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Research article Special Issues

Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields

  • Received: 31 December 2022 Revised: 20 March 2023 Accepted: 04 April 2023 Published: 22 April 2023
  • 35L72, 35L05, 83C05

  • In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.

    Citation: Senhao Duan, Yue MA, Weidong Zhang. Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields[J]. Communications in Analysis and Mechanics, 2023, 15(2): 111-131. doi: 10.3934/cam.2023007

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  • In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.



    We introduce and study a coupled system of nonlinear third-order ordinary differential equations on an arbitrary domain:

    u(t)=f(t,u(t),v(t),w(t)),t[a,b],v(t)=g(t,u(t),v(t),w(t)),t[a,b],w(t)=h(t,u(t),v(t),w(t)),t[a,b], (1.1)

    supplemented with nonlocal multi-point anti-periodic type coupled boundary conditions of the form:

    u(a)+u(b)=mj=1αjv(ηj),u(a)+u(b)=ml=1βlv(ηl),u(a)+u(b)=mn=1γnv(ηn),v(a)+v(b)=me=1δew(ηe),v(a)+v(b)=mq=1ρqw(ηq),v(a)+v(b)=mr=1σrw(ηr),w(a)+w(b)=mk=1ξku(ηk),w(a)+w(b)=mp=1ζpu(ηp),w(a)+w(b)=md=1κdu(ηd), (1.2)

    where f,g,andh:[a,b]×R3R are given continuous functions, a<η1<η2<<ηm<b, andαj,βl,γn,δe,ρq,σr,ξk,ζp andκdR+(j,l,n,e,q,r,k,p andd=1,2,,m).

    Boundary value problems arise in the mathematical modeling of several real world phenomena occurring in diverse disciplines such as fluid mechanics, mathematical physics, etc. [1]. The available literature on the topic deals with the existence and uniqueness of solutions, analytic and numerical methods, stability properties of solutions, etc., for instance, see [2,3,4,5]. Classical boundary conditions cannot cater the complexities of the physical and chemical processes occurring within the specified domain. In order to resolve this issue, the concept of nonlocal boundary conditions was introduced. The details on theoretical development of nonlocal boundary value problems can be found in the articles [6,7,8,9,10] and the references cited therein. For some recent works on the topic, we refer the reader to the articles [11,12,13,14,15,16] and the references cited therein.

    Nonlinear third-order ordinary differential equations appear in the study of many applied and technical problems. In [2], third-order nonlinear boundary value problems associated with nano-boundary layer fluid flow over stretching-surfaces were investigated. Systems of third order nonlinear ordinary differential equations are involved in the study of magnetohydrodynamic flow of second-grade nanofluid over a nonlinear stretching-sheet [17] and in the analysis of magneto Maxwell nano-material by a surface of variable thickness [18]. In heat conduction problems, the boundary conditions of the form (1.2) help to accommodate the nonuniformities occurring at nonlocal positions on the heat sources (finite many segments separated by points of discontinuity). Moreover, the conditions (1.2) are also helpful in modeling finitely many edge-scattering problems. For engineering applications, see [19,20,21]. It is expected that the results presented in this work will help establish the theoretical aspects of nonlinear coupled systems occurring in the aforementioned applications.

    The main objective of the present paper is to establish the existence theory for the problems (1.1) and (1.2). We arrange the rest of the paper as follows. In Section 2, we present an auxiliary lemma, while the main results for the given problem are presented in Section 3. The paper concludes with some interesting observations.

    The following lemma plays a key role in the study of the problems (1.1) and (1.2).

    Lemma 2.1. Let f1,g1,h1C[a,b]. Then the solution of the following linear system of differential equations:

    u(t)=f1(t),v(t)=g1(t),w(t)=h1(t),t[a,b], (2.1)

    subject to the boundary conditions (1.2) is equivalent to the system of integral equations:

    u(t)=ta(ts)22f1(s)ds+1Λ{ba[2Λ1(bs)2+G1(t)(bs)+P1(t)]f1(s)dsba[Λ1mj=1αj(bs)2+G2(t)(bs)+P2(t)]g1(s)dsba[Λ1S11(bs)22+G3(t)(bs)+P3(t)]h1(s)ds+P3(t)(md=1κdηdaf1(s)ds)+P1(t)(mn=1γnηnag1(s)ds)+P2(t)(mr=1σrηrah1(s)ds)+G3(t)(mp=1ζpηpa(ηps)f1(s)ds)+G1(t)(ml=1βlηla(ηls)g1(s)ds)+G2(t)(mq=1ρqηqa(ηqs)h1(s)ds)+Λ1S11(mk=1ξkηka(ηks)22f1(s)ds)+2Λ1(mj=1αjηja(ηjs)2g1(s)ds)+Λ1mj=1αj(me=1δeηea(ηes)2h1(s)ds)}, (2.2)
    v(t)=ta(ts)22g1(s)ds+1Λ{ba[Λ1S12(bs)22+G4(t)(bs)+P4(t)]f1(s)dsba[2Λ1(bs)2+G5(t)(bs)+P5(t)]g1(s)dsba[Λ1me=1δe(bs)2+G6(t)(bs)+P6(t)]h1(s)ds+P6(t)(md=1κdηdaf1(s)ds)+P4(t)(mn=1γnηnag1(s)ds)+P5(t)(mr=1σrηrah1(s)ds)+G6(t)(mp=1ζpηpa(ηps)f1(s)ds)+G4(t)(ml=1βlηla(ηls)g1(s)ds)+G5(t)(mq=1ρqηqa(ηqs)h1(s)ds)+Λ1me=1δe(mk=1ξkηka(ηks)2f1(s)ds)+Λ1S12(mj=1αjηja(ηjs)22g1(s)ds)+2Λ1(me=1δeηea(ηes)2h1(s)ds)}, (2.3)
    w(t)=ta(ts)22h1(s)ds+1Λ{ba[Λ1mk=1ξk(bs)22+G7(t)(bs)+P7(t)]f1(s)dsba[Λ1S13(bs)22+G8(t)(bs)+P8(t)]g1(s)dsba[2Λ1(bs)2+G9(t)(bs)+P9(t)]h1(s)ds+P9(t)(md=1κdηdaf1(s)ds)+P7(t)(mn=1γnηnag1(s)ds)+P8(t)(mr=1σrηrah1(s)ds)+G9(t)(mp=1ζpηpa(ηps)f1(s)ds)+G7(t)(ml=1βlηla(ηls)g1(s)ds)+G8(t)(mq=1ρqηqa(ηqs)h1(s)ds)+2Λ1(mk=1ξkηka(ηks)2f1(s)ds)+Λ1mk=1ξk(mj=1αjηja(ηjs)2g1(s)ds)+Λ1S13(me=1δeηea(ηes)22h1(s)ds)}, (2.4)

    where

    G1(t)=(8B1)(μ1+4Ω(t)),G2(t)=(8B1)(μ2+2Ω(t)ml=1βl),G3(t)=(8B1)(μ3+S6Ω(t)),G4(t)=(8B1)(μ4+S8Ω(t)),G5(t)=(8B1)(μ5+4Ω(t)),G6(t)=(8B1)(μ6+2Ω(t)mq=1ρq),G7(t)=(8B1)(μ7+2Ω(t)mp=1ζp),G8(t)=(8B1)(μ8+S7Ω(t)),G9(t)=(8B1)(μ9+4Ω(t)),P1(t)=L1+A1Ω(t)+2Λ2(ta)2,P2(t)=L2+A2Ω(t)+Λ2(ta)2mn=1γn,P3(t)=L3+A3Ω(t)+S1Λ2(ta)22,P4(t)=L4+A7Ω(t)+S3Λ2(ta)22,P5(t)=L5+A8Ω(t)+2Λ2(ta)2,P6(t)=L6+A9Ω(t)+Λ2(ta)2mr=1σr,P7(t)=L7+A4Ω(t)+Λ2(ta)2md=1κd,P8(t)=L8+A5Ω(t)+S2Λ2(ta)22,P9(t)=L9+A6Ω(t)+2Λ2(ta)2,Ω(t)=(8B3)(ta), (2.5)
    S1=(mr=1σr)(mn=1γn),S2=(mn=1γn)(md=1κd),S3=(mr=1σr)(md=1κd),S4=(ml=1βl)(md=1κd),S5=(mr=1σr)(ml=1βl),S6=(ml=1βl)(mq=1ρq),S7=(ml=1βl)(mp=1ζp),S8=(mp=1ζp)(mq=1ρq),S9=(md=1κd)(mq=1ρq),S10=(mn=1γn)(mq=1ρq),S11=(me=1δe)(mj=1αj),S12=(mk=1ξk)(me=1δe),S13=(mk=1ξk)(mj=1αj),E1=mj=1αj(ηja),E2=mj=1αj(ηja)22,E3=ml=1βl(ηla),E4=me=1δe(ηea),E5=me=1δe(ηea)22,E6=mq=1ρq(ηqa),E7=mk=1ξk(ηka),E8=mk=1ξk(ηka)22,E9=mp=1ζp(ηpa), (2.6)
    A1=2(ba)[8+S6(md=1κd)+S3(ml=1βl)]+4S6E9+4S3E3+4S4E6,A2=(ba)[S2S6+8(mn=1γn)+8(ml=1βl)]+2S6E9(mn=1γn)+16E3+2S2E6(ml=1βl),A3=4(ba)[S6+S1+S5]+S1S6E9+8E3(mr=1σr)+8E6(ml=1βl),A4=(ba)[8(md=1κd)+8(mp=1ζp)+S3S7]+16E9+2S3E3(mp=1ζp)+2S4E6(mp=1ζp),A5=4(ba)[S2+(mn=1γn)(mp=1ζp)+S7]+8E9(mn=1γn)+8E3(mp=1ζp)+S2S7E6,A6=2(ba)[8+S1(mp=1ζp)+S5(mp=1ζp)]+4S1E9+4E3(mr=1σr)(mp=1ζp)+4S7E6,A7=4(ba)[S8+S3+S9]+S3S8E3+8E6(md=1κd)+8E9(mq=1ρq),A8=2(ba)[8+S8(mn=1γn)+S2(mq=1ρq)]+4S8E3+4S2E6+4S10E9,A9=(ba)[S1S8+8(mr=1σr)+8(mq=1ρq)]+2S8E3(mr=1σr)+16E6+2S1E9(mq=1ρq), (2.7)
    J1=E1A7A1(ba)+(8B2)(S3E22(ba)2),J2=E1A8A2(ba)+(8B2)(4E2mn=1γn(ba)2),J3=E1A9A3(ba)+(8B2)(2E2mr=1σrS1(ba)22),J4=E4A4A7(ba)+(8B2)(2E5md=1κdS3(ba)22),J5=E4A5A8(ba)+(8B2)(E5S22(ba)2),J6=E4A6A9(ba)+(8B2)(4E5mr=1σr(ba)2),J7=E7A1A4(ba)+(8B2)(4E8md=1κd(ba)2),J8=E7A2A5(ba)+(8B2)(2E8mn=1γnS2(ba)22),J9=E7A3A6(ba)+(8B2)(S1E82(ba)2), (2.8)
    μ1=4S8E1(ba)[16+2(mj=1αj)S8+2S11(mp=1ζp)]+4S11E7+4E4(mp=1ζp)(mj=1αj),μ2=16E1(ba)[8(ml=1βl)+8(mj=1αj)+S11S7]+2E4S7(mj=1αj)+2S11E7(ml=1βl),μ3=8E1(mq=1ρq)+8E4(mj=1αj)+S6S11E7,4(ba)[S6+S11+(mq=1ρq)(mj=1αj)], (2.9)
    μ4=S8S12E14(ba)[S12+S8+(mp=1ζp)(me=1δe)]+8E7(me=1δe)+8E4(mp=1ζp),μ5=4S12E12(ba)[8+S12(ml=1βl)+S7(me=1δe)]+4S7E4+4E7(ml=1βl)(me=1δe),μ6=2(mq=1ρq)S12E1(ba)[S6S12+8(mq=1ρq)+8(me=1δe)]+16E4+2(me=1δe)S6E7,μ7=2(mp=1ζp)S13E4(ba)[S8S13+8(mk=1ξk)+8(mp=1ζp)]+2S8E1(mk=1ξk)+16E7,μ8=S7S13E44(ba)[S13+(ml=1βl)(mk=1ξk)+S7]+8E1(mk=1ξk)+8E7(ml=1βl),μ9=4S13E42(ba)[8+S13(mq=1ρq)+S6(mk=1ξk)]+4E1(mq=1ρq)(mk=1ξk)+4S6E7,L1=4J1+J7S11+2J4mj=1αj,L2=4J2+J8S11+2J5mj=1αj,L3=4J3+J9S11+2J6mj=1αj,L4=4J4+J1S12+2J7me=1δe,L5=4J5+J2S12+2J8me=1δe,L6=4J6+J3S12+2J9me=1δe,L7=4J7+J4S13+2J1mk=1ξk,L8=4J8+J5S13+2J2mk=1ξk,L9=4J9+J6S13+2J3mk=1ξk, (2.10)

    and it is assumed that

    Λ=(8B1)(8B2)(8B3)0, (2.11)
    Λ1=Λ/(8B3),Λ2=Λ/(8B1),B1=(mr=1σr)(md=1κd)(mn=1γn),B2=(mp=1ζp)(ml=1βl)(mq=1ρq),B3=(mk=1ξk)(mj=1αj)(me=1δe). (2.12)

    Proof. We know that the general solution of the linear differential equations (2.1) can be written as

    u(t)=c0+c1(ta)+c2(ta)22+ta(ts)22f1(s)ds, (2.13)
    v(t)=c3+c4(ta)+c5(ta)22+ta(ts)22g1(s)ds, (2.14)
    w(t)=c6+c7(ta)+c8(ta)22+ta(ts)22h1(s)ds, (2.15)

    where ciR,i=1,,8 are arbitrary real constants. Using the boundary conditions (1.2) in (2.13), (2.14) and (2.15), we obtain

    2c0+(ba)c1+(ba)22c2(mj=1αj)c3(mj=1αj(ηja))c4(mj=1αj(ηja)22)c5=ba(bs)22f1(s)ds+mj=1αjηja(ηjs)22g1(s)ds, (2.16)
    2c1+(ba)c2(ml=1βl)c4(ml=1βl(ηla))c5=ba(bs)f1(s)ds+ml=1βlηla(ηls)g1(s)ds, (2.17)
    2c2(mn=1γn)c5=baf1(s)ds+mn=1γnηnag1(s)ds, (2.18)
    2c3+(ba)c4+(ba)22c5(me=1δe)c6(me=1δe(ηea))c7(me=1δe(ηea)22)c8=ba(bs)22g1(s)ds+me=1δeηea(ηes)22h1(s)ds, (2.19)
    2c4+(ba)c5(mq=1ρq)c7(mq=1ρq(ηqa))c8=ba(bs)g1(s)ds+mq=1ρqηqa(ηqs)h1(s)ds, (2.20)
    2c5(mr=1σr)c8=bag1(s)ds+mr=1σrηrah1(s)ds, (2.21)
    (mk=1ξk)c0(mk=1ξk(ηka))c1(mk=1ξk(ηka)22)c2+2c6+(ba)c7+(ba)22c8=ba(bs)22h1(s)ds+mk=1ξkηka(ηks)22f1(s)ds, (2.22)
    (mp=1ζp)c1(mp=1ζp(ηpa))c2+2c7+(ba)c8=ba(bs)h1(s)ds+mp=1ζpηpa(ηps)f1(s)ds, (2.23)
    (md=1κd)c2+2c8=bah1(s)ds+md=1κdηdaf1(s)ds. (2.24)

    Solving (2.18), (2.21) and (2.24) for c2,c5 and c8, together with the notations S1,S2 and S3 given by (2.6), we get

    c2=18B1{4baf1(s)ds2(mn=1γn)bag1(s)dsS1bah1(s)ds+S1(md=1κdηdaf1(s)ds)+4(mn=1γnηnag1(s)ds)+2(mn=1γn)(mr=1σrηrah1(s)ds)},c5=18B1{S3baf1(s)ds4bag1(s)ds2(mr=1σr)bah1(s)ds+2(mr=1σr)(md=1κdηdaf1(s)ds)+S3(mn=1γnηnag1(s)ds)+4(mr=1σrηrah1(s)ds)},c8=18B1{2(md=1κd)baf1(s)dsS2bag1(s)ds4bah1(s)ds+4(md=1κdηdaf1(s)ds)+2(md=1κd)(mn=1γnηnag1(s)ds)+S2(mr=1σrηrah1(s)ds)}.

    Inserting the values of c2,c5 and c8 in (2.17), (2.20) and (2.23), and using (2.6), we obtain

    2c1(ml=1βl)c4=18B1{ba[(bs)(8B1)+S3E34(ba)]f1(s)dsba[4E32(ba)(mn=1γn)]g1(s)dsba[2E3(mr=1σr)S1(ba)]h1(s)ds+md=1κdηda[2E3mr=1σrS1(ba)]f1(s)ds+mn=1γnηna[S3E34(ba)]g1(s)ds+mr=1σrηra[4E32(ba)(mn=1γn)]h1(s)ds}+ml=1βlηla(ηls)g1(s)ds, (2.25)
    2c4(mq=1ρq)c7=18B1{ba[2E6(md=1κd)S3(ba)]f1(s)dsba[(bs)(8B1)+S2E64(ba)]g1(s)dsba[4E62(ba)(mr=1σr)]h1(s)ds+md=1κdηda[4E62(ba)(mr=1σr)]f1(s)ds+mn=1γnηna[2E6(md=1κd)S3(ba)]g1(s)ds+mr=1σrηra[S2E64(ba)]h1(s)ds}+mq=1ρqηqa(ηqs)h1(s)ds, (2.26)
    (mp=1ζp)c1+2c7=18B1{ba[4E92(md=1κd)(ba)]f1(s)dsba[2E9(mn=1γn)S2(ba)]g1(s)dsba[(bs)(8B1)+S1E94(ba)]h1(s)ds+md=1κdηda[S1E94(ba)]f1(s)ds+mn=1γnηna[4E92(md=1κd)(ba)]g1(s)ds+mr=1σrηra[2E9(mn=1γn)S2(ba)]h1(s)ds}+mp=1ζpηpa(ηps)f1(s)ds. (2.27)

    Solving the systems (2.25)(2.27) for c1,c4 and c7 together with the notations (2.7) we find that

    c1=1Λ1{ba[4(8B1)(bs)+A1]f1(s)dsba[2(8B1)(bs)(ml=1βl)+A2]g1(s)dsba[S6(8B1)(bs)+A3]h1(s)ds+A3md=1κdηdaf1(s)ds+A1mn=1γnηnag1(s)ds+A2mr=1σrηrah1(s)ds+S6(8B1)(mp=1ζpηpa(ηps)f1(s)ds)+4(8B1)(ml=1βlηla(ηls)g1(s)ds)+2(8B1)(ml=1βl)(mq=1ρqηqa(ηqs)h1(s)ds)},c4=1Λ1{ba[S8(8B1)(bs)+A7]f1(s)dsba[4(8B1)(bs)+A8]g1(s)dsba[2(mq=1ρq)(8B1)(bs)+A9]h1(s)ds+A9md=1κdηdaf1(s)ds+A7mn=1γnηnag1(s)ds+A8mr=1σrηrah1(s)ds+2(8B1)(mq=1ρq)(mp=1ζpηpa(ηps)f1(s)ds)+S8(8B1)(ml=1βlηla(ηls)g1(s)ds)+4(8B1)(mq=1ρqηqa(ηqs)h1(s)ds)},c7=1Λ1{ba[2(mp=1ζp)(8B1)(bs)+A4]f1(s)dsba[S7(8B1)(bs)+A5]g1(s)dsba[4(8B1)(bs)+A6]h1(s)ds+A6md=1κdηdaf1(s)ds+A4mn=1γnηnag1(s)ds+A5mr=1σrηrah1(s)ds+4(8B1)(mp=1ζpηpa(ηps)f1(s)ds)+2(8B1)(mp=1ζp)(ml=1βlηla(ηls)g1(s)ds)+S7(8B1)(mq=1ρqηqa(ηqs)h1(s)ds)}.

    Substituting the values of c1,c2,c4,c5,c7 and c8 in (2.16), (2.19) and (2.22), together with the notations (2.6) and (2.8) yields

    2c0(mj=1αj)c3=1Λ1{ba[Λ1(bs)22+((8B1)(bs))(S8E14(ba))+J1]f1(s)dsba[((8B1)(bs))(4E12ml=1βl(ba))+J2]g1(s)dsba[((8B1)(bs))(2E1mq=1ρqS6(ba))+J3]h1(s)ds+J3md=1κdηdaf1(s)ds+J1mn=1γnηnag1(s)ds+J2mr=1σrηrah1(s)ds+(8B1)(mp=1ζpηpa(ηps)[2E1mq=1ρqS6(ba)]f1(s)ds)+(8B1)(ml=1βlηla(ηls)[S8E14(ba)]g1(s)ds)+(8B1)(mq=1ρqηqa(ηqs)[4E12ml=1βl(ba)]h1(s)ds)+Λ1(mj=1αjηja(ηjs)22g1(s)ds)}, (2.28)
    2c3(me=1δe)c6=1Λ1{ba[((8B1)(bs))(2E4mp=1ζpS8(ba))+J4]f1(s)dsba[Λ1(bs)22+((8B1)(bs))(S6E44(ba))+J5]g1(s)dsba[((8B1)(bs))(4E42mq=1ρq(ba))+J6]h1(s)ds+J6md=1κdηdaf1(s)ds+J4mn=1γnηnag1(s)ds+J5mr=1σrηrah1(s)ds+(8B1)(mp=1ζpηpa(ηps)[4E42mq=1ρq(ba)]f1(s)ds)+(8B1)(ml=1βlηla(ηls)[2E4mp=1ζpS8(ba)]g1(s)ds)+(8B1)(mq=1ρqηqa(ηqs)[S6E44(ba)]h1(s)ds)+Λ1(me=1δeηea(ηes)22h1(s)ds)}, (2.29)
    (mk=1ξk)c0+2c6=1Λ1{ba[((8B1)(bs))(4E72mp=1ζp(ba))+J7]f1(s)dsba[((8B1)(bs))(2E7ml=1βlS6(ba))+J8]g1(s)dsba[Λ1(bs)22+((8B1)(bs))(S6E74(ba))+J9]h1(s)ds+J9md=1κdηdaf1(s)ds+J7mn=1γnηnag1(s)ds+J8mr=1σrηrah1(s)ds+(8B1)(mp=1ζpηpa(ηps)[S6E74(ba)]f1(s)ds)+(8B1)(ml=1βlηla(ηls)[4E72mp=1ζp(ba)]g1(s)ds)+(8B1)(mq=1ρqηqa(ηqs)[2E7ml=1βlS6(ba)]h1(s)ds)+Λ1(mk=1ξkηka(ηks)22f1(s)ds)}. (2.30)

    Next, solving the system of Eqs (2.28)(2.30) for c0,c3 and c6 together with the notations (2.9), we obtain

    c0=1Λ{ba[2Λ1(bs)2+μ1(8B1)(bs)+L1]f1(s)dsba[Λ1(mj=1αj)(bs)2+μ2(8B1)(bs)+L2]g1(s)dsba[Λ1S11(bs)22+μ3(8B1)(bs)+L3]h1(s)ds+L3md=1κdηdaf1(s)ds+L1mn=1γnηnag1(s)ds+L2mr=1σrηrah1(s)ds+μ3(8B1)(mp=1ζpηpa(ηps)f1(s)ds)+μ1(8B1)(ml=1βlηla(ηls)g1(s)ds)+μ2(8B1)(mq=1ρqηqa(ηqs)h1(s)ds)+Λ1S11(mk=1ξkηka(ηks)22f1(s)ds)+2Λ1(mj=1αjηja(ηjs)2g1(s)ds)+Λ1(mj=1αj)(me=1δeηea(ηes)2h1(s)ds)},c3=1Λ{ba[Λ1S12(bs)22+μ4(8B1)(bs)+L4]f1(s)dsba[2Λ1(bs)2+μ5(8B1)(bs)+L5]g1(s)dsba[Λ1(me=1δe)(bs)2+μ6(8B1)(bs)+L6]h1(s)ds+L6md=1κdηdaf1(s)ds+L4mn=1γnηnag1(s)ds+L5mr=1σrηrah1(s)ds+μ6(8B1)(mp=1ζpηpa(ηps)f1(s)ds)+μ4(8B1)(ml=1βlηla(ηls)g1(s)ds)+μ5(8B1)(mq=1ρqηqa(ηqs)h1(s)ds)+Λ1(me=1δe)(mk=1ξkηka(ηks)2f1(s)ds)+Λ1S12(mj=1αjηja(ηjs)22g1(s)ds)+2Λ1(me=1δeηea(ηes)2h1(s)ds)},c6=1Λ{ba[Λ1(mk=1ξk)(bs)2+μ7(8B1)(bs)+L7]f1(s)dsba[Λ1S13(bs)22+μ8(8B1)(bs)+L8]g1(s)dsba[2Λ1(bs)2+μ9(8B1)(bs)+L9]h1(s)ds+L9md=1κdηdaf1(s)ds+L7mn=1γnηnag1(s)ds+L8mr=1σrηrah1(s)ds+μ9(8B1)(mp=1ζpηpa(ηps)f1(s)ds)+μ7(8B1)(ml=1βlηla(ηls)g1(s)ds)+μ8(8B1)(mq=1ρqηqa(ηqs)h1(s)ds)+2Λ1(mk=1ξkηka(ηks)22f1(s)ds)+Λ1(mk=1ξk)(mj=1αjηja(ηjs)22g1(s)ds)+Λ1S13(me=1δeηea(ηes)22h1(s)ds)}.

    Inserting the values of ci(i=1,,8) in (2.13), (2.14) and (2.15), we get the solutions (2.2), (2.3) and (2.4)). The converse follows by direct computation. This completes the proof.

    Let us introduce the space X={u(t)|u(t)C([a,b])} equipped with norm u=sup{|u(t)|,t [a,b]}. Obviously (X,.) is a Banach space and consequently, the product space (X×X×X,(u,v,w)) is a Banach space with norm (u,v,w)=u+v+w for (u,v,w)X3. In view of Lemma 2.1, we transform the problems (1.1) and (1.2) into an equivalent fixed point problem as

    (u,v,w)=H(u,v,w), (3.1)

    where H:X3X3 is defined by

    H(u,v,w)(t)=(H1(u,v,w)(t),H2(u,v,w)(t),H3(u,v,w)(t)), (3.2)
    H1(u,v,w)(t)=ta(ts)22ˆf(s)ds+1Λ{ba[2Λ1(bs)2+G1(t)(bs)+P1(t)]ˆf(s)dsba[Λ1mj=1αj(bs)2+G2(t)(bs)+P2(t)]ˆg(s)dsba[Λ1S11(bs)22+G3(t)(bs)+P3(t)]ˆh(s)ds+P3(t)(md=1κdηdaˆf(s)ds)+P1(t)(mn=1γnηnaˆg(s)ds)+P2(t)(mr=1σrηraˆh(s)ds)+G3(t)(mp=1ζpηpa(ηps)ˆf(s)ds)+G1(t)(ml=1βlηla(ηls)ˆg(s)ds)+G2(t)(mq=1ρqηqa(ηqs)ˆh(s)ds)+Λ1S11(mk=1ξkηka(ηks)22ˆf(s)ds)+2Λ1(mj=1αjηja(ηjs)2ˆg(s)ds)+Λ1mj=1αj(me=1δeηea(ηes)2ˆh(s)ds)}, (3.3)
    H2(u,v,w)(t)=ta(ts)22ˆg(s)ds+1Λ{ba[Λ1S12(bs)22+G4(t)(bs)+P4(t)]ˆf(s)dsba[2Λ1(bs)2+G5(t)(bs)+P5(t)]ˆg(s)dsba[Λ1me=1δe(bs)2+G6(t)(bs)+P6(t)]ˆh(s)ds+P6(t)(md=1κdηdaˆf(s)ds)+P4(t)(mn=1γnηnaˆg(s)ds)+P5(t)(mr=1σrηraˆh(s)ds)+G6(t)(mp=1ζpηpa(ηps)ˆf(s)ds)+G4(t)(ml=1βlηla(ηls)ˆg(s)ds)+G5(t)(mq=1ρqηqa(ηqs)ˆh(s)ds)+Λ1me=1δe(mk=1ξkηka(ηks)2ˆf(s)ds)+Λ1S12(mj=1αjηja(ηjs)22ˆg(s)ds)+2Λ1(me=1δeηea(ηes)2ˆh(s)ds)}, (3.4)
    H3(u,v,w)(t)=ta(ts)22ˆh(s)ds+1Λ{ba[Λ1mk=1ξk(bs)22+G7(t)(bs)+P7(t)]ˆf(s)dsba[Λ1S13(bs)22+G8(t)(bs)+P8(t)]ˆg(s)dsba[2Λ1(bs)2+G9(t)(bs)+P9(t)]ˆh(s)ds+P9(t)(md=1κdηdaˆf(s)ds)+P7(t)(mn=1γnηnaˆg(s)ds)+P8(t)(mr=1σrηraˆh(s)ds)+G9(t)(mp=1ζpηpa(ηps)ˆf(s)ds)+G7(t)(ml=1βlηla(ηls)ˆg(s)ds)+G8(t)(mq=1ρqηqa(ηqs)ˆh(s)ds)+2Λ1(mk=1ξkηka(ηks)2ˆf(s)ds)+Λ1mk=1ξk(mj=1αjηja(ηjs)2ˆg(s)ds)+Λ1S13(me=1δeηea(ηes)22ˆh(s)ds)}, (3.5)
    ˆf(s)=f(s,u(s),v(s),w(s)),ˆg(s)=g(s,u(s),v(s),w(s)),ˆh(s)=h(s,u(s),v(s),w(s)).

    In order to establish the main results, we need the following assumptions:

    (N1) (Linear growth conditions) There exist real constants mi,ˉmi,ˆmi0,(i=1,2,3) and m0>0,ˉm0>0,ˆm0>0 such that u,v,wR, we have

    |f(t,u,v,w)|m0+m1|u|+m2|v|+m3|w|,
    |g(t,u,v,w)|ˉm0+ˉm1|u|+ˉm2|v|+ˉm3|w|,
    |h(t,u,v,w)|ˆm0+ˆm1|u|+ˆm2|v|+ˆm3|w|.

    (N2) (Sub-growth conditions) There exist nonnegative functions ϕ(t),ψ(t) and χ(t)L(a,b) and ϵi>0,0<λi<1,(i=1,,9) such that u,v,wR, we have

    |f(t,u,v,w)|ϕ(t)+ϵ1|u|λ1+ϵ2|v|λ2+ϵ3|w|λ3,
    |g(t,u,v,w)|ψ(t)+ϵ4|u|λ4+ϵ5|v|λ5+ϵ6|w|λ6,
    |h(t,u,v,w)|χ(t)+ϵ7|u|λ7+ϵ8|v|λ8+ϵ9|w|λ9.

    (N3) (Lipschitz conditions) For all t[a,b] and ui,vi,wiR,i=1,2 there exist i>0(i=1,2,3) such that

    |f(t,u1,v1,w1)f(t,u2,v2,w2)|1(|u1u2|+|v1v2|+|w1w2|),
    |g(t,u1,v1,w1)g(t,u2,v2,w2)|2(|u1u2|+|v1v2|+|w1w2|),
    |h(t,u1,v1,w1)h(t,u2,v2,w2)|3(|u1u2|+|v1v2|+|w1w2|).

    For the sake of computational convenience, we set

    Θ1=Δ1+Δ4+Δ7,Θ2=Δ2+Δ5+Δ8,Θ3=Δ3+Δ6+Δ9, (3.6)

    where

    Δ1=(ba)36+13|8B3|[2(ba)3+S11(mk=1ξk(ηka)32)]+1|Λ|[Q1(ba)22+Υ1(ba)+Υ3(md=1κd(ηda))+Q3(mp=1ζp(ηpa)22)], (3.7)
    Δ2=mj=1αj3|8B3|[(ba)3+2(ηja)3]+1|Λ|[Q2(ba)22+Υ2(ba)+Υ1(mn=1γn(ηna))+Q1(ml=1βl(ηla)22)], (3.8)
    Δ3=13|8B3|[S11(ba)32+(mj=1αj)(me=1δe(ηea)3)]+1|Λ|[Q3(ba)22+Υ3(ba)+Υ2(mr=1σr(ηra))+Q2(mq=1ρq(ηqa)22)], (3.9)
    Δ4=13|8B3|[S12(ba)32+(me=1δe)(mk=1ξk(ηka)3)]+1|Λ|[Q4(ba)22+Υ4(ba)+Υ6(md=1κd(ηda))+Q6(mp=1ζp(ηpa)22)], (3.10)
    Δ5=(ba)36+13|8B3|[2(ba)3+S12(mj=1αj(ηja)32)]+1|Λ|[Q5(ba)22+Υ5(ba)+Υ4(mn=1γn(ηna))+Q4(ml=1βl(ηla)22)], (3.11)
    Δ6=me=1δe3|8B3|[(ba)3+2(ηea)3)]+1|Λ|[Q6(ba)22+Υ6(ba)+Υ5(mr=1σr(ηra))+Q5(mq=1ρq(ηqa)22)], (3.12)
    Δ7=mk=1ξk3|8B3|[(ba)32+2(ηka)3)]+1|Λ|[Q7(ba)22+Υ7(ba)+Υ9(md=1κd(ηda))+Q9(mp=1ζp(ηpa)22)], (3.13)
    Δ8=13|8B3|[S13(ba)32+(mk=1ξk)(mj=1αj(ηja)3)]+1|Λ|[Q8(ba)22+Υ8(ba)+Υ7(mn=1γn(ηna))+Q7(ml=1βl(ηla)22)], (3.14)
    Δ9=(ba)36+13|8B3|[2(ba)3+S13(me=1δe(ηea)32)]+1|Λ|[Q9(ba)22+Υ9(ba)+Υ8(mr=1σr(ηra))+Q8(mq=1ρq(ηqa)22)], (3.15)

    Qi=maxt[a,b]|Gi(t)|, and Υi=maxt[a,b]|Pi(t)|,(i=1,,9). Also, we set

    Θ=min{1(Θ1m1+Θ2ˉm1+Θ3ˆm1),1(Θ1m2+Θ2ˉm2+Θ3ˆm2),1(Θ1m3+Θ2ˉm3+Θ3ˆm3)}, (3.16)

    where mi,ˉmi,ˆmi are given in (N1).

    Firstly, we apply Leray-Schauder alternative [22] to prove the existence of solutions for the problems (1.1) and (1.2).

    Lemma 3.1. (Leray-Schauder alternative). Let Y be a Banach space, and T:YY be a completely continuous operator (i.e., a map restricted to any bounded set in Y is compact). Let Ξ(T)={xY:x=φT(x)for some0<φ<1}. Then either the set Ξ(T) is unbounded, or T has at least one fixed point.

    Theorem 3.1. Assume that the condition (N1) holds and that

    Θ1m1+Θ2ˉm1+Θ3ˆm1<1,Θ1m2+Θ2ˉm2+Θ3ˆm2<1andΘ1m3+Θ2ˉm3+Θ3ˆm3<1, (3.17)

    where Θ1,Θ2 and Θ3 are given by (3.6). Then there exists at least one solution for the problem (1.1) and (1.2) on [a,b].

    Proof. First of all, we show that the operator H:X3X3 defined by (3.2) is completely continuous. Notice that H1,H2 and H3 are continuous in view of continuity of the functions f,g and h. So the operator H is continuous. Let ΦX3 be a bounded set. Then there exist positive constants ϱf,ϱg and ϱh such that |ˆf(t)|=|f(t,u(t),v(t),w(t))|ϱf,|ˆg(t)|=|g(t,u(t),v(t),w(t))|ϱg and |ˆh(t)|=|h(t,u(t),v(t),w(t))|ϱh,(u,v,w)Φ. Then, for any (u,v,w)Φ, we obtain

    |H1(u,v,w)(t)|=|ta(ts)22ˆf(s)ds+1Λ{ba[2Λ1(bs)2+G1(t)(bs)+P1(t)]ˆf(s)dsba[Λ1mj=1αj(bs)2+G2(t)(bs)+P2(t)]ˆg(s)dsba[Λ1S11(bs)22+G3(t)(bs)+P3(t)]ˆh(s)ds+P3(t)(md=1κdηdaˆf(s)ds)+P1(t)(mn=1γnηnaˆg(s)ds)+P2(t)(mr=1σrηraˆh(s)ds)+G3(t)(mp=1ζpηpa(ηps)ˆf(s)ds)+G1(t)(ml=1βlηla(ηls)ˆg(s)ds)+G2(t)(mq=1ρqηqa(ηqs)ˆh(s)ds)+Λ1S11(mk=1ξkηka(ηks)22ˆf(s)ds)+2Λ1(mj=1αjηja(ηjs)2ˆg(s)ds)+Λ1mj=1αj(me=1δeηea(ηes)2ˆh(s)ds)}|ϱf{(ba)36+13|8B3|[2(ba)3+S11(mk=1ξk(ηka)32)]+1|Λ|[Q1(ba)22+Υ1(ba)+Υ3(md=1κd(ηda))+Q3(mp=1ζp(ηpa)22)]}+ϱg{mj=1αj3|8B3|[(ba)3+2(ηja)3]+1|Λ|[Q2(ba)22+Υ2(ba)+Υ1(mn=1γn(ηna))+Q1(ml=1βl(ηla)22)]}+ϱh{13|8B3|[S11(ba)32+(mj=1αj)(me=1δe(ηea)3)]+1|Λ|[Q3(ba)22+Υ3(ba)+Υ2(mr=1σr(ηra))+Q2(mq=1ρq(ηqa)22)]}ϱfΔ1+ϱgΔ2+ϱhΔ3,

    which implies that

    H1(u,v,w)ϱfΔ1+ϱgΔ2+ϱhΔ3,

    where we have used the notations (3.7),(3.8) and (3.9). In a similar manner, it can be shown that

    H2(u,v,w)ϱfΔ4+ϱgΔ5+ϱhΔ6,

    and

    H3(u,v,w)ϱfΔ7+ϱgΔ8+ϱhΔ9,

    where Δi(i=4,,9) are given by (3.10)(3.15). In consequence, we get

    H(u,v,w)ϱfΘ1+ϱgΘ2+ϱhΘ3,

    where Θ1, Θ2 and Θ3 are given by (3.6). From the foregoing arguments, it follows that the operator H is uniformly bounded. Next, we prove that H is equicontinuous. For a<t<τ<b, and (u,v,w)Φ, we have

    |H1(u,v,w)(τ)H1(u,v,w)(t)||ta[(τs)22(ts)22]ˆf(s)ds+τt(τs)22ˆf(s)dsba[(τt)(4(bs)(8B2)+A1Λ1)+2(8B1)(τ2t2)]ˆf(s)dsba[(τt)(2ml=1βl(8B2)(bs)+A2Λ1)+mn=1γn(8B1)(τ2t2)]ˆg(s)dsba[(τt)(S6(8B2)(bs)+A3Λ1)+S12(8B1)(τ2t2)]ˆh(s)ds+md=1κdηda[A3Λ1(τt)+S12(8B1)(τ2t2)]ˆf(s)ds+mn=1γnηna[A1Λ1(τt)+2(8B1)(τ2t2)]ˆg(s)ds+mr=1σrηra[A2Λ1(τt)+mn=1γn(8B1)(τ2t2)]ˆh(s)ds+S6(8B2)(τt)(mp=1ζpηpa(ηps)ˆf(s)ds)+4(8B2)(τt)(ml=1βlηla(ηls)ˆg(s)ds)+2ml=1βl(8B2)(τt)(mq=1ρqηqa(ηqs)ˆh(s)ds)|ϱf[(τt)33+13!|(τa)3(ta)3|]+(τt)|8B2|[(ba)2(2ϱf+ϱgml=1βl+12ϱhS6)+ϱfS6(mp=1ζp(ηpa)22)+2ϱg(ml=1βl(ηla)2)+ϱh(ml=1βl)(mq=1ρq(ηqa)2)]+(τt)|Λ1|[(ba)(ϱf|A1|+ϱg|A2|+ϱh|A3|)+ϱf|A3|(md=1κd(ηda))+ϱg|A1|(mn=1γn(ηna))+ϱh|A2|(mr=1σr(ηra))]+(τ2t2)|8B1|[(ba)(2ϱf+ϱgmn=1γn+12ϱhS1)+12ϱfS1(md=1κd(ηda))+2ϱg(mn=1γn(ηna))+ϱh(mn=1γn)(mr=1σr(ηra))]0independent of(u,v,w)Φasτt0.

    Similarly, it can be established that

    |H2(u,v,w)(τ)H2(u,v,w)(t)|ϱg[(τt)33+13!|(τa)3(ta)3|]+(τt)|8B2|[(ba)2(12ϱfS8+2ϱg+ϱhmq=1ρq)+ϱf(mq=1ρq)(mp=1ζp(ηpa)2)+ϱgS8(ml=1βl(ηla)22)+2ϱh(mq=1ρq(ηqa)2)]+(τt)|Λ1|[(ba)(ϱf|A7|+ϱg|A8|+ϱh|A9|)+ϱf|A9|(md=1κd(ηda))+ϱg|A7|(mn=1γn(ηna))+ϱh|A8|(mr=1σr(ηra))]+(τ2t2)|8B1|[(ba)(12ϱfS3+2ϱg+ϱhmr=1σr)+ϱf(mr=1σr)(md=1κd(ηda))+12ϱgS3(mn=1γn(ηna))+2ϱh(mr=1σr(ηra))]0independent of(u,v,w)Φasτt0,

    and

    |H3(u,v,w)(τ)H3(u,v,w)(t)|ϱh[(τt)33+13!|(τa)3(ta)3|]+(τt)|8B2|[(ba)2(ϱfmp=1ζp+12ϱgS7+2ϱh)+2ϱf(mp=1ζp(ηpa)2)+ϱg(mp=1ζp)(ml=1βl(ηla)2)+ϱhS7(mq=1ρq(ηqa)22)]+(τt)|Λ1|[(ba)(ϱf|A4|+ϱg|A5|+ϱh|A6|)+ϱf|A6|(md=1κd(ηda))+ϱg|A4|(mn=1γn(ηna))+ϱh|A5|(mr=1σr(ηra))]+(τ2t2)|8B1|[(ba)(ϱfmd=1κd+12ϱgS2+2ϱh)+2ϱf(md=1κd(ηda))+ϱg(md=1κd)(mn=1γn(ηna))+12ϱhS2(mr=1σr(ηra))]0independent of(u,v,w)Φasτt0.

    In view of the foregoing steps, the Arzelá-Ascoli theorem applies and hence the operator H is completely continuous. Finally, it will be verified that the set Ξ={(u,v,w)X3|(u,v,w)=φH(u,v,w),0<φ<1} is bounded. Let (u,v,w)Ξ. Then (u,v,w)=φH(u,v,w) and for any t[a,b], we have

    u(t)=φH1(u,v,w)(t),v(t)=φH2(u,v,w)(t),w(t)=φH3(u,v,w)(t).

    Thus, we get

    |u(t)|Δ1(m0+m1u+m2v+m3w)+Δ2(ˉm0+ˉm1u+ˉm2v+ˉm3w)+Δ3(ˆm0+ˆm1u+ˆm2v+ˆm3w)Δ1m0+Δ2ˉm0+Δ3ˆm0+(Δ1m1+Δ2ˉm1+Δ3ˆm1)u+(Δ1m2+Δ2ˉm2+Δ3ˆm2)v+(Δ1m3+Δ2ˉm3+Δ3ˆm3)w,
    |v(t)|Δ4(m0+m1u+m2v+m3w)+Δ5(ˉm0+ˉm1u+ˉm2v+ˉm3w)+Δ6(ˆm0+ˆm1u+ˆm2v+ˆm3w)Δ4m0+Δ5ˉm0+Δ6ˆm0+(Δ4m1+Δ5ˉm1+Δ6ˆm1)u+(Δ4m2+Δ5ˉm2+Δ6ˆm2)v+(Δ4m3+Δ5ˉm3+Δ6ˆm3)w,

    and

    |w(t)|Δ7m0+Δ8ˉm0+Δ9ˆm0+(Δ7m1+Δ8ˉm1+Δ9ˆm1)u+(Δ7m2+Δ8ˉm2+Δ9ˆm2)v+(Δ7m3+Δ8ˉm3+Δ9ˆm3)w.

    Therefore, we can deduce that

    u+v+wΘ1m0+Θ2ˉm0+Θ3ˆm0+(Θ1m1+Θ2ˉm1+Θ3ˆm1)u+(Θ1m2+Θ2ˉm2+Θ3ˆm2)v+(Θ1m3+Θ2ˉm3+Θ3ˆm3)w.

    Using (3.17) together with the value of Θ given by (3.16), we find that

    (u,v,w)Θ1m0+Θ2ˉm0+Θ3ˆm0Θ,

    which shows that the set Ξ is bounded. Hence, by Lemma 2, the operator H has at least one fixed point. Therefore, the problems (1.1) and (1.2) have at least one solution on [a, b]. This completes the proof.

    Secondly, we apply the sub-growth condition (N2) under Schauder's fixed point theorem to show the existence of solutions for the problems (1.1) and (1.2).

    Theorem 3.2. Assume that (N2) holds. Then there exists at least one solution for the problems (1.1) and (1.2) on [a,b].

    Proof. Define a set Γ in the Banach space X3 as follows Γ={(u,v,w)X3:(u,v,w)x}, where

    xmax{12Θ1ϕ,12Θ2ψ,12Θ3χ,(12Θ1ϵ1)11λ1,(12Θ1ϵ2)11λ2,(12Θ1ϵ3)11λ3,(12Θ2ϵ4)11λ4,(12Θ2ϵ5)11λ5,(12Θ2ϵ6)11λ6,(12Θ3ϵ7)11λ7,(12Θ3ϵ8)11λ8,(12Θ3ϵ9)11λ9}

    Firstly, we prove that H:ΓΓ. For that, we consider

    |H1(u,v,w)(t)|=|ta(ts)22ˆf(s)ds+1Λ{ba[2Λ1(bs)2+G1(t)(bs)+P1(t)]ˆf(s)dsba[Λ1mj=1αj(bs)2+G2(t)(bs)+P2(t)]ˆg(s)dsba[Λ1S11(bs)22+G3(t)(bs)+P3(t)]ˆh(s)ds+P3(t)(md=1κdηdaˆf(s)ds)+P1(t)(mn=1γnηnaˆg(s)ds)+P2(t)(mr=1σrηraˆh(s)ds)+G3(t)(mp=1ζpηpa(ηps)ˆf(s)ds)+G1(t)(ml=1βlηla(ηls)ˆg(s)ds)+G2(t)(mq=1ρqηqa(ηqs)ˆh(s)ds)+Λ1S11(mk=1ξkηka(ηks)22ˆf(s)ds)+2Λ1(mj=1αjηja(ηjs)2ˆg(s)ds)+Λ1mj=1αj(me=1δeηea(ηes)2ˆh(s)ds)}|(ϕ(t)+ϵ1|u|λ1+ϵ2|v|λ2+ϵ3|w|λ3)Δ1+(ψ(t)+ϵ4|u|λ4+ϵ5|v|λ5+ϵ6|w|λ6)Δ2+(χ(t)+ϵ7|u|λ7+ϵ8|v|λ8+ϵ9|w|λ9)Δ3,

    which, on taking the norm

    ||H1(u,v,w)||(ϕ+ϵ1|u|λ1+ϵ2|v|λ2+ϵ3|w|λ3)Δ1+(ψ+ϵ4|u|λ4+ϵ5|v|λ5+ϵ6|w|λ6)Δ2+(χ+ϵ7|u|λ7+ϵ8|v|λ8+ϵ9|w|λ9)Δ3,

    where we have used the notations (3.7)(3.9). Analogously, we have

    ||H2(u,v,w)||(ϕ+ϵ1|u|λ1+ϵ2|v|λ2+ϵ3|w|λ3)Δ4+(ψ+ϵ4|u|λ4+ϵ5|v|λ5+ϵ6|w|λ6)Δ5+(χ+ϵ7|u|λ7+ϵ8|v|λ8+ϵ9|w|λ9)Δ6,

    and

    ||H3(u,v,w)||(ϕ+ϵ1|u|λ1+ϵ2|v|λ2+ϵ3|w|λ3)Δ7+(ψ+ϵ4|u|λ4+ϵ5|v|λ5+ϵ6|w|λ6)Δ8+(χ+ϵ7|u|λ7+ϵ8|v|λ8+ϵ9|w|λ9)Δ9,

    where Δi(i=4,,9) are given by (3.10)(3.15). Consequently,

    ||H(u,v,w)||(ϕ+ϵ1|u|λ1+ϵ2|v|λ2+ϵ3|w|λ3)Θ1+(ψ+ϵ4|u|λ4+ϵ5|v|λ5+ϵ6|w|λ6)Θ2+(χ+ϵ7|u|λ7+ϵ8|v|λ8+ϵ9|w|λ9)Θ3x,

    where Θ1,Θ2 and Θ3 are given by (3.6). Therefore, we conclude that H:ΓΓ, where H1(u,v,w)(t),H2(u,v,w)(t)andH3(u,v,w)(t) are continuous on [a,b].

    As in Theorem 3.1, one can show that the operator H is completely continuous. So, by Schauder's fixed point theorem, there exists a solution for the problems (1.1) and (1.2) on [a,b].

    Here we apply Banach's contraction mapping principle to show the existence of a unique solution for the problems (1.1) and (1.2).

    Theorem 3.3. Assume that (N3) holds. In addition, we suppose that

    Θ11+Θ22+Θ33<1, (3.18)

    where Θ1,Θ2 and Θ3 are given by (3.6). Then the problems (1.1) and (1.2) have a unique solution on [a,b].

    Proof. Let us set supt[a,b]|f(t,0,0,0)|=M1,supt[a,b]|g(t,0,0,0)|=M2, supt[a,b]|h(t,0,0, 0)|=M3, and show that HBςBς, where Bς={(u,v,w)X3:(u,v,w)ς} with

    ςΘ1M1+Θ2M2+Θ3M31(Θ11+Θ22+Θ33).

    For any (u,v,w)Bς,t[a,b], we find that

    |f(s,u(s),v(s),w(s))|=|f(s,u(s),v(s),w(s))f(s,0,0,0)+f(s,0,0,0)||f(s,u(s),v(s),w(s))f(s,0,0,0)|+|f(s,0,0,0)|1(u+v+w)+M11(u,v,w)+M11ς+M1.

    In a similar manner, we have

    |g(s,u(s),v(s),w(s))|2ς+M2,|h(s,u(s),v(s),w(s)|3ς+M3.

    Then, for (u,v,w)Bς, we obtain

    |H1(u,v,w)(t)|=|ta(ts)22ˆf(s)ds+1Λ{ba[2Λ1(bs)2+G1(t)(bs)+P1(t)]ˆf(s)dsba[Λ1mj=1αj(bs)2+G2(t)(bs)+P2(t)]ˆg(s)dsba[Λ1S11(bs)22+G3(t)(bs)+P3(t)]ˆh(s)ds+P3(t)(md=1κdηdaˆf(s)ds)+P1(t)(mn=1γnηnaˆg(s)ds)+P2(t)(mr=1σrηraˆh(s)ds)+G3(t)(mp=1ζpηpa(ηps)ˆf(s)ds)+G1(t)(ml=1βlηla(ηls)ˆg(s)ds)+G2(t)(mq=1ρqηqa(ηqs)ˆh(s)ds)+Λ1S11(mk=1ξkηka(ηks)22ˆf(s)ds)+2Λ1(mj=1αjηja(ηjs)2ˆg(s)ds)+Λ1mj=1αj(me=1δeηea(ηes)2ˆh(s)ds)}|(1ς+M1){(ba)36+13|8B3|[2(ba)3+S11(mk=1ξk(ηka)32)]+1|Λ|[Q1(ba)22+Υ1(ba)+Υ3(md=1κd(ηda))+Q3(mp=1ζp(ηpa)22)]}+(2ς+M2){mj=1αj3|8B3|[(ba)3+2(ηja)3]+1|Λ|[Q2(ba)22+Υ2(ba)+Υ1(mn=1γn(ηna))+Q1(ml=1βl(ηla)22)]}+(3ς+M3){13|8B3|[S11(ba)32+(mj=1αj)(me=1δe(ηea)3)]+1|Λ|[Q3(ba)22+Υ3(ba)+Υ2(mr=1σr(ηra))+Q2(mq=1ρq(ηqa)22)]}(1ς+M1)Δ1+(2ς+M2)Δ2+(3ς+M3)Δ3,

    which, on taking the norm for t[a,b], yields

    H1(u,v,w)(1ς+M1)Δ1+(2ς+M2)Δ2+(3ς+M3)Δ3.

    Similarly, we can find that

    H2(u,v,w)(1ς+M1)Δ4+(2ς+M2)Δ5+(3ς+M3)Δ6,

    and

    H3(u,v,w)(1ς+M1)Δ7+(2ς+M2)Δ8+(3ς+M3)Δ9,

    where Δi(i=1,,9) are defined in (3.7)(3.15). In consequence, it follows that

    H(u,v,w)(1ς+M1)Θ1+(2ς+M2)Θ2+(3ς+M3)Θ3ς.

    Next we show that the operator H is a contraction. For (u1,v1,w1),(u2,v2,w2)X3, we have

    |H1(u1,v1,w1)(t)H1(u2,v2,w2)(t)|ta(ts)22|f(s,u1(s),v1(s),w1(s))f(s,u2(s),v2(s),w2(s))|ds+1|Λ|{ba[2|Λ1|(bs)2+|G1(t)|(bs)+|P1(t)|]×|f(s,u1(s),v1(s),w1(s))f(s,u2(s),v2(s),w2(s))|ds+ba[|Λ1|mj=1αj(bs)2+|G2(t)|(bs)+|P2(t)|]×|g(s,u1(s),v1(s),w1(s))g(s,u2(s),v2(s),w2(s))|ds+ba[|Λ1|S11(bs)22+|G3(t)|(bs)+|P3(t)|]×|h(s,u1(s),v1(s),w1(s))h(s,u2(s),v2(s),w2(s))|ds+|P3(t)|(md=1κdηda|f(s,u1(s),v1(s),w1(s))f(s,u2(s),v2(s),w2(s))|ds)+|P1(t)|(mn=1γnηna|g(s,u1(s),v1(s),w1(s))g(s,u2(s),v2(s),w2(s))|ds)+|P2(t)|(mr=1σrηra|h(s,u1(s),v1(s),w1(s))h(s,u2(s),v2(s),w2(s))|ds)+|G3(t)|(mp=1ζpηpa(ηps)|f(s,u1(s),v1(s),w1(s))f(s,u2(s),v2(s),w2(s))|ds)+|G1(t)|(ml=1βlηla(ηls)|g(s,u1(s),v1(s),w1(s))g(s,u2(s),v2(s),w2(s))|ds)+|G2(t)|(mq=1ρqηqa(ηqs)|h(s,u1(s),v1(s),w1(s))h(s,u2(s),v2(s),w2(s))|ds)+|Λ1|S11(mk=1ξkηka(ηks)22|f(s,u1(s),v1(s),w1(s))f(s,u2(s),v2(s),w2(s))|ds)+2|Λ1|(mj=1αjηja(ηjs)2|g(s,u1(s),v1(s),w1(s))g(s,u2(s),v2(s),w2(s))|ds)+|Λ1|mj=1αj(me=1δeηea(ηes)2|h(s,u1(s),v1(s),w1(s))h(s,u2(s),v2(s),w2(s))|ds)}1(|u1u2|+|v1v2|+|w1w2|){(ba)36+13|8B3|[2(ba)3+S11(mk=1ξk(ηka)32)]+1|Λ|[Q1(ba)22+Υ1(ba)+Υ3(md=1κd(ηda))+Q3(mp=1ζp(ηpa)22)]}+2(|u1u2|+|v1v2|+|w1w2|){mj=1αj3|8B3|[(ba)3+2(ηja)3]+1|Λ|[Q2(ba)22+Υ2(ba)+Υ1(mn=1γn(ηna))+Q1(ml=1βl(ηla)22)]}+3(|u1u2|+|v1v2|+|w1w2|){13|8B3|[S11(ba)32+(mj=1αj)(me=1δe(ηea)3)]+1|Λ|[Q3(ba)22+Υ3(ba)+Υ2(mr=1σr(ηra))+Q2(mq=1ρq(ηqa)22)]}(1Δ1+2Δ2+3Δ3)(|u1u2|+|v1v2|+|w1w2|),

    which implies that

    H1(u1,v1,w1)H1(u2,v2,w2)(1Δ1+2Δ2+3Δ3)(|u1u2|+|v1v2|+|w1w2|),

    where Δ1Δ2 and Δ3 are given by (3.7),(3.8) and (3.9) respectively. In a similar fashion, one can find that

    H2(u1,v1,w1)H2(u2,v2,w2)(1Δ4+2Δ5+3Δ6)(|u1u2|+|v1v2|+|w1w2|),

    and

    H3(u1,v1,w1)H3(u2,v2,w2)(1Δ7+2Δ8+3Δ9)(|u1u2|+|v1v2|+|w1w2|),

    where Δi,(i=4,,9) are given by (3.10)(3.15). Thus we have

    H(u1,v1,w1)H(u2,v2,w2)(Θ11+Θ22+Θ33)(u1u2+v1v2+w1w2), (3.19)

    where Θ1,Θ2 and Θ3 are given by (3.6). By the assumption (3.18) it follows from (3.19) that the operator H is a contraction. Thus, by Banach contraction mapping principle, we deduce that the operator H has a fixed point, which corresponds to a unique solution of the problems (1.1) and (1.2) on [a,b].

    Example 3.1. Consider the following coupled system of third-order ordinary differential equations

    u(t)=531t3+24+|u(t)|2204(1+|u(t)|)+3342sinv(t)+1t2+97w(t),t[1,3],v(t)=e(t1)12(15+t)+1798πsin(7πu)+|v(t)|396(1+|v(t)|2)+4(t+7)3w(t),t[1,3],w(t)=12(4+t)2cost+264356tu(t)+w(t)|v(t)|810(1+|v(t)|),t[1,3], (3.20)

    supplemented to the following boundary conditions

    u(1)+u(3)=4j=1αjv(ηj),u(1)+u(3)=4l=1βlv(ηl),u(1)+u(3)=4n=1γnv(ηn),v(1)+v(3)=4e=1δew(ηe),v(1)+v(3)=4q=1ρqw(ηq),v(1)+v(3)=4r=1σrw(ηr),w(1)+w(3)=4k=1ξku(ηk),w(1)+w(3)=4p=1ζpu(ηp),w(1)+w(3)=4d=1κdu(ηd), (3.21)

    where

    a=1,b=3,m=4,η1=4/3,η2=5/3,η3=2,η4=7/3,α1=1/4,α2=1/2,α3=3/4,α4=1,β1=0.2,β2=8/15,β3=13/15,β4=6/5,γ1=1/8,γ2=9/40,γ3=13/40,γ4=17/40,δ1=2/11,δ2=3/11,δ3=4/11,δ4=5/11,ρ1=1/6,ρ2=7/24,ρ3=5/12,ρ4=13/24,σ1=1/9,σ2=2/9,σ3=1/3,σ4=4/9,ξ1=1/7,ξ2=2/7,ξ3=3/7,ξ4=4/7,ζ1=2/15,ζ2=1/3,ζ3=8/15,ζ4=11/15,κ1=1/3,κ2=4/9,κ3=5/9,κ4=2/3.

    By direct substitution, we get B12.4444448,B26.8755568,B34.5454528, and Λ21.580256 (Λ is given by (2.11)). Also, Δ121.294227,Δ222.603176,Δ311.800813,Δ47.983258,Δ512.996835,Δ68.497948,Δ710.977544,Δ814.165941 and Δ912.745457 (Δi(i=1,,9) are defined in (3.7)(3.15)). Furthermore we obtain Θ140.255029,Θ249.765952 and Θ333.044218(Θ1,Θ2 and Θ3 are given by (3.6)). Evidently,

    |f(t,u,v,w)|131+1204u+1114v+198w,|g(t,u,v,w)|1192+1114u+196v+1128w,|h(t,u,v,w)|150+1198u+1810w.

    Clearly, m0=1/31,m1=1/204,m2=1/114,m3=1/98,ˉm0=1/192,ˉm1=1/114,ˉm2=1/96,ˉm3=1/128, and ˆm0=1/50,ˆm1=1/198,ˆm2=0,ˆm3=1/810. Using (3.17), we find that Θ1m1+Θ2ˉm1+Θ3ˆm10.800762<1,Θ1m2+Θ2ˉm2+Θ3ˆm20.871509<1 and Θ1m3+Θ2ˉm3+Θ3ˆm30.840357<1. Also, from (3.16) we obtain Θ=0.128491. Hence, all the conditions of Theorem 3.1 are satisfied and consequently the problems (3.20) and (3.21) has at least one solution on [1, 3].

    Example 3.2. Consider the following system

    u(t)=39(t3+72)(tan1(u(t))+v(t)+|w|1+|w|)+e(t1),t[1,3],v(t)=1610πsin(2πu)+42t+1218sin(v(t))+73+1305w(t),t[1,3],w(t)=322999+90t(u(t)+|v(t)|1+|v(t)|+tan1(w(t)))+cos(t1),t[1,3], (3.22)

    subject to the coupled boundary conditions (3.21). It is easy to see that 1=1/219,2=1/305 and 3=1/242 as

    |f(t,u1,v1,w1)f(t,u2,v2,w2)|1219(|u1u2|+|v1v2|+|w1w2|),|g(t,u1,v1,w1)g(t,u2,v2,w2)|1305(|u1u2|+|v1v2|+|w1w2|),|h(t,u1,v1,w1)h(t,u2,v2,w2)|1242(|u1u2|+|v1v2|+|w1w2|).

    Using the values obtained in Example 3.1, we find that Θ11+Θ22+Θ330.483526<1, where Θ1,Θ2 and Θ3 are given by (3.6). Therefore, by Theorem 3.3, the system (3.22) equipped with the boundary conditions (3.21) has a unique solution on [1, 3].

    In this paper, we discussed the existence and uniqueness of solutions for a coupled system of nonlinear third order ordinary differential equations supplemented with nonlocal multi-point anti-periodic type boundary conditions on an arbitrary domain with the aid of modern fixed point theorems. Our results are new and enrich the literature on third-order boundary value problems. As a special case, our results correspond to the ones for an anti-periodic boundary value problem of nonlinear third order ordinary differential equations by fixing all αj=βl=γn=δe=ρq=σr=ξk=ζp=κd=0 in (1.2).

    We thank the reviewers for their useful remarks on our work.

    All authors declare no conflicts of interest in this paper.



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