# Existence of Solutions for a Coupled System of p-Laplacian Caputo–Hadamard Fractional Sturm–Liouville–Langevin Equations with Antiperiodic Boundary Conditions

Authors: Jinbo Ni; Jifeng Zhang; Wei Zhang
Journal: Journal of Mathematics
 (2022)

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## Abstract

Fractional Sturm–Liouville and Langevin equations have recently attracted much attention. In this paper, we investigate a coupled system of fractional Sturm–Liouville–Langevin equations with antiperiodic boundary conditions in the framework of Caputo–Hadamard fractional derivative. Comparing with the existing literature, we study fractional differential equations with a p-Laplacian operator, which will enrich and generalize the previous work. Based on the Leray–Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem, some interesting existence results are obtained. Finally, an example is constructed to illustrate our main results.


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## 1. Introduction

On the one hand, the Sturm–Liouville problem plays an important role in various fields of sciences and engineering [1]. A standard form of the second-order Sturm–Liouville differential equation is given by(1)−ddtptddtut=ft,ut,t∈a,b,where the function f:a,b×ℝ⟶ℝ is a continuous function.On the other hand, Langevin equation was first formulated by Langevin in 1908 which is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [2]. The classical Langevin equation as follows:(2)d2xdt2+γdxdt=Γt,where x is the coordinate of the particle at time t, γ=α/m is the resistance coefficient per unit mass, m is the mass of particle, and Γt represents lifting force per unit mass.During the past two decades, fractional differential equations have gained prominence and attention by many researchers due to their wide applications as a modeling tool in various research fields, such as physics, biomedical, signal processing, control theory, electric circuits, viscoelasticity, diffusion equations, and electromagnetic waves. For further details, refer [3–6].Recently, by means of different tools, such as the Banach contraction principle, Schaefer’s fixed point, Leray–Schauder nonlinear alternative, Krasnoselskii’s fixed point theorem, coupled fixed-point theorems in partially ordered boundary value problems (BVPs) for fractional Langevin equations have extensively been studied in the papers [7–15].Due to the comprehensive practical application background of Sturm–Liouville and Langevin equations, the Sturm–Liouville–Langevin differential equations have been under consideration by many researchers; for example, in 2016, Kiataramkul et al. [16] discussed the following fractional Sturm–Liouville–Langevin equations with antiperiodic boundary conditions:(3)DβptDα+rtxt=gt,xt,1<t<T,x1=−xT,Dαx1=−DαxT,where Dρ denotes the Caputo–Hadamard fractional derivative of order ρ,ρ∈α,β with 0<α,β<1, p∈C1,T,ℝ and pt≥K>0,r∈C1,T,ℝ, g∈C1,T×ℝ,ℝ. The existence and uniqueness results are obtained by using the Banach contraction mapping principle, Leray–Schauder nonlinear alternative, and Krasnoselskii’s fixed point theorem, respectively.More recently, coupled systems of fractional differential equations are of significant importance as such systems appear in a variety of problems of interdisciplinary fields such as synchronization phenomena, nonlocal thermoelasticity, and bioengineering [17], thus attracted scholars to research on the fractional coupled system of Sturm–Liouville–Langevin equations with various boundary conditions [18–20].In 2020, Muensawat et al. [18] investigated the following fractional coupled Sturm–Liouville–Langevin equations with antiperiodic boundary value conditions.(4)Dα2ptDα1+rtxt=ft,xt,yt,0<t<T,Dβ2qtDβ1+styt=gt,xt,yt,0<t<T,x0=−xT,Dα1x0=−Dα1x(T),y0=−yT,Dβ1y0=−Dβ1y(T),where Dθ denotes Caputo fractional derivative of order θ∈α1,α2,β1,β2,f,g∈C0,T×ℝ2,ℝ,p,q∈C0,T,ℝ\0 and pt≥K1,qt≥K2,K1,K2>0,r,s∈C0,T,ℝ. The existence and uniqueness results are obtained by using the Banach contraction mapping principle and Leray–Schauder nonlinear alternative.In [19], Berhail et al. discussed the following fractional coupled Sturm–Liouville–Langevin equations with the help of the Banach contraction mapping principle and Leray–Schauder nonlinear alternative:(5)DαipitDβi+rituit=fit,u1t,u2t,t∈1,T,i=1,2,ui1=0,DβiuiT+riTpiTuiT=0,where 0<αi,βi<1,Dv denotes the Hadamard fractional derivative of order v∈αi,βi,pi∈C1,T,ℝ\0,ri∈C1,T,ℝ\0,fi:1,T×ℝ×ℝ⟶ℝ that are continuous functions for i=1,2.Motivated by the above results, in this paper, we study the existence results for the following fractional p-Laplacian coupled Sturm–Liouville–Langevin equations with antiperiodic boundary conditions:(6)DHC1+βϕpb1tDHC1+α+r1tx1t=f1t,x2t,1<t<T,DHC1+βϕpb2tDHC1+α+r2tx2t=f2t,x1t,1<t<T,x11=−x1T,DHC1+αx1(1)=−DHC1+αx1T,x21=−x2T,DHC1+αx21=−DHC1+αx2T,where DHC1+θ denotes Caputo–Hadamard fractional derivative of order θ∈α,β with 0<α,β<1, ϕps=sp−2s, p>1 is a p-Laplacian operator, and its inverse is ϕp−1=ϕq, where 1/p+1/q=1, q>1, bi∈C1,T,ℝ,bit≥K>0,ri∈C1,T,ℝ,bi1=biT,ri1=riT,i=1,2,f1,f2:1,T×ℝ⟶ℝ are two continuous functions.We note that system (6) is a generalization of Sturm–Liouville and Langevin fractional differential systems. For the special case:(1)
Ifrit≡0,i=1,2 for t∈1,T, then the problem (6) is reduced to fractional p-Laplacian coupled Sturm–Liouville equations with antiperiodic boundary conditions.(7)DHC1+βϕpb1tDHC1+αx1t=f1t,x2t,1<t<T,DHC1+βϕpb2tDHC1+αx2t=f2t,x1t,1<t<T,x11=−x1T,DHC1+αx11=−DHC1+αx1T,x21=−x2T,DHC1+αx21=−DHC1+αx2T.(2)
Ifbit≡1,rit≡λ,i=1,2,λ∈ℝ,t∈1,T, then the problem (6) is reduced to fractional p-Laplacian coupled Langevin equations with antiperiodic boundary conditions.(8)DHC1+βϕpDHC1+α+λx1t=f1t,x2t,1<t<T,DHC1+βϕpDHC1+α+λx2t=f2t,x1t,1<t<T,x11=−x1T,DHC1+αx11=−DHC1+αx1T,x21=−x2T,DHC1+αx21=−DHC1+αx2T.The rest of the paper is organized as follows. In Section2, we recall some useful preliminaries. Section 3 contains two existence results which are obtained by applying Leray–Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem, respectively. In Section 4, we present a concrete example to illustrate the theoretical results. Finally, some concluding remarks are given in Section 5.


## 2. Preliminaries

In this section, we recall some preliminary definitions, lemma, and theorems that will be used to prove our main results.Definition 1.
(see [3]). The Hadamard fractional integral of order αα>0 is defined as(9)IH1+αft=1Γα∫1tlntsα−1fsdss,provided that the integral exists.Definition 2.
(see [21]). For an at least n-times differentiable function f:[1,∞)⟶ℝ, the Caputo–Hadamard derivative of fractional order α is defined as(10)DHC1+αft=1Γn−α∫1tlntsn−α−1δnfsdss,n−1<α<n,where δ=td/dt,n=α+1,α denotes the integer part of the real number α.Lemma 1 (see [21]).
Letu∈ACδna,b or Cδna,b and α>0, where Xδna,b=u:a,b⟶ℝ:δn−1ut∈Xa,b. Then,(11)IH1+αDHC1+αut=ut−∑k=0n−1cklntk,where ck∈ℝ,k=0,1,2,…,n−1,n=α+1.Theorem 1 (see [22]).
(Leray–Schauder nonlinear alternative) LetE be a Banach space, C be a closed and convex subset of E, U be a relatively open subset of C such that 0∈U, and S:U¯⟶C be completely continuous. Then, either(i)
S has a fixed point in U¯, or(ii)
there existx∈∂U (the boundary of U in C) and ν∈0,1 such that x=νSx.Theorem 2 (see [23]).
(Krasnoselskii’s fixed point theorem). LetΩ be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A,B be two operators such that(i)
Ax+By∈Ω, where x,y∈Ω(ii)
A is compact and continuous(iii)
B is a contraction mapping
Then, there existsz∈Ω such that z=Az+Bz.


## 3. Main Results

It is defined that the spaceX=xt|xt∈C1,T,ℝ endowed with the norm x∞=maxxt,t∈1,T. It is obvious that X,⋅∞ is a Banach space. Then, the product space X×X,⋅ is also a Banach space with the norm x1,x2=maxx1∞,x2∞.Lemma 2.
LetF1,F2∈C1,T,ℝ. Then, the solution of the coupled system(12)DHC1+βϕpb1tDHC1+α+r1tx1t=F1t,t∈1,T,DHC1+βϕpb2tDHC1+α+r2tx2t=F2t,t∈1,T,x11=−x1T,DHC1+αx11=−DHC1+αx1T,x21=−x2T,DHC1+αx21=−DHC1+αx2T,is given by(13)x1t=−IH1+αr1tb1tx1t+IH1+α1b1tϕqIH1+βF1t−12IH1+βF1t|t=T+12IH1+αr1tb1tx1t|t=T−12IH1+α1b1tϕqIH1+βF1t−12IH1+βF1t|t=T,(14)x2t=−IH1+αr2tb2tx2t+IH1+α1b2tϕqIH1+βF2t−12IH1+βF2t|t=T+12IH1+αr2tb2tx2t|t=T−12IH1+α1b2tϕqIH1+βF2t−12IH1+βF2t|t=Tt=T.Proof.
Applying the operatorIH1+β on both sides of the first equation in (12) and then applying Lemma 1, we obtain(15)ϕp(b1tDHC1+α+r1t)x1t=IH1+βF1t+c0,c0∈ℝ.
Sinceϕp−1⋅=ϕq⋅, we can find that(16)b1tDHC1+αx1t+r1tx1t=ϕqIH1+βF1t+c0.
By using the boundary conditionsx11=−x1T,DHC1+αx11=−DHC1+αx1T, we get(17)b11DHC1+αx11+r11x11=ϕqc0,(18)b1TDHC1+αx1T+r1Tx1T=ϕqIH1+βF1T|t=T+c0.
Combining formulas (17) and (18), we find that(19)c0=−12IH1+βF1t|t=T.
Substituting the value ofc0 into (16), we obtain(20)DHC1+αx1t+r1tb1tx1t=1b1tϕqIH1+βFt−12IH1+βFt|t=T.
Taking the operatorIH1+α to the both sides of (20), and using Lemma 1 yields(21)x1t+IH1+αr1tb1tx1t=IH1+α1b1tϕqIH1+βF1t−12IH1+βF1t|t=T+c1,c1∈ℝ.
Applying the boundary conditionx11=−x1T in (21), it follows that(22)c1=12IH1+αr1tb1tx1t|t=T−12IH1+α1b1tϕqIH1+βF1t−12IH1+βF1t|t=Tt=T.
Substituting the valuec1 into (21), we obtain the solution (13). Similarly, using the same way in second equation of the system (12), we obtain (14). The converse follows by direct computation. This completes the proof.
Based on Lemma2, we define the operator S:X×X⟶X×X as follows:(23)Sx1,x2t=S1x1,x2t,S2x1,x2t,∀x1,x2∈X×X,where(24)S1x1,x2t=−1Γα∫1tlntsα−1r1sb1sx1sdss+12Γα∫1TlnTsα−1r1sb1sx1sdss+1Γα∫1tlntsα−11b1sϕq1Γβ∫1slnsτβ−1f1τ,x2τdττ−12Γβ∫1TlnTτβ−1f1τ,x2τdττdss−12Γα∫1TlnTsα−11b1sϕq1Γβ∫1slnsτβ−1f1τ,x2τdττ−12Γβ∫1TlnTτβ−1f1τ,x2τdττdss,S2x1,x2t=−1Γα∫1tlntsα−1r2sb2sx2sdss+12Γα∫1TlnTsα−1r2sb2sx2sdss+1Γα∫1tlntsα−11b2sϕq1Γβ∫1slnsτβ−1f2τ,x1τdττ−12Γβ∫1TlnTτβ−1f2τ,x1τdττdss−12Γα∫1TlnTsα−11b2sϕq1Γβ∫1slnsτβ−1f2((τ,x1τ)dττ−12Γ(β)∫1TlnTτβ−1f2(τ,x1(τ))dττdss.
We observe that the solutions of system (6) are equivalent operator S that has fixed points.
For the sake of computational convenience, we make the following notations:(25)bi∗=mint∈1,Tbit,ri∗=maxt∈1,Trit,i=1,2,Δ1=3lnTαr1∗2Γα+1b1∗,Δ1′=3qlnTα+βq−12qb1∗Γα+1Γβ+1q−1,Δ2=3lnTαr2∗2Γα+1b2∗,Δ2′=3qlnTα+βq−12qb2∗Γα+1Γβ+1q−1,Δ′=maxΔ1′,Δ2′=3qlnTα+βq−12qb∗Γα+1Γβ+1q−1,Δ=maxΔ1,Δ2=3lnTαr∗2Γα+1b∗,b∗=minb1∗,b2∗,r∗=maxr1∗,r2∗.
Now, we give an existence result for BVP (6) via Leray–Schauder nonlinear alternative.Theorem 3.
We assume thatf1,f2:1,T×ℝ2⟶ℝ are two continuous functions and the following conditions hold:(i)
H1. there exist continuous nondecreasing function ψ:[0,∞)⟶[0,∞) and functions φ1,φ2∈C1,T,ℝ+ such that(26)f1t,x2t≤ϕpφ1tψx2t,f2t,x1t≤ϕpφ2tψx1t.(ii)
H2. There exists a constant M>0 such that(27)1−ΔMΔ′φ∞ψM>1,whereφ∞=maxφ1∞,φ2∞. Then, coupled system (6) has at least one solution on 1,T, provided that Δ<1.Proof.
We divide the proof in to the following two steps:Step 1.
We prove the operatorS:X×X⟶X×X is completely continuous. From the continuity of the functions f1 and f2, we can easily obtain that the operator S is continuous. We now show that the operator S is compact. To this end, let ρ>0, and we define Bρ=x1,x2∈X×X:x1,x2≤ρ. Firstly, we claim that S is uniformly bounded on Bρ. In fact, by using condition H1, for any t∈1,T and x1,x2∈Bρ, we have(28)S1x1,x2t≤1Γα∫1TlnTsα−1r1sb1sx1sdss+12Γα∫1TlnTsα−1r1sb1sx1sdss+1Γα∫1TlnTsα−11b1sϕq1Γβ∫1TlnTτβ−1f1τ,x2τdττ+12Γβ∫1TlnTτβ−1f1τ,x2τdττdss+12Γα∫1TlnTsα−11b1sϕq1Γβ∫1TlnTτβ−1f1τ,x2τdττ+12Γβ∫1TlnTτβ−1f1τ,x2τdττdss≤3lnTαr1∗ρ2Γα+1b1∗+3lnTα2Γα+1b1∗ϕq3lnTβ2Γβ+1ϕpφ1∞ψρ=3lnTαr1∗2Γα+1b1∗ρ+3lnTα2Γα+1b1∗3lnTβ2Γβ+1q−1φ1∞ψρ=Δ1ρ+Δ1′φ1∞ψρ.
Thus,S1x1,x2∞≤Δ1ρ+Δ1′φ1∞ψρ. In a similar way, we can obtain(29)S2x1,x2∞≤Δ2ρ+Δ2′φ2∞ψρ.
Hence,(30)Sx1,x2=maxS1x1,x2∞,S2x1,x2∞≤Δρ+Δ′φ∞ψρ.
Consequently, the operatorS is uniformly bounded on Bρ. Secondly, we claim that S is equicontinuous on Bρ. In fact, for any x1,x2∈Bρ, t1,t2∈1,T with t1<t2, we have(31)S1x1,x2t2−S1x1,x2t1≤r1∗ρΓαb1∗∫1t2lnt2sα−1dss−∫1t1lnt1sα−1dss+1Γαb1∗∫1t2lnt2sα−1dss−∫1t1lnt1sα−1dssϕq3lnTβ2Γβ+1ϕpφ1ψρ≤r1∗ρΓα+1b1∗lnt1α−lnt2α+2lnt2t1α+3qlnTβq−1φ1ψρ2q−1Γα+1(Γβ+1q−1b1∗lnt1α−lnt2α+2lnt2t1α≤r1∗ρΓα+1b1∗+3qlnTβq−1φ1ψρ2q−1Γα+1(Γβ+1q−1b1∗lnt1α−lnt2α+2lnt2t1α.
This yields(32)S1x1,x2t2−S1x1,x2t2⟶0,as t1⟶t2. In a similar manner, we deduce that(33)S2x1,x2t2−S2x1,x2t1⟶0,as t1⟶t2. Thus, the operator S is equicontinuous. From the Arzelá–Ascoli theorem, it follows that the operator S is completely continuous.Step 2.
We shall show that the operatorS:X×X⟶X×X has fixed points. To this end, we define a set U=x1,x2∈X×X,x1,x2<M. First, we show that SU¯⊂U¯. In fact, for any x1,x2∈U¯ and t∈0,1, we shall adopt the same procedure as in the proof of formula (30), and it follows(34)Sx1,x2=maxS1x1,x2∞,S2x1,x2∞≤ΔM+Δ′φ∞ψM,and then combined with H2 yields that(35)Sx1,x2<M,which means that SU¯⊂U¯. Next, we claim that for any x1,x2∈∂U and ν∈0,1, then x1,x2≠νSx1,x2. By using reduction to absurdity, we suppose that there exists x1,x2∈∂U such that x1,x2=νSx1,x2 for some ν∈0,1. Then, we have(36)M=x1,x2=νSx1,x2≤maxS1x1,x2,S2x1,x2≤ΔM+Δ′φ∞ψM<M.
This leads to a contradiction. Consequently, the operatorS has a fixed point x1,x2∈U¯ by the virtue of the Leray–Schauder nonlinear alternative theorem. Hence, the given problem (6) has at least one solution on 1,T. The proof is completed.Corollary 1.
We suppose that the conditionH1 holds. If there exists constant M>0 such that(37)MΔ′φ∞ψM>1,then the problem (7) has at least one solution on 1,T.Corollary 2.
We assume that the conditionH1 is satisfied. If there exists constant M>0 such that(38)M1−3lnTαλ2Γα+13qlnTα+βq−1‖φ|∞ψM2qΓα+1Γβ+1q−1−1>1,then the problem (8) has at least one solution on 1,T.Our last existence result is derived by Krasnoselskii’s fixed point theorem. To this end, we define the operatorsE and G by(39)E,G:X×X⟶X×X,S=E+G,with(40)Ex1,x2t=E1x1t,E2x2t,Gx1,x2t=G1x2t,G2x1t,where the operators Ei,Gi:X⟶X are defined by(41)Eixit=−1Γα∫1tlntsα−1risbisxisdss+12Γα∫1TlnTsα−1risbisxisdss,i=1,2,Gixjt=1Γα∫1tlntsα−11bisϕq1Γβ∫1slnsτβ−1fiτ,xjτdττ−12Γβ∫1TlnTτβ−1fiτ,xjτdττdss−12Γα∫1TlnTsα−11bisϕq1Γβ∫1slnsτβ−1fiτ,xjτdττ−12Γβ∫1TlnTτβ−1fiτ,xjτdττdss,i,j=1,2,i≠j.Theorem 4.
We assume thatf1,f2:1,T×ℝ⟶ℝ are continuous functions and the following condition holds:(i)
H3. There exists a function wt∈C1,T,ℝ+ such that fi≤ϕpwt,i=1,2.
Then, BVP (6) has at least one solution, provided that Δ<1.Proof.
We define a setBσ by(42)Bσ=x1,x2∈X×X:x1,x2≤σ,where(43)σ≥Δ′w∞1−Δ.
Clearly,Bσ⊂X×X is a closed, convex, and nonempty subset. Now, the proof will be given in three steps:Step 1.
For anyx=x1,x2,y=y1,y2∈Bσ,Ex+Gy∈Bσ. In fact, taking into account assumption H3, we have(44)Eixi+Giyj≤1Γα∫1TlnTsα−1risbisxisdss+12Γα∫1TlnTsα−1risbisxisdss+1Γα∫1TlnTsα−11bisϕq1Γβ∫1TlnTτβ−1fiτ,yjτdττ+12Γβ∫1TlnTτβ−1fiτ,yjτdττdss+12Γα∫1TlnTsα−11bisϕq1Γβ∫1slnsτβ−1fiτ,yjτdττ+12Γβ∫1TlnTτβ−1fiτ,yjτdττdss≤3lnTαri∗xi∞2Γα+1bi∗+3lnTα2Γα+1bi∗ϕq3lnTβ2Γβ+1ϕpw∞≤3lnTαri∗xi∞2Γα+1bi∗+3qlnTα+βq−1w∞2qbi∗Γα+1Γβ+1q−1,i,j=1,2,i≠j,which on taking the norm for t∈1,T, yields(45)Eixi+Giyi∞≤3lnTαri∗σ2Γα+1bi∗+3qlnTα+βq−1w∞2qbi∗Γα+1Γβ+1q−1,i,j=1,2,i≠j.
It follows that(46)Ex+Gy=maxE1x1+G1y2∞,E2x2+G2y1∞≤3lnTαr∗σ2Γα+1b∗+3qlnTα+βq−1‖w‖∞2qb∗Γα+1Γβ+1q−1=Δσ+Δ′w∞≤σ.
Thus,Ex+Gy∈Bσ.Step 2.
It is to show thatE is a contraction mapping. In fact, for each x=x1,x2,y=y1,y2∈Bσ and t∈1,T, we have(47)Eixit−Eiyit≤1Γα∫1tlntsα−1risbisxis−yisdss+12Γα∫1TlnTsα−1risbisxis−yisdss≤3ri∗2Γαbi∗∫1TlnTsα−1dssxi−yi∞≤3ri∗lnTα2Γα+1bi∗xi−yi∞,i=1,2.
Evaluating the maximum values, fort∈1,T, one has(48)Eixi−Eiyj∞≤3ri∗lnTα2Γα+1bi∗xi−yi∞.
Using the definition of the norm in the Banach spaceX×X, we obtain(49)Ex−Ey≤3r∗lnTα2Γα+1b∗x−y=Δx−y.
In view ofΔ<1, we can conclude that E is a contraction mapping.Step 3.
On the one hand, we prove thatG is compact and continuous. Indeed, the continuity of f1, f2, and ϕq⋅ implies that the operator G is continuous. On the other hand, by Step 1, it is not hard to see that G is uniformly bounded on Bσ. Finally, we have to show that the operator G is equicontinuous; for t1,t2∈1,T with t1<t2 and x=x1,x2∈Bσ, we have(50)Gixjt2−Gixjt1≤1Γαbi∗∫1t2lnt2sα−1dss−∫1t1lnt1sα−1dss×ϕq1Γβ∫1slnsτβ−1fiτ,xjτdττ+12Γβ∫1TlnTτβ−1fiτ,xjτdττ≤1Γαb∗ϕq3lnTβ2Γβ+1ϕpw∞∫1t2lnt2sα−1dss−∫1t1lnt1sα−1dss≤3q−1lnTβq−1w∞2q−1b∗Γα+1Γβ+1q−12lnt2t1α+lnt2α−lnt1α,i,j=1,2,i≠j,which is independent of x∈Bσ and tends to zero as t1⟶t2. Thus, Gii=1,2 are equicontinuous, which implies that G is equicontinuous. By Arzelá–Ascoli theorem, G is compact on Bσ. Hence, by Theorem 2, there exists a fixed point x=x1,x2∈X×X such that Sx=x, which is a solution to the BVP (6) on 1,T. This completes the proof.Corollary 3.
We suppose that the conditionH3 is satisfied. Then, the BVP (7) has at least one solution.Corollary 4.
We suppose that the conditionH3 is satisfied. If(51)3lnTαλ2Γα+1<1,then the BVP (8) has at least one solution.


## 4. Example

Example 1.
Considering the following coupled Strum–Liouville–Langevin equations via Caputo–Hadamard derivative with antiperiodic boundary condition involving p-Laplacian operator(52)DHC1+2/3ϕ3/2t−22+1DHC1+1/2+t−22/2x1t=tcosx2t/40,t∈1,3,DHC1+2/3ϕ3/2t−22+2DHC1+1/2+t−22x2t=tcosx1t/40,t∈1,3,x11=−x13,DHC1+1/2x11=−DHC1+1/2x13,x21=−x23,DHC1+1/2x21=−DHC1+1/2x23.
Corresponding to problem (6), here,(53)α=12,β=23,p=32,T=3,f1t,x2t=tcosx2t40,f2t,x1t=tcosx1t40,b1t=t−22+1,b2t=t−22+2,r1t=t−222,r2t=t−22.
We choose(54)φ1t=φ2t=t24,M=6,ψx=x2400acs,x∈[0,∞).
One can easily find thatH1 is satisfied. A simple calculation gives(55)q=3,b∗=maxb1∗,b2∗=2,r∗=maxr1∗,r2∗=1,φ∞=94,ψM=9100,Δ=3ln31/24Γ3/2≈0.887<1,Δ′=33ln311/624Γ3/2Γ5/32≈2.776.
Moreover,(56)1−ΔMΔ′φ∞ψM=1−0.887×62.776×9/4×9/100≈1.206>1.
This implies thatH2 is satisfied. By Theorem 3, the problem (52) has at least one solution.
Letwt=t/16; then, the assumption H3 holds. In view of Δ<1, then from Theorem 4, it follows that problem (52) has at least one solution.


## 5. Conclusion

In the present article, we investigated the existence results for a kind of coupled system of p-Laplacian fractional Sturm–Liouville–Langevin equations involving Caputo–Hadamard factional derivative and supplemented with antiperiodic boundary conditions. By using the Leray–Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem, the existence of solutions for the underlying problem is established under certain assumptions on the righthand side functionsf1 and f2. In the end, we provided an example to illustrate the applicability of our results. Comparing our work with the available literature on the topic [16, 18–20], we considered the fractional Sturm–Liouville–Langevin equations with p-Laplacian operator (ϕps=sp−2s, p>1). We notice that ϕp⋅ is a quasilinear operator, which brings difficulties to the study of the problem (6). For instance, since ϕp(s) is a quasilinear operator, which brings the difficulties to prove the compactness and the priori estimations of the operators S and G in Theorems 3 and 4, etc. In the special case if p=2, then ϕps=s, which means that (6) can be reduced to the problems studied in [18–20]. Therefore, our results are new, and we extend some previous results. Last but not least, an interesting extension of our studies would be to consider the fractional p-Laplacian Sturm–Liouville–Langevin equations on graph.



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