Positive Solutions of a Generalized Nonautonomous Fractional Differential System

Department of Mathematics, Hodeidah University, Al-Hudaydah, Yemen Department of Physics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia Research Center for Advanced Materials Science (RCAMS), King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia Nanoscience Laboratory for Environmental and Biomedical Applications (NLEBA), Department of Physics, Faculty of Education, Ain Shams University, Roxy, Cairo 11757, Egypt Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia Department of Physics, College of Sciences, University of Bisha, PO Box 344, Bisha 61922, Saudi Arabia Physics Department, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt


Introduction
Fractional-order systems (FOSs) are viewed as more satisfactory than integer-order systems in some real-world issues, as fractional derivatives (FDs) supply a brilliant tool for the description of memory and hereditary properties of di erent processes and materials. Numerous applications of FOSs in various elds of engineering, chemistry [1], physics [2], aerodynamics [3,4], and biological sciences [5] were presented. As a result, the topic of fractional di erential equations (FDEs) has acquired great importance and unparalleled interest in recent times (see [6][7][8]). D-Gejji and his coauthor [9] have introduced a brief analysis of a system of FDEs. Once upon a time, Zhang [10] has discussed the existence of positive solutions for the next FDE: D s 0 + ω(ϰ) f(ϰ, ω(ϰ)), 0 < s < 1.
Since the beginning of 1695 fractional calculus was presented, over these years, there have been various local and nonlocal derivatives and amazing developments of FDs and integrals which have various kernels and applications, and the most famous of these derivatives were introduced by Riemann-Liouville, Caputo, Hadamard, and Erdélyi-Kober (see [8]). Recently, Caputo and Fabrizio [13] suggested a new definition of FD without a singular kernel based on exponential law.
en, some properties of this operator are proved by Losada in [14]. In this regard, Atangana and Baleanu [15] contributed to sitting a novel FD with nonlocal and nonsingular kernel based on Mittag-Leffler law. Some recent applications and results of these derivatives can be found in [16][17][18].
It was necessary to present a FD with respect to another function, by utilizing the FD in the concept of Riemann-Liouville given by [8].
where m − 1 < s < m, m � [s] + 1 for s ∉ N. Like this definition is restricted to the possible FDs that contain the differentiation operator following up on the integral operator.
Very recently, Almeida [19], using the notion of the FD with respect to another function, proposed a new FD called the ψ-Caputo; in this framework, Sousa and Oliveira [20], based on previous ideas, proposed a derivative that generalizes a class of FDs, the so-called ψ-Hilfer. A perfect number of recent literature on the existence of solutions to various fractional problems that include some of the recent fractional operators can be found in [21][22][23][24][25]. For instance, some existence of positive solutions to the ψ-weighted Caputo FDE: is have been obtained by using lower and upper solutions along with fixed point approach (see [23]).
To the best of our knowledge, there is no contribution, regarding the existence of positive solutions for a system of FDEs involving ψ-Riemann-Liouville FDs. In pursuance of this, in our current paper, we transact with the existence and uniqueness of positive solutions for the following system: where 0 < s i < 1, and for all i � 1, . . . , m, D Our work is organized as follows. In Section 2, some primary ordnances on ψ-fractional calculus are given. e main results are acquired by applying Guo-Krasnoselskii's and Banach's fixed point theorems in Section 3. In section 4, we attempt to present some examples of the proposed system. In the last section, concluding results are provided.
If ω is continuous on (11) takes the type Proof. We first use Lemma 1 if ρ ≤ s, or the relation D ρ;ψ if ρ ≥ s, and then property (10). is gives where we used the next fact: To prove the part two, let M � max|ω(ϰ)|; then, using Definition 2, we get ϰ a D ρ;ψ Hence, (11) reduces to (12).
Theorem 2 (see [26]). Let X be a Banach space and let the operator Q: X ⟶ X be a contraction. en, Q admits a unique fixed point x ∈ X with Qx � x.

Main Results
Here, we demonstrate the existence of positive solutions to the following system of FDEs: Denote by X � C([) the space of all continuous real functions on [. Let X m � X × · · · × X √√√√ √ √√√√ √ m− times denote the Banach space with the following norm: As per Lemma 4, the system (17)  Let K ⊂ X m be a cone defined by and the pair (X m , K) represents an ordered Banach space. Let Q: X m ⟶ X m be the operator defined as Qω(ϰ): � (Q 1 ω 1 (ϰ), . . . , Q m ω m (ϰ)) such that which implies us, Q(S) is bounded. Next, we show that Q: K ⟶ K is continuous. Let ω∈ K where ‖ω‖ ≤ κ and let P � ϖ ∈ K: ‖ω − ϖ‖ ≤ r 1 .

Journal of Mathematics
Proof. Let ω, ϖ ∈ K. en, As per condition (33) along with eorem 2, Q has unique fixed point in K, which is the unique positive solution.