System of fractional boundary value problem with p-Laplacian and advanced arguments

Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 https://doi.org/10.1186/s13662-021-03508-4 RESEARCH Open Access System of fractional boundary value problem with p-Laplacian and advanced arguments Amina Mahdjouba1,JuanJ.Nieto 2* and Abdelghani Ouahab1 *Correspondence: [email protected] 2Departamento de Estatística, Análise Matemática, e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, 15782, Spain Full list of author information is available at the end of the article Abstract In this paper, we discuss the existence and multiplicity of positive solutions for a system of fractional differential equations with boundary condition and advanced arguments. The existence result is proved via Leray–Schauder’s fixed point theorem type in a vector Banach space. Further, by using a new fixed point theorem in order Banach space, we study the multiplicity of positive solutions. Finally, some examples are given to illustrate our results. MSC: 34A08; 34B15; 34B18 Keywords: Fractional differential equations; p-Laplacian operator; Cone; Fixed point theorem; Positive solutions; Multiplicity of solutions 1 Introduction Fractionalcalculusanddifferentialequationshavenowprovedtobeimportanttoolsmodeling many real world phenomena like chemistry and physics [11,22,23,25]). For the description of hereditary properties of fractional calculus, see [20,24,32,37]andthereferences therein. Recently, there have been some papers dealing with the existence and multiplicity of solution (or positive solution) of nonlinear initial fractional differential equation by the use of techniques of nonlinear analysis, see [2–7,9,33,35,38]. Forexample,Chaiobtainedin[10]theexistenceofatleastonenonnegativesolutionand two positive solutions by using fixed point theorem on cone for the following problem: ⎧ ⎨ ⎩ (ϕp(u(t))+a(t)f(u(t))=0, 0<t<1, u(0)=u(1)=0. (1.1) Su et al. [31] studied the existence of one and two positive solutions by using the fixed point index theory of the following boundary values problems: ⎧ ⎨ ⎩ (ϕp(u(t))+a(t)f(u(t))=0, 0<t<1, αϕp(u(0))–βϕp(u(ξ))=0, γϕ p(u(1))+δϕp(u(η))=0. (1.2) ©The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 2 of 22 Tang etal.[34]studiedtheexistenceofpositivesolutionsoffractionaldifferentialequation with p-Laplacian of the following type (1.3) by using the coincidence degree theory. ⎧ ⎨ ⎩ Dα 0+(φ(Dβ 0+u(t)))(t)=f(t,u(t),Dβ 0+u(t)), 0<t<1, Dα 0+u(0)=0, Dβ 0+u(1)=γDβ 0+u(1). (1.3) In this work, we study the existence and multiplicity of positive solutions of the following problem: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (ϕp(Dα 0+u(t)))+a1(t)f(u(θ1(t)),v(θ2(t)))=0, 0<t<1, (ϕ˜ p(Dα 0+v(t)))+a2(t)g(u(θ1(t)),v(θ2(t)))=0, 0<t<1, Dα 0+u(0)=u(0)=u(0) =0, Dβ 0+u(1)=γDβ 0+u(η), Dα 0+v(0)=v(0)=v(0)=0, Dβ 0+v(1)=γDβ 0+v(η), (1.4) where η∈(0,1),γ∈(0, 1 ηα–β–1 ),Dα 0+,Dβ 0+are the standard Riemann–Liouville fractional derivatives with α∈(2,3),β∈(1,2) such that α≥β+1,p-Laplacian operator is defined as ϕp(s)=|s|p–2s,p> 1, and the functions f,g∈C(R2,R). Inrecentyears,manyauthorsstudiedtheexistenceofsolutionsforsystemsofdifference and differential equations with and without fractional derivative by using the vector version of the fixed point theorem (see [1,8,13,15–19,21,26–28], the monograph of Graef et al. [12], and the references therein). For establishing the existence and multiplicity of positive solutions of problem (1.4), let us list the following assumptions: (H1)ai∈L1[0,1] is nonnegative and ai(t)≡0on any subinterval of [0,1] for i=1,2. (H2)The advanced argument θ∈C((0,1),(0,1]) and 0≤θ(t)≤1,∀t∈(0,1). This work is organized as follows: In Sect. 2, we introduce all the background material used in this paper such as fractional calculus analysis and some results from fixed point theory. In Sects. 3,4, the existence and multiplicity results of solutions for a system of fractional p-Laplacedifferentialequations(1.4)arediscussedbyusingthefixedpointtheorems in the generalized Banach space. We end the paper with two examples to illustrate our main results. 2Preliminaries Inthissection,weintroducesomepreliminaryfactswhichareusedthroughoutthispaper. Definition2.1([14]) Let Xbe a real Banach space. A nonempty closed convex set P⊂X is called cone if (1) x∈P,λ≥0,thenλx∈P; (2) x∈P,–x∈P,thenx=0. If x,y∈Rnwith x=(x1,...,xn)andy=(y1,...,yn), then by x≤ywe mean xi≤yifor alli=1,...,n.Alsoweset|x|=(|x1|,...,|xn|),max(x,y)=(max(x1,y1),...,max(xn,yn)),and Rn +={x∈Rn:xi>0}.Ifc∈R,thenx≤cmeans xi≤cfor each i=1,...,n. Definition 2.2 Let Xbe a nonempty set, and consider the space Rm +endowed with the usual component-wise partial order. The mapping d:X×X→Rm +, which satisfies all the Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 3 of 22 usual axioms of the metric, is called a generalized metric in Perov’s sense and (X,d)is called a generalized metric space. Let (X,d) be a generalized metric space in Perov’s sense. For r:= (r1,...,rm)∈Rm +,we will denote by B(x0,r)=x∈X:d(x0,x)<r the open ball centered in x0with radius r,andby B(x0,r)=x∈X:d(x0,x)≤r the closed ball centered at x0with radius r. Theorem 2.1 ([12,36]) Let X be a generalized Banach space,and let N :X→Xbea completely continuous operator.Then either (i) the equation N(x)=xhas at least one solution,or (ii) the set M={x∈X|μN(x)=x,μ∈(0,1)}is unbounded. Theorem 2.2 ([30]) Let (X,·)be a normed space,P1,P2⊂X be two cones;P:= P1×P2;r,R∈R2 +,Pr,R:= {u∈Pi:ri≤ui≤Ri}with 0<r<R;and let N :Pr,R→P,N= (N1,N2)beacompactmap.Assumethat,for each i ∈{1,2},one of the following conditions is satisfied in Pr,R: (1) Ni(ui)⊀uiif ui=ri,and Ni(ui)uiif ui=Ri; (2) Ni(ui)uiif ui=ri,and Ni(ui)⊀uiif ui=Ri. Then N has a fixed point u in P with ri≤ui≤Rifor i ∈{1,2},where ,namely uvif and only if v–u∈P.We shall say that u ≺vifv–u∈P\{0}. Remark 2.1 ([30]) In Theorem (2.2) four cases are possible for u∈pr,R: (c1)N1(u)⊀u1if u1=r1,andN1(u)u1if u1=R1,N2(u)⊀u2if u2=r2,and N2(u)u2if u2=R2; (c2)N1(u)⊀u1if u1=r1,andN1(u)u1if u1=R1,N2(u)u2if u2=r2,and N2(u)⊀u2if u2=R2; (c3)N1(u)u1if u1=r1,andN1(u)⊀u1if u1=R1,N2(u)⊀u2if u2=r2,and N2(u)u2if u2=R2; (c4)N1(u)u1if u1=r1,andN1(u)⊀u1if u1=R1,N2(u)u2if u2=r2,and N2(u)⊀u2if u2=R2. Theorem 2.3 ([29]) Let (X,·)be a Banach space,P1,P2⊂X be two cones,and P := P1×P2be the corresponding cone of X2=X×X,and let αi,βi>0.We denote Uαi=u∈Pi:u<αiand Vβi=u∈Pi:u<βi, with αi=βi,ri=min{αi,βi}and Ri=max{αi,βi}for i =1,2.Assume that N :W1×W2→ P,N=(N1,N2)is a compact map (where Wi=Uαi∪Vβifor i =1,2)and there exist hi∈ Pi\{0},i=1,2,such that for each i ∈{1,2}the following condition is satisfied in W1×W2: λui=Niuforui=αiand λ≥1; (2.1) Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 4 of 22 ui=Niu+μhifor ui=βiand μ≥0. (2.2) Then (1) Nhas at least one fixed point u=(u1,u2)in Psuch that ui∈Uαi\Vβifor i=1,2if αi>βifor i=1,2; (2) Nhas at least two fixed points located in (Uα1\Vβ1)×Uα2and (Uα1\Vβ1)×(Vβ2\Uα2)if β1<α1and β2>α2; (3) Nhas at least two fixed points located in Uα1×(Uα2\Vβ2)and (Vβ1\Uα1)×(Uα2\Vβ2)if β1>α1and β2<α2; (4) Nhas at least four (three nontrivial)fixed points in Uα1×Uα2,Uα1×(Vβ2\Uα2),(Vβ1\Uα1)×Uα2,and (Vβ1\Uα1)×(Vβ2\Uα2)if αi<βifor i=1,2. Remark 2.2 ([29]) Our previous results can be easily generalized to systems of noperator equations. Definition 2.3 ([7]) The fractional integral of Riemann–Liouville of the function h∈ L1((0,∞),R)oforderα>0isdefinedby Iα 0+h(t)= 1 (α)t 0(t–s)α–1h(s)ds, where (α) is the Euler gamma function defined by (α)=∞ 0tα–1e–tdt. Definition2.4 Forafunctionh∈ACn(J),theRiemann–Liouvillefractionalorderderivative of order α>0ofhis defined by Dα 0+h(t)= 1 (n–α)dn dtnt 0 h(s) (t–s)α–n+1 ds, where n=[α]+1and[α] denotes the integer part of the real number α. Remark 2.3 ([7]) (1) If λ>–1 Dα 0+tλ=(λ+1) (λ–α+1tλ–α, and Dα 0+tα–m=0,m=1,2,...,n,wheren=[α]+1. (2) Dα 0+Iα 0+u(t)=u(t)for all u∈C(0,1)∩L1(0,1). (3) If u∈L1(0,1),α>β>0,then Dβ 0+Iα 0+u(t)=Iα–β 0+u(t). Lemma 2.1 ([7]) If we assume that u ∈C(0,1) ∩L1(0,1), then the fractional differential equation Dα 0+u(t)=0, α>0 Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 5 of 22 has u(t)=C1tα–1 +C2tα–2 +···+Cntα–n,Ci∈R,i=1,2,...,n,as a unique solution,where n=[α]+1. Lemma 2.2 ([7]) Suppose that u ∈C(0,1)∩L1(0,1) is such that Dα 0+u∈C(0,1)∩L1(0,1). Then Iα 0+Dα 0+u(t)=u(t)+C1tα–1 +C2tα–2 +···+Cntα–n for some Ci∈R,i=1,2,...,n,where n=[α]+1. Lemma2.3([10]) If x,y≥0,γ>0,then (x+y)γ≤max2γ–1,1xγ+yγ. Lemma2.4([10]) Let c>0,γ>0.For any x,y∈[0,c], we have that (1) if γ>1,then |xγ–yγ|≤γcγ–1|x–y|; (2) if 0<γ≤1,then |xγ–yγ|≤|x–y|γ. 3 Existence result Denote by C([0,1]) the Banach space of all continuous functions from [0,1] into Rwith the norm u=maxu(t):t∈[0,1]. Define the cone Pin C([0,1]2)asP={u∈C([0,1]):u(t)≥0,t∈[0,1]}.Letq>1and˜ q>1 satisfy the relation 1 p+1 q=1,1 ˜ p+1 ˜ q=1,wherep,˜ pare given by (1.4). To prove the existence of solutions to (1.4), we need the following auxiliary lemma. Lemma 3.1 Given h1,h2∈C[0,1],η∈(0,1),γ∈(0, 1 ηα–β–1 ), and α≥β+1,theuniquesolution of C boundary value problem for a coupled system ϕpDα 0+u(t)+h1(t)=0, 0<t<1, (3.1) ϕ˜ pDα 0+v(t)+h2(t)=0, 0<t<1, (3.2) Dα 0+u(0)=u(0)=u(0)=0, Dβ 0+u(1)= γDβ 0+u(η), (3.3) Dα 0+v(0)=v(0)=v(0)=0, Dβ 0+v(1)=γDβ 0+v(η), (3.4) is u(t)=1 0G1(t,s)ϕqs 0h1(τ)dτds +γtα–1 1–γηα–β–1 1 0G2(η,s)ϕqs 0h1(τ)dτds (3.5) Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 6 of 22 and v(t)=1 0G1(t,s)ϕ˜ qs 0h2(τ)dτds +γtα–1 1–γηα–β–1 1 0G2(η,s)ϕ˜ qs 0h2(τ)dτds, (3.6) where G1(t,s)=⎧ ⎨ ⎩ tα–1(1–s)α–β–1–(t–s)α–1 (α),0≤s≤t≤1, tα–1(1–s)α–β–1 (α),0≤t≤s≤1, G2(η,s)=⎧ ⎨ ⎩ [η(1–s)]α–β–1–(η–s)α–β–1 (α),0≤s≤η≤1, [η(1–s)]α–β–1 (α),0≤η≤s≤1. Proof Integrating equation (3.1)from0tot,wehave ϕpDα 0+u(t)–ϕpDα 0+u(0)=t 0h1(s)ds and so, Dα 0+u(t)=–ϕqt 0h1(s)ds. From Lemma 2.2, u(t)=–Iα 0+ϕqt 0h1(s)ds+Atα–1 +Btα–2 +Ctα–3 =– 1 (α)t 0(t–s)α–1ϕqs 0h1(τ)dτds+Atα–1 +Btα–2 +Ctα–3. From (3.3), B=C=0,andso u(t)=–Iα 0+ϕqt 0h1(s)ds+Atα–1. (3.7) Now, from Remark 2.3 Dβ 0+u(t)=–Iα–β 0+ϕqt 0h1(s)ds+A(α) (α–β)tα–β–1 =– 1 (α–β)t 0(t–s)α–β–1ϕqs 0h1(τ)dτds+A(α) (α–β)tα–β–1. Therefore Dβ 0+u(1)=– 1 (α–β)1 0(1–s)α–β–1ϕqs 0h1(τ)dτds+A(α) (α–β), γDβ 0+u(η)=– γ (α–β)η 0(η–s)α–β–1ϕqs 0h1(τ)dτds+A(α)γ (α–β)tα–β–1ηα–β–1. Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 7 of 22 By boundary condition (3.3), we have A=1 1–γηα–β–1 1 0 (1–s)α–β–1 (α)ϕqs 0h1(τ)dτds –γ 1–γηα–β–1 η 0 (η–s)α–β–1 (α)ϕqs 0h1(τ)dτds, and replacing in (3.7), we obtain u(t)=– 1 (α)t 0(t–s)α–1ϕqs 0h1(τ)dτds +tα–1 1–γηα–β–1 1 0 (1–s)α–β–1 (α)ϕqs 0h1(τ)dτds +γtα–1 1–γηα–β–1 1 0 (η–s)α–β–1 (α)ϕqs 0h1(τ)dτds. Splitting the second integral in two parts of the form tα–1 +k 1–γηα–β–1 =tα–1 1–γηα–β–1 , we have k=γηα–β–1tα–1,andthus u(t)=– 1 (α)t 0(t–s)α–1ϕqs 0h1(τ)dτds +tα–1 1 0 (1–s)α–β–1 (α)ϕqs 0h1(τ)dτds +γηα–β–1tα–1 1–γηα–β–1 1 0 (1–s)α–β–1 (α)ϕqs 0h1(τ)dτds –γtα–1 1–γηα–β–1 η 0 (η–s)α–β–1 (α)ϕqs 0h1(τ)dτds =t 0 tα–1(1–s)α–β–1 –(t–s)α–1 (α)ϕqs 0h1(τ)dτds +1 t tα–1(1–s)α–β–1 (α)ϕqs 0h1(τ)dτds +γtα–1 1–γηα–β–1 η 0 [η(1–s)]α–β–1 –(η–s)α–β–1 (α)×ϕqs 0h1(τ)dτds +γtα–1 1–γηα–β–1 η 0 [η(1–s)]α–β–1 (α)×ϕqs 0h1(τ)dτds =1 0G1(t,s)ϕqs 0h1(τ)dτds+γtα–1 1–γηα–β–1 ×1 0G2(η,s)ϕqs 0h1(τ)dτds. This completes the proof.  Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 8 of 22 Lemma3.2 ([35]) Let ρ∈(0,1) be fixed.The kernel G1(t,s)satisfies the following properties: (1) G1(t,s)∈C([0,1]×[0,1]) and G1(t,s)>0for all s,t∈(0,1); (2) G1(t,s)≤G1(1,s)for all s∈(0,1); (3) minρ≤t≤1G1(t,s)≥ρα–1G1(1,s)for all s∈[0,1]. We are now ready to present our main result. In this section we give an existence result based on the nonlinear alternative of Leray–Schauder type. Theorem3.1 Assume (H1)–(H2)and that the following condition holds: (H3)There exist functions p,q,h,˘ p,˘ q,and ¯ h∈L1([0,1],R+)and constants α1,α2,α3,and α4∈[0,1) such that f(u,v)≤p(t)|u|α1+q(t)|v|α2+h(t)for each t∈[0,1] and u,v∈R and g(u,v)≤˘ p(t)|u|α3+˘ q(t)|v|α4+˘ h(t)for each t∈[0,1] and u,v∈R. If α1p,α2p,α3q,and α4q∈[0,1),then system (1.4)has at least one solution. Proof Let Nbe the operator N:C(0,1)×C(0,1)→C(0,1)×C(0,1) defined by N(u,v)=N1(u,v),N2(u,v), where N1(u,v)(t) =1 0G1(t,s)ϕqs 0a1(τ)fuθ1(τ),vθ2(τ)dτds +γtα–1 1–γηα–β–1 1 0G2(η,s)ϕqs 0a1(τ)fuθ1(τ),vθ2(τ)dτds, (3.8) and N2(u,v)=1 0G1(t,s)ϕ˜ qs 0a2(τ)guθ1(τ),vθ2(τ)dτds +γtα–1 1–γηα–β–1 1 0G2(η,s)ϕ˜ qs 0a2(τ)guθ1(τ),vθ2(τ)dτds. (3.9) We shall use the Leray–Schauder fixed point theorem to prove that Nhas a fixed point. The proof will be given in several steps. Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 9 of 22 Step 1. To show that Nis continuous, let (un,vn)beasequencesuchthat(un,vn)→ (u,v)∈C[0,1]×C[0,1] as n→∞.Thenwehave N1(un,vn)(t)–N1(u,v)(t) =1 0G1(t,s)ϕqs 0a1(τ)funθ1(τ),vnθ2(τ)dτds +γtα–1 1–γηα–β–1 1 0G2(η,s)ϕqs 0a1(τ)funθ1(τ),vnθ2(τ)dτds –1 0G1(t,s)ϕqs 0a1(τ)fuθ1(τ),vθ2(τ)dτds +1 0 γtα–1G2(η,s) 1–γηα–β–1 ϕqs 0a1(τ)fuθ1(τ),vθ2(τ)dτds. By Lemma 3.2 and t∈[0,1], N1(un,vn)(t)–N1(u,v)(t) ≤1 0G1(1,s)s 0a1(τ)funθ1(τ),vnθ2(τ)q–1 –a1(τ)fuθ1(τ),vθ2(τ)q–1 dτds +γ 1–γηα–β–1 1 0G2(η,s)s 0a1(τ)funθ1(τ),vnθ2(τ)dτq–1 –a1(τ)fuθ1(τ),vθ2(τ)q–1 dτds. On the other hand, since fis a continuous function combined with the fact that un–u→0, as n→∞, then there exists N≥1 such that for all τ∈[0,1] the following estimate funθ1(τ),vnθ2(τ)dτ)–fuθ1(τ),vθ2(τ)<, holds for n≥N. By the Lebesgue dominated convergence theorem, we have N1(un,vn)–N1(u,v)→0, as n→∞. Similarly, N2(un,vn)–N2(u,v)→0, as n→∞. Consequently, Nis continuous. Step 2. Nmaps bounded sets into bounded sets in C[0,1] ×C[0,1], it suffices to show thatforanyr>0thereexistsapositiveconstantvectorl=(l1,l2)suchthat,foreach(u,v)∈ Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 16 of 22 Theorem4.1 Assume that there exist αi,βi>0with αi=βi,i=1,2,such that Bq–1 1≤α1,Aγq–1 1≥β1, Bq–1 2≤α2,Aγq–1 2≥β2.(4.1) Then (1.4)has a positive solution u =(u1,u2)with ri≤ui≤Ri,i=1,2,where ri= min{αi,βi},Ri=max{αi,βi}.Moreover,the corresponding orbit of u is included in the rectangle [ρr1,R1]×[ρr2,R2]. Proof First note that if u∈Pr,R,thenr1≤|u1≤R1and r2≤u2≤R2,andbythedefinition of P, ρα–1r1≤u1(t)≤R1and ρα–1r2≤u2(t)≤R2 for all t, showing that the orbit of ufor t∈[ρ,1]isincludedintherectangle[ρr1,R1]× [ρr2,R2]. Also, if we know for example that u1=α1,then ρα–1α1≤u1(t)≤α1. We now prove that, for every u∈Pr,Rand i∈{1,2}, the following properties hold: ui=αiimplies ui⊀Ni(u), ui=βiimplies uiNi(u), (4.2) guaranteeing the applicability of Theorem 2.2. Indeed, if u1=α1and we would have u1≺N1(u), then u1(t)<N1(u)(t) ≤1 0 maxG1(t,s)maxa1(t)fuθ1(t),vθ2(t)q–1 dt +γtα–1 1–γηα–β–1 1 0G2(η,s)maxa1(t)fuθ1(t),vθ2(t)q–1 dt ≤Bq–1 1+q–1 1 γ 1–γηα–β–1 1 0G2(η,s) ≤Bq–1 1 ≤α1 for all t. This yields the contradiction α1<α1. Now, if u1=β1and u1N1(u), then for t∈[ρ,1]weobtain u1(t)>N1(u)(t) ≥1 0 minG1(t,s)mina1(t)fuθ1(t),vθ2(t)q–1 dt Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 17 of 22 +γtα–1 1–γηα–β–1 1 0G2(η,s)mina1(t)fuθ1(t),vθ2(t)q–1 dt ≥Aγq–1 1 ≥β1. Thenwededucethatβ1>β1,whichisacontradiction.Hence(4.2)holdsfori=1.Similarly, (4.2)istruefori=2.ByTheorem2.2,weseethatNhas at least one fixed point in P. Therefore, system (1.4) has at least one positive solution.  Now we study the existence of multiple positive solutions for the systems of fractional boundary value problem with p-Laplacian boundary conditions. (H4)f,gare positive and increasing, i.e., 0≤u≤x,0≤v≤yimply 0≤f(u,v)≤f(x,y),0≤g(u,v)≤g(x,y). We present the following general existence, multiplicity, and localization result. Theorem 4.2 Let conditions (H1)–(H2)–(H4)hold,and assume that the norm ·is monotonewithrespecttoeachconePi(i=1,2).Moreover,supposethatthereexistαi,βi>0, with αi=βi,i=1,2,such that N1α1ρα–1,R2ρα–1<α1,N2R1ρα–1,α2ρα–1<α2, (4.3) N1β1ρα–1,0>β1,N20,β2ρα–1<β2, (4.4) where Ri=max{αi,βi}(i=1,2). Then problem (1.4)has at least (1) one solution u=(u1,u2)such that βi<ui<αifor i=1,2,if αi>βifor i=1,2; (2) two solutions (u1,u2)and (v1,v2)such that β1<u1<α1,β2<u2<α2, β1<v1<α1,and v2<α2if α1>β1and α2<β2; (3) two solutions (u1,u2)and (v1,v2)such that α1<u1<β1,α2<u2<β2,v1<α1, and β2<v2<α2if α1<β1and α2>β2; (4) four solutions (u1,u2),(v1,v2),(w1,w2),and (z1,z2)such that βi<ui<αi,α1<v1<β1,and v2<α2,w1<α1,α2<w1<β2,and zi<αi, if αi<βifor i=1,2. Proof We shall apply Theorem 2.3 to the operator N=(N1,N2)definedasin(3.8)and (3.9). Let us see that it satisfies conditions (2.1)(2.2). First we prove that λu1=N1(u)foreveryu∈kwith u1=α1,u2≤R2and all λ≥1. (4.5) Indeed,ifnot, λu1=λα1=N1(u). Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 18 of 22 From 0 ≤u1≤α1ρα–1 and 0≤u2≤R2ρα–1,by(H1),(H4) it follows that 0≤f(u1,u2)≤fα1ρα–1,R2ρα–1, 0≤ϕqs 0a1(τ)f(u1,u2)dτ≤ϕqs 0a1(τ)fα1ρα–1,R2ρα–1dτds. By Lemma (3.2)weobtain 0≤N1(u1,u2)≤N1α1ρα–1,R2ρα–1, and the norm of Xbeing monotone, N1(u1,u2)≤N1α1ρα–1,R2ρα–1. By assumption (4.3), N1α1ρα–1,R2ρα–1<α1, so we obtain the contradiction λα1<α1for some λ≥1. Hence (4.5)holds. Now, we provethat u1=N1(u)+μρα–1 for every u∈Pwithu1=β1,u2≤R2andall μ≥0. Assume the contrary, i.e., u1=N1(u)+μρα–1 for some u∈Pwith u1=β1,u2≤ R2and some μ≥0. Then u1–N1(u)∈P1,so0≤N1(u)≤u1,andthenormofXbeing monotone N1(u)≤u1=β1. (4.6) Also, from condition (H4), 0≤β1ρα–1 ≤u1and 0≤u2,soweobtain 0≤f1β1ρα–1,0≤f(u1,u2), then by (H1)weobtain 0≤ϕqs 0a1(τ)f1β1ρα–1,0dτ≤ϕqs 0a1(τ)f(u1,u2)dτ, andbyLemma3.2weconclude0≤N1(β1ρα–1,0)≤N1(u1,u2).Hence,bymonotonicityof the norm, N1β1ρα–1,0≤N1(u1,u2). Now, from (4.6)wehave N1β1ρα–1,0≤β1, Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 19 of 22 which contradicts assumption (4.4). Therefore, conditions (2.1)–(2.2) hold for i = 1. Similarly, they can be verified for i=2.  5 Applications Example 5.1 Consider the fractional differential equation with advanced argument for p-Laplacian: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ϕ3/2(D5/2 0+u(t))+t–1/2 4t 1+t(|u(θ(t))|1 4+|v(θ(t))|1 5)=0, 0<t<1, ϕ3/2(D5/2 0+v(t))+7t–1/2 2t2 1+t2(13+|v(θ(t))|1/4 +|u(θ(t))|1 6)=0, 0<t<1, D5/2 0+u(0)=u(0)=u(0)=0, D7/6 0+u(1)= 7 10D7/6 0+u(1 2), D5/2 0+v(0)=v(0)=v(0)=0, D7/6 0+v(1)= 7 10D7/6 0+v(1 2), (5.1) where α=5 2,β=7 6,η=7 10,p=˜ p=3 2,q=˜ q=3,a1(t)=t–1/2 4,a2(t)=7t–1/2 2,ϕ3(1 0a1(t)dt)= 1 4,ϕ3(1 0a2(t)dt)=√7 2,α1p=α3˜ p=3 8∈(0,1), α2p=3 10 ∈(0,1),α4˜ p=3 12 ∈(0,1) fuθ(t),vθ(t)=t 1+tuθ(t)1 4+vθ(t)1 5,θ(t)=tγ,γ∈(0,1), and guθ(t),vθ(t)=t2 1+t213+vθ(t)1/4 +uθ(t)1 6,θ(t)=tγ,γ∈(0,1). It is clear that, for all (t,u,v)∈[0,1]×R2, ⎧ ⎨ ⎩|f(u,v)|≤t(|u|1 4+|v|1 5), |g(u,v)|≤t2(13+|v|1/4 +|u|1 6). HencealltheconditionsofTheorem3.1hold,thisimpliesthatproblem(5.1)hasatleast one solution. Example 5.2 Consider the fractional differential equation with advanced argument for p-Laplacian: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ϕ3/2(D5/2 0+u(t))+t–1/2 4f(u(θ(t)),v(θ(t)))=0, 0< t<1, ϕ3/2(D5/2 0+v(t))+7t–1/2 2g(u(θ(t)),v(θ(t)))=0, 0< t<1, D5/2 0+u(0)=u(0)=u(0)=0, D7/6 0+u(1)= 7 10D7/6 0+u(1 2), D5/2 0+v(0)=v(0)=v(0)=0, D7/6 0+v(1)= 7 10D7/6 0+v(1 2), (5.2) where f,g∈C(R2,R+) are nondecreasing in uand v,θ(t)=tγ,γ∈(0,1). Assume that lim z→∞ f(z,z) z=lim z→∞ g(z,z) z=0 (5.3) and lim z→0 f(z,z) z=lim z→∞ g(z,z) z=∞. (5.4) Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 20 of 22 Fromconditions(5.3)and(5.4),wecanprovethatthereexistα1,α2,β1,β2>0,α1<β1,α2= β1,andβ2=α1such that f(ρα–1β1,ρα–1α1) ρα–1β1≥1 ρα–1A, g(ρα–1α1,ρα–1α1) ρα–1β2≥1 ρα–1A (5.5) and f(α1,β1) α1≤1 B,g(β1,β1) α2≤1 B. (5.6) Then we set ri=α1,Ri=β1for i∈{1,2}, 1=f(α1,β1), 2=g(β1,β1), and γ1=fρα–1β1,ρα–1α1,γ2=gρα–1α1,ρα–1α1. We concluded that (5.5)and(5.6)guarantee(4.1).Hence,byTheorem4.2,problem(5.2) has a positive solution. Since f,gare positive and increasing, we can easily show that f(ρα–1β1,0) β1≥1 A,f(ρα–1α1,ρα–1R2) α1<1 B, g(,0,ρα–1β2) β2≤1 A,g(ρα–1R1,α2ρα–1) α2<1 B. Thusconditions(4.3)and(4.4)hold.Then,byTheorem4.2,problem(5.2)hasmultiplicity of solutions. 6Conclusions In this present work, we discussed some existence multiplicity results for system of fractional differential equations, under various assumptions on the right-hand side nonlinearity. The main assumptions on the nonlinearity are the continuity and some Nagumo– Bernsteintypegrowthconditions.Wehaveusedfixedpointtheoryinvectormetricspaces with general properties from functional analysis. Also the positivity result for a fractional systemofdifferentialequationswasconsidered.Wehopethatthispapercanprovidecontributions to the questions of existence, positivity, and multiplicity of solutions for fractional differential equations on bounded domains. Funding The research of J.J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under Grant MTM2016-75140-P, co-financed by the European Community fund FEDER, and by Xunta de Galicia under grant ED431C 2019/02. Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 21 of 22 Availability of data and materials All data generated or analysed during this study are included in this published article. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Author details 1Laboratory of Mathematics, Sidi-Bel-Abbès University, PoBox 89, 22000, Sidi-Bel-Abbès, Algeria. 2Departamento de Estatística, Análise Matemática, e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, 15782, Spain. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 17 March 2021 Accepted: 14 July 2021 References 1. Abdeljawad, T., Agarwal, R.P., Karapinar, E., Kumari, P.S.: Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space. Symmetry 11(5, 686 (2019) 2. Adigzel, R.S., Aksoy, Ü., Karapinar, E., Erhan, I.: On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6652 3. Adguzel, R.S., Aksoy, Ü., Karapinar, E., Erhan, I.M.: On the solutions of fractional differential equations via Geraghty type hybrid contractions. Appl. Comput. Math. 20(2), 313–333 (2021) 4. Adguzel, R.S., Aksoy, Ü., Karapinar, E., Erhan, I.M.: Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions. RACSAM 115, 155 (2021). https://doi.org/10.1007/s13398-021-01095-3 5. Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional differential equations via coupled fixed point. Electron. J. Differ. Equ. 2015, 286 (2015) 6. Alqahtani, B., Aydi, H., Karapinar, E., Rakoˇ cevi´ c, V.: A solution for Volterra fractional integral equations by hybrid contractions. Mathematics 7, 694 (2019) 7. Bai, Z., Lü, H.: Positive solutions for a boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005) 8. Bolojan-Nica, O., Infante, G., Precup, R.: Existence results for systems with coupled nonlocal initial conditions. Nonlinear Anal. 4, 231–242 (2014) 9. Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403–411 (2012) 10. Chai, G.: Positive solutions for boundary value problems of fractional differential equation with p-Laplacian. Bound. Value Probl. (2012) 11. Gaul, L., Klein, P., Kempfle, S.: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81–88 (1991) 12. Graef, J.R., Henderson, J., Ouahab, A.: Topological Methods for Differential Equations and Inclusions. Monographs and Research Notes in Mathematics Series Profile. CRC Press, Boca Raton (2019) 13. Graef, J.R., Kadari, H., Ouahab, A., Oumansour, A.: Existence results for systems of second-order impulsive differential equations. Acta Math. Univ. Comen. 88, 51–66 (2019) 14. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988) 15. Guendouz, C., Lazreg, J.E., Nieto, J.J., Ouahab, A.: Existence and compactness results for a system of fractional differential equations. J. Funct. Spaces 2020, Article ID 5735140 (2020) 16. Henderson, J., Ouahab, A., Slimani, M.: Existence results of system of semilinear discrete equations. Int. J. Difference Equ. 12, 235–253 (2017) 17. Karapinar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric spaces with an application to nonlinear fractional differential equations. Mathematics 7(4), 444 (2019) 18. Kadari, H., Nieto, J.J., Ouahab, A., Oumansour, A.: Implicit first order impulsive differential systems. Int. J. Difference Equ. 15, 429–451 (2020) 19. Lazreg, J.E., Abbas, S., Benchohra, M., Karapinar, E.: Impulsive Caputo–Fabrizio fractional differential equations in b-metric spaces. Open Math. 19, 363–372 (2021) 20. Liang, R., Peng, J., Shen, J.: Double positive solutions for a nonlinear four-point boundary value problem with a p-Laplacian operator. Nonlinear Anal. 68, 1881–1889 (2008) 21. Malki, Z., Berhoun, F., Ouahab, A.: System of boundary random fractional differential equations via Hadamard derivative. Ann. Univ. Paedagog. Crac. Stud. Math. 20, 17–41 (2021) 22. Malinowska, A.B., Torres, D.F.M.: Towards a combined fractional mechanics and quantization. Fract. Calc. Appl. Anal. 15, 407–417 (2012) 23. Metzler, F., Schick, W., Kilian, H.G., Nonnenmacher, T.F.: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995) 24. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) 25. Nasholm, S.P., Holm, S.: On a fractional Zener elastic wave equation. Fract. Calc. Appl. Anal. 16, 26–50 (2013) Mahdjouba et al. Advances in Difference Equations (2021) 2021:352 Page 22 of 22 26. Nica, O.: Initial-value problems for first-order differential systems with general nonlocal conditions. Electron. J. Differ. Equ. 2012, 74 (2012) 27. Nica, O., Precup, R.: On the nonlocal initial value problem for first order differential systems. Stud. Univ. Babe¸s–Bolyai, Math. 56(3), 125–137 (2011) 28. Nieto, J.J., Ouahab, A., Slimani, M.A.: Existence and boundedness of solutions for systems of difference equations with infinite delay. Glas. Mat. 53(73), 123–141 (2018) 29. Precup, R., Rodríguez-López, J.: Multiplicity results for operator systems via fixed point index. Results Math. 74,25 (2019) 30. Precup, R.: A vector version of Krasnosel’ski˘ ı’s fixed point theorem in cones and positive periodic solutions of nonlinear systems. J. Fixed Point Theory Appl. 2, 141–151 (2007) 31. Su, H., Wei, Z., Wang, B.: The existence of positive solutions for a nonlinear four-point singular boundary value problem with a p-Laplacian operator. Nonlinear Anal. 66, 2204–2217 (2007) 32. El-Sayed, A.M.A.: Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33, 181–186 (1998) 33. Su, Y., Feng, Z.: Existence theory for an arbitrary order fractional differential equation with deviating argument. Acta Appl. Math. 118, 81–105 (2012) 34. Tang, X., Yan, C., Liu, Q.: Existence of solutions of two point boundary value problems for fractional p-Laplace differential equations at resonance. J. Appl. Math. Comput. 41, 119–131 (2013) 35. Torres, F.: Positive solutions for fractional boundary value problem with p-Laplacian. Int. J. Math. Anal. 10, 469–484 (2018) 36. Viorel, A.: Contributions to the Study of Nonlinear Evolution Equations. Ph.D. thesis, Babe¸s-Bolyai University Cluj-Napoca, Department of Mathematics (2011) 37. Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results and problems II. Appl. Anal. 81, 435–493 (2002) 38. Zhao, D., Wang, H., Ge, W.: Existence of triple positive solutions to a class of p-Laplacian boundary value problems. J. Math. Anal. Appl. 328, 972–983 (2007)