On existence–uniqueness results for proportional fractional differential equations and incomplete gamma functions

*Correspondence: tabdeljawad@psu.edu.sa 2Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia 3Department of Medical Research, China Medical University, 40402 Taichung, Taiwan Full list of author information is available at the end of the article Abstract In this article, we employ the lower regularized incomplete gamma functions to demonstrate the existence and uniqueness of solutions for fractional differential equations involving nonlocal fractional derivatives (GPF derivatives) generated by proportional derivatives of the form


Introduction
Over the last decades there has been an extensive use of fractional dynamic equations in modeling and describing complex and chaotic systems [1][2][3][4][5][6]. This fact has motivated researchers to discover new fractional operators. After Riemann-Liouville fractional derivatives, Caputo fractional derivatives were defined to transform constant functions to zero and hence have similar properties as ordinary derivatives. On the other hand, some researchers have spent their efforts to define more general classes of fractional operators by inserting more general or new kernels. Since the appearance of the concept of conformable derivatives, which allow the derivation up to arbitrary order and resemble ordinary derivatives, in [7] and their modifications in [8,9], several researchers realized that conformable type derivatives can be used to produce nonlocal more generalized fractional derivatives [10,11]. Indeed, in [10], the authors used the conformable derivatives presented in [8] to present a class of generalized nonlocal fractional derivatives, called conformable fractional derivatives, slightly different from the so-called Katugampola [12,13]. In fact, the derivatives in [10] and the Katugampola are characterized as fractional derivatives of a function with respect to another function g(t), with g(t) = t ρ ρ and g(t) = (t-a) ρ ρ , respectively [14]. Also, the authors in [11] used particular versions of the proportional derivatives presented in [9], called modified conformable derivatives, to present the fractional counterpart pro-portional derivatives and integrals. Later, the authors in [15,16] generalized proportional derivatives and used them to generate more generalized classes of nonlocal fractional integrals and derivatives, and in [17] the authors discussed a new type of fractional operators combining proportional and classical derivatives/integrals. Besides, there have been many attempts to generate fractional operators with more complicated kernels with the hope to describe complex systems more accurately. Some authors thought of replacing the singular kernels with power law by nonsingular kernels with either exponential law [18] or Mittag-Leffler law [19] via ML kernel and via generalized ML kernel. For the interest of readers, we attract their attention to the recent work where the author studied the relationships between the model of Prabhakar depending on fractional integrals with generalized ML kernels and Atangana-Baleanu model in [19] and its extension in [20].
Fixed point techniques are always used to prove existence and uniqueness of ordinary and fractional dynamic equations [21][22][23][24][25][26][27]. It turns out that the structure of the kernel of fractional operator affects the applied analysis technique in proving the existence and uniqueness of solution or its stability criteria due to the natural appearance of the exponential function in the kernel of proportional fractional point technique in proving the existence and uniqueness of solutions for fractional differential equations in the setting of GPF derivatives. Indeed, we investigate the following Cauchy problem: where ρ ∈ (0, 1], n = -[-α], -∞ < a < b < +∞, b k ∈ R, and c D α,ρ a denotes the Caputo proportional fractional derivative of order α, f : [a, b] × R − → R is a given continuous nonlinear function.
Further, we will also obtain a result for the following fractional differential equation involving proportional fractional order with initial conditions: where n = -[-α], R D α,ρ a denotes the generalized proportional integral of Riemann-Liouville type of order α.
The paper is organized as follows: Sect. 2 presents some definitions and results needed in the rest of the article. Sect. 3 discusses new lemmas needed for the proofs of the existence and uniqueness of the Cauchy problems proposed in Sect. 4. I α
Proof The proof follows by writing D ρ = (1ρ) + ρD and applying the binomial theorem.
Remark 2.8 Let ρ ∈ (0, 1] and α > 0 with n -1 < α ≤ n, n ∈ N. By using Lemma 2.7, we can write the definition of the left generalized proportional derivative of Caputo type of the function ∈ C (n) [a, b] as follows: we have the following properties: Definition 2.10 ( [28,29]) Let α ∈ C ( (α) > 0), we have the following definitions: The upper incomplete gamma function is defined by The lower incomplete gamma function is defined by The upper regularized incomplete gamma function is defined by The lower regularized incomplete gamma function is defined by The functions P and Q are also called "Incomplete gamma function ratios".
For all t ≥ 0, we have the following properties:

Incomplete gamma functions vs fractional proportional integrals
In this section, we present new essential lemmas, which allow us to proceed in proving our main results about the existence and uniqueness of solutions for GPF differential equations.
Proof The proof follows just by following the same steps as in Lemma 3.1.
, and α > 0. Then where the function P is defined by (22).
Using property (26) in Lemma 2.11, we get The proof is completed.
Proof To calculate the above limit, the sign to the term inside the absolute value must be studied.
For ρ = 1, we look at the three cases α = 1, α < 1, and α > 1 as follows: hence the integral has the value zero as t 2 → t 1 .

Some Cauchy problems in the frame of fractional proportional derivatives
This section is devoted to applying the above proven essential lemmas to study the initial value problem (2), and then we deduce the results of problem (3). The proof of the next result follows by Theorem 5.3 in [11] or (17) in Proposition 2.9 and Lemma 2.7.

Lemma 4.1 For ∈ C([a, b], R), the solution of the following linear problem
is given by the integral equation where ϕ k = k j=0 k j ρ j (1ρ) k-j b k for ρ ∈ (0, 1) and ϕ k = b k for ρ = 1.
Let X = C([a, b], R) be a Banach space of all continuous functions from [a, b] to R endowed with the norm u = sup t∈ [a,b] |u(t)|.
Associated with problem (2), we define a fixed point operator T : X → X by Consider the following hypothesis: where P is defined by means of (22).
where h k is defined in Lemma 3.5 with k ∈ {0, 1, . . . , n -1}, and consider M R = {u ∈ X : u ≤ R}. We first show that TM R ⊂ M R . For u ∈ M R and t ∈ [a, b], we have Using Lemma 3.5 and Lemma 3.2, we obtain which implies that Tu ≤ R for any u ∈ M R . We get TM R ⊆ M R . Next we prove that the operator T is a contraction mapping. For u, v ∈ X, for all t ∈ [a, b], we have Taking the supremum over all t ∈ [a, b] yields By condition (41) the operator T is a contraction. Hence, by the Banach fixed point theorem, problem (2) has a unique solution on [a, b]. The proof is completed.
Now, based on Leray-Schauder alternative fixed point theorem [30], we present the following result about the existence of solutions for the investigated problem (2).
Consider the following hypothesis: (H 2 ) f : [a, b] × R → R is a continuous functions, and there exist real positive constants ς 0 , ς 1 such that

Theorem 4.3
Let ρ ∈ (0, 1), and assume that (H 2 ) holds. If then the initial value problem (2) has at least one solution on [a, b].
Proof We first show that the operator T is completely continuous.
The continuity of f implies the continuity of the operator T. Let ϒ be any nonempty bounded subset of X. Then there exists ξ > 0 such that, for any u ∈ ϒ, u ≤ ξ . Notice that from condition (H 2 ), for all u ∈ ϒ, we have Next we prove that T(ϒ) is uniformly bounded. Let u ∈ ϒ. Then, for any t ∈ [a, b], we have Consequently, u < +∞ for any u ∈ ϒ. Therefore, T(ϒ) is uniformly bounded.
Then, by using Lemma 3.4, we obtain Thus, the operator T is equicontinuous. Hence, by Arzela-Ascoli theorem, we deduce that the operator T is completely continuous. Finally, we will verify that the set M(T) = {u ∈ X : u = mTu for some 0 < m < 1} is bounded.
For all u ∈ M(T), and for any t ∈ [a, b], we have which yields u ≤ n-1 k=0 , which proves that M is bounded. Thus, by Leray-Schauder alternative theorem, the operator T has at least one fixed point. Hence, the initial value problem (2) has at least one solution on [a, b]. The proof is completed. Because P(α, x) ∈ [0, 1] for all α, x ∈ R + , we obtain the following results.
Corollary 4.6 Let ρ ∈ (0, 1), and assume that (H 2 ) holds. If then the initial value problem (2) has at least one solution on [a, b]. Now, concerning the study of the initial value problem of Riemann-Liouville type (3), we present the following results.
Then inequality (43) is satisfied. Hence, by Remark 4.8 and Theorem 4.3, we conclude that the GPF problem (53) has at least one solution on [0, 1].