# Positive Solutions of a Generalized Nonautonomous Fractional Differential System

Authors: Mohammed S. Abdo; Heba Y. Zahran; El Sayed Yousef; Emad E. Mahmoud; Abdel-Haleem Abdel-Aty
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this work, we investigate the existence and uniqueness of positive solutions to a system of nonautonomous fractional differential equations. The fractional derivative of the system at hand isψ- Riemann–Liouville fractional derivative. Some significant properties of this derivative that assist in the analysis of our results substantial are proved. The analysis is based on Guo-Krasnoselskii’s and Banach’s fixed point theorem. Finally, we give two examples to verify our results.


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## Body

## 1. 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 different processes and materials. Numerous applications of FOSs in various fields of engineering, chemistry [1], physics [2], aerodynamics [3, 4], and biological sciences [5] were presented. As a result, the topic of fractional differential equations (FDEs) has acquired great importance and unparalleled interest in recent times (see [6–8]). 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:(1)D0+sωϰ=fϰ,ωϰ,0<s<1.The existence result of positive solutions to the multiterm FDE is given by(2)LDωϰ=fϰ,ωϰ,where LD=Dsm−am−1Dsm−1−⋯−a1Ds1, 0<s1<⋯<sm−1<sm<1, ai>0 have been presented by Babakhani and Daftardar-Gejji [11]. In this regard, Daftardar-Gejji in [12] have studied the existence of positive solutions for the following system of FDE:(3)D0+siωiϰ=fiϰ,ω¯ϰ,ωi0=0,i=1,…,m,where ω¯ϰ:=ω1ϰ,…,ωmϰ.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. Then, 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 [16–18].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].(4)Da+s;ψωϰ=1ψ′ϰddϰmℐ0+m−s;ψωϰ,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 [21–25]. For instance, some existence of positive solutions to the ψ-weighted Caputo FDE:(5)D0+s;ψ;wωϰ=fϰ,ωϰ,ω0=ω0.This 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 involvingψ-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:(6)D0+si;ψωiϰ=fiϰ,ω1ϰ,…,ωmϰ,ωi0=0,i=1,…,m,where 0<si<1, and for all i=1,…,m, D0+s1;ψ,D0+s2;ψ,…,D0+sm;ψ denote generalized Riemann–Liouville FDs of order s1,s2,…,sm, respectively, and fi:℧×ℝ+m⟶ℝ+, ψ:℧⟶ℝ+ are given functions with ψ is an increasing function satisying ψ′ϰ>0 for all ϰ∈℧.Our work is organized as follows. In Section2, some primary ordnances on ψ-fractional calculus are given. The 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.


## 2. Preliminaries

Definition 1 (see [8]).
Lets>0, and ω:a,b⟶ℝ such that ω∈ℒ1a,b. Then, the ψ -Riemann–Liouville FI of order s is defined by(7)ℐa+s;ψωϰ=1Γs∫aϰψ′ζψϰ−ψζ1−sωζdζ,a<ϰ<b.Definition 2 (see [8]).
Letm=s+1∈ℕ, and ω:a,b⟶ℝ. Then, the ψ-Riemann–Liouville FD of order s is given by(8)Da+s;ψωϰ=1Γm−s1ψ′ϰddϰm∫aϰψ′ζψϰ−ψζs−m+1ωζdζ=Dm;ψℐ0+m−s;ψωϰ,a<ϰ<b,where Dm;ψ=1/ψ′ϰd/dϰm.In particular, if m=1, then Da+s;ψωϰ=D1;ψℐ0+1−s;ψωϰ, where D1;ψωϰ=ω′ϰ/ψ′ϰ.Lemma 1 (see [8]).
Ifs,ρ>0, then ℐa+s;ψDa+ρ;ψωϰ=ℐa+s+ρ;ψωϰ, a.e. on. [a, b]Lemma 2.
Let0<s<1. If ω∈ℒ1J1,ℝ, then Da+s;ψℐa+s;ψωϰ=ωϰ, a.e. on [a, b].Proof.
Letω∈ℒ1a,b. Then, from Definitions 1, 2, and Lemma 1, we have(9)Da+s;ψℐa+s;ψωϰ=D1;ψℐa+1−s;ψℐa+s;ψωϰ=D1;ψℐa+1;ψωϰ=1ψ′ϰddϰ∫aϰψ′ζωζdζ,=ddϰ∫aϰωζdζ=ωζ.Lemma 3.
Let0<s<1, if Da+s;ψωϰ∈ℒ1a,b, then(10)ℐa+s;ψDa+s;ψωϰ=ωϰ−ℐa+1−s;ψωϰ|ϰ=aψϰ−ψas−1Γs.Proof.
The proof is deduced directly from Definitions1 and 2, then the integration by part.Lemma 4.
LetDa+ρ;ψωϰ is integrable, then(11)ℐa+s;ψDa+ρ;ψωϰ=ℐa+s−ρ;ψωϰ−ℐa+1−ρ;ψωϰϰ=aψϰ−ψas−1Γs,0<ρ≤s<1.
Ifω is continuous on a,b, then Da+ρ;ψωϰ is integrable, ℐa+1−ρ;ψωϰϰ=a=0 and (11) takes the type(12)ℐa+s;ψDa+ρ;ψωϰ=ℐa+s−ρ;ψωϰ,0<ρ≤s<1.Proof.
We first use Lemma1 if ρ≤s, or the relation Da+ρ;ψℐa+s;ψωϰ=Da+ρ−s;ψωϰ if ρ≥s, and then property (10). This gives(13)ℐa+s;ψDa+ρ;ψωϰ=Da+ρ−s;ψℐa+ρ;ψDa+ρ;ψωϰ=Da+ρ−s;ψωϰ−ℐa+1−ρ;ψωϰ|ϰ=aψϰ−ψaρ−1Γρ=ℐa+s−ρ;ψωϰ−ℐa+1−ρ;ψωϰ|ϰ=aℐa+s−ρ;ψψϰ−ψaρ−1Γρ=ℐa+s−ρ;ψωϰ−ℐa+1−ρ;ψωϰ|ϰ=aψϰ−ψas−1Γs,where we used the next fact:(14)ℐa+s−ρ;ψψϰ−ψaρ−1=ΓρΓsψϰ−ψas−1.
To prove the part two, letM=maxωϰ; then, using Definition 2, we get(15)∫aϰDa+ρ;ψωζdζ=∫aϰD1;ψℐa+1−ρ;ψωζdζ=∫aϰ1ψ′ζddζℐa+1−ρ;ψωζdζ≤M∫aϰ1ψ′ζddζ1Γ1−ρ∫aζψ′τψζ−ψτ−ρdτdζ=M∫aϰ1ψ′ζddζψζ−ψa1−ρΓ2−ρdζ=MΓ2−ρψϰ−ψa1−ρ.
SoDa+ρ;ψωϰ is integrable. Also,(16)ℐa+1−ρ;ψωϰϰ=a≤MΓ1−ρ∫aϰψ′ζψϰ−ψζ−ρdζϰ=a=MΓ2−ρψϰ−ψa1−ρϰ=a=0.
Hence, (11) reduces to (12).Theorem 1 (see [26]).
LetX,K be a Banach space, and let Ω1,Ω2⊂X with 0∈Ω1 and Ω1¯⊂Ω2. Assume Q:K∩Ω2¯/Ω1⟶K be completely continuous s.t., either(i)
Qω≤ω,forω∈K∩∂Ω1andQω≥ω,forω∈K∩∂Ω2,or(ii)
Qω≥ω,forω∈K∩∂Ω1andQω≤ω,forω∈K∩∂Ω2.
Then,Q has a fixed point in K∩Ω2¯/Ω1.Theorem 2 (see [26]).
LetX be a Banach space and let the operator Q:X⟶X be a contraction. Then, Q admits a unique fixed point x∈X with Qx=x.


## 3. Main Results

Here, we demonstrate the existence of positive solutions to the following system of FDEs:(17)D0+si;ψωiϰ=fiϰ,ω¯ϰ,ωi0=0,i=1,…,m,where 0<si<1, ϰ∈℧:=0,b, ω¯=ω1,ω2,…,ωm, D0+si;ψ denote ψ-Riemann–Liouville FDs of order si, ∀i, and fi:℧×ℝ+m⟶ℝ+ where ℝ+m=ℝ+×⋯×ℝ+︸m−times.Denote byX=C℧ the space of all continuous real functions on ℧. Let Xm=X×⋯×X︸m−times denote the Banach space with the following norm:(18)ω¯=max1≤i≤mmaxϰ∈℧ωiϰ,forωi∈X,ω¯∈Xm.As per Lemma4, the system (17) is equivalent to(19)ωiϰ=ℐ0+si;ψfiϰ,ω¯ϰ,i=1,…,mandϰ∈℧.LetK⊂Xm be a cone defined by(20)K=ω¯∈Xm:ωiϰ≥0,∀i=1,…,m,ϰ∈℧,and the pair Xm,K represents an ordered Banach space.LetQ:Xm⟶Xm be the operator defined as Qω¯ϰ:=Q1ω1ϰ,…,Qmωmϰ such that(21)Qiωiϰ=ℐ0+si;ψfiϰ,ω¯ϰ,i=1,…,mandϰ∈℧.Lemma 5.
It is assumed thatfi:℧×ℝ+m⟶ℝ+ be continuous ∀i=1,…,m. Then, Q:K⟶K is completely continuous mapping.Proof.
LetS⊂K be bounded. Then, there exists κ>0 such that ω¯≤κ, for all ω¯∈S. Now, we prove that QS is bounded. It is noted that QK⊂K, since fi are nonnegative for all i.
LetM=maxϰ∈℧1+f¯ϰ,ω¯:0<ω¯<κ. For ϰ∈℧ and ω¯∈S, we have(22)Qiωiϰ≤ℐ0+si;ψfiϰ,ω¯ϰ=1Γsi∫0ϰψ′ζψϰ−ψζsi−1fiζ,ω¯ζdζ≤1Γsi∫0ϰψ′ζψϰ−ψζsi−1f¯ζ,ω¯dζ≤MΓsi+1ψϰ−ψ0si≤MΓsi+1ψb−ψ0si,which implies(23)Qiωi=max1≤i≤mmaxϰ∈℧Qiωiϰ=maxϰ∈℧Q1ω1ϰ,…,Qmωmϰ≤maxϰ∈℧MΓs1+1ψb−ψ0s1,…,MΓsm+1ψb−ψ0sm.
Thus,QS is bounded.
Next, we show thatQ:K⟶K is continuous. Let ω¯∈K where ω¯≤κ and let P=ϖ¯∈K:ω¯−ϖ¯≤r1. Then, ϖ¯≤κ+r1:=r, for each ϖ¯∈P. As fi are continuous on ℧×0,rm, they are uniformly continuous in ℧×0,rm. Consequently, let ε>0, there exists a 0<δ<r1 such that f¯ϰ,ω¯−f¯ϰ,ϖ¯<εΓs+1/ψb−ψ0s, where ϰ∈℧, s=maxs1,…,sm, for ω¯−ϖ¯<δ. If ω¯−ϖ¯<δ, then ϖ¯∈P and ϖ¯≤r. Since ϖ¯∈P⊂K, ϖi≤r. Hence,(24)Qω¯−Qϖ¯=max1≤i≤mmaxϰ∈℧Qiωiϰ−Qiϖiϰ≤max1≤i≤mmaxϰ∈℧1Γsi∫0ϰψ′ζψϰ−ψζsi−1fiζ,ω¯ζ−fiζ,ϖ¯ζdζ=1Γs∫0ϰψ′ζψϰ−ψζs−1f¯ζ,ω¯−f¯ζ,ϖ¯dζ≤εΓs+1ψb−ψ0s1Γs∫0ϰψ′ζψϰ−ψζs−1dζ≤εΓs+1ψb−ψ0sψb−ψ0sΓs+1=ε.
Thus,Q is continuous on K.
Letω¯∈S, ϰ1,ϰ2∈℧ with ϰ1<ϰ2. Then, for ε>0 we can select δ=δε=ερΓs/2M1/s where s=maxs1,…,sm, and ρ=mins1,…,sm, and 0<ψϰ2−ψϰ1<δ, we have(25)Qiωiϰ1−Qiωidϰ2=1Γsi∫0ϰ1ψ′ζψϰ1−ψζsi−1fiζ,ω¯ζdζ−1Γsi∫0ϰ2ψ′ζψϰ2−ψζsi−1fiζ,ω¯ζdζ=1Γsi∫0ϰ1ψ′ζψϰ1−ψζsi−1−ψϰ2−ψζsi−1fiζ,ω¯ζdζ−1Γsi∫ϰ1ϰ2ψ′ζψϰ2−ψζsi−1fiζ,ω¯ζdζ≤2MsiΓsiψϰ2−ψϰ1si+ψϰ1−ψ0si−ψϰ2−ψ0si≤2MsiΓsiψϰ2−ψϰ1si≤2MsiΓsiδsi≤2MρΓsδs=ε.
As0<si<s<1, Γs<Γsi. Therefore, Qiωiϰ2−Qiωiϰ1<ε, ∀i, which implies Qω¯ϰ2−Qω¯ϰ1<ε. Consequently, QS is equicontinuous, and as per Arzela–Ascoli theorem, the operator Q:K⟶K is completely continuous.Theorem 3.
Letfi:℧×ℝ+m⟶ℝ+i=1,…,m be continuous. Suppose that there exist d1,d2>0d1≠d2 such that d1≤fiϰ,ω¯≤d2, i=1,…,m. Then, the system (17) has at least one positive solution.Proof.
We set(26)V1=ω¯∈K:ω¯≤d1ψb−ψ0ssΓs,V2=ω¯∈K:ω¯≤d2ψb−ψ0sρΓs,where s=maxs1,…,sm, and ρ=mins1,…,sm. For ω¯∈K∩∂V2, and ϰ∈℧, we have(27)0≤ωiϰ≤d2ψb−ψ0sΓs+1,∀i.
Asfiϰ,ω¯≤d2, we get(28)Qiωiϰ=1Γsi∫0ϰψ′ζψϰ−ψζsi−1fiζ,ω¯ζdζ≤d2Γsi∫0ϰψ′ζψϰ−ψζsi−1dζ≤d2ψb−ψ0siΓsi+1.
Hence,(29)Qω¯=max1≤i≤mQiωi=max1≤i≤mmaxϰ∈℧Qiωiϰ≤d2ψb−ψ0siΓsi+1≤d2ψb−ψ0sρΓs=ω¯.
In this regard, forω¯∈K∩∂V1, and ϰ∈℧, we have(30)0≤ωiϰ≤d1ψb−ψ0sΓs+1,∀i.
Asfiϰ,ω¯≥d1, we get(31)Qiωiϰ=1Γsi∫0ϰψ′ζψϰ−ψζsi−1fiζ,ω¯ζ≥d1Γsi∫0ϰψ′ζψϰ−ψζsi−1dζ≥d1ψϰ−ψ0siΓsi+1≥d1ψϰ−ψ0sisΓsi.
Hence,Qiωiϰ≥d1ψb−ψ0si/sΓsi, fori=1,…,m. This means that(32)Qω¯≥d1ψb−ψ0sΓs+1=ω¯,ω¯∈K∩∂V1.
Therefore, Theorem1 shows that Q has a fixed point in K∩V2¯/V1, which is consistent with the system’s positive solution (17).Theorem 4.
Letfi:℧×ℝ+m⟶ℝ+i=1,…,m be continuous. Suppose that fiϰ,ω¯−fiϰ,ϖ¯≤λω¯−ϖ¯,∀i,ω¯,ϖ¯∈ℝ+m and ϰ∈℧. If(33)λψb−ψ0sminΓs1+1,…,Γsm+1<1.Then, the system (17) has unique positive solution.Proof.
Letω¯,ϖ¯∈K. Then,(34)Qω¯−Qϖ¯=max1≤i≤mQiωi−Qiϖi=max1≤i≤mmaxϰ∈℧Qiωiϰ−Qiϖiϰ≤max1≤i≤mmaxϰ∈℧1Γsi∫0ϰψ′ζψϰ−ψζsi−1fiζ,ω¯ζ−fiζ,ϖ¯ζdζ≤max1≤i≤m1Γsi∫0ϰψ′ζψϰ−ψζsi−1fiζ,ω¯−fiζ,ϖ¯dζ≤max1≤i≤mλω¯−ϖ¯Γdsi∫0ϰψ′ζψϰ−ψζsi−1dζ≤max1≤i≤mλω¯−ϖ¯ψb−ψ0siΓsi+1≤λψb−ψ0smin1≤i≤mΓs1+1,…,Γsm+1ω¯−ϖ¯.
As per condition (33) along with Theorem 2, Q has unique fixed point in K, which is the unique positive solution.


## 4. Examples

Example 1.
Consider the fractional model as(35)D0+s1;ψω1ϰ=c1+ω121+ω12+⋯+ωm2,ω10=0,c1>0,D0+s2;ψω2ϰ=c2+ω221+ω12+⋯+ωm2,ω20=0,c2>0,⋮D0+sm;ψωmϰ=cm+ωm21+ω12+⋯+ωm2,ωm0=0,cm>0.
Setfiϰ,ω1ϰ,…,ωmϰ=fiϰ,ω¯=ci+ωi2/1+ωi2+ωm2, for i=1,…,m, where(36)ω¯=ω121+ω12+⋯+ωm2,ω221+ω12+⋯+ωm2,…,ωm21+ω12+⋯+ωm2.
Due tolimω¯⟶∞fiϰ,ω¯=ci+1 and fiϰ,ω¯≥ci for all i and ω¯∈0,∞m. Therefore, as per Theorem 1, there exists a positive solution for model (35).Example 2.
Letm=2. Then, following coupled system(37)D0+s1;ψω1ϰ=d1+ψϰ−ψ0tan−1ω2ϰ,d1>0,D0+s2;ψω2ϰ=d2+ψϰ−ψ0tan−1ω1ϰ,d2>0,has positive solution in view of Theorem 1. To verify Theorem 4, let us choose s1=1/2, s2=1/3, ψϰ=ϰ/3 forϰ∈0,1, and(38)f1ϰ,ω1,ω2=d1+ψϰ−ψ0tan−1ω2ϰ,f2ϰ,ω1,ω2=d2+ψϰ−ψ0tan−1ω1ϰ,where for i=1,2, ωi,ϖi∈ℝ+2 and ϰ∈0,1, we obtain(39)f1ϰ,ω1,ω2−f1ϰ,ϖ1,ϖ2≤ψϰ−ψ0tan−1ω2ϰ−tan−1ϖ2ϰ≤13ω2−ϖ2,f2ϰ,ω1,ω2−f2ϰ,ϖ1,ϖ2≤ψϰ−ψ0tan−1ω1ϰ−tan−1ϖ1ϰ≤13ω1−ϖ1.
Moreover, the condition (33) holds, i.e.,(40)λψb−ψ0maxs1,s2minΓs1+1,Γs2+1=λb/3s1Γs1+1=233π<1.
Therefore, Theorem4 can be applied to the problem (37).


## 5. Conclusions

In the theory of FDEs, the investigation and analysis of the existence and uniqueness of solutions and positive solutions are critical. In the past few years, many researchers have had great interest in qualitative analysis of FDEs, but the literature focused on Caputo FDs. In this work, utilizing new techniques, the existence and uniqueness of positive solutions for a system of FDEs underψ-Riemann–Liouville derivatives have been successfully obtained. The analysis has been based on Guo-Krasnoselskii’s and Banach’s fixed point theorem. The results were consistent with Daftardar-Gejji’s results [12] when ψϰ=ϰ, and they have been new even for the special case: ψϰ=logϰ or ψϰ=ϰρ.The suggested techniques can be extended to otherψ-Hilfer FDEs.



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