# Investigation of the Generalized Proportional Langevin and Sturm–Liouville Fractional Differential Equations via Variable Coefficients and Antiperiodic Boundary Conditions with a Control Theory Application Arising from Complex Networks

Authors: Abdellatif Boutiara; Mohammed K. A. Kaabar; Zailan Siri; Mohammad Esmael Samei; Xiao-Guang Yue
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

This article studies the existence theory of an innovation type of generalized proportional fractional differential equations with the assistance of the technique of Kuratowski measure on noncompactness combined with the fixed-point theorem of Mönch. Also, we use Lebesgue’s dominated convergence theorem and Arzelá–Ascoli fixed point theorem on existence and uniqueness results. An application is also presented by employing two illustrative iks, which enrich our outcomes.


---

## Body

## 1. Introduction

The idea of a measure of noncompactnessMℕℂ was presented by Kuratowski in 1958 for the first time. In fact, for any bounded subset M of a metric space C, the Mℕℂ denoted is defined to be the infimum of numbers ω>0 such that C can be covered by a finite number of sets of diameter less than ω.The first who used the indexα to the nonlinear analysis was Darbo [1]. The technique is a very useful tool for the existence solution of integral equations, which is well presented by Banaś et al. [2]. In 2008, Benchohra et al. considered the IVℙ for a nonlinear fractional differential equation(1)Gcσqι=℘ι,qι,∀0≤ι≤τ∗,where G​cσ is the Caputo fractional derivative of order 1<σ<2 and ℘ is a given function [3]. Some authors have been utilized comparative strategies through K Mℕℂ technique to diverse sorts of FBVℙs [4,5]. In 2016, Kiataramkul et al studied the generalized Langevin and Sturm–Liouville fractional differential equations (GL-SLFDs) of Hadamard type, with antiperiodic boundary conditions ( AℙBℂs) of the form(2)HGcσyιHGcσ+wιqι=℘ι,qι,ι∈1,τ∗,q1=−qτ∗,HGcσq1=−HGcσqτ∗,where GcηH denotes the Caputo-type Hadamard fractional derivative of order η∈0,1, ℘:1,τ∗×ℝ⟶ℝ is continuous, y∈C1,τ∗,ℝ with yι≥K⌣>0, and w∈C=C1,τ∗,ℝ [6].In this manuscript, we consider an innovation class ofBVℙs by combining Langevin and Sturm–Liouville fractional differential equations (L-SL FDs). This work discusses the existence theory of the generalized proportional fractional differential equations with the help of the technique of K Mℕℂ combined with the Mönch fixed-point theorem. To be more precise, we initiate the study of the existence, uniqueness, and different types of Ulam stability US for the following generalized proportional Langevin and Sturm–Liouville fractional differential problems (Gℙ L-SL FDℙ s) with variable coefficients (Vℂs) and AℙBℂs(3)Gcσ;ℏyιGcσ′;ℏ+wιqι=℘ι,qι,ι∈J≕0,τ∗,q0=−qτ∗,Gcσ′;ℏq0=−Gcσ′;ℏqτ∗.So thatGcη;ℏ denotes the generalized proportional Caputo fractional derivatives GℙCFD of order 0<η≤1, 1<σ+σ´≤2, ℏ>0, ℘:J×ℝ⟶ℝ is continuous, y∈CJ,ℝ/0 with yι≥K⌣>0, and w∈C=CJ,ℝ. Our proposed method is essentially based on the result given by Banaś et al. [2]. Therefore, it is emphasized that the K Mℕℂ technique is implemented for the first time on the Gℙ L-SL FDℙs.This article is arranged as follows: in Section2, we provide the definitions and preliminary lemmas that we will use to verify our main theorems. In Section 3, we establish the existence and uniqueness of solutions to problem (3) by the K Mℕℂ -method. In Section 4, we discuss some types of fractional US. We give an application by presenting two examples to illustrate numerical findings in Section 5. Finally, we conclude our exposition.


## 2. Preliminary Notions

We consider the spaceC with the norm(4)q∞=supqι:ι∈J.And denote byCB the class of all bounded mappings in C. Regard ℒ=L1J,ℝ as a Banach space of all measurable Bochner integrable mappings like as q:J⟶ℝ which is endowed with the norm(5)qℒ=∫Jqτdτ.Now we look at the specifications of the concept of KMℕℂ.Definition 1.
(see [2, 7]). The mapping ϒ:CB⟶0,∞ denoted by ϒM for M∈CB, is named as the KMℕℂ, if(6)ϒM≔infω>0:∃finitely many setsMi∋M=∪i=1mMiandDMi≤ω,where(7)DMi=supq−q^:q,q^∈Mi.Proposition 1 (see [2, 7]).
TheKMℕℂ fulfills the following:(1)
ϒM≤ϒM′ whenever M⊂M′, ∀M,M′∈CB;(2)
ϒM=0 iff M is relatively compact;(3)
ϒM=ϒM¯=ϒconvM, where M¯ and convM represent the closure and the convex hull of M, respectively;(4)
ϒM1+M2≤ϒM1+ϒM2;(5)
ϒyM=yϒM,∀y∈ℝ.
LetΣ stands for the set of functions q:J⟶ℝ, ∀ι∈J, set Σι=qι:q∈Σ, and(8)ΣJ=qι:q∈Σ,ι∈J.Theorem 1 (see [8–10]).
The continuous self functionϱ on convex bounded 0∈X⊆C admits a fixed-point whenever(9)Σ=conv¯ϱΣ,orΣ=ϱΣ∪0⇒ϒΣ=0,∀Σ⊂X.Lemma 1 (see [10]).
LetM⊂C be convex bounded and closed, ℘:J×X⟶C be Carathéodory, and ∃y∈L1J,ℝ+ so that ∀ι∈J and every bounded set E⊂M,(10)limδ⟶0+ϒ℘Jι,δ×E≤yιϒE;Jι,δ=ι−δ,ι∩J.
From equicontinuous ofJ⟶X, we have(11)ϒ∫JG^ι,ξ℘ξ,qξdξ:q∈Σ≤∫JG^ι,ξyξϒΣξdξ.
HereG∈CJ×J.Definition 2.
(see [11]) The GℙF integral of q of order σ∈ℂ,Reσ>0 is(12)Iaσ,ℏGℙFqι=∫aιι−sσ−1ℏσΓσexpℏ−1ℏι−ξqξdξ=ℏ−σexpℏ−1ℏιIaσexp1−ℏℏιqι.
For0<ℏ≤1, where Iaσ,ℏGℙF is Riemann–Liouville fractional integral and the GℙF Caputo type derivative of order σ is(13)Gcσ;ℏqι=∫aιι−ξn−σ−1ℏn−σΓn−σexpℏ−1ℏι−ξDn,ℏqξdξ,where n=Reσ+1.Lemma 2 (see [11]).
For the mapq, we have(14)expℏ−1ℏι−ξDn,ℏqξdξ,(15)Iaσ,ℏGℙFGcσ;ℏqι=qι−∑k=0n−1Dn,ℏqaℏkk!ι−akexpℏ−1ℏι−ξ.Proposition 2 (see [11]).
Letσ,σ´∈ℂ be such that Reσ′>0 and n=Reσ+1. Then, for any 0<ℏ≤1, we have(i)
Iaσ,ℏGℙFexpℏ−1/ℏιι−aσ′−1ι=Γσ′/ℏσΓσ+σ′expℏ−1/ℏιι−aσ+σ′−1,Reσ>0;(ii)
Gcσ;ℏexpℏ−1/ℏιι−aσ′−1ι=ℏσΓσ′/Γσ′−σexpℏ−1/ℏι−aσ′−σ−1,Reσ>n;(iii)
Gcσ;ℏexpℏ−1/ℏιι−akι=0,Reσ>n,k=0,1,…,n−1.


## 3. Main Results

We investigate the givenGℙ L-SL FDs (3) and present the solution’s characterization in relation to it.Definition 3.
By a solution of theGℙ L-SL FDs with Vℂs and AℙBℂs (3), we mean a measurable function q∈C such that q0=−qτ∗,(16)Gcσ;ℏq0=−Gcσ;ℏqτ∗.
AndFDs(17)Gcσ;ℏyιGcσ′;ℏ+wιqι=℘qι,ι∈J,are fulfilled on J.Lemma 3.
Let℘qι∈C and 0<σ,σ´≤1 with 1<σ+σ´≤2. Then, the solution of the linear Gℙ L-SL FDs with Vℂs and AℙBℂs(18)Gcσ;ℏyιGcσ′;ℏ+wιqι=℘qι,ι∈J,q0=−qτ∗,Gcσ′;ℏq0=−Gcσ′;ℏqτ∗.
∀ℏ∈CJ,ℝ/0 and w∈C is given by(19)qι=GℙFIaσ′,ℏ1yGℙFIaσ,ℏ℘qι−wιyιGℙFIaσ′,ℏqι+−1Δ1yτ∗GℙFIaσ,ℏ℘qτ∗−Δ2Δ1qτ∗ισ´expℏ−1/ℏιwιℏσ′Γσ′+1+−GℙFIaσ′,ℏ1Δ3yGℙFIaσ,ℏ℘qτ∗−Δ2Δ3GℙFIaσ′,ℏqτ∗−Δ4Δ31Δ1yτ∗GℙFIaσ,ℏ℘qτ∗−Δ4Δ3Δ2Δ1qτ∗expℏ−1ℏι,where(20)Δ1=1y0+1yτ∗expℏ−1ℏτ∗,Δ2=wτ∗yτ∗−w0y0≠0,Δ3=1+expℏ−1ℏτ∗,Δ4=τ∗σ′expℏ−1/ℏτ∗yτ∗ℏσ′Γσ′+1.Proof.
Taking theσth-GℙF integral to FDq of (3.1), we get(21)Gcσ′;ℏqι=Iaσ,ℏGℙF℘qι−wιqι+c0expℏ−1/ℏιyι,c0∈ℝ.
From theBℂs of system (3.1), we obtain(22)c0=−Iaσ,ℏGℙF℘qτ∗Δ1yτ∗−Δ2Δ1qτ∗.
Taking theσ´th-GℙF integral to (3.4), we obtain(23)qι=GℙFIaσ′,ℏ1yGℙFIaσ,ℏqιι−wιyιGℙFIaσ+σ′,ℏqι+c0ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+c1expℏ−1ℏι,c1∈ℝ.
Using the conditionq0=−qτ∗ of (18), we have(24)c1=−Iaσ′,ℏGℙFc0=−℘qτ∗Δ1yτ∗−Δ2Δ1qτ∗.1Δ3yIaσ,ℏGℙF℘qτ∗−Δ2Δ3Iaσ′,ℏGℙFqτ∗−Δ4Δ3Iaσ,ℏGℙF℘qτ∗Δ1yτ∗−Δ4Δ3Δ2Δ4qτ∗.
So, we derive (19) by substituting (24), and the proof is ended.
Clearly, by using Lemma 2.7, the system (18) is immediately established whenever we apply the σth-GℙF-derivative and σ´th-GℙF-derivative to both sides of (19). On the basis of Lemma 3, the solutions of (3) are corresponding to GℙF-integral equation in the following format:(25)qι=GℙFIaσ′,ℏ1yGℙFIaσ,ℏ℘qι−wιyιGℙFIaσ′,ℏqι+−1Δ1yτ∗GℙFIaσ,ℏ℘qτ∗−Δ2Δ1qτ∗ισ′expℏ−1/ℏιwιℏσ´Γσ´+1+−GℙFIaσ′,ℏ1Δ3yGℙFIaσ,ℏ℘qτ∗−Δ2Δ3GℙFIaσ′,ℏqτ∗−Δ4Δ31Δ1yτ∗GℙFIaσ,ℏ℘qτ∗−Δ4Δ3Δ2Δ1qτ∗expℏ−1ℏι.


### 3.1. Existence Result viaKMℕℂ-Method

We consider the following hypotheses:(H1)℘q:J×C⟶C fulfills the Carathéodory criterion;(H2)∃p∈CJ,ℝ+ s.t.(26)℘ι,qι≤pιq,∀ι∈J,∀q∈C.(H3) For anyι∈J and each bounded measurable set E⊂C,(27)limℏ⟶0+ϒ℘Jι,ℏ×E,0≤pιιϒB,whereϒ is the KMℕℂ and Jι,ℏ=ι−ℏ,ι∩J.Then, let us set(28)y∗=infι∈Jyι,w∗=supι∈Jwι,p∗=supι∈Jpι.Theorem 2.
Let us (H1)-(H3) holds. If(29)ϑp∗+ν≔Λ1<1,where(30)ϑ=1y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗Δ1τ∗σℏσ+σ′Γσ+1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+1y∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+Δ4y∗Δ1Δ3τ∗σℏσΓσ+1expℏ−1ℏτ∗.(31)ν=w∗y∗τ∗σ′ℏσ′Γσ′+1−Δ2Δ1τ∗σ′expℏ−1/ℏτ∗yτ∗ℏσ′Γσ′+1+−Δ2Δ3τ∗σℏσΓσ+1−Δ4Δ3Δ2Δ1expℏ−1ℏτ∗.
Then the system of GPFDEs (3) possesses a solution on J.Proof.
We define the self-operatorN∗ on C by(32)N∗qι=Iaσ′,ℏGℙF1yIaσ,ℏGℙF℘qι−wιyιIaσ′,ℏGℙFqι+−Iaσ,ℏGℙF℘qτ∗Δ1yτ∗−Δ2Δ1℘τ∗ισ′expℏ−1/ℏιyιισ′Γσ′+1+−Iaσ′,ℏGℙF1Δ3yIaσ,ℏGℙF℘qτ∗−Δ2Δ3Iaσ′,ℏGℙFqτ∗−Δ4Δ3Iaσ,ℏGℙF℘qτ∗Δ1yτ∗−Δ4Δ3Δ2Δ1qτ∗expℏ−1ℏι.
∀q∈C. Evidently, the fixed-point N∗ is a solution of the Gℙ L-SL FDℙs (3). We define(33)ΩM=q∈C:q≤M.
We shall follow the proof in three steps.Step 1.
Letqnn be a sequence with qn⟶q in C. Then ∀ι∈J, one may write(34)N∗qnι−N∗qι≤Iaσ+σ′,ℏGℙF1yIaσ,ℏGℙF℘ξ,qnξ−℘ξ,qξι−wιpιIα,ρGℙFwns−wsι+−1Δ1yτ∗Iaσ,ℏGℙF℘ξ,qnξ−℘ξ,qξτ∗−Δ2Δ1qnξ−qξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−1Δ3yτ∗GℙFIaσ+σ′,ℏ℘ξ,qnξ−℘ξ,qξτ∗−Δ2Δ3GℙFIaσ+σ′,ℏqnξ−qξτ∗−Δ4Δ31Δ1yτ∗GℙFIaσ′,ℏ℘ξ,qnξ−℘ξ,qξτ∗−Δ4Δ3Δ2Δ1qnξ−qξτ∗expℏ−1ℏι.
By continuityq and from (H1), the function ℘ι,qnι tends uniformly to ℘ι,qι. In accordance with Lebesgue’s dominated convergence theorem, N∗qnι tends uniformly to N∗qι, that is N∗qn⟶N∗q. Hence N∗ is sequentially continuous.Step 2.
Let is takeq¯∈ΩM, by (H2), and assume that N∗q¯ι≠0, we obtain(35)N∗q¯ι≤Iaσ′,ℏGℙF1yIaσ,ℏGℙF℘q¯ι+wιyιIaσ′,ℏGℙFq¯ι+Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗+Δ2Δ1q¯τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+Iaσ′,ℏGℙF1Δ3yIaσ,ℏGℙF℘q¯τ∗+Δ2Δ3Iaσ′,ℏGℙFq¯τ∗+Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗+Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι≤GℙFIaσ′,ℏ1yIaσ,ℏGℙF℘q¯ι+wιyιIaσ′,ℏGℙFq¯ι+1Δ1yτ∗Iaσ,ℏGℙF℘q¯τ∗+Δ2Δ1q¯τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗Iaσ+σ′,ℏGℙF℘q¯τ∗+Δ2Δ3Iaσ′,ℏGℙFq¯τ∗+Δ4Δ31Δ1yτ∗Iaσ,ℏGℙF℘q¯τ∗+Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι≤1yιIaσ+σ′,ℏGℙFq¯pξι+wιyιIaσ′,ℏGℙFq¯ξι+1Δ1yτ∗Iaσ,ℏGℙFq¯pξτ∗+Δ2Δ1q¯ξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗Iaσ+σ′,ℏGℙFq¯pξτ∗+Δ2Δ3Iaσ′,ℏGℙFq¯ξτ∗+Δ4Δ31Δ1yτ∗Iaσ,ℏGℙFq¯pξτ∗+Δ4Δ3Δ2Δ1q¯ξτ∗expℏ−1ℏι.
∀ι∈J. By using the property(36)0<expσ′−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗, we have(37)N∗q¯ι≤p∗MyιGℙFIaσ+σ′,ℏ1ι+MwιyιGℙFIaσ′,ℏ1ι+p∗MΔ1yτ∗GℙFIaσ,ℏ1τ∗+MΔ2Δ1ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+p∗MΔ3yτ∗GℙFIaσ+σ′,ℏ1τ∗+MΔ2Δ3GℙFIaσ′,ℏ1τ∗+Δ4Δ3p∗MΔ1yτ∗GℙFIaσ,ℏ1τ∗+MΔ4Δ3Δ2Δ1expℏ−1ℏι≤Mp∗y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗τ∗σ′ℏσ′Γσ′+1+p∗Δ1yτ∗τ∗σℏσΓσ+1+Δ2Δ1τ∗σ´expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+p∗yτ∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+Δ2Δ3τ∗σ′ℏσ′Γσ′+1+Δ4Δ3p∗yτ∗Δ1τ∗σℏσΓσ+1+Δ4Δ3Δ2Δ1expℏ−1ℏτ∗=Mϑp∗+ν.
Hence, we get(38)N∗qC≤Mϑp∗+ν≤M.
IndeedN∗ΩM⊆ΩM.Step 3.
By Step2, the N∗ΩM is bounded. Let ι1,ι2∈J, ι1<ι2 and q¯∈ΩM, so(39)N∗q¯ι2−N∗q¯ι1≠0.
Then(40)=N∗q¯ι2−N∗q¯ι1≤Iaσ,ℏGℙF1ℏ∗Iaσ+σ′,ℏGℙF℘ξ,q¯ξι2−℘ξ,q¯ξι1+Iaσ,ℏGℙFw∗ℏ∗q¯ξι2−q¯ξι1+−Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ2Δ1q¯τ∗ι2σ′expℏ−1/ℏι2−ι1σ′expℏ−1/ℏι2yιℏσ′Γσ′+1+−Iaσ+σ′,ℏGℙF℘q¯τ∗Δ3yτ∗−Δ2Δ3GℙFIaσ′,ℏq¯ι−Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι2−expℏ−1ℏι1≤p∗MIaσ,ℏGℙF1y∗Iaσ+σ′,ℏGℙF1ι2−Iaσ+σ′,ℏGℙF1ι1+Mw∗y∗Iaσ,ℏGℙF1ι2−Iaσ,ℏGℙF1ι1+−Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ2Δ1q¯τ∗ι2σ′expℏ−1/ℏι2−ι1σ′expℏ−1/ℏι1yιℏσ′Γσ′+1+−Iaσ+σ′,ℏGℙF℘q¯τ∗Δ3yτ∗−Δ2Δ3GℙFIaσ′,ℏq¯τ∗−Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι2−expℏ−1ℏι1.
By using the property(41)0<expσ′−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗, we get(42)N∗q¯ι2−N∗q¯ι1≤p∗Mℏσ+σ′y∗Γσ+σ′∫aι1expℏ−1ℏι2−ξι2−ξσ+σ′−1−expℏ−1ℏι1−ξι1−ξσ+σ′−1dξ+p∗Mℏσ+σ′y∗Γσ+σ′∫ι1ι2expy−1ℏι2−ξι2−ξσ+σ′−1dξ+w∗y∗Mℏσ′Γσ′∫aι1expℏ−1ℏι2−ξι2−ξσ′−1−expy−1ℏι1−ξι1−ξσ′−1dξ+w∗y∗Mℏσ′Γσ′∫ι1ι2expy−1ℏι2−ξι2−ξσ′−1q¯ξdξ+−Iaσ,ℏGℙF℘q¯τ∗Δ1w∗−Δ2Δ1q¯τ∗ι2σ′expℏ−1/ℏι2−ι1σ′expℏ−1/ℏι1y∗ℏσ′Γσ′+1+−Iaσ+σ′,ℏGℙF℘q¯τ∗Δ3y∗−Δ2Δ3GℙFIaσ′,ℏq¯τ∗−Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1y∗−Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι2−expℏ−1ℏι1.
Ast1⟶t2, the right-hand side of inequality (42) tends to zero. This means that N∗ΩM is equicontinuous. Now, let Ω^⊂ΩM with(43)Ω^=conv¯N∗Ω^∪0.
SinceΩ^ is bounded and equicontinuous, the function ι⟶ωι=ϒΩ^ι is continuous on J. By assumption (H2) and using the property(44)0<expℏ−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗, we get.(45)ωι≤ϒN∗Ω^ι∪0≤ϒN∗Ω^ι≤GℙFIaσ,ℏ1ℏIaσ´,ℏGℙFpξϒΩ^ξι+Iaσ,ℏGℙFwℏϒΩ^ξι+−Iaσ,ℏGℙFwΔ1yϒΩ^ξτ∗−Δ2Δ1ϒΩ^ξτ∗ισ′expℏ−1/ℏιyιισ′Γσ′+1+−Iaσ,ℏGℙF1Δ3yIaσ´,ℏGℙFpξϒΩ^ξτ∗−Iaσ,ℏGℙFΔ2Δ3ϒΩ^ξτ∗−Δ4Δ31Δ1yτ∗GℙFIaσ,ℏpξϒΩ^ξτ∗−Δ4Δ3Δ2Δ1ϒΩ^ξτ∗expℏ−1ℏι≤p∗ωIaσ,ℏGℙF1ℏIaσ´,ℏGℙF1ι−Iaσ,ℏGℙFwΔ1y1τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−Iaσ,ℏGℙF1Δ3yIaσ´,ℏGℙF1τ∗−Δ4Δ31Δ1yτ∗GℙFIaσ,ℏ1τ∗expℏ−1ℏι+ωIaσ,ℏGℙFwℏ1ι−Δ2Δ1ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−Iaσ,ℏGℙFΔ2Δ31τ∗−Δ4Δ3Δ2Δ1expℏ−1ℏι≤p∗ω1y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗Δ1τ∗σℏσ+σ′Γσ+1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+1y∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+Δ4y∗Δ1Δ3τ∗σℏσΓσ+1expℏ−1ℏτ∗+ωw∗y∗τ∗σℏσ′Γσ′+1−Δ2Δ1τ∗σ′expℏ−1/ℏτ∗yτ∗ℏσ′Γσ′+1+−Δ2Δ3τ∗σℏσΓσ+1−Δ4Δ3Δ2Δ1expℏ−1ℏτ∗≤p∗ωϑ+ων.
Indeed,(46)ω1−p∗ϑ−ν≤0.
Equation (29) implies that ω=0 or ωι=0, ∀ι∈J. Hence Ω^ι is relatively compact in ℝ and so Arzelá–Ascoli theorem implies that Ω^ is relatively compact in ΩM. Now, from Theorem 3, we conclude that the problem (3) has a solution which is a fixed point of operator N∗.


### 3.2. Uniqueness Criterion

Theorem 3.
Let the following assumptions hold:(H4)℘:J×ℝ⟶ℝ is a continuous function.(H5) there exists a nonnegative constantM>0, such that(47)℘ι,q−℘ι,q′≤Mq−q′,∀ι∈J,∀q,q′∈ℝ.
Then, problem (3) has a unique solution on J, provided that(48)Λ2=ϑM+ν<1,where ϑ and ν are given by (30) and (31), respectively.Proof.
In the first step, we show thatN∗ΩM⊂ΩM, where the operator N∗:C⟶CJ is defined by Equation (32), ΩM=q∈C:q≤M, and we choose a real number M>, such that(49)M≥ϑ℘01−Λ2,℘0=sup1≤ι≤τ∗℘ι,0.
For anyq¯∈ΩM, using (H5), we occur(50)N∗qι≤Iaσ+σ′,ℏGℙF℘qιyι+wιyιGℙFIaσ′,ℏq¯ι+Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗+Δ2Δ1q¯τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+Iaσ+σ′,ℏGℙF℘q¯Δ3yτ∗+Δ2Δ3GℙFIaσ′,ℏq¯τ∗+Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗+Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι.
By using the property(51)0<expℏ−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗ and (H5) implies that(52)N∗qι≤GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏ℘ξ,q¯ξ−℘ξ,0+℘ξ,0ι+GℙFIaσ,ℏw∗y∗q¯ξι+1Δ1yτ∗GℙFIaσ,ℏ℘ξ,q¯ξ−℘ξ,0+℘ξ,0τ∗+Δ2Δ1q¯ξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗GℙFIaσ+σ′,ℏ℘ξ,q¯ξ−℘ξ,0+℘ξ,0τ∗+Δ2Δ3GℙFIaσ´,ℏq¯ξτ∗+Δ4Δ31Δ1yτ∗GℙFIaσ,ℏ℘ξ,q¯ξ−℘ξ,0+℘ξ,0τ∗+Δ4Δ3Δ2Δ1q¯ξτ∗expℏ−1ℏι≤GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏMq¯+℘0ι+w∗y∗GℙFIaσ,ℏq¯ι+1Δ1yτ∗GℙFIaσ,ℏMq¯+℘0τ∗+Δ2Δ1q¯τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗GℙFIaσ+σ′,ℏMq¯+℘0τ∗+Δ2Δ3GℙFIaσ′,ℏq¯τ∗+Δ4Δ31Δ1yτ∗GℙFIaσ,ℏMq¯+℘0τ∗+Δ4Δ3Δ2Δ1q¯expℏ−1ℏι≤Mq¯+℘0y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+q¯w∗y∗τ∗σ′ℏσ′Γσ′+1+Mq¯+℘0Δ1yτ∗τ∗σℏσΓσ+1+MΔ2Δ1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+Mq¯+℘0yτ∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+q¯Δ2Δ3τ∗σ′ℏσ′Γσ′+1+Δ4Δ3Mq¯+℘0yτ∗Δ1τ∗σ′ℏσ′Γσ′+1+q¯Δ4Δ3Δ2Δ1expℏ−1ℏτ∗≤ϑMq¯+℘0+νq¯≤M.
Which impliesN∗q¯≤M after taking the norm on J. Thus, N∗ maps ΩM into itself. For q,q´∈C, we obtain by using the notations in (30) and (31) that(53)N∗qι−N∗q′ι≤GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏ℘ξ,qξ−℘ξ,q′ξι+GℙFIaσ,ℏw∗y∗qξ−q′ξι+1Δ1yτ∗GℙFIaσ,ℏ℘ξ,qξ−℘ξ,q′ξτ∗+Δ2Δ1qξ−q′ξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗GℙFIaσ+σ′,ℏ℘ξ,qξ−℘ξ,q′ξτ∗+Δ2Δ3GℙFIaσ′,ℏqξ−q′ξτ∗+Δ4Δ31Δ1yτ∗GℙFIaσ,ℏ℘ξ,qξ−℘ξ,q′ξτ∗+Δ4Δ3Δ2Δ1qξ−q′ξτ∗expℏ−1ℏι≤GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏMqξ−q′ξι+GℙFIaσ,ℏw∗y∗qξ−q′ξι+1Δ1yτ∗GℙFIaσ,ℏMqξ−q′ξτ∗+Δ2Δ1qξ−q′ξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗GℙFIaσ+σ′,ℏMqξ−q′ξτ∗+Δ2Δ3GℙFIaσ′,ℏqξ−q′ξτ∗+Δ4Δ31Δ1yτ∗GℙFIaσ,ℏMqξ−q′ξτ∗+Δ4Δ3Δ2Δ1qξ−q′ξτ∗expℏ−1ℏι.
By using the property(54)0<expℏ′−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗, we have(55)N∗qι−N∗q′ι≤My∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗τ∗σ′ℏσ′Γσ′+1+MΔ1yτ∗τ∗σℏσΓσ+1+Δ2Δ1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+Myτ∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+Δ2Δ3τ∗σ′ℏσ′Γσ′+1+Δ4Δ3Myτ∗Δ1τ∗σℏσΓσ+1+Δ4Δ3Δ2Δ1expℏ−1ℏτ∗qξ−q′ξ≤ϑM+νq−q′.
Consequently, we get(56)N∗q−N∗q′≤ϑM+νq−q′.
Therefore,N∗ is a contraction. Hence, the operator N∗ has a unique fixed point, which corresponds to a unique solution of the (3) on J.


## 3.1. Existence Result viaKMℕℂ-Method

We consider the following hypotheses:(H1)℘q:J×C⟶C fulfills the Carathéodory criterion;(H2)∃p∈CJ,ℝ+ s.t.(26)℘ι,qι≤pιq,∀ι∈J,∀q∈C.(H3) For anyι∈J and each bounded measurable set E⊂C,(27)limℏ⟶0+ϒ℘Jι,ℏ×E,0≤pιιϒB,whereϒ is the KMℕℂ and Jι,ℏ=ι−ℏ,ι∩J.Then, let us set(28)y∗=infι∈Jyι,w∗=supι∈Jwι,p∗=supι∈Jpι.Theorem 2.
Let us (H1)-(H3) holds. If(29)ϑp∗+ν≔Λ1<1,where(30)ϑ=1y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗Δ1τ∗σℏσ+σ′Γσ+1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+1y∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+Δ4y∗Δ1Δ3τ∗σℏσΓσ+1expℏ−1ℏτ∗.(31)ν=w∗y∗τ∗σ′ℏσ′Γσ′+1−Δ2Δ1τ∗σ′expℏ−1/ℏτ∗yτ∗ℏσ′Γσ′+1+−Δ2Δ3τ∗σℏσΓσ+1−Δ4Δ3Δ2Δ1expℏ−1ℏτ∗.
Then the system of GPFDEs (3) possesses a solution on J.Proof.
We define the self-operatorN∗ on C by(32)N∗qι=Iaσ′,ℏGℙF1yIaσ,ℏGℙF℘qι−wιyιIaσ′,ℏGℙFqι+−Iaσ,ℏGℙF℘qτ∗Δ1yτ∗−Δ2Δ1℘τ∗ισ′expℏ−1/ℏιyιισ′Γσ′+1+−Iaσ′,ℏGℙF1Δ3yIaσ,ℏGℙF℘qτ∗−Δ2Δ3Iaσ′,ℏGℙFqτ∗−Δ4Δ3Iaσ,ℏGℙF℘qτ∗Δ1yτ∗−Δ4Δ3Δ2Δ1qτ∗expℏ−1ℏι.
∀q∈C. Evidently, the fixed-point N∗ is a solution of the Gℙ L-SL FDℙs (3). We define(33)ΩM=q∈C:q≤M.
We shall follow the proof in three steps.Step 1.
Letqnn be a sequence with qn⟶q in C. Then ∀ι∈J, one may write(34)N∗qnι−N∗qι≤Iaσ+σ′,ℏGℙF1yIaσ,ℏGℙF℘ξ,qnξ−℘ξ,qξι−wιpιIα,ρGℙFwns−wsι+−1Δ1yτ∗Iaσ,ℏGℙF℘ξ,qnξ−℘ξ,qξτ∗−Δ2Δ1qnξ−qξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−1Δ3yτ∗GℙFIaσ+σ′,ℏ℘ξ,qnξ−℘ξ,qξτ∗−Δ2Δ3GℙFIaσ+σ′,ℏqnξ−qξτ∗−Δ4Δ31Δ1yτ∗GℙFIaσ′,ℏ℘ξ,qnξ−℘ξ,qξτ∗−Δ4Δ3Δ2Δ1qnξ−qξτ∗expℏ−1ℏι.
By continuityq and from (H1), the function ℘ι,qnι tends uniformly to ℘ι,qι. In accordance with Lebesgue’s dominated convergence theorem, N∗qnι tends uniformly to N∗qι, that is N∗qn⟶N∗q. Hence N∗ is sequentially continuous.Step 2.
Let is takeq¯∈ΩM, by (H2), and assume that N∗q¯ι≠0, we obtain(35)N∗q¯ι≤Iaσ′,ℏGℙF1yIaσ,ℏGℙF℘q¯ι+wιyιIaσ′,ℏGℙFq¯ι+Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗+Δ2Δ1q¯τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+Iaσ′,ℏGℙF1Δ3yIaσ,ℏGℙF℘q¯τ∗+Δ2Δ3Iaσ′,ℏGℙFq¯τ∗+Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗+Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι≤GℙFIaσ′,ℏ1yIaσ,ℏGℙF℘q¯ι+wιyιIaσ′,ℏGℙFq¯ι+1Δ1yτ∗Iaσ,ℏGℙF℘q¯τ∗+Δ2Δ1q¯τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗Iaσ+σ′,ℏGℙF℘q¯τ∗+Δ2Δ3Iaσ′,ℏGℙFq¯τ∗+Δ4Δ31Δ1yτ∗Iaσ,ℏGℙF℘q¯τ∗+Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι≤1yιIaσ+σ′,ℏGℙFq¯pξι+wιyιIaσ′,ℏGℙFq¯ξι+1Δ1yτ∗Iaσ,ℏGℙFq¯pξτ∗+Δ2Δ1q¯ξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗Iaσ+σ′,ℏGℙFq¯pξτ∗+Δ2Δ3Iaσ′,ℏGℙFq¯ξτ∗+Δ4Δ31Δ1yτ∗Iaσ,ℏGℙFq¯pξτ∗+Δ4Δ3Δ2Δ1q¯ξτ∗expℏ−1ℏι.
∀ι∈J. By using the property(36)0<expσ′−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗, we have(37)N∗q¯ι≤p∗MyιGℙFIaσ+σ′,ℏ1ι+MwιyιGℙFIaσ′,ℏ1ι+p∗MΔ1yτ∗GℙFIaσ,ℏ1τ∗+MΔ2Δ1ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+p∗MΔ3yτ∗GℙFIaσ+σ′,ℏ1τ∗+MΔ2Δ3GℙFIaσ′,ℏ1τ∗+Δ4Δ3p∗MΔ1yτ∗GℙFIaσ,ℏ1τ∗+MΔ4Δ3Δ2Δ1expℏ−1ℏι≤Mp∗y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗τ∗σ′ℏσ′Γσ′+1+p∗Δ1yτ∗τ∗σℏσΓσ+1+Δ2Δ1τ∗σ´expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+p∗yτ∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+Δ2Δ3τ∗σ′ℏσ′Γσ′+1+Δ4Δ3p∗yτ∗Δ1τ∗σℏσΓσ+1+Δ4Δ3Δ2Δ1expℏ−1ℏτ∗=Mϑp∗+ν.
Hence, we get(38)N∗qC≤Mϑp∗+ν≤M.
IndeedN∗ΩM⊆ΩM.Step 3.
By Step2, the N∗ΩM is bounded. Let ι1,ι2∈J, ι1<ι2 and q¯∈ΩM, so(39)N∗q¯ι2−N∗q¯ι1≠0.
Then(40)=N∗q¯ι2−N∗q¯ι1≤Iaσ,ℏGℙF1ℏ∗Iaσ+σ′,ℏGℙF℘ξ,q¯ξι2−℘ξ,q¯ξι1+Iaσ,ℏGℙFw∗ℏ∗q¯ξι2−q¯ξι1+−Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ2Δ1q¯τ∗ι2σ′expℏ−1/ℏι2−ι1σ′expℏ−1/ℏι2yιℏσ′Γσ′+1+−Iaσ+σ′,ℏGℙF℘q¯τ∗Δ3yτ∗−Δ2Δ3GℙFIaσ′,ℏq¯ι−Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι2−expℏ−1ℏι1≤p∗MIaσ,ℏGℙF1y∗Iaσ+σ′,ℏGℙF1ι2−Iaσ+σ′,ℏGℙF1ι1+Mw∗y∗Iaσ,ℏGℙF1ι2−Iaσ,ℏGℙF1ι1+−Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ2Δ1q¯τ∗ι2σ′expℏ−1/ℏι2−ι1σ′expℏ−1/ℏι1yιℏσ′Γσ′+1+−Iaσ+σ′,ℏGℙF℘q¯τ∗Δ3yτ∗−Δ2Δ3GℙFIaσ′,ℏq¯τ∗−Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι2−expℏ−1ℏι1.
By using the property(41)0<expσ′−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗, we get(42)N∗q¯ι2−N∗q¯ι1≤p∗Mℏσ+σ′y∗Γσ+σ′∫aι1expℏ−1ℏι2−ξι2−ξσ+σ′−1−expℏ−1ℏι1−ξι1−ξσ+σ′−1dξ+p∗Mℏσ+σ′y∗Γσ+σ′∫ι1ι2expy−1ℏι2−ξι2−ξσ+σ′−1dξ+w∗y∗Mℏσ′Γσ′∫aι1expℏ−1ℏι2−ξι2−ξσ′−1−expy−1ℏι1−ξι1−ξσ′−1dξ+w∗y∗Mℏσ′Γσ′∫ι1ι2expy−1ℏι2−ξι2−ξσ′−1q¯ξdξ+−Iaσ,ℏGℙF℘q¯τ∗Δ1w∗−Δ2Δ1q¯τ∗ι2σ′expℏ−1/ℏι2−ι1σ′expℏ−1/ℏι1y∗ℏσ′Γσ′+1+−Iaσ+σ′,ℏGℙF℘q¯τ∗Δ3y∗−Δ2Δ3GℙFIaσ′,ℏq¯τ∗−Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1y∗−Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι2−expℏ−1ℏι1.
Ast1⟶t2, the right-hand side of inequality (42) tends to zero. This means that N∗ΩM is equicontinuous. Now, let Ω^⊂ΩM with(43)Ω^=conv¯N∗Ω^∪0.
SinceΩ^ is bounded and equicontinuous, the function ι⟶ωι=ϒΩ^ι is continuous on J. By assumption (H2) and using the property(44)0<expℏ−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗, we get.(45)ωι≤ϒN∗Ω^ι∪0≤ϒN∗Ω^ι≤GℙFIaσ,ℏ1ℏIaσ´,ℏGℙFpξϒΩ^ξι+Iaσ,ℏGℙFwℏϒΩ^ξι+−Iaσ,ℏGℙFwΔ1yϒΩ^ξτ∗−Δ2Δ1ϒΩ^ξτ∗ισ′expℏ−1/ℏιyιισ′Γσ′+1+−Iaσ,ℏGℙF1Δ3yIaσ´,ℏGℙFpξϒΩ^ξτ∗−Iaσ,ℏGℙFΔ2Δ3ϒΩ^ξτ∗−Δ4Δ31Δ1yτ∗GℙFIaσ,ℏpξϒΩ^ξτ∗−Δ4Δ3Δ2Δ1ϒΩ^ξτ∗expℏ−1ℏι≤p∗ωIaσ,ℏGℙF1ℏIaσ´,ℏGℙF1ι−Iaσ,ℏGℙFwΔ1y1τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−Iaσ,ℏGℙF1Δ3yIaσ´,ℏGℙF1τ∗−Δ4Δ31Δ1yτ∗GℙFIaσ,ℏ1τ∗expℏ−1ℏι+ωIaσ,ℏGℙFwℏ1ι−Δ2Δ1ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−Iaσ,ℏGℙFΔ2Δ31τ∗−Δ4Δ3Δ2Δ1expℏ−1ℏι≤p∗ω1y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗Δ1τ∗σℏσ+σ′Γσ+1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+1y∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+Δ4y∗Δ1Δ3τ∗σℏσΓσ+1expℏ−1ℏτ∗+ωw∗y∗τ∗σℏσ′Γσ′+1−Δ2Δ1τ∗σ′expℏ−1/ℏτ∗yτ∗ℏσ′Γσ′+1+−Δ2Δ3τ∗σℏσΓσ+1−Δ4Δ3Δ2Δ1expℏ−1ℏτ∗≤p∗ωϑ+ων.
Indeed,(46)ω1−p∗ϑ−ν≤0.
Equation (29) implies that ω=0 or ωι=0, ∀ι∈J. Hence Ω^ι is relatively compact in ℝ and so Arzelá–Ascoli theorem implies that Ω^ is relatively compact in ΩM. Now, from Theorem 3, we conclude that the problem (3) has a solution which is a fixed point of operator N∗.


## 3.2. Uniqueness Criterion

Theorem 3.
Let the following assumptions hold:(H4)℘:J×ℝ⟶ℝ is a continuous function.(H5) there exists a nonnegative constantM>0, such that(47)℘ι,q−℘ι,q′≤Mq−q′,∀ι∈J,∀q,q′∈ℝ.
Then, problem (3) has a unique solution on J, provided that(48)Λ2=ϑM+ν<1,where ϑ and ν are given by (30) and (31), respectively.Proof.
In the first step, we show thatN∗ΩM⊂ΩM, where the operator N∗:C⟶CJ is defined by Equation (32), ΩM=q∈C:q≤M, and we choose a real number M>, such that(49)M≥ϑ℘01−Λ2,℘0=sup1≤ι≤τ∗℘ι,0.
For anyq¯∈ΩM, using (H5), we occur(50)N∗qι≤Iaσ+σ′,ℏGℙF℘qιyι+wιyιGℙFIaσ′,ℏq¯ι+Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗+Δ2Δ1q¯τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+Iaσ+σ′,ℏGℙF℘q¯Δ3yτ∗+Δ2Δ3GℙFIaσ′,ℏq¯τ∗+Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗+Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι.
By using the property(51)0<expℏ−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗ and (H5) implies that(52)N∗qι≤GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏ℘ξ,q¯ξ−℘ξ,0+℘ξ,0ι+GℙFIaσ,ℏw∗y∗q¯ξι+1Δ1yτ∗GℙFIaσ,ℏ℘ξ,q¯ξ−℘ξ,0+℘ξ,0τ∗+Δ2Δ1q¯ξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗GℙFIaσ+σ′,ℏ℘ξ,q¯ξ−℘ξ,0+℘ξ,0τ∗+Δ2Δ3GℙFIaσ´,ℏq¯ξτ∗+Δ4Δ31Δ1yτ∗GℙFIaσ,ℏ℘ξ,q¯ξ−℘ξ,0+℘ξ,0τ∗+Δ4Δ3Δ2Δ1q¯ξτ∗expℏ−1ℏι≤GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏMq¯+℘0ι+w∗y∗GℙFIaσ,ℏq¯ι+1Δ1yτ∗GℙFIaσ,ℏMq¯+℘0τ∗+Δ2Δ1q¯τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗GℙFIaσ+σ′,ℏMq¯+℘0τ∗+Δ2Δ3GℙFIaσ′,ℏq¯τ∗+Δ4Δ31Δ1yτ∗GℙFIaσ,ℏMq¯+℘0τ∗+Δ4Δ3Δ2Δ1q¯expℏ−1ℏι≤Mq¯+℘0y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+q¯w∗y∗τ∗σ′ℏσ′Γσ′+1+Mq¯+℘0Δ1yτ∗τ∗σℏσΓσ+1+MΔ2Δ1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+Mq¯+℘0yτ∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+q¯Δ2Δ3τ∗σ′ℏσ′Γσ′+1+Δ4Δ3Mq¯+℘0yτ∗Δ1τ∗σ′ℏσ′Γσ′+1+q¯Δ4Δ3Δ2Δ1expℏ−1ℏτ∗≤ϑMq¯+℘0+νq¯≤M.
Which impliesN∗q¯≤M after taking the norm on J. Thus, N∗ maps ΩM into itself. For q,q´∈C, we obtain by using the notations in (30) and (31) that(53)N∗qι−N∗q′ι≤GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏ℘ξ,qξ−℘ξ,q′ξι+GℙFIaσ,ℏw∗y∗qξ−q′ξι+1Δ1yτ∗GℙFIaσ,ℏ℘ξ,qξ−℘ξ,q′ξτ∗+Δ2Δ1qξ−q′ξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗GℙFIaσ+σ′,ℏ℘ξ,qξ−℘ξ,q′ξτ∗+Δ2Δ3GℙFIaσ′,ℏqξ−q′ξτ∗+Δ4Δ31Δ1yτ∗GℙFIaσ,ℏ℘ξ,qξ−℘ξ,q′ξτ∗+Δ4Δ3Δ2Δ1qξ−q′ξτ∗expℏ−1ℏι≤GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏMqξ−q′ξι+GℙFIaσ,ℏw∗y∗qξ−q′ξι+1Δ1yτ∗GℙFIaσ,ℏMqξ−q′ξτ∗+Δ2Δ1qξ−q′ξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+1Δ3yτ∗GℙFIaσ+σ′,ℏMqξ−q′ξτ∗+Δ2Δ3GℙFIaσ′,ℏqξ−q′ξτ∗+Δ4Δ31Δ1yτ∗GℙFIaσ,ℏMqξ−q′ξτ∗+Δ4Δ3Δ2Δ1qξ−q′ξτ∗expℏ−1ℏι.
By using the property(54)0<expℏ′−1ℏs−ξ≤1.
Fora≤ξ<s<ι≤τ∗, we have(55)N∗qι−N∗q′ι≤My∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗τ∗σ′ℏσ′Γσ′+1+MΔ1yτ∗τ∗σℏσΓσ+1+Δ2Δ1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+Myτ∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+Δ2Δ3τ∗σ′ℏσ′Γσ′+1+Δ4Δ3Myτ∗Δ1τ∗σℏσΓσ+1+Δ4Δ3Δ2Δ1expℏ−1ℏτ∗qξ−q′ξ≤ϑM+νq−q′.
Consequently, we get(56)N∗q−N∗q′≤ϑM+νq−q′.
Therefore,N∗ is a contraction. Hence, the operator N∗ has a unique fixed point, which corresponds to a unique solution of the (3) on J.


## 4. UH and UHR Stability of System

In the recent section, we are interested to study UH and UHR stability of the system (3). The system (3) is UH stable if ∃cq>0 such that, for each ϵ∈ℝ+ and for each q∈C satisfying(57)Gcσ;ℏyιGcσ′;ℏ+wιqι−℘ι,qι≤ε,ι∈J,q0=−qτ∗,Gcσ′;ℏq0=−Gcσ′;ℏqτ∗.There exists a unique solutionq¯∈C of (3) with(58)q−q¯≤cqε.The system (3) is generalized UH stable (GUH) if there exists cq∈Cℝ+,ℝ+ and cq0=0 such that for each ϵ∈ℝ+ and for each q∈C satisfying (57), there exists a unique solution q¯∈C of (3) with q−q¯≤cqϵ.Remark 1.
A functionq¯∈C is a solution of inequality (57) if ∃q1∈C (which depends on solution q¯) such that, i) q1ι≤ε,ι∈J; ii)(59)Gcσ;ℏyιGcσ;ℏ+qιqι=℘ι,qι+q1ι,ι∈J.Theorem 4.
Suppose that the conditionH5 and inequality (48) are fulfilled. Then, the solution of (3) is UH and GUH stable.Proof.
Letϵ>0 and let q¯∈C be a function that satisfies the inequality (57) and let q∈C the unique solution of the problem(60)Gcσ;ℏyιGcσ′;ℏ+wιqι=℘ι,qι,ι∈J,q0=−qτ∗,Gcσ′;ℏq0=−Gcσ′;ℏqτ∗.
Lemma3 implies that(61)qι=Iaσ+σ′,ℏGℙF℘qιyι−wιyιGℙFIaσ′,ℏqι+−Iaσ,ℏGℙF℘qτ∗Δ1yτ∗−Δ2Δ1qτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−Iaσ+σ′,ℏGℙF℘qτ∗Δ3yτ∗−Δ2Δ3GℙFIaσ′,ℏqτ∗−Δ4Δ3Iaσ,ℏGℙF℘qτ∗Δ1yτ∗−Δ4Δ3Δ3Δ1qτ∗expℏ−1ℏι.
Hence by Remark1, we have(62)Gcσ′;ℏyιGcσ;ℏ+wιq¯ι=℘ι,q¯ι+q1ι,ι∈J,q¯0=−q¯τ∗,G​cσ′;ℏq¯0=−G​cσ′;ℏq¯τ∗.
Again Lemma3 implies that(63)q¯ι=Iaσ+σ′,ℏGℙF℘q¯ιyι−wιyιGℙFIaσ′,ℏq¯ι+−Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ2Δ1q¯τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−Iaσ+σ′,ℏGℙF℘q¯q¯Δ3yτ∗−Δ2Δ3GℙFIaσ′,ℏq¯τ∗−Δ4Δ3Iaσ,ℏGℙF℘q¯τ∗Δ1yτ∗−Δ4Δ3Δ2Δ1q¯τ∗expℏ−1ℏι+Iaσ+σ′,ℏGℙFq1ιyι+−Iaσ,ℏGℙFq1τ∗Δ1yτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−Iaσ+σ′,ℏGℙFq1τ∗Δ3yτ∗−Δ4Δ3Iaσ,ℏGℙFq1τ∗Δ1yτ∗expℏ−1ℏι.
On the other hand, we have, for eachι∈J,(64)q¯ι=1yιGℙFIaσ,ℏ1y∗GℙFIaσ′,ℏ℘ξ,q¯ξ−℘ξ,qξι+GℙFIaσ,ℏw∗y∗q¯ξ−qξι+−1Δ1yτ∗GℙFIaσ′,ℏ℘ξ,GℙFIaσ′,ℏξ−℘ξ,qξτ∗−Δ2Δ1τ∗ξ−qξτ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−1Δ3yτ∗GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏ℘ξ,q¯ξ−℘ξ,qξτ∗−Δ2Δ3GℙFIaσ′,ℏq¯ξ−qξτ∗−Δ4Δ31Δ1yτ∗GℙFIaσ,ℏ℘ξ,q¯ξ−℘ξ,qξτ∗−Δ4Δ3Δ2Δ1q¯ξ−qξτ∗expℏ−1ℏι+1yιGℙFIaσ,ℏ1y∗GℙFIaσ′,ℏq1ξι+−1Δ1yτ∗GℙFIaσ,ℏq1τ∗ισ′expℏ−1/ℏιyιℏσ′Γσ′+1+−1Δ3yτ∗GℙFIaσ,ℏ1y∗GℙFIaσ′,ℏq1ξτ∗−Δ4Δ31Δ1yτ∗GℙFIaσ,ℏq1τ∗expℏ−1ℏι.
Hence using part 1 of Remark1 and (H5), we can get(65)q¯−q≤ϑε+Λ2q¯−q.
In consequence, it follows that(66)q¯−q|≤ϑ1−Λ2ε.
If we letcq=ϑ/1−Λ2, then, the UH stability condition is satisfied. More generally, for(67)cqε=ϑ1−Λ2ε.
cq0=0 the generalized UH stability condition is also satisfied. This completes the proof.


## 5. Application and Numerical Examples

The notations and terminologies in this section are adopted from [11–13]. In control theory, a proportional derivative controller ℙDℂ for controller output q at time ι has the following equation:(68)qι=ϒpEι+ϒdddιEι,where ϒp, ϒd, and E are the proportional gain, the derivative gain, and the input deviation (or the error between the state variable and the process variable), respectively. Recent investigations have shown that ℙDℂ has direct incorporation in the control of complex networks models (see [14] for more details).Let us consider the continuous functionsϒ0,ϒ1:0,1×ℝ⟶0,∞ such that, ∀ι∈ℝ(69)limℏ⟶0+ϒ0ℏ,ι=limℏ⟶0−ϒ1ℏ,ι=0,limℏ⟶0−ϒ0ℏ,ι=limℏ⟶0+ϒ1ℏ,ι=1.Forℏ∈0,1 and ϒ0ℏ,t≠0, ϒ1ℏ,ι≠0 for ℏ∈0,1, ℏ∈0,1, respectively. Then, Anderson et al. [21] defined the proportional derivative of order ℏ by(70)Dℏqι=ϒ0ℏ,ιq′ι+ϒ1ℏ,ιqι,∀ι∈ℝ.Here q′≔d/dιq, provided that the right-hand side exists and ϒ1 is a type of proportional gain ϒp,ϒ0 is a type of derivative gain ϒd, q is the error, and Dℏq is the controller output (for more details refer to [15]). We next restrict ourselves to the case that(71)ϒ0ℏ,ι=ℏ,ϒ1ℏ,ι=1−ℏ.Therefore,(72)Dℏqι=ℏq′ι+1−ℏqι.Clearly,(73)limℏ⟶0+Dℏqι=qι,limℏ⟶1−Dℏqι=q′ι.Thus, equation (72) is considered to be more general than the conformable derivative, which evidently does not tend to the original functions as ℏ tends to 0.Now, we present some examples to illustrate our results.Example 1.
Based on (3), we consider an Gℙ L-SL FDℙs as(74)Gc0.92;2.5153+sin22πι2+sin22πιGc0.43;2.5+1101+expι/2qι=10+ι25+ι2tan−1qι1+qι,q0+q1.85=0,Gc0.43;2.5q0+Gc0.43;2.5q1.85=0.
Forι∈J≕0,1.85. Clearly(75)τ∗=1.85>0,σ=0.92∈0,1,σ′=0.43∈0,1,ℏ=2.5.
By taking(76)yι=153+sin22πι2+sin22πι,wι=1101+expι/2.(77)℘ι,qι=10+ι25+ι2tan−1qι1+qι.
Hypothesis (H1) is held and we have(78)10+ι25+ι2tan−1qι1+qι≤10+ι25+ι2tan−1qι1+qι≤pιqι,where(79)pι=10+ι25+ι2.
By using Equations (20), (30), and (31), we got(80)Δ1=1y0+1yτ∗expℏ−1ℏτ∗≃3.91977,Δ2=wτ∗yτ∗−w0y0≃−0.001704≠0,Δ3=1+expℏ−1ℏτ∗≃3.944679,Δ4=τ∗σ′expℏ−1/ℏτ∗yτ∗ℏσ′Γσ′+1≃0.05168,ϑ=1y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗Δ1τ∗σℏσ+σ′Γσ+1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+1y∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+Δ4y∗Δ1Δ3τ∗σℏσΓσ+1expℏ−1ℏτ∗=1151.851.352.51.35Γ2.35+0.0515×3.9197791.850.922.51.35Γ2.351.850.43exp1.5×1.85/2.515×2.50.43Γ2.43+115×3.944671.851.352.51.35Γ2.35+0.051681315×3.919779×3.9446791.850.922.50.92Γ1.92×exp1.5×1.852.5≃0.062698,here y∗=15, w∗=0.05, and p∗=1.3155, and(81)ν=w∗y∗τ∗σ′ℏσ′Γσ′+1−Δ2Δ1τ∗σ′expℏ−1/ℏτ∗yτ∗ℏσ′Γσ′+1+−Δ2Δ3τ∗σℏσΓσ+1−Δ4Δ3Δ2Δ1expℏ−1ℏτ∗=0.05151.850.432.50.43Γ1.43−−0.0017043.9197791.850.43exp1.5/2.51.8515×2.50.43Γ1.43+−−0.0017043.9446791.850.922.50.92Γ1.92−0.05168133.944679−0.0017043.91977×exp1.52.51.85≃0.003619.
We can see the results ofϑ, ν, and Λ1 for ι∈0,τ∗ in Table 1. These results are plotted in Figure 1. By using the MATLAB program (Algorithm 1) according to (29), we find(82)Λ1=ϑp∗+ν≃0.171934.
Thus from Theorem2, the system of Gℙ L-SL FDs (74) possesses a solution on J=0,1.85.Algorithm 1: MATLAB lines for calculating values of Δi, ϑ, v, and Λ1 in Example 1 for ι∈J=0,τ∗ and τ∗=1.85.
(1)
clear;(2)
format long;(3)
symsv e;(4)
sigma = 0.92; accutesigma = 0.43; hslash = 2.5;(5)
a = 0; tauast = 1.85;(6)
y = 15∗(3sin(2 ∗ pi∗v)^2)/(2sin(2 ∗ pi∗v));(7)
yast = 15;(8)
w = 1/(10∗(1exp(v/2)));(9)
wast = 0.05;(10)
wp = log(v);(11)
mathfrakp= (10v^2)/(5v^2);(12)
mathfrakpast=(10+tauast^2)/(5a^2);(13)(14)
n = floor(sigma)+1;(15)
iota =a;(16)
column = 1;(17)
nn = 1;(18)
while iota ≤ tauast(19)
MI(nn,column) = nn;(20)
MI(nn,column+1) = iota;(21)
Delta_1 = 1/(eval(subs(y, {v}, {a}))) ...(22)
+ 1/(eval(subs(y, {v}, {iota}))) ...(23)
+ exp ((hslash-1)/2 ∗ iota);(24)
MI(nn,column+2) = Delta_1;(25)
Delta_2 = eval(subs(w, {v}, {iota}))/eval(subs(y, {v}, {iota})) ...(26)
- eval(subs(w, {v}, {a}))/eval(subs(y, {v}, {a}));(27)
MI(nn,column+3) = Delta_2;(28)
Delta_3 = 1 + exp((hslash-1)/hslash ∗ iota);(29)
MI(nn,column+4) = Delta_3;(30)
Delta_4 = (iota^accutesigma ∗ exp((hslash-1)/hslash ∗ iota))...(31)
/(eval(subs(y, {v}, {iota}))  ∗  hslash^accutesigma ∗ gamma(accutesigma+1));(32)
MI(nn,column+5) = Delta_4;(33)
vartheta = iota^(sigma + accutesigma)/(yast ∗ hslash^(sigma + accutesigma)...(34)
∗ gamma(sigma + accutesigma+1)) + wast ∗ iota^(sigma)...(35)
∗  iota^(accutesigma)∗exp((hslash-1)/hslash ∗ iota)...(36)
/(yast × Delta_1 ∗ hslash^(sigma + accutesigma)...(37)
∗ gamma(sigma+1)∗ yast ∗ hslash^(accutesigma)...(38)
∗  gamma (accutesigma+1)) + (iota^(sigma + accutesigma)...(39)
/(yast ∗ Delta_3 ∗ hslash^(sigma + accutesigma) ...(40)
∗ gamma(sigma + accutesigma+1)) + Delta_4 ∗ iota^(sigma)...(41)
/(yast ∗ Delta_1 ∗ Delta_3 ∗ hslash^(sigma) ∗  gamma(sigma+1)))...(42)
∗ exp((hslash-1)/hslash ∗ iota);(43)
MI(nn,column+6) = vartheta;(44)
nu = wast ∗ iota^(accutesigma)/(yast ∗ hslash^(accutesigma)...(45)
∗  gamma(accutesigma+1)) - Delta_2 ∗ iota^(accutesigma)...(46)
∗  exp((hslash-1)/hslash ∗ ...iota)/(Delta_1 ∗ yast ∗ hslash^(accutesigma+1))...(47)
+ (-Delta_2 ∗ iota^(sigma)/(Delta_3 ∗ hslash ∗ gamma(sigma+1))...(48)
- Delta_4 ∗ Delta_2/(Delta_3 ∗ Delta_1) ∗ exp((hslash-1)/hslash ∗ iota));(49)
MI(nn,column+7) = nu;(50)
MI(nn,column+8) = vartheta ∗ mathfrakpast + nu;(51)
iota = iota + 0.1;(52)
nn = nn + 1;(53)
end;Table 1
Numerical results ofϑ, ν, and Λ1 for t∈J=0,τ∗ in Example 1.
ιΔ1Δ2Δ3Δ4ϑνΛ1<10.001.088 890.000 002.000 000.000 000.000 000.000 000.000 000.101.173 900.000 292.061 840.015 480.001 120.000 930.003 920.201.256 670.000 172.127 500.021 640.002 870.001 260.008 960.301.347 150.000 112.197 220.027 360.005 010.001 500.014 950.401.445 870.000 102.271 250.033 650.007 460.001 700.021 730.501.543 88−0.000282.349 860.033 890.010 150.001 920.029 160.601.640 90−0.001032.433 330.024 650.013 010.002 170.037 090.701.752 81−0.001482.521 960.017 800.016 090.002 380.045 580.801.884 47−0.001502.616 070.020 010.019 440.002 530.054 720.902.03662−0.001132.716 010.035 130.023 100.002 610.064 611.002.20589−0.000542.822 120.061 640.027 050.002 630.075 241.102.377 89−0.000342.934 790.079 110.031 110.002 710.086 221.202.554 44−0.000443.054 430.085 210.035 250.002 830.097 461.302.746 00−0.000493.181 470.093 650.039 580.002 940.109 181.402.953 66−0.000513.316 370.105 060.044 090.003 040.121 401.503.169 11−0.000803.459 600.099 050.048 600.003 190.133 651.603.392 70−0.001353.611 700.068 470.053 010.003 390.145 701.703.641 06−0.001693.773 200.047 490.057 660.003 530.158 331.803.919 78−0.001713.944 680.051 680.062 700.003 620.171 93Figure 1
Graphical representation ofϑ, ν, and Λ for ι∈0,τ∗ in Example 1. (a) ϑ and ν, (b) θ.
(a)(b)Example 2.
By takingσ=0.51∈0,1, σ′=0.86∈0,1, and ℏ=1.25, let us have the following system of Gℙ L-SL FDℙs:(83)Gc0.51;1.25ι+2Gc0.86;1.25+ι225qι=316+ιqι128+qι,q0+q0.95=0,Gc0.86;2.5q0+Gc0.86;2.5q0.95=0.
Forι∈J≕0,0.95. We choose(84)yι=ι+2,wι=ι225.
And(85)℘ι,qι=316+ιqι128+qι.
So, forι∈J and q,q´, we have(86)316+ιqι123+qι−316+ιq′ι123+q′ι≤Mqι−q′ι,where M=1/8. By using Equations (20), (30), and (31), we got(87)Δ1=1y0+1yτ∗expℏ−1ℏτ∗≃1.96225,Δ2=wτ∗yτ∗−w0y0≃0.012135≠0,Δ3=1+expℏ−1ℏτ∗≃2.20924,Δ4=τ∗σ′expℏ−1/ℏτ∗yτ∗ℏσ′Γσ′+1≃0.33841,ϑ=1y∗τ∗σ+σ′ℏσ+σ′Γσ+σ′+1+w∗y∗Δ1τ∗σℏσ+σ′Γσ+1τ∗σ′expℏ−1/ℏτ∗y∗ℏσ′Γσ′+1+1y∗Δ3τ∗σ+σ′ℏσ+σ′Γσ+σ´+1+Δ4y∗Δ1Δ3τ∗σℏσΓσ+1expℏ−1ℏτ∗≃0.486044.Here y∗=2, w∗=0.0361 and(88)ν=w∗y∗τ∗σ′ℏσ′Γσ′+1−Δ2Δ1τ∗σ′expℏ−1/ℏτ∗yτ∗ℏσ′Γσ′+1+−Δ2Δ3τ∗σℏσΓσ+1−Δ4Δ3Δ2Δ1expℏ−1ℏτ∗≃0.006689.
Table2 shows these results. Also, we can see from the graphical representation of ϑ, ν, and Λ1 in Figures 2 and 3 whenever ι∈0,0.95. By using the MATLAB program (Algorithm 2) according to (48), we find(89)Λ2=ϑM+ν≃0.06744.
Thus, from Theorem3 we conclude that the system of Gℙ L-SL FDℙs (83) has a unique solution on J.Algorithm 2: MATLAB lines for calculating values of Δi, ϑ, ν, and Λ2 in Example 2 for ι∈J=0,τ∗ and τ∗=0.95.
(1)
clear;(2)
format long;(3)
symsv e;(4)
sigma = 0.51; accutesigma = 0.86; hslash = 1.25;(5)
a = 0; tauast = 1.0;(6)
y = sqrt(v)+2;(7)
yast = 2;(8)
w = v^2/25;(9)
wast = 0.0361;(10)
wp = 3/16 +v ∗ abs(e)/(12 ∗ (3abs(e)));(11)
wpzero = 3/16;(12)
M = 1/8;(13)
n = floor(sigma)+1;(14)
iota =a;(15)
column = 1;(16)
nn = 1;(17)
while iota ≤ tauast(18)
MI(nn,column) = nn;(19)
MI(nn,column+1) = iota;(20)
Delta_1 = 1/(eval(subs(y, {v}, {a}))) ...(21)
+ 1/(eval(subs(y, {v}, {iota}))) ...(22)
+ exp ((hslash-1)/2 ∗ iota);(23)
MI(nn, column+2) = Delta_1;(24)
Delta_2 = eval(subs(w, {v}, {iota}))/eval(subs(y, {v}, {iota})) ...(25)
- eval(subs(w, {v}, {a}))/eval(subs(y, {v}, {a}));(26)
MI(nn, column+3) = Delta_2;(27)
Delta_3 = 1+ exp((hslash-1)/hslash ∗ iota);(28)
MI(nn, column+4) = Delta_3;(29)
Delta_4 = (iota∧accutesigma ∗ exp((hslash-1)/hslash ∗ iota))...(30)
/(eval(subs(y, {v}, {iota})) ∗ hslash∧accutesigma ∗ gamma(accutesigma+1));(31)
MI(nn, column+5) = Delta_4;(32)
vartheta = iota∧(sigma + accutesigma)/(yast ∗ hslash∧(sigma + accutesigma)...(33)
∗ gamma(sigma + accutesigma+1)) + wast ∗ iota∧(sigma)...(34)
∗  iota∧(accutesigma)∗exp((hslash-1)/hslash ∗ iota)...(35)
/(yast ∗ Delta_1 ∗ hslash∧(sigma + accutesigma)...(36)
∗ gamma(sigma+1) ∗  yast ∗ hslash∧(accutesigma)...(37)
∗ gamma (accutesigma+1)) + (iota∧(sigma + accutesigma)...(38)
/(yast ∗ Delta_3 ∗ hslash∧(sigma + accutesigma) ...(39)
∗ gamma(sigma + accutesigma+1)) + Delta_4 ∗ iota∧(sigma)...(40)
/(yast ∗  Delta_1 ∗ Delta_3 ∗ hslash∧(sigma) ∗  gamma(sigma+1)))...(41)
∗  exp((hslash-1)/hslash  ∗  iota);(42)
MI(nn, column+6) = vartheta;(43)
nu = wast ∗ iota∧(accutesigma)/(yast ∗ hslash∧(accutesigma)...(44)
∗ gamma(accutesigma+1)) - Delta_2 ∗  iota∧(accutesigma)...(45)
∗ exp((hslash-1)/hslash  ∗  iota)/(Delta_1 ∗ yast ∗ hslash∧(accutesigma+1))...(46)
+ (-Delta_2  ∗  iota∧(sigma)/(Delta_3 ∗ hslash ∗ gamma(sigma+1))...(47)
- Delta_4 ∗ Delta_2/(Delta_3 ∗ Delta_1) ∗ exp((hslash-1)/hslash ∗ iota));(48)
MI(nn, column+7) = nu;(49)
Lamda_2 = vartheta  ∗M + nu;(50)
MI(nn, column+8) = Lamda_2;(51)
MI(nn, column+9) = vartheta  ∗  wpzero/(1-Lamda_2);(52)
iota = iota+0.05;(53)
nn = nn+1;(54)
end;Table 2
Numerical results ofϑ, ν, and Λ2 for t∈J=0,τ∗ in Example 2.
ιΔ1Δ2Δ3Δ4ϑνΛ2<1M≥ϑ℘0/1−Λ20.002.000 000.000 002.000 000.000 000.000 000.000 000.000 000.000 000.051.955 990.000 052.010 050.030 060.008 400.001 190.002 240.001 580.101.944 320.000 172.020 200.052 900.021 690.002 140.004 850.004 090.151.937 810.000 382.030 460.073 470.037 820.002 990.007 720.007 150.201.933 940.000 652.040 810.092 710.056 130.003 760.010 780.010 640.251.931 740.001 002.051 270.111 060.076 280.004 470.014 000.014 510.301.930 720.001 412.061 840.128 760.098 050.005 100.017 360.018 710.351.930 580.001 892.072 510.145 970.121 260.005 670.020 830.023 220.401.931 150.002 432.083 290.162 810.145 820.006 170.024 400.028 020.451.932 280.003 032.094 170.179 370.171 610.006 600.028 050.033 110.501.933 890.003 692.105 170.195 690.198 570.006 960.031 780.038 450.551.935 920.004 412.116 280.211 840.226 640.007 250.035 580.044 060.601.938 300.005 192.127 500.227 860.255 760.007 460.039 430.049 920.651.940 990.006 022.138 830.243 770.285 890.007 590.043 330.056 030.701.943 970.006 912.150 270.259 610.316 990.007 650.047 280.062 390.751.947 200.007 852.161 830.275 390.349 030.007 630.051 260.068 980.801.950 660.008 852.173 510.291 150.381 980.007 520.055 270.075 810.851.954 340.009 892.185 310.306 900.415 810.007 330.059 310.082 880.901.958 210.010 992.197 220.322 650.450 510.007 060.063 370.090 190.950 001.962 260.012 142.209 250.338 420.486 050.006 690.067 450.097 72Figure 2
Graphical representation ofϑ, ν, and Λ2 for ι∈0,τ∗ in Example 2. (a) ϑ and ν, (b) θ.
(a)(b)Figure 3
Numerical results ofM, ν, and ϑ℘0/1−Λ2 for t∈J=0,τ∗ in Example 2.


## 6. Conclusion

With respect to the oddity of the display composition, no commitments exist, as far as we know, concerning the existence theory of theGℙFDs (3) with the assistance of the strategy of K Mℕℂ combined with the Mönch fixed-point theorem. As a result, the objective of this paper is to enhance this scholastic zone by means of modern procedures based on an uncommon idea of Kuratowski measures. Subsequently, the Uℍ and Uℍℝ stability were established for the proposed nonlinear Gℙ L-SL FDℙs (3). We initiate the study of the existence, uniqueness, and different types of US for the Gℙ L-SL FDℙs (3) with Vℂs and AℙBℂs. Moreover, two examples were presented as an illustration of the obtained theory. With respect to another research venture, we are attending to proceed the investigation of such combined structures of physical and mathematical models by utilizing nonsingular fractional operators which provide more exact numerical results.



---

*Source: 7018170-2022-04-22.xml*