# Multiple Positive Solutions for a System of Caputo Fractionalp-Laplacian Boundary Value Problems

Authors: Zhi-Wei Lv; Jie Liu; Jiafa Xu
Journal: Complexity
 (2020)

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## Abstract

In this work, we are pleased to investigate multiple positive solutions for a system of Caputo fractionalp-Laplacian boundary value problems, and we also provide an example for illustrating our main results.


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## Body

## 1. Introduction

In this work, we are pleased to discuss the positive solutions for the following system of Caputo fractionalp-Laplacian boundary value problems:(1)Dtν1cφpDtμ1cut+g1t,ut,vt=0,0<t<1,Dtν2cφpDtμ2cvt+g2t,ut,vt=0,0<t<1,α1u0−β1u1=0,γ1u′0−δ1u′1=0,Dtμ1cu0=0,Dtμ1cu1=b1Dtμ1cuξ1,α2v0−β2v1=0,γ2v′0−δ2v′1=0,Dtμ2cv0=0,Dtμ2cv1=b2Dtμ2cvξ2,where Dtμic,Dtνici=1,2 are the fractional derivatives of Caputo sense with μi,νi∈1,2; φp is the p-Laplacian, i.e., φpx=xp−2x with p>1,x∈ℝ; and the constants αi,βi,γi,δi, bi,ξii=1,2 and the functions gii=1,2 satisfy the following conditions:(H0)αi>βi>0, γi>δi>0, bi≥0,ξi∈0,1 with bip−1ξi∈0,1(H1)gii=1,2 are nonnegative continuous functions on 0,1×ℝ+×ℝ+, where ℝ+≔0,+∞As a generalization of integer-order equations, fractional-order equations can effectively describe various materials and physical processes with memory and genetic properties. It has a large number of applications in our society, such as biology, chemical kinetics, electromagnetics, transmission and diffusion, and automatic control. Recently, there many researchers pay their attentions to studying the existence of solutions for various types of fractional-order equations by use of fixed point theorems, upper and lower solution methods, and monotone iterative techniques. For instance, we refer the readers to [1–25] and references therein.In [1], by using the method of upper and lower solutions and the Schauder fixed point theorem, Vong investigated the positive solutions for the following nonlocal fractional boundary value problem:(2)Dtαcut+ft,ut=0,0<t<1,u′0=⋯=un−10=0,u1=∫01usdμs,where n≥2,α∈n−1,n, and μs is a function of bounded variation. f may be singular at t=1.In [2], Wang and Yang studied the integral boundary value problem of Caputo sense:(3)Dtqcχt=ft,χt,0<t<1,αχ0−βχ1=∫01htχtdt,γχ′0−δχ′1=∫01gtχtdt.By the Leggett–Williams fixed point theorem, they obtained multiple positive solutions when the nonlinearityf is bounded from below. Moreover, when f is asymptotically linear at infinity, they also obtained an existence theorem.In [3], Wang et al. adopted the theory of mixed monotone operators to obtain a unique positive solution for the mixed fractional boundary value problems involving the p-Laplacian:(4)DtαcφpD0+βvt+λft,vt,D0+β1vt,…,D0+βn−2vt=0,0<t<1,v0=D0+νiv0=D0+βv0=φpD0+βv0′=0,i=1,…,n−2,D0+r0v1=λ1∫01h1sD0+r1vsdA1s+λ2∫0ηh2sD0+r2vsdA2s+λ3∑j=1∞ρjD0+r3vηj,where Dtαc is the Caputo fractional derivative and D0+β,D0+βi,D0+νi,andD0+ri are the Riemann–Liouville fractional derivatives.Recently, coupled systems of fractional differential equations have also been investigated by many authors. Some results on the direction can be found in a series of papers [11–25] and the references cited therein.In [11], Wang utilized the Guo–Krasnosel’skii fixed point theorem to investigate the multiple positive solutions for the mixed fractional p-Laplacian differential system:(5)D0+β1φp1Dtα1cut+f1t,ut,vt=0,0<t<1,D0+β2φp2Dtα2cvt+f2t,ut,vt=0,0<t<1,u′0=u″0=⋯=un−10=0,u1=μ1∫01asvsdA1s,v′0=v″0=⋯=vm−10=0,v1=μ2∫01bsusdA2s,Dtα1cu0=0,Dtα1cu1=ε1Dtα1cuη1,Dtα2cv0=0,Dtα2cv1=ε2Dtα2cvη2,where D0+βi are the Riemann–Liouville fractional derivatives, Dtαic the Caputo fractional derivatives, and ∫01asvsdA1s and ∫01bsusdA2s are the Riemann–Stieltjes integrals.In [12], Rao studied the system of fractional p-Laplacian differential equations:(6)D0+β1φpD0+α1ut+f1t,ut,vt=0,a<t<b,D0+β2φpD0+α2vt+f2t,ut,vt=0,a<t<b,ξua−ηu′a=0,γub+δu′b=0,D0+α1ua=0,ξva−ηv′a=0,γvb+δv′b=0,D0+α2va=0.When the nonlinearitiesfi satisfy some appropriate conditions, the author made use of the Avery–Henderson fixed point theorem and the six functionals’ fixed point theorem to obtain some existence theorems of multiple positive solutions.Inspired by the aforementioned results, in this work, we study the solvability for (1) and establish the existence results of multiple positive solutions via the six functional fixed point theorem under some bounded conditions for gii=1,2. Finally, we also provide an example to illustrate our main results.


## 2. Preliminaries

In this section, we only recall the definition of Caputo fractional derivative, for more details, see the book [26].Definition 1.
The fractional derivative off in the Caputo sense is defined as(7)Dtμcft=1Γn−μ∫0tt−sn−μ−1fnsds,n−1<μ<n,where n=μ+1, μ denotes the integer part of the number μ.
Now, we calculate Green’s functions associated with (1). Let(8)φpDtμ1cut=xt,0<t<1,φpDtμ2cvt=yt,0<t<1,and then by the boundary conditions in (1), we have(9)x0=0,x1=b1p−1xξ1,y0=0,y1=b2p−1xξ2.
Consequently, substituting (8) and (9) into (1), we obtain(10)Dtν1cxt+g1t,ut,vt=0,0<t<1,Dtν2cyt+g2t,ut,vt=0,0<t<1,x0=0,x1=b1p−1xξ1,y0=0,y1=b2p−1xξ2.
We next translate (10) into an equivalent system of integral equations. By the similar arguments as in [2], we have the following result.Lemma 1.
Problem (10) is equivalent to the following system of Hammerstein-type integral equations:(11)xtyt=∫01H1t,sg1s,us,vsds∫01H2t,sg2s,us,vsds,where(12)Hit,s=H¯it,s+bip−1t1−bip−1ξiH¯iξi,s,t,s∈0,1,H¯it,s=1Γνit1−sνi−1−t−sνi−1,0≤s≤t≤1,t1−sνi−1,0≤t≤s≤1.Proof.
We only need to consider the casei=1 (by the similar method, the case i=2 can be easily proved). Using Lemma 2.5 of [2], we have(13)xt=c0+c1t−∫0tt−sν1−1Γν1g1s,us,vsds,where ci∈ℝ,i=0,1. x0=0 implies that c0=0, and by x1=b1p−1xξ1, we have(14)c1−∫011−sν1−1Γν1g1s,us,vsds=b1p−1ξ1c1−b1p−1∫0ξ1ξ1−sν1−1Γν1g1s,us,vsds,c1=11−b1p−1ξ1∫011−sν1−1Γν1g1s,us,vsds−b1p−11−b1p−1ξ1∫0ξ1ξ1−sν1−1Γν1g1s,us,vsds.
Therefore,(15)xt=11−b1p−1ξ1∫01t1−sν1−1Γν1g1s,us,vsds−b1p−1t1−b1p−1ξ1∫0ξ1ξ1−sν1−1Γν1g1s,us,vsds−∫0tt−sν1−1Γν1g1s,us,vsds=11−b1p−1ξ1∫01t1−sν1−1Γν1g1s,us,vsds−b1p−1t1−b1p−1ξ1∫0ξ1ξ1−sν1−1Γν1g1s,us,vsds−∫01t1−sν1−1Γν1g1s,us,vsds+∫01t1−sν1−1Γν1g1s,us,vsds−∫0tt−sν1−1Γν1g1s,us,vsds=∫01H¯1t,sg1s,us,vsds+b1p−1ξ11−b1p−1ξ1∫01t1−sν1−1Γν1g1s,us,vsds−b1p−1t1−b1p−1ξ1∫0ξ1ξ1−sν1−1Γν1g1s,us,vsds=∫01H¯1t,sg1s,us,vsds+b1p−1t1−b1p−1ξ1∫01H¯1ξ1,sg1s,us,vsds=∫01H1t,sg1s,us,vsds.
This completes the proof.
Note from (8) and Lemma 1, we have(16)φpDtμ1cut=xt=∫01H1t,sg1s,us,vsds,0<t<1,φpDtμ2cvt=yt=∫01H2t,sg2s,us,vsds,0<t<1,and(17)Dtμ1cut=φp−1∫01H1t,sg1s,us,vsds,0<t<1,Dtμ2cvt=φp−1∫01H2t,sg2s,us,vsds,0<t<1,with the boundary conditions(18)α1u0−β1u1=0,γ1u′0−δ1u′1=0,α2v0−β2v1=0,γ2v′0−δ2v′1=0.
By Lemma 2.5 of [2], we have the result:Lemma 2.
Problems (17) and (18) is equivalent to the following Hammerstein-type integral equation:(19)utvt=∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds,where(20)Git,s=t−sμi−1Γμi+βi1−sμi−1αi−βiΓμi+βiδiμi−11−sμi−2αi−βiγi−δiΓμi+δiμi−1t1−sμi−2γi−δiΓμi,0≤s≤t≤1,βi1−sμi−1αi−βiΓμi+βiδiμi−11−sμi−2αi−βiγi−δiΓμi+δiμi−1t1−sμi−2γi−δiΓμi,0≤t≤s≤1.Lemma 3 (see [2], Lemma 2.8).
The functionsGi,Hii=1,2 have the following properties:(i)
Gi,Hi∈C0,1×0,1,ℝ+(ii)
ℳis≤Git,s≤αi/βiℳis,∀t,s∈0,1, where ℳis=βi1−sμi−1/αi−βiΓμi+βiδiμi−11−sμi−2/αi−βiγi−δiΓμiLetE≔C0,1,u≔maxt∈0,1utandP≔u∈E:ut≥0,∀t∈0,1. Then, E,⋅ is a real Banach space and P is a cone on E. Moreover, E×E is a Banach space with the norm u,v=u+v, and P×P a cone on E×E. From Lemmas 1 and 2, we define operators Tii=1,2 and T as follows:(21)T1u,vt=∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds,T2u,vt=∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds,Tu,vt=T1,T2u,vt,fort∈0,1.Then, we obtain thatTii=1,2:P×P⟶P, T:P×P⟶P×P are completely continuous operators, and if there exists u¯,v¯∈P×P\0 such that Tu¯,v¯=u¯,v¯, i.e., T1u¯,v¯=u¯, T2u¯,v¯=v¯, then u¯,v¯ is a positive solution for (1).Lemma 4 (see [27]).
LetP be a cone in a real Banach space E. Suppose that α,ψ, and ζ are nonnegative continuous concave functionals on P, β,σ,andθ are nonnegative continuous convex functionals on P, and there are nonnegative constants l,l′,r,r′,R, and R′ such that A:Qβ,R⟶P is a completely continuous operator and(B1)Qβ,R is a bounded set(B2)Qσ,l and Qα,β,r,R are disjoint subset of Qβ,R(B3)u∈P:θu<r′,r<αu,R′<ψu,βu<R≠∅(B4)u∈P:l′<ζu,σu<l≠∅(B5)u∈P:l<σu,αu<r≠∅If the following claims hold:(C1)αAu>r,∀u∈P with αu=r,βu≤R, and r′<θAu(C2)αAu>r,∀u∈P with αu=r,βu≤R, and θu≤r′(C3)βAu<R,∀u∈P with r≤αu,βu=R, and ψAu<R′(C4)βAu<R,∀u∈P with r≤αu,βu=R, and R′≤ψu(C5)σAu<l,∀u∈P with σu=l and ζAu<l′(C6)σAu<l,∀u∈P with σu=l and l′≤ζuThen,A has at least three fixed points u1,u2, and u3 in Qβ,R such that σu1≤l,r≤u2 with βu2≤R, and l<σu3 with αu3<r.


## 3. Main Results

We first define some notations:(22)Qβ,R=u,v∈P×P:βu,v≤R,Qα,β,r,R=u,v∈P×P:r≤αu,vandβu,v≤R,αu,v=mint∈0,1ut+vt,ψu,v=mint∈Iut+vt,ζu,v=mint∈I1ut+vt,θu,v=maxt∈0,1ut+vt,βu,v=maxt∈Iut+vt,σu,v=maxt∈I1ut+vt,where I=1/4,3/4 and I1=1/8,7/8. Then, α,ψ, and ζ are the concave functionals on P, and θ,β, and σ are the convex functionals on P.Theorem 1.
Suppose that there exist positive real numbersl,l′,r,r′,R, and R′ such that gii=1,2 satisfy the following conditions:(H2)g1t,ut,vt<φplβ1/2α1ρ1,g2t,ut,vt<φplβ2/2α2ρ2 for all t∈0,1 and u,v∈0,r/3.(H3)g1t,ut,vt<φpRβ1/2α1ρ1,g2t,ut,vt<φpRβ2/2α2ρ2 for all t∈0,1 and u,v∈0,R.(H4)g1t,ut,vt<φpr/2ρ1,g2t,ut,vt<φpr/2ρ2 for all t∈0,1 and u,v∈r,r/η, where η=minβ1/α1,β2/α2:(23)ρ1=∫01ℳ1sφp−1∫01H1s,τdτds,ρ2=∫01ℳ2sφp−1∫01H2s,τdτds.Then, (1) has at least three positive solutions u1,v1,u2,v2, and u3,v3 in Qβ,R.Proof.
Note, from a standard calculus argument, we obtain the setQβ,R is bounded. Since if u,v∈Qβ,R, and then maxt∈Iut+vt<R. Let r′=r/η,R=R′/η,l=l′/η, and l=r/3 with R′∈2η2r′/η+1,2ηr′/η+1. Then, if u,v∈Qσ,l, we have maxt∈I1ut+vt<l. This means that(24)αu,v=mint∈0,1ut+vt≤mint∈I1ut+vt≤maxt∈I1ut+vt<l<r,and Qσ,l∩Qα,β,r,R=∅. Also, it can easily be shown that(25)R′2,R2∈u,v∈P:θu,v<r′,r<αu,v,R′<ψu,vandβu,v<R,ηl2,l2∈u,v∈P:l′<ζu,vandσu,v<l,r2,l2∈u,v∈P:l<σu,vandαu,v<r.
Hence, these sets are nonempty. As a result, (B1)–(B5) of Lemma4 are satisfied.
Now, we verify the functional conditions.Claim 1.
αTu,v>r, for all u,v∈Qα,β,r,R with αu,v=r and r′<θTu,v. Note, from Lemma 3 (ii), we have(26)αTu,v=mint∈0,1T1u,vt+T2u,vt=mint∈0,1∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds≥∫01ℳ1sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01ℳ2sφp−1∫01H2s,τg2τ,uτ,vτdτds=β1α1∫01α1β1ℳ1sφp−1∫01H1s,τg1τ,uτ,vτdτds+β2α2∫01α2β2ℳ2sφp−1∫01H2s,τg2τ,uτ,vτdτds≥ηmaxt∈0,1∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds=ηθTu,v>ηr′=r.Claim 2.
αTu,v>r, for all u,v∈u,v∈Qα,β,r,R:θu,v≤r′ with αu,v=r. At this case, we have r≤uτ+vτ≤r/η, for τ∈0,1, and from (H4), we obtain(27)αTu,v=mint∈0,1T1u,vt+T2u,vt=mint∈0,1∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds≥∫01ℳ1sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01ℳ2sφp−1∫01H2s,τg2τ,uτ,vτdτds>∫01ℳ1sφp−1∫01H1s,τφpr2ρ1dτds+∫01ℳ2sφp−1∫01H2s,τφpr2ρ2dτds=r2ρ1∫01ℳ1sφp−1∫01H1s,τdτds+r2ρ2∫01ℳ2sφp−1∫01H2s,τdτds=r2+r2=r.Claim 3.
βTu,v<R, for all u,v∈Qα,β,r,R,βu,v=R, and ψTu,v<R′. Then, we have(28)βTu,v=maxt∈IT1u,vt+T2u,vt=maxt∈I∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds≤∫01α1β1ℳ1sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01α2β2ℳ2sφp−1∫01H2s,τg2τ,uτ,vτdτds≤1η∫01ℳ1sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01ℳ2sφp−1∫01H2s,τg2τ,uτ,vτdτds≤1ηmint∈I∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds=1ηψu,v<R′η=R.Claim 4.
βTu,v<R, for all u,v∈u,v∈Qα,β,r,R:R′≤ψu,v with βu,v=R. Therefore, by Lemma 3 (ii) and (H3), we have(29)βTu,v=maxt∈IT1u,vt+T2u,vt=maxt∈I∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds≤∫01α1β1ℳ1sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01α2β2ℳ2sφp−1∫01H2s,τg2τ,uτ,vτdτds<∫01α1β1ℳ1sφp−1∫01H1s,τφpRβ12α1ρ1dτds+∫01α2β2ℳ2sφp−1∫01H2s,τφpRβ22α2ρ2dτds=Rβ12α1ρ1∫01α1β1ℳ1sφp−1∫01H1s,τdτds+Rβ22α2ρ2∫01α2β2ℳ2sφp−1∫01H2s,τdτds=R2+R2=R.Claim 5.
σTu,v<l, for all u,v∈Qσ,l with σu,v=l, and ζTu,v<l′. Then, we obtain(30)σTu,v=maxt∈I1T1u,vt+T2u,vt=maxt∈I1∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds≤∫01α1β1ℳ1sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01α2β2ℳ2sφp−1∫01H2s,τg2τ,uτ,vτdτds≤1η∫01ℳ1sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01ℳ2sφp−1∫01H2s,τg2τ,uτ,vτdτds≤1ηmint∈I1∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds=1ηζTu,v<l′η=l.Claim 6.
σTu,v<l, for all u,v∈Qσ,l:l′≤ζu,v with σTu,v<l. Therefore, by Lemma 3 (ii) and (H2), we have(31)σTu,v=maxt∈I1T1u,vt+T2u,vt=maxt∈I1∫01G1t,sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01G2t,sφp−1∫01H2s,τg2τ,uτ,vτdτds≤∫01α1β1ℳ1sφp−1∫01H1s,τg1τ,uτ,vτdτds+∫01α2β2ℳ2sφp−1∫01H2s,τg2τ,uτ,vτdτds<∫01α1β1ℳ1sφp−1∫01H1s,τφplβ12α1ρ1dτds+∫01α2β2ℳ2sφp−1∫01H2s,τφplβ22α2ρ2dτds=lβ12α1ρ1∫01α1β1ℳ1sφp−1∫01H1s,τdτds+lβ22α2ρ2∫01α2β2ℳ2sφp−1∫01H2s,τdτds=l2+l2=l.Remark 1.
Our Green’s functionsGii=1,2 satisfy some special inequalities (see Lemma 3 (ii)). So, in the Claims 1 and 2 of Theorem 1, αTu,v=mint∈0,1T1u,vt+T2u,vt cannot reach zero. Therefore, we use the Caputo derivative for the considered scheme. This also explains the reason that we do not use other forms of fractional derivatives (including the Riemann–Liouville type and the Hadamard type).Example 1.
Consider the example:(32)Dt3/2cφpDt3/2cut+g1t,ut,vt=0,0<t<1,Dt3/2cφpDt3/2cvt+g2t,ut,vt=0,0<t<1,2u0−u1=0,6u′0−5u′1=0,Dt3/2cu0=0,Dt3/2cu1=Dt3/2cu14,4v0−3v1=0,8v′0−7v′1=0,Dt3/2cv0=0,Dt3/2cv1=2Dt3/2cv18,where p=2.
In order to obtain the constants in Theorem1, we need to calculate(33)∫01H¯it,sds=1Γνi∫01t1−sνi−1ds−1Γνi∫0tt−sνi−1ds=tνiΓνi−1−sνi01+1νiΓνit−sνi0t=t−tνiΓνi+1,and we have(34)∫01Hit,sds=∫01H¯it,sds+bip−1t1−bip−1ξi∫01H¯iξi,sds=t−tνiΓνi+1+bip−1t1−bip−1ξiξi−ξiνiΓνi+1=t−tνi1−bip−1ξi+bip−1tξi−ξiνi1−bip−1ξiΓνi+1=t1−bip−1ξiνi−tνi1−bip−1ξi1−bip−1ξiΓνi+1.
Moreover, we also obtain thatη=0.5,bip−1ξi=0.25,b1p−1ξ1ν1=0.125,b2p−1ξ2ν2=0.088, ∫01H1t,sds=0.878t−0.752t1.5,∫01H2t,sds=0.915t−0.752t1.5, ℳ1s=1.131−s0.5+2.821−s−0.5, ℳ2s=3.391−s0.5+11.851−s−0.5, ρ1=0.9, and ρ2=4.29. If we put l=2,l′=1,r=6,r′=12, R=15.6, R′=7.8, and(35)g1t,u,v=t60+0.25u+v,0≤u+v≤2,t∈0,1,t60+0.875u+v−1.25,2≤u+v≤6,t∈0,1,t60+4,6≤u+v≤12,t∈0,1,t60−0.14u+v+5.68,12≤u+v≤15.6,t∈0,1,g2t,u,v=t70+0.08u+v,0≤u+v≤2,t∈0,1,t70+0.21u+v−0.26,2≤u+v≤6,t∈0,1,t70+1,6≤u+v≤12,t∈0,1,t70+0.056u+v+0.328,12≤u+v≤15.6,t∈0,1.
Then,gii=1,2 satisfies the following conditions:(i)
g1t,ut,vt<0.56,g2t,ut,vt<0.18 for all t∈0,1 and u,v∈0,2(ii)
g1t,ut,vt<4.33,g2t,ut,vt<1.37 for all t∈0,1 and u,v∈0,15.6(iii)
g1t,ut,vt>3.33,g2t,ut,vt>0.7 for all t∈0,1 and u,v∈6,12
Therefore, all assumptions in Theorem1 are satisfied, and the system of (32) has at least triple positive solutions.


## 4. Conclusion

In this paper, we use the six functional fixed point theorem to study the system of (1) under some bounded conditions for gii=1,2. We first transform the original fractional differential system into the equivalent system of Hammerstein-type integral equations, and then with the help of properties of Green’s functions, we provide some sufficient conditions to guarantee the existence of multiple positive solutions for our system. Finally, an example is presented to illustrate the effectiveness of the main result.



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*Source: 6723791-2020-06-02.xml*