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Review

New encoding concepts for shape recognition are needed

  • Received: 08 February 2018 Accepted: 26 February 2018 Published: 01 July 2018
  • Models designed to explain how shapes are perceived and stored by the nervous system commonly emphasize encoding of contour features, especially orientation, curvature, and linear extent. A number of experiments from my laboratory provide evidence that contours deliver a multitude of location markers, and shapes can be identified when relatively few of the markers are displayed. The emphasis on filtering for orientation and other contour features has directed attention away from full and effective examination of how the location information is registered and used for summarizing shapes. Neural network (connectionist) models try to deal with location information by modifying linkage among neuronal populations through training trials. Connections that are initially diffuse and not useful in achieving recognition get eliminated or changed in strength, resulting in selective response to a given shape. But results from my laboratory, reviewed here, demonstrate that unknown shapes that are displayed only once can be identified using a matching task. These findings show that our visual system can immediately encode shape information with no requirement for training trials. This encoding might be accomplished by neuronal circuits in the retina.

    Citation: Ernest Greene. New encoding concepts for shape recognition are needed[J]. AIMS Neuroscience, 2018, 5(3): 162-178. doi: 10.3934/Neuroscience.2018.3.162

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  • Models designed to explain how shapes are perceived and stored by the nervous system commonly emphasize encoding of contour features, especially orientation, curvature, and linear extent. A number of experiments from my laboratory provide evidence that contours deliver a multitude of location markers, and shapes can be identified when relatively few of the markers are displayed. The emphasis on filtering for orientation and other contour features has directed attention away from full and effective examination of how the location information is registered and used for summarizing shapes. Neural network (connectionist) models try to deal with location information by modifying linkage among neuronal populations through training trials. Connections that are initially diffuse and not useful in achieving recognition get eliminated or changed in strength, resulting in selective response to a given shape. But results from my laboratory, reviewed here, demonstrate that unknown shapes that are displayed only once can be identified using a matching task. These findings show that our visual system can immediately encode shape information with no requirement for training trials. This encoding might be accomplished by neuronal circuits in the retina.


    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, \; \text{and}\; h:[a, b]\times\mathbb{R}^3\rightarrow\mathbb{R} $ are given continuous functions, $ a < \eta_1 < \eta_2 < \dots < \eta_m < b, \; \text{ and}\; \alpha_j, \; \beta_l, \; \gamma_n, \; \delta_e, \; \rho_q, \; \sigma_r, \; \xi_k, \; \zeta_p \; \text{ and}\; \kappa_d \; \in \mathbb{R}^+ \; (j, l, n, e, q, r, k, p\; \text{ and}\; d = 1, 2, \dots, 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 $ f_1, g_1, h_1 \in C[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 $ c_i \in \mathbb{R}, \; i = 1, \dots, 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 $ c_2, c_5 $ and $ c_8, $ together with the notations $ S_1, S_2 $ and $ S_3 $ 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 $ c_2, c_5 $ and $ c_8 $ 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 $ c_1, c_4 $ and $ c_7 $ 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 $ c_1, \; c_2, \; c_4, \; c_5, \; c_7 $ and $ c_8 $ 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 $ c_0, c_3 $ and $ c_6 $ 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 $ c_i \; (i = 1, \dots, 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 $ {\mathcal{X}} = \{u(t)|u(t) \in C([a, b])\} $ equipped with norm $ \|u\| = \sup \{|u(t)|, t $ $ \in [a, b]\}. $ Obviously $ (\mathcal{X}, \|.\|) $ is a Banach space and consequently, the product space $ (\mathcal{X}\times\mathcal{X}\times \mathcal{X}, \|(u, v, w)\|) $ is a Banach space with norm $ \|(u, v, w)\| = \|u\|+\|v\|+\|w\| $ for $ (u, v, w)\in \mathcal{X}^{3} $. 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 $ \mathcal{H}:\mathcal{X}^{3}\rightarrow \mathcal{X}^{3} $ 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)
    $ \widehat{f}(s) = f(s, u(s), v(s), w(s)), \; \widehat{g}(s) = g(s, u(s), v(s), w(s)), \; \widehat{h}(s) = h(s, u(s), v(s), w(s)). $

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

    ($N_1$) (Linear growth conditions) There exist real constants $ m_i, \bar{m}_i, \widehat{m}_i\geq 0, \; (i = 1, 2, 3) $ and $ m_0 > 0, \; \bar{m}_0 > 0, \; \widehat{m}_0 > 0 $ such that $ \forall \; u, v, w\in \mathbb{R}, $ we have

    $ |f(t, u, v, w)|\leq m_0 + m_1 |u|+ m_2|v|+ m_3|w|, $
    $ |g(t, u, v, w)|\leq \bar{m}_0+ \bar{m}_1 |u|+ \bar{m}_2|v|+ \bar{m}_3|w|, $
    $ |h(t, u, v, w)|\leq \widehat{m}_0+ \widehat{m}_1 |u|+ \widehat{m}_2|v|+ \widehat{m}_3|w|. $

    ($N_2$) (Sub-growth conditions) There exist nonnegative functions $ \phi(t), \; \psi(t) $ and $ \chi(t) \in L(a, b) $ and $ \epsilon_i > 0, \; 0 < \lambda_i < 1, (i = 1, \dots, 9) $ such that $ \forall \; u, v, w\in \mathbb{R}, $ we have

    $ |f(t, u, v, w)|\leq \phi(t) + \epsilon_1 |u|^{\lambda_1}+ \epsilon_2|v|^{\lambda_2}+ \epsilon_3|w|^{\lambda_3}, $
    $ |g(t, u, v, w)|\leq \psi(t) + \epsilon_4 |u|^{\lambda_4}+ \epsilon_5|v|^{\lambda_5}+ \epsilon_6 |w|^{\lambda_6}, $
    $ |h(t, u, v, w)|\leq \chi(t) + \epsilon_7 |u|^{\lambda_7}+ \epsilon_8|v|^{\lambda_8}+ \epsilon_9 |w|^{\lambda_9}. $

    ($N_3$) (Lipschitz conditions) For all $ t\in[a, b] $ and $ u_i, v_i, w_i\in \mathbb{R}, \; i = 1, 2 $ there exist $ \ell_i > 0 \; (i = 1, 2, 3) $ such that

    $ |f(t, u_1, v_1, w_1)-f(t, u_2, v_2, w_2)|\leq \ell_1 \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big), $
    $ |g(t, u_1, v_1, w_1)-g(t, u_2, v_2, w_2)|\leq \ell_2 \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big), $
    $ |h(t, u_1, v_1, w_1)-h(t, u_2, v_2, w_2)|\leq \ell_3 \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big). $

    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)

    $ Q_i = \max_{t\in[a, b]}|G_i(t)|, $ and $ \Upsilon_i = \max_{t\in[a, b]}|P_i(t)|, \; (i = 1, \dots, 9). $ Also, we set

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

    where $ m_i, \bar{m}_i, \widehat{m}_i $ are given in $ (N_1) $.

    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 $ \mathcal{Y} $ be a Banach space, and $ T:\mathcal{Y} \to \mathcal{Y} $ be a completely continuous operator (i.e., a map restricted to any bounded set in $ \mathcal{Y} $ is compact). Let $ \Xi(T) = \{x\in \mathcal{Y}: x = \varphi T(x)\; \mathit{\text{for some}}\; 0 < \varphi < 1\} $. Then either the set $ \Xi(T) $ is unbounded, or $ T $ has at least one fixed point.

    Theorem 3.1. Assume that the condition $ (N_1) $ holds and that

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

    where $ \Theta_1, \; \Theta_2 $ and $ \Theta_3 $ 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 $ \mathcal{H}:\mathcal{X}^{3}\rightarrow \mathcal{X}^{3} $ defined by $ (3.2) $ is completely continuous. Notice that $ \mathcal{H}_1, \; \mathcal{H}_2 $ and $ \mathcal{H}_3 $ are continuous in view of continuity of the functions $ f, \; g $ and $ h $. So the operator $ \mathcal{H} $ is continuous. Let $ \Phi \subset \mathcal{X}^{3} $ be a bounded set. Then there exist positive constants $ \varrho_f, \; \varrho_g $ and $ \varrho_h $ such that $ |\widehat{f}(t)| = |f(t, u(t), v(t), w(t))|\leq \varrho_f, \; |\widehat{g}(t)| = |g(t, u(t), v(t), w(t))|\leq \varrho_g $ and $ |\widehat{h}(t)| = |h(t, u(t), v(t), w(t))|\leq \varrho_h, \; \; \forall(u, v, w)\in \Phi. $ Then, for any $ (u, v, w)\in \Phi, $ 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

    $ \|\mathcal{H}_1(u, v, w)\|\le \varrho_f \Delta_1 +\varrho_g \Delta_2 + \varrho_h \Delta_3, $

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

    $ \|\mathcal{H}_2(u, v, w)\|\le \varrho_f \Delta_4 +\varrho_g \Delta_5 + \varrho_h \Delta_6 , $

    and

    $ \|\mathcal{H}_3(u, v, w)\|\le \varrho_f \Delta_7 +\varrho_g \Delta_8 + \varrho_h \Delta_9 , $

    where $ \Delta_i\; (i = 4, \dots, 9) $ are given by $ (3.10)-(3.15). $ In consequence, we get

    $ \|\mathcal{H}(u, v, w)\|\le \varrho_f \Theta_1 +\varrho_g \Theta_2 + \varrho_h \Theta_3 , $

    where $ \Theta_1, $ $ \Theta_2 $ and $ \Theta_3 $ are given by $ (3.6). $ From the foregoing arguments, it follows that the operator $ \mathcal{H} $ is uniformly bounded. Next, we prove that $ \mathcal{H} $ is equicontinuous. For $ a < t < \tau < b, $ and $ (u, v, w)\in \Phi, $ 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 $ {\mathcal{H}} $ is completely continuous. Finally, it will be verified that the set $ \Xi = \{(u, v, w)\in\mathcal{X}^{3}|(u, v, w) = \varphi \mathcal{H}(u, v, w), 0 < \varphi < 1\} $ is bounded. Let $ (u, v, w)\in \Xi. $ Then $ (u, v, w) = \varphi\mathcal{H}(u, v, w) $ and for any $ t\in[a, b], $ we have

    $ u(t) = \varphi\mathcal{H}_1(u, v, w)(t), \; \; v(t) = \varphi\mathcal{H}_2(u, v, w)(t), \; \; w(t) = \varphi\mathcal{H}_3(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 $ \Theta $ given by $ (3.16), $ we find that

    $ \|(u, v, w)\|\leq\frac{\Theta_1 m_0 +\Theta_2 \bar{m}_0 + \Theta_3 \widehat{m}_0}{\Theta}, $

    which shows that the set $ \Xi $ is bounded. Hence, by Lemma 2, the operator $ \mathcal{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 $ (N_2) $ 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 $ (N_2) $ holds. Then there exists at least one solution for the problems $ (1.1) $ and $ (1.2) $ on $ [a, b]. $

    Proof. Define a set $ \Gamma $ in the Banach space $ \mathcal{X}^{3} $ as follows $ \Gamma = \{(u, v, w) \in \mathcal{X}^{3}: \|(u, v, w)\|\le x\}, $ where

    $ x $

    Firstly, we prove that $ \mathcal{H}:\Gamma \rightarrow \Gamma. $ For that, we consider

    $ \begin{eqnarray*} &&|\mathcal{H}_1(u, v, w)(t)| \\ & = & \Big|\int_a^t \frac{(t-s)^2}{2}\widehat{f}(s)ds +\frac{1}{\Lambda}\Big\{-\int_a^b \Big[2 \Lambda_1 (b-s)^2 +G_1(t) (b-s)+ P_1(t) \Big]\widehat{f}(s)ds \\ && -\int_a^b \Big[ \Lambda_1 \sum\limits_{j = 1}^m \alpha_j (b-s)^2 +G_2(t) (b-s)+ P_2(t) \Big] \widehat{g}(s)ds \\ && -\int_a^b \Big[ \Lambda_1 S_{11} \frac{(b-s)^2}{2} +G_3(t) (b-s)+ P_3(t) \Big] \widehat{h}(s)ds \\ && +P_3(t)\Big(\sum\limits_{d = 1}^m \kappa_d \int_a^{\eta_d} \widehat{f}(s)ds \Big)+P_1(t)\Big(\sum\limits_{n = 1}^m \gamma_n \int_a^{\eta_n} \widehat{g}(s)ds \Big)\\ && +P_2(t)\Big(\sum\limits_{r = 1}^m \sigma_r \int_a^{\eta_r} \widehat{h}(s)ds \Big)+G_3(t)\Big(\sum\limits_{p = 1}^m \zeta_p \int_a^{\eta_p} (\eta_p-s)\widehat{f}(s)ds \Big) \\ && +G_1(t)\Big(\sum\limits_{l = 1}^m \beta_l \int_a^{\eta_l} (\eta_l-s)\widehat{g}(s)ds \Big)+G_2(t)\Big(\sum\limits_{q = 1}^m \rho_q \int_a^{\eta_q} (\eta_q-s)\widehat{h}(s)ds \Big)\\ && +\Lambda_1 S_{11}\Big(\sum\limits_{k = 1}^m \xi_k \int_a^{\eta_k} \frac{(\eta_k-s)^2}{2}\widehat{f}(s)ds \Big)+2\Lambda_1 \Big(\sum\limits_{j = 1}^m \alpha_j \int_a^{\eta_j} (\eta_j-s)^2 \widehat{g}(s)ds \Big)\\ && +\Lambda_1 \sum\limits_{j = 1}^m \alpha_j \Big(\sum\limits_{e = 1}^m \delta_e \int_a^{\eta_e} (\eta_e-s)^2 \widehat{h}(s)ds \Big)\Big\} \Big| \\ & \le &\Big(\phi(t) + \epsilon_1 |u|^{\lambda_1}+ \epsilon_2|v|^{\lambda_2}+ \epsilon_3|w|^{\lambda_3}\Big) \Delta_1 +\Big(\psi(t) + \epsilon_4 |u|^{\lambda_4}+ \epsilon_5 |v|^{\lambda_5}+ \epsilon_6 |w|^{\lambda_6}\Big) \Delta_2 \\ && + \Big(\chi(t) + \epsilon_7 |u|^{\lambda_7} + \epsilon_8 |v|^{\lambda_8} + \epsilon_9|w|^{\lambda_9}\Big) \Delta_3, \end{eqnarray*} $

    which, on taking the norm

    $ \begin{eqnarray*} || \mathcal{H}_1(u, v, w)|| &\le& \Big(\phi + \epsilon_1 |u|^{\lambda_1}+ \epsilon_2|v|^{\lambda_2}+ \epsilon_3|w|^{\lambda_3}\Big) \Delta_1 \\ && +\Big(\psi + \epsilon_4 |u|^{\lambda_4}+ \epsilon_5|v|^{\lambda_5}+ \epsilon_6|w|^{\lambda_6}\Big) \Delta_2\\ && + \Big(\chi + \epsilon_7 |u|^{\lambda_7}+ \epsilon_8|v|^{\lambda_8}+ \epsilon_9|w|^{\lambda_9}\Big) \Delta_3, \end{eqnarray*} $

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

    $ \begin{eqnarray*} || \mathcal{H}_2(u, v, w)|| &\le& \Big(\phi + \epsilon_1 |u|^{\lambda_1}+ \epsilon_2|v|^{\lambda_2}+ \epsilon_3|w|^{\lambda_3}\Big) \Delta_4 \\ && +\Big(\psi + \epsilon_4 |u|^{\lambda_4}+ \epsilon_5|v|^{\lambda_5}+ \epsilon_6|w|^{\lambda_6}\Big) \Delta_5 \\ &&+ \Big(\chi + \epsilon_7 |u|^{\lambda_7}+ \epsilon_8|v|^{\lambda_8}+ \epsilon_9|w|^{\lambda_9}\Big) \Delta_6, \end{eqnarray*} $

    and

    $ \begin{eqnarray*} || \mathcal{H}_3(u, v, w)|| &\le& \Big(\phi + \epsilon_1 |u|^{\lambda_1}+ \epsilon_2|v|^{\lambda_2}+ \epsilon_3|w|^{\lambda_3}\Big) \Delta_7 \\ && +\Big(\psi + \epsilon_4 |u|^{\lambda_4}+ \epsilon_5|v|^{\lambda_5}+ \epsilon_6|w|^{\lambda_6}\Big) \Delta_8 \\ &&+ \Big(\chi + \epsilon_7 |u|^{\lambda_7}+ \epsilon_8|v|^{\lambda_8}+ \epsilon_9|w|^{\lambda_9}\Big) \Delta_9, \end{eqnarray*} $

    where $ \Delta_i\; (i = 4, \dots, 9) $ are given by $ (3.10)- (3.15). $ Consequently,

    $ \begin{eqnarray*} || \mathcal{H}(u, v, w)|| &\le& \Big(\phi + \epsilon_1 |u|^{\lambda_1}+ \epsilon_2|v|^{\lambda_2}+ \epsilon_3|w|^{\lambda_3}\Big) \Theta_1 \\ && +\Big(\psi + \epsilon_4 |u|^{\lambda_4}+ \epsilon_5|v|^{\lambda_5}+ \epsilon_6|w|^{\lambda_6}\Big) \Theta_2\\ && + \Big(\chi + \epsilon_7 |u|^{\lambda_7}+ \epsilon_8|v|^{\lambda_8}+ \epsilon_9|w|^{\lambda_9}\Big) \Theta_3 \le x, \end{eqnarray*} $

    where $ \Theta_1, \; \Theta_2 $ and $ \Theta_3 $ are given by $ (3.6). $ Therefore, we conclude that $ \mathcal{H}:\Gamma \rightarrow \Gamma, $ where $ \mathcal{H}_1(u, v, w)(t), \; \mathcal{H}_2(u, v, w)(t)\; \text {and} \; \mathcal{H}_3(u, v, w)(t) $ are continuous on $ [a, b]. $

    As in Theorem $ 3.1, $ one can show that the operator $ \mathcal{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 $ (N_3) $ holds. In addition, we suppose that

    $ \begin{equation} \Theta_1 \ell_1 + \Theta_2 \ell_2 +\Theta_3 \ell_3 \lt 1, \end{equation} $ (3.18)

    where $ \Theta_1, \Theta_2 $ and $ \Theta_3 $ are given by $ (3.6). $ Then the problems $ (1.1) $ and $ (1.2) $ have a unique solution on $ [a, b]. $

    Proof. Let us set $ \sup_{t \in [a, b]}|f(t, 0, 0, 0)| = M_1, \; \sup_{t \in [a, b]}|g(t, 0, 0, 0)| = M_2, $ $ \sup_{t \in [a, b]}|h(t, 0, 0, $ $ 0)| = M_3, $ and show that $ \mathcal{H} B_\varsigma \subset B_\varsigma, $ where $ B_\varsigma = \{(u, v, w) \in \mathcal{X}^{3} : \|(u, v, w)\|\le \varsigma \} $ with

    $ \varsigma \ge \frac{\Theta_1 M_1 + \Theta_2 M_2 + \Theta_3 M_3}{1- (\Theta_1 \ell_1 + \Theta_2 \ell_2 +\Theta_3 \ell_3)}. $

    For any $ (u, v, w)\in B_\varsigma, \; \; t\in[a, b] $, we find that

    $ \begin{eqnarray*} |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)|\\ &\le& |f(s, u(s), v(s), w(s))-f(s, 0, 0, 0)|+|f(s, 0, 0, 0)|\\ &\le& \ell_1 (\|u\|+\|v\|+\|w\|)+M_1 \le \ell_1 \|(u, v, w)\|+M_1 \le \ell_1 \varsigma +M_1. \end{eqnarray*} $

    In a similar manner, we have

    $ |g(s, u(s), v(s), w(s))| \le \ell_2 \varsigma +M_2, \; \; |h(s, u(s), v(s), w(s)| \le \ell_3 \varsigma +M_3. $

    Then, for $ (u, v, w) \in B_\varsigma, $ we obtain

    $ \begin{eqnarray*} &&|\mathcal{H}_1(u, v, w)(t)|\\ & = & \Big|\int_a^t \frac{(t-s)^2}{2}\widehat{f}(s)ds +\frac{1}{\Lambda}\Big\{-\int_a^b \Big[2 \Lambda_1 (b-s)^2 +G_1(t) (b-s)+ P_1(t) \Big]\widehat{f}(s)ds \\ && -\int_a^b \Big[ \Lambda_1 \sum\limits_{j = 1}^m \alpha_j (b-s)^2 +G_2(t) (b-s)+ P_2(t) \Big] \widehat{g}(s)ds \\ && -\int_a^b \Big[ \Lambda_1 S_{11} \frac{(b-s)^2}{2} +G_3(t) (b-s)+ P_3(t) \Big] \widehat{h}(s)ds \\ && +P_3(t)\Big(\sum\limits_{d = 1}^m \kappa_d \int_a^{\eta_d} \widehat{f}(s)ds \Big)+P_1(t)\Big(\sum\limits_{n = 1}^m \gamma_n \int_a^{\eta_n} \widehat{g}(s)ds \Big)\\ && +P_2(t)\Big(\sum\limits_{r = 1}^m \sigma_r \int_a^{\eta_r} \widehat{h}(s)ds \Big)+G_3(t)\Big(\sum\limits_{p = 1}^m \zeta_p \int_a^{\eta_p} (\eta_p-s)\widehat{f}(s)ds \Big) \\ && +G_1(t)\Big(\sum\limits_{l = 1}^m \beta_l \int_a^{\eta_l} (\eta_l-s)\widehat{g}(s)ds \Big)+G_2(t)\Big(\sum\limits_{q = 1}^m \rho_q \int_a^{\eta_q} (\eta_q-s)\widehat{h}(s)ds \Big)\\ && +\Lambda_1 S_{11}\Big(\sum\limits_{k = 1}^m \xi_k \int_a^{\eta_k} \frac{(\eta_k-s)^2}{2}\widehat{f}(s)ds \Big)+2\Lambda_1 \Big(\sum\limits_{j = 1}^m \alpha_j \int_a^{\eta_j} (\eta_j-s)^2 \widehat{g}(s)ds \Big)\\ && +\Lambda_1 \sum\limits_{j = 1}^m \alpha_j \Big(\sum\limits_{e = 1}^m \delta_e \int_a^{\eta_e} (\eta_e-s)^2 \widehat{h}(s)ds \Big)\Big\} \Big| \\ & \le& (\ell_1 \varsigma +M_1) \Big\{\frac{(b-a)^{3}}{6}+\frac{1}{3|8-B_3|} \Big[ 2(b-a)^3 +S_{11} \Big(\sum\limits_{k = 1}^m \xi_k \frac{(\eta_k-a)^3}{2}\Big) \Big]\\ && + \frac{1}{|\Lambda|} \Big[Q_1 \frac{(b-a)^{2}}{2} + \Upsilon_1 (b-a) +\Upsilon_3 \Big(\sum\limits_{d = 1}^m \kappa_d (\eta_d-a)\Big) \\ && +Q_3 \Big(\sum\limits_{p = 1}^m \zeta_p \frac{(\eta_p-a)^2}{2}\Big) \Big]\Big\}+ (\ell_2 \varsigma +M_2) \Big\{\frac{\sum_{j = 1}^m \alpha_j}{3|8-B_3|} \Big[(b-a)^3 +2 (\eta_j-a)^3 \Big]\\ && + \frac{1}{|\Lambda|} \Big[Q_2 \frac{(b-a)^{2}}{2} + \Upsilon_2 (b-a) +\Upsilon_1 \Big(\sum\limits_{n = 1}^m \gamma_n (\eta_n-a)\Big) \\ && +Q_1 \Big(\sum\limits_{l = 1}^m \beta_l \frac{(\eta_l-a)^2}{2}\Big) \Big]\Big\} +(\ell_3 \varsigma +M_3) \Big\{\frac{1}{3|8-B_3|} \Big[S_{11} \frac{(b-a)^3}{2}\\ && +\Big(\sum\limits_{j = 1}^m \alpha_j \Big)\Big(\sum\limits_{e = 1}^m \delta_e (\eta_e-a)^3\Big) \Big] + \frac{1}{|\Lambda|} \Big[Q_3 \frac{(b-a)^{2}}{2}+ \Upsilon_3 (b-a) \\ &&+\Upsilon_2 \Big(\sum\limits_{r = 1}^m \sigma_r (\eta_r-a)\Big) +Q_2 \Big(\sum\limits_{q = 1}^m \rho_q \frac{(\eta_{q}-a)^2}{2}\Big) \Big]\Big\} \\ & \le& (\ell_1 \varsigma +M_1) \Delta_1 +(\ell_2 \varsigma +M_2) \Delta_2 + (\ell_3 \varsigma +M_3) \Delta_3, \end{eqnarray*} $

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

    $ \|\mathcal{H}_1(u, v, w)\|\le (\ell_1 \varsigma +M_1) \Delta_1 +(\ell_2 \varsigma +M_2) \Delta_2 + (\ell_3 \varsigma +M_3) \Delta_3. $

    Similarly, we can find that

    $ \|\mathcal{H}_2(u, v, w)\|\le (\ell_1 \varsigma +M_1) \Delta_4 +(\ell_2 \varsigma +M_2) \Delta_5 + (\ell_3 \varsigma +M_3) \Delta_6, $

    and

    $ \|\mathcal{H}_3(u, v, w)\|\le (\ell_1 \varsigma +M_1) \Delta_7 +(\ell_2 \varsigma +M_2) \Delta_8 + (\ell_3 \varsigma +M_3) \Delta_9, $

    where $ \Delta_i\; (i = 1, \dots, 9) $ are defined in $ (3.7)-(3.15). $ In consequence, it follows that

    $ \|\mathcal{H}(u, v, w)\|\le (\ell_1 \varsigma +M_1) \Theta_1 +(\ell_2 \varsigma +M_2) \Theta_2 + (\ell_3 \varsigma +M_3) \Theta_3\le \varsigma . $

    Next we show that the operator $ \mathcal{H} $ is a contraction. For $ (u_1, v_1, w_1), \; (u_2, v_2, w_2) \in \mathcal{X}^{3}, $ we have

    $ \begin{eqnarray*} &&\big|\mathcal{H}_1(u_1, v_1, w_1)(t) - \mathcal{H}_1(u_2, v_2, w_2)(t)\big|\\ & \le& \int_a^t \frac{(t-s)^{2}}{2} \Big|f(s, u_1(s), v_1(s), w_1(s)) - f(s, u_2(s), v_2(s), w_2(s))\Big|ds \\ && +\frac{1}{|\Lambda|}\Big\{\int_a^b \Big[2 |\Lambda_1| (b-s)^2 +|G_1(t)| (b-s)+ |P_1(t)| \Big]\\ && \times \Big|f(s, u_1(s), v_1(s), w_1(s)) - f(s, u_2(s), v_2(s), w_2(s))\Big|ds \\ && +\int_a^b \Big[ |\Lambda_1| \sum\limits_{j = 1}^m \alpha_j (b-s)^2 +|G_2(t)| (b-s)+ |P_2(t)| \Big] \\ && \times \Big|g(s, u_1(s), v_1(s), w_1(s)) - g(s, u_2(s), v_2(s), w_2(s))\Big|ds \\ && +\int_a^b \Big[ |\Lambda_1| S_{11} \frac{(b-s)^2}{2} +|G_3(t)| (b-s)+ |P_3(t)| \Big]\\ && \times \Big|h(s, u_1(s), v_1(s), w_1(s)) - h(s, u_2(s), v_2(s), w_2(s))\Big|ds \\ && +|P_3(t)|\Big(\sum\limits_{d = 1}^m \kappa_d \int_a^{\eta_d} \Big|f(s, u_1(s), v_1(s), w_1(s)) - f(s, u_2(s), v_2(s), w_2(s))\Big|ds \Big)\\ && +|P_1(t)|\Big(\sum\limits_{n = 1}^m \gamma_n \int_a^{\eta_n}\Big|g(s, u_1(s), v_1(s), w_1(s)) - g(s, u_2(s), v_2(s), w_2(s))\Big|ds \Big)\\ && +|P_2(t)|\Big(\sum\limits_{r = 1}^m \sigma_r \int_a^{\eta_r} \Big|h(s, u_1(s), v_1(s), w_1(s)) - h(s, u_2(s), v_2(s), w_2(s))\Big|ds \Big)\\ && +|G_3(t)|\Big(\sum\limits_{p = 1}^m \zeta_p \int_a^{\eta_p} (\eta_p-s)\Big|f(s, u_1(s), v_1(s), w_1(s)) - f(s, u_2(s), v_2(s), w_2(s))\Big|ds \Big)\\ && +|G_1(t)|\Big(\sum\limits_{l = 1}^m \beta_l \int_a^{\eta_l} (\eta_l-s)\Big|g(s, u_1(s), v_1(s), w_1(s)) - g(s, u_2(s), v_2(s), w_2(s))\Big|ds \Big)\\ && +|G_2(t)|\Big(\sum\limits_{q = 1}^m \rho_q \int_a^{\eta_q} (\eta_q-s)\Big|h(s, u_1(s), v_1(s), w_1(s)) - h(s, u_2(s), v_2(s), w_2(s))\Big|ds \Big)\\ && +|\Lambda_1| S_{11} \Big(\sum\limits_{k = 1}^m \xi_k \int_a^{\eta_k} \frac{(\eta_k-s)^2}{2}\Big|f(s, u_1(s), v_1(s), w_1(s)) - f(s, u_2(s), v_2(s), w_2(s))\Big|ds \Big)\\ && +2|\Lambda_1| \Big(\sum\limits_{j = 1}^m \alpha_j \int_a^{\eta_j} (\eta_j-s)^2 \Big|g(s, u_1(s), v_1(s), w_1(s)) - g(s, u_2(s), v_2(s), w_2(s))\Big|ds \Big)\\ && +|\Lambda_1| \sum\limits_{j = 1}^m \alpha_j \Big(\sum\limits_{e = 1}^m \delta_e \int_a^{\eta_e} (\eta_e-s)^2 \Big|h(s, u_1(s), v_1(s), w_1(s)) - h(s, u_2(s), v_2(s), w_2(s))\Big|ds \Big)\Big\} \\ & \le& \ell_1 \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big) \Big\{\frac{(b-a)^{3}}{6}+\frac{1}{3|8-B_3|} \Big[ 2(b-a)^3 \\ && +S_{11} \Big(\sum\limits_{k = 1}^m \xi_k \frac{(\eta_k-a)^3}{2}\Big) \Big]+ \frac{1}{|\Lambda|} \Big[Q_1 \frac{(b-a)^{2}}{2} + \Upsilon_1 (b-a) +\Upsilon_3 \Big(\sum\limits_{d = 1}^m \kappa_d (\eta_d-a)\Big) \\ && +Q_3 \Big(\sum\limits_{p = 1}^m \zeta_p \frac{(\eta_p-a)^2}{2}\Big) \Big]\Big\}+\ell_2 \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big) \Big\{\frac{\sum_{j = 1}^m \alpha_j}{3|8-B_3|} \Big[(b-a)^3 \\ && +2 (\eta_j-a)^3 \Big]+ \frac{1}{|\Lambda|} \Big[Q_2 \frac{(b-a)^{2}}{2} + \Upsilon_2 (b-a) +\Upsilon_1 \Big(\sum\limits_{n = 1}^m \gamma_n (\eta_n-a)\Big)\\ && +Q_1 \Big(\sum\limits_{l = 1}^m \beta_l \frac{(\eta_l-a)^2}{2}\Big) \Big]\Big\} \\ && +\ell_3 \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big) \Big\{\frac{1}{3|8-B_3|} \Big[S_{11} \frac{(b-a)^3}{2} \\ && +\Big(\sum\limits_{j = 1}^m \alpha_j \Big)\Big(\sum\limits_{e = 1}^m \delta_e (\eta_e-a)^3\Big) \Big]+ \frac{1}{|\Lambda|} \Big[Q_3 \frac{(b-a)^{2}}{2} + \Upsilon_3 (b-a) +\Upsilon_2 \Big(\sum\limits_{r = 1}^m \sigma_r (\eta_r-a)\Big) \\ && +Q_2 \Big(\sum\limits_{q = 1}^m \rho_q \frac{(\eta_q-a)^2}{2}\Big) \Big]\Big\}\\ & \le& (\ell_1 \Delta_1 + \ell_2 \Delta_2+ \ell_3 \Delta_3) \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big), \end{eqnarray*} $

    which implies that

    $ \big \|\mathcal{H}_1(u_1, v_1, w_1) - \mathcal{H}_1(u_2, v_2, w_2)\big \|\le (\ell_1 \Delta_1 + \ell_2 \Delta_2+ \ell_3 \Delta_3) \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big), $

    where $ \Delta_1\; \Delta_2 $ and $ \Delta_3 $ are given by $ (3.7), (3.8) $ and $ (3.9) $ respectively. In a similar fashion, one can find that

    $ \big \|\mathcal{H}_2(u_1, v_1, w_1)- \mathcal{H}_2(u_2, v_2, w_2)\big \|\le (\ell_1 \Delta_4 + \ell_2 \Delta_5+ \ell_3 \Delta_6) \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big), $

    and

    $ \big \|\mathcal{H}_3(u_1, v_1, w_1) - \mathcal{H}_3(u_2, v_2, w_2)\big \|\le (\ell_1 \Delta_7 + \ell_2 \Delta_8+ \ell_3 \Delta_9) \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big), $

    where $ \Delta_i, \; (i = 4, \dots, 9) $ are given by $ (3.10)- (3.15). $ Thus we have

    $ \begin{equation} \|\mathcal{H}(u_1, v_1, w_1) - \mathcal{H}(u_2, v_2, w_2)\big \|\le (\Theta_1 \ell_1+ \Theta_2 \ell_2+ \Theta_3 \ell_3)\big(\|u_1 - u_2\|+\|v_1 - v_2\|+\|w_1 - w_2\|\big), \end{equation} $ (3.19)

    where $ \Theta_1, \; \Theta_2 $ and $ \Theta_3 $ are given by $ (3.6). $ By the assumption $ (3.18) $ it follows from $ (3.19) $ that the operator $ \mathcal{H} $ is a contraction. Thus, by Banach contraction mapping principle, we deduce that the operator $ \mathcal{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

    $ \begin{equation} \begin{array}{ll} u'''(t) = \frac{5}{31 \sqrt{t^{3}+24}} + \frac{|u(t)|^2}{204 (1+|u(t)|)}+ \frac{3}{342} \sin{v(t)}+\frac{1}{t^2 +97}w(t), \; t\in [1, 3], \\[0.5cm] v'''(t) = \frac{e^{-(t-1)}}{12(15+t)}+\frac{1}{798 \pi} \sin(7\pi u)+ \frac{|v(t)|^3}{96 (1+|v(t)|^2)}+\frac{4}{(t +7)^3}w(t), \; \; t\in [1, 3], \\[0.5cm] w'''(t) = \frac{1}{2(4+t)^2} \cos{t}+\frac{2}{6 \sqrt{4356 t}} u(t)+\frac{w(t)|v(t)|}{810 (1+|v(t)|)}, \; t\in [1, 3], \end{array} \end{equation} $ (3.20)

    supplemented to the following boundary conditions

    $ \begin{equation} \begin{array}{ll} u(1)+u(3) = \sum\limits_{j = 1}^4 \alpha_j v(\eta_j), \; \; u'(1)+u'(3) = \sum\limits_{l = 1}^4 \beta_l v'(\eta_l), \; \; u''(1)+u''(3) = \sum\limits_{n = 1}^4 \gamma_n v''(\eta_n), \\ v(1)+v(3) = \sum\limits_{e = 1}^4 \delta_e w(\eta_e), \; \; v'(1)+v'(3) = \sum\limits_{q = 1}^4 \rho_q w'(\eta_q), \; \; v''(1)+v''(3) = \sum\limits_{r = 1}^4 \sigma_r w''(\eta_r), \\ w(1)+w(3) = \sum\limits_{k = 1}^4 \xi_k u(\eta_k), \; \; w'(1)+w'(3) = \sum\limits_{p = 1}^4 \zeta_p u'(\eta_p), \; \; w''(1)+w''(3) = \sum\limits_{d = 1}^4 \kappa_d u''(\eta_d), \end{array} \end{equation} $ (3.21)

    where

    $ a = 1, \, \, b = 3, \, m = 4, \, \, \eta_{1} = 4/3, \, \,\eta_{2} = 5/3, \, \, \eta_{3} = 2, \, \, \eta_{4} = 7/3, \, \, \alpha_{1} = 1/4, \, \, \alpha_{2} = 1/2, \, \, \alpha_{3} = 3/4, \, \, \alpha_{4} = 1, \, \, \beta_{1} = 0.2, \, \, \beta_{2} = 8/15, \, \, \beta_{3} = 13/15, \, \, \beta_{4} = 6/5, \, \, \gamma_{1} = 1/8, \, \, \gamma_{2} = 9/40, \, \, \gamma_{3} = 13/40, \, \, \gamma_{4} = 17/40, \, \, \delta_{1} = 2/11, \, \, \delta_{2} = 3/11, \, \, \delta_{3} = 4/11, \, \, \delta_{4} = 5/11, \, \, \rho_{1} = 1/6, \, \, \rho_{2} = 7/24, \, \, \rho_{3} = 5/12, \, \, \rho_{4} = 13/24, \, \, \sigma_{1} = 1/9, \, \, \sigma_{2} = 2/9, \, \, \sigma_{3} = 1/3, \, \, \sigma_{4} = 4/9, \, \, \xi_{1} = 1/7, \, \, \xi_{2} = 2/7, \, \, \xi_{3} = 3/7, \, \, \xi_{4} = 4/7, \, \, \zeta_{1} = 2/15, \, \, \zeta_{2} = 1/3, \, \, \zeta_{3} = 8/15, \, \, \zeta_{4} = 11/15, \, \, \kappa_{1} = 1/3, \, \, \kappa_{2} = 4/9, \, \, \kappa_{3} = 5/9, \, \, \kappa_{4} = 2/3. $

    By direct substitution, we get $ B_1\approx 2.444444\neq8, \; \; B_2\approx 6.875556\neq8, \; \; B_3\approx 4.545452 \neq8, $ and $ \Lambda\approx 21.580256 $ ($ \Lambda $ is given by $ (2.11) $). Also, $ \Delta_1\approx 21.294227, \; \; \Delta_2\approx 22.603176, \; \; \Delta_3\approx 11.800813, \; \; \Delta_4\approx 7.983258, \; \; \Delta_5\approx 12.996835, \; \; \Delta_6\approx 8.497948, \; \; \Delta_7\approx 10.977544, \; \; \Delta_8\approx 14.165941 $ and $ \Delta_9\approx 12.745457 $ ($ \Delta_i \; (i = 1, \dots, 9) $ are defined in $ (3.7)-(3.15) $). Furthermore we obtain $ \Theta_1\approx 40.255029, \; \; \Theta_2\approx 49.765952 $ and $ \Theta_3\approx 33.044218 \; (\Theta_1, \; \Theta_2 $ and $ \Theta_3 $ are given by $ (3.6) $). Evidently,

    $ \begin{eqnarray*} &&|f(t, u, v, w)|\le \frac{1}{31}+ \frac{1}{204} \|u\|+ \frac{1}{114}\|v\|+\frac{1}{98}\|w\|, \\ && |g(t, u, v, w)|\le \frac{1}{192}+ \frac{1}{114} \|u\|+\frac{1}{96}\|v\|+\frac{1}{128}\|w\|, \\ &&|h(t, u, v, w)|\le \frac{1}{50}+ \frac{1}{198} \|u\|+\frac{1}{810}\|w\|. \end{eqnarray*} $

    Clearly, $ m_0 = 1/31, \; m_1 = 1/204, \; m_2 = 1/114, \; m_3 = 1/98, \; \bar{m}_0 = 1/192, \; \bar{m}_1 = 1/114, \; \bar{m}_2 = 1/96, \; \bar{m}_3 = 1/128, $ and $ \widehat{m}_0 = 1/50, \; \widehat{m}_1 = 1/198, \; \widehat{m}_2 = 0, \; \widehat{m}_3 = 1/810. $ Using $ (3.17), $ we find that $ \Theta_1 m_1+ \Theta_2 \bar{m}_1+\Theta_3 \widehat{m}_1 \approx 0.800762 < 1, \; \; \Theta_1 m_2+ \Theta_2 \bar{m}_2+\Theta_3 \widehat{m}_2\approx 0.871509 < 1 $ and $ \Theta_1 m_3+ \Theta_2 \bar{m}_3+\Theta_3 \widehat{m}_3 \approx 0.840357 < 1. $ Also, from $ (3.16) $ we obtain $ \Theta = 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

    $ \begin{equation} \begin{array}{ll} u'''(t) = \frac{3}{9(t^3 +72)}\Big(\tan^{-1}(u(t)) + v(t)+\frac{|w|}{1+|w|}\Big) +e^{-(t-1)}, \; \; \; \; \; \; t\in [1, 3], \\[0.5cm] v'''(t) = \frac{1}{610 \pi} \sin(2\pi u)+ \frac{4}{2t+1218} \sin(v(t))+ \frac{7}{3}+ \frac{1}{305}w(t), \; t\in [1, 3], \\[0.5cm] w'''(t) = \frac{3}{22 \sqrt{999+90t}} \Big(u(t)+ \frac{|v(t)|}{1+|v(t)|}+\tan^{-1}(w(t))\Big)+\cos{(t-1)}, \; t\in [1, 3], \end{array} \end{equation} $ (3.22)

    subject to the coupled boundary conditions $ (3.21). $ It is easy to see that $ \ell_1 = 1/219, \ell_2 = 1/305 $ and $ \ell_3 = 1/242 $ as

    $ \begin{eqnarray*} &&|f(t, u_1, v_1, w_1)-f(t, u_2, v_2, w_2)|\leq \frac{1}{219} \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big), \\ &&|g(t, u_1, v_1, w_1)-g(t, u_2, v_2, w_2)|\leq \frac{1}{305} \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big), \\ &&|h(t, u_1, v_1, w_1)-h(t, u_2, v_2, w_2)|\leq \frac{1}{242} \big(|u_1-u_2|+|v_1-v_2|+|w_1-w_2|\big). \end{eqnarray*} $

    Using the values obtained in Example $ 3.1 $, we find that $ \Theta_1 \ell_1+ \Theta_2 \ell_2+ \Theta_3 \ell_3 \approx 0.483526 < 1, $ where $ \Theta_1, \; \Theta_2 $ and $ \Theta_3 $ 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 $ \alpha_j = \beta_l = \gamma_n = \delta_e = \rho_q = \sigma_r = \xi_k = \zeta_p = \kappa_d = 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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