed Bougara, Boumerdes, Dynamic Systems Laboratory, Faculty of Mathematics, U.S.T.H.B., Algeria 3 Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand 4 Department of Mathematics, Sukkur IBA University, 65200, Sukkur-Pakistan


Background and preliminaries
The fractional calculus is nowadays considered as an important branch of mathematics, with a positive impact on several applied sciences; see, for example, the classical monograph by Samko et al. [1] and Kilbas et al. [2]. In [3], Kiryakova proposed a theory of a generalized fractional calculus (generalizations of fractional integrals and derivatives) and their applications. One of the proposed generalizations of the fractional calculus operators which has wide applications is the ψ-fractional operator. This notion is referred to as the fractional derivative and integral of a function with respect to another function ψ. Several properties of this operator could be found in [1,2,[4][5][6][7]. For some new developments on this topic; see [8][9][10][11][12] and references therein.
Inequalities play a vital role in both pure and applied mathematics. In particular, inequalities involving the derivative and integral of functions are very captivating for researchers [13]. Integral inequalities have many applications in the theory of differential equations, theory of approximations, transform theory, probability, and statistical problems and many others. Therefore, in the literature we found several extensions and significant developments for the forms of classical integral inequalities. Furthermore, the study of qualitative and quantitative properties of solution of fractional differential and integral equations requires the use of various types of integral inequalities.
As our concern is Gronwall's inequality, we state its classical form as follows.
Theorem 1.1. [14] Let u (t) , g (t) be nonnegative functions for any t ∈ [a, T ] and a, T and v be nonnegative constants such that We review some recent results for the sake of comparison. In [15], Bellman generalized Theorem 1.1 by letting v be a nonnegative and nondecreasing function, which is stated in many references such as [13,16]. In [17], Pachpatte also established the following inequality In [19], Jiang and Meng discussed the following integral inequality u r (t) ≤ v (t) + g 1 (t) t a g 2 (τ) u (τ) p dτ + g 1 (t) t a g 3 (τ) u (τ) q dτ, r, p, q > 0, (1.5) under the same initial condition. For further detail on Gronwall-type inequalities involving the Riemann-Liouville fractional integrals [20,21], for the Hadamard fractional integrals [22,23] and for the Katugampola fractional integrals [24,25], where other formulations of the Gronwall's inequality can be found via fractional integrals [26,27]. As one of the objectives of this article is to propose a generalized Gronwall's inequality, we state the inequality of Gronwall which was first introduced in fractional settings in [28] u (t) ≤ v (t) + g (t) t a (t − τ) α−1 u (τ) dτ, α > 0, (1.6) where u (t) , v (t) are nonnegative functions and g (t) is a nonnegative and nondecreasing function for t ∈ [0, T ] . In [12], the Gronwall's inequality (1.6) was generalized as under where ψ ∈ C 1 [a, T ] is an increasing function such that ψ (t) 0, ∀ t ∈ [a, T ]. Further in [29], Willett discussed the linear inequality The following generalizations of the Gronwall type inequality were given in [30,31] Oriented by above discussion, some other generalizations for the inequalities (1.1) and (1.6) have been elaborated. For relevant results; see [32][33][34][35][36][37] and the references cited therein.
The main objective of this paper is to extend Theorem 1.1, Gronwall-type inequalities (1.6), (1.7), (1.9) and (1.10) to the general case by the implementation of ψ-fractional operator. We claim that the results of this paper are obtained within a general platform that includes all previous forms as particular cases. As applications, we prove the existence and uniqueness of solutions for ψ-fractional initial value problem and study the Ulam-Hyers stability of solutions for ψ-fractional differential equations. Particular examples are given to confirm the proposed results.
We continue with the definitions and properties of the fractional derivative and integral of a function u with respect to given function ψ. These definitions are referred to as ψ-fractional operators.
The standard Riemann-Liouville fractional integral of order α > 0, namely (1.11) The left-sided factional integrals and fractional derivatives of a function u with respect to another function ψ in the sense of Riemann-Liouville are defined as follows [2] J α,ψ (1.12) and respectively, where n = [α] + 1 and u, ψ ∈ C n [a, T ] are two functions such that ψ is increasing and ψ (t) 0, for all t ∈ [a, T ].
We propose the remarkable paper [38] in which some generalizations using ψ-fractional integrals and derivatives are described. In particular, we have (1.14) where J α a+,t , H J α a+,t and ρ J α a+,t are the classical Riemann-Liouville, Hadamard and Katugampola fractional operators, respectively.
Let α, β > 0. Then, we have the following For α, β > 0, the following properties are valid The Mittag-Leffler function is given by the series where Re (α) > 0 and Γ (z) is a Gamma function. In particular where er f (z) error function.
We outline the structure of the paper as follows: Section 2 is devoted to the new generalizations for the ψ-Gronwall-type inequality. Meanwhile, two remarks are addressed to show that the obtained forms of Gronwall-type inequality include other results as particular cases. Section 3 provides applications for the proposed results. Firstly, we demonstrate that the new inequalities can be used as handy tools in the study of existence and uniqueness of solutions of ψ-fractional initial value problem. Secondly, we use the the new inequalities to investigate the Ulam-Hyers stability of ψ-fractional differential equations. We also give some interesting examples to illustrate the effectiveness of our main results in Section 4. At last, the paper is concluded in Section 5.

New generalized ψ-Gronwall's inequality
By the same arguments of [30, Lemma 2.1], we can easily obtain the following result, which plays a very important role in proving the main results.
Proof. Clearly, U (a) ≥ 0. If the proposition is false, that is where φ is an empty set, then there exists a point t 0 on [a, T ) which satisfies U| [a,t 0 ] ≥ 0, U (t 0 ) = 0. The function U is a strictly monotonic decreasing function on (t 0 , t 0 + ε) ⊂ [a, T ). Let ε > 0. Hence, for each t ∈ (t 0 , t 0 + ε) , we have U (t) < 0 and Let t −→ t 0 , then we have a contradiction, that is, 0 ≥ 1. The proof of Lemma 2.1 is completed.
In light of the approach introduced in [30], we generalize Gronwall's inequality as follows. If By Dirichlet's formula and using the definition of Beta function, the following equality is given From the fact that g i (i = 0, 1, . . . , n) are monotonic increasing functions on [a, T ) and g i (s) ≤ g i (t), for all s ≤ t, we obtain By using (2.5), the inequality (2.6) can be rewritten as which implies that Corollary 2.1. Under the hypotheses of Theorem 2.1, assume further that u (t) is a nondecreasing function for t ∈ [a, T ), then where E α i is the Mittag-Leffler function.
Proof. From (2.4) and v (t) is a nondecreasing function for t ∈ [a, T ) , we have Then, with the help of (1.12) and Lemma 1.1, it follows that The proof is completed.  With the help of this Theorem 2.1, we have the following results.

13)
Here ε is an arbitrary given positive number.

Some applications
In this section, we present some applications of Theorem 2.1 and Theorem 2.2 to obtain the existence and uniqueness of the solution for ψ-fractional initial value problem. Further, we apply the main results of this work to study the stability of the ψ-fractional differential equations.

Existence and uniqueness
Consider the initial value problems with the ψ-fractional derivative where 0 < α 1 < α 2 < · · · < α n < 1, D α i ,ψ a+,t , J α i ,ψ a+,t denote the left-sided of fractional derivative and fractional integral operators of a function u with respect to another function ψ in the sense of Riemann-Liouville, f ∈ C([a, T ] × R, R) and δ ∈ R.
The following lemma presents the uniqueness of solution for the initial value problem (3.1). For simplicity of presentation, we set f u ≡ f (t, u (t)) . Lemma 3.1. For each t ∈ [a, T ), suppose that γ (t) ≥ 0 is a bounded and monotonic increasing function and If the initial value problem (3.1) has a solution, then the solution is unique.
Proof. The proof will be given in two claims. Claim 1. Since 0 < α 1 < α 2 < . . . < α n < 1, then according to Lemma 1.2, we get where c i , (i = 1, 2, . . . , n) are some real numbers. By (3.3) and (1.15) we also have Applying the fractional integral operator J 1−α n ,ψ a+,t to both sides of (3.4), we get Hence, we have Since Claim 2. Let u 1 and u 2 be two solutions of (3.1). Then from (3.6) and (3.2), we get Therefore, we can conclude that Then the initial value problem (3.1) has at most one solution. The proof is completed.
We have the following lemma.
Lemma 3.2. Let f, g : [a, T ] × R → R be two continuous functions and let u, v be solutions of the two systems (3.9). Assume that the following assumptions hold: • (A 1 ) There exists a positive constant c such that (A 2 ) There exists a continuous function χ : Then, for all t ∈ [a, T ], we have the following inequality:

10)
where the function w : [a, T ] → R is defined by Proof. With the help of (A 1 ) and (A 2 ), it follows that Setting By applying Theorem 2.1 to (3.12), the desired inequality (3.10) is obtained. This completes the proof.
Remark 3.1. In particular, when f = g then χ (t) ≡ 0, we obtain a simpler formula (3.10) with (3.14) In view of inequality (3.10) with (3.14), we see that the solution of system (3.9) is unique.

Ulam-Hyers stability
In this subsection, we study the Ulam-Hyers stability of the initial value problem (3.1).

Remark 3.2.
For every > 0, a function u ε ∈ C([a, T ], R) is a solution of the inequality 25) if and only if there exists a function h ∈ C([a, T ], R), (which depends on u ε ) such that Proof. If u ε is a solution to (3.25), then u ε is a solution to the problem For each t ∈ [a, T ], one has In virtue of (2.3), one has where By using Corollary 2.1, we obtain Therefore This shows that (3.26) holds. The proof is completed.

Particular examples
In this section, we provide some particular examples that validate and confirm the proposed theorems.

Conclusions
In this paper, we introduced new generalizations for Gronwall's inequality within the ψ-fractional integral operators. The results of this paper provide general forms of Gronwall's inequality that include the forms obtained in [12,30,31]. Furthermore, Gronwall's inequalities involving fractional integrals of Riemann-Liouville, Hadamard and Katugampola types as well as fractional integrals of a function with respect to another function are recovered for particular cases of function ψ.
To examine the validity and applicability of our results, we discussed the existence and uniqueness of solutions of ψ-fractional initial and boundary value problems which are an important and useful contributions to the existing theory. On the other hand, the stability of ψ-fractional differential equations was studied via the obtained generalized ψ-Gronwall's inequality. Interesting examples are discussed at the end for the sake of confirming the results.
Reported results in this paper can be considered as a promising contribution to the theory of fractional integral inequalities. These results can be used to study and develop further quantitative and qualitative properties of generalized fractional differential equations.