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)=m∑j=1αjv(ηj),u′(a)+u′(b)=m∑l=1βlv′(ηl),u″(a)+u″(b)=m∑n=1γnv″(ηn),v(a)+v(b)=m∑e=1δew(ηe),v′(a)+v′(b)=m∑q=1ρqw′(ηq),v″(a)+v″(b)=m∑r=1σrw″(ηr),w(a)+w(b)=m∑k=1ξku(ηk),w′(a)+w′(b)=m∑p=1ζpu′(ηp),w″(a)+w″(b)=m∑d=1κdu″(ηd), | (1.2) |
where f,g,andh:[a,b]×R3→R are given continuous functions, a<η1<η2<⋯<ηm<b, andαj,βl,γn,δe,ρq,σr,ξk,ζp andκd∈R+(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,h1∈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(t−s)22f1(s)ds+1Λ{−∫ba[2Λ1(b−s)2+G1(t)(b−s)+P1(t)]f1(s)ds−∫ba[Λ1m∑j=1αj(b−s)2+G2(t)(b−s)+P2(t)]g1(s)ds−∫ba[Λ1S11(b−s)22+G3(t)(b−s)+P3(t)]h1(s)ds+P3(t)(m∑d=1κd∫ηdaf1(s)ds)+P1(t)(m∑n=1γn∫ηnag1(s)ds)+P2(t)(m∑r=1σr∫ηrah1(s)ds)+G3(t)(m∑p=1ζp∫ηpa(ηp−s)f1(s)ds)+G1(t)(m∑l=1βl∫ηla(ηl−s)g1(s)ds)+G2(t)(m∑q=1ρq∫ηqa(ηq−s)h1(s)ds)+Λ1S11(m∑k=1ξk∫ηka(ηk−s)22f1(s)ds)+2Λ1(m∑j=1αj∫ηja(ηj−s)2g1(s)ds)+Λ1m∑j=1αj(m∑e=1δe∫ηea(ηe−s)2h1(s)ds)}, | (2.2) |
v(t)=∫ta(t−s)22g1(s)ds+1Λ{−∫ba[Λ1S12(b−s)22+G4(t)(b−s)+P4(t)]f1(s)ds−∫ba[2Λ1(b−s)2+G5(t)(b−s)+P5(t)]g1(s)ds−∫ba[Λ1m∑e=1δe(b−s)2+G6(t)(b−s)+P6(t)]h1(s)ds+P6(t)(m∑d=1κd∫ηdaf1(s)ds)+P4(t)(m∑n=1γn∫ηnag1(s)ds)+P5(t)(m∑r=1σr∫ηrah1(s)ds)+G6(t)(m∑p=1ζp∫ηpa(ηp−s)f1(s)ds)+G4(t)(m∑l=1βl∫ηla(ηl−s)g1(s)ds)+G5(t)(m∑q=1ρq∫ηqa(ηq−s)h1(s)ds)+Λ1m∑e=1δe(m∑k=1ξk∫ηka(ηk−s)2f1(s)ds)+Λ1S12(m∑j=1αj∫ηja(ηj−s)22g1(s)ds)+2Λ1(m∑e=1δe∫ηea(ηe−s)2h1(s)ds)}, | (2.3) |
w(t)=∫ta(t−s)22h1(s)ds+1Λ{−∫ba[Λ1m∑k=1ξk(b−s)22+G7(t)(b−s)+P7(t)]f1(s)ds−∫ba[Λ1S13(b−s)22+G8(t)(b−s)+P8(t)]g1(s)ds−∫ba[2Λ1(b−s)2+G9(t)(b−s)+P9(t)]h1(s)ds+P9(t)(m∑d=1κd∫ηdaf1(s)ds)+P7(t)(m∑n=1γn∫ηnag1(s)ds)+P8(t)(m∑r=1σr∫ηrah1(s)ds)+G9(t)(m∑p=1ζp∫ηpa(ηp−s)f1(s)ds)+G7(t)(m∑l=1βl∫ηla(ηl−s)g1(s)ds)+G8(t)(m∑q=1ρq∫ηqa(ηq−s)h1(s)ds)+2Λ1(m∑k=1ξk∫ηka(ηk−s)2f1(s)ds)+Λ1m∑k=1ξk(m∑j=1αj∫ηja(ηj−s)2g1(s)ds)+Λ1S13(m∑e=1δe∫ηea(ηe−s)22h1(s)ds)}, | (2.4) |
where
G1(t)=(8−B1)(μ1+4Ω(t)),G2(t)=(8−B1)(μ2+2Ω(t)m∑l=1βl),G3(t)=(8−B1)(μ3+S6Ω(t)),G4(t)=(8−B1)(μ4+S8Ω(t)),G5(t)=(8−B1)(μ5+4Ω(t)),G6(t)=(8−B1)(μ6+2Ω(t)m∑q=1ρq),G7(t)=(8−B1)(μ7+2Ω(t)m∑p=1ζp),G8(t)=(8−B1)(μ8+S7Ω(t)),G9(t)=(8−B1)(μ9+4Ω(t)),P1(t)=L1+A1Ω(t)+2Λ2(t−a)2,P2(t)=L2+A2Ω(t)+Λ2(t−a)2m∑n=1γn,P3(t)=L3+A3Ω(t)+S1Λ2(t−a)22,P4(t)=L4+A7Ω(t)+S3Λ2(t−a)22,P5(t)=L5+A8Ω(t)+2Λ2(t−a)2,P6(t)=L6+A9Ω(t)+Λ2(t−a)2m∑r=1σr,P7(t)=L7+A4Ω(t)+Λ2(t−a)2m∑d=1κd,P8(t)=L8+A5Ω(t)+S2Λ2(t−a)22,P9(t)=L9+A6Ω(t)+2Λ2(t−a)2,Ω(t)=(8−B3)(t−a), | (2.5) |
S1=(m∑r=1σr)(m∑n=1γn),S2=(m∑n=1γn)(m∑d=1κd),S3=(m∑r=1σr)(m∑d=1κd),S4=(m∑l=1βl)(m∑d=1κd),S5=(m∑r=1σr)(m∑l=1βl),S6=(m∑l=1βl)(m∑q=1ρq),S7=(m∑l=1βl)(m∑p=1ζp),S8=(m∑p=1ζp)(m∑q=1ρq),S9=(m∑d=1κd)(m∑q=1ρq),S10=(m∑n=1γn)(m∑q=1ρq),S11=(m∑e=1δe)(m∑j=1αj),S12=(m∑k=1ξk)(m∑e=1δe),S13=(m∑k=1ξk)(m∑j=1αj),E1=m∑j=1αj(ηj−a),E2=m∑j=1αj(ηj−a)22,E3=m∑l=1βl(ηl−a),E4=m∑e=1δe(ηe−a),E5=m∑e=1δe(ηe−a)22,E6=m∑q=1ρq(ηq−a),E7=m∑k=1ξk(ηk−a),E8=m∑k=1ξk(ηk−a)22,E9=m∑p=1ζp(ηp−a), | (2.6) |
A1=−2(b−a)[8+S6(m∑d=1κd)+S3(m∑l=1βl)]+4S6E9+4S3E3+4S4E6,A2=−(b−a)[S2S6+8(m∑n=1γn)+8(m∑l=1βl)]+2S6E9(m∑n=1γn)+16E3+2S2E6(m∑l=1βl),A3=−4(b−a)[S6+S1+S5]+S1S6E9+8E3(m∑r=1σr)+8E6(m∑l=1βl),A4=−(b−a)[8(m∑d=1κd)+8(m∑p=1ζp)+S3S7]+16E9+2S3E3(m∑p=1ζp)+2S4E6(m∑p=1ζp),A5=−4(b−a)[S2+(m∑n=1γn)(m∑p=1ζp)+S7]+8E9(m∑n=1γn)+8E3(m∑p=1ζp)+S2S7E6,A6=−2(b−a)[8+S1(m∑p=1ζp)+S5(m∑p=1ζp)]+4S1E9+4E3(m∑r=1σr)(m∑p=1ζp)+4S7E6,A7=−4(b−a)[S8+S3+S9]+S3S8E3+8E6(m∑d=1κd)+8E9(m∑q=1ρq),A8=−2(b−a)[8+S8(m∑n=1γn)+S2(m∑q=1ρq)]+4S8E3+4S2E6+4S10E9,A9=−(b−a)[S1S8+8(m∑r=1σr)+8(m∑q=1ρq)]+2S8E3(m∑r=1σr)+16E6+2S1E9(m∑q=1ρq), | (2.7) |
J1=E1A7−A1(b−a)+(8−B2)(S3E2−2(b−a)2),J2=E1A8−A2(b−a)+(8−B2)(4E2−m∑n=1γn(b−a)2),J3=E1A9−A3(b−a)+(8−B2)(2E2m∑r=1σr−S1(b−a)22),J4=E4A4−A7(b−a)+(8−B2)(2E5m∑d=1κd−S3(b−a)22),J5=E4A5−A8(b−a)+(8−B2)(E5S2−2(b−a)2),J6=E4A6−A9(b−a)+(8−B2)(4E5−m∑r=1σr(b−a)2),J7=E7A1−A4(b−a)+(8−B2)(4E8−m∑d=1κd(b−a)2),J8=E7A2−A5(b−a)+(8−B2)(2E8m∑n=1γn−S2(b−a)22),J9=E7A3−A6(b−a)+(8−B2)(S1E8−2(b−a)2), | (2.8) |
μ1=4S8E1−(b−a)[16+2(m∑j=1αj)S8+2S11(m∑p=1ζp)]+4S11E7+4E4(m∑p=1ζp)(m∑j=1αj),μ2=16E1−(b−a)[8(m∑l=1βl)+8(m∑j=1αj)+S11S7]+2E4S7(m∑j=1αj)+2S11E7(m∑l=1βl),μ3=8E1(m∑q=1ρq)+8E4(m∑j=1αj)+S6S11E7,−4(b−a)[S6+S11+(m∑q=1ρq)(m∑j=1αj)], | (2.9) |
μ4=S8S12E1−4(b−a)[S12+S8+(m∑p=1ζp)(m∑e=1δe)]+8E7(m∑e=1δe)+8E4(m∑p=1ζp),μ5=4S12E1−2(b−a)[8+S12(m∑l=1βl)+S7(m∑e=1δe)]+4S7E4+4E7(m∑l=1βl)(m∑e=1δe),μ6=2(m∑q=1ρq)S12E1−(b−a)[S6S12+8(m∑q=1ρq)+8(m∑e=1δe)]+16E4+2(m∑e=1δe)S6E7,μ7=2(m∑p=1ζp)S13E4−(b−a)[S8S13+8(m∑k=1ξk)+8(m∑p=1ζp)]+2S8E1(m∑k=1ξk)+16E7,μ8=S7S13E4−4(b−a)[S13+(m∑l=1βl)(m∑k=1ξk)+S7]+8E1(m∑k=1ξk)+8E7(m∑l=1βl),μ9=4S13E4−2(b−a)[8+S13(m∑q=1ρq)+S6(m∑k=1ξk)]+4E1(m∑q=1ρq)(m∑k=1ξk)+4S6E7,L1=4J1+J7S11+2J4m∑j=1αj,L2=4J2+J8S11+2J5m∑j=1αj,L3=4J3+J9S11+2J6m∑j=1αj,L4=4J4+J1S12+2J7m∑e=1δe,L5=4J5+J2S12+2J8m∑e=1δe,L6=4J6+J3S12+2J9m∑e=1δe,L7=4J7+J4S13+2J1m∑k=1ξk,L8=4J8+J5S13+2J2m∑k=1ξk,L9=4J9+J6S13+2J3m∑k=1ξk, | (2.10) |
and it is assumed that
Λ=(8−B1)(8−B2)(8−B3)≠0, | (2.11) |
Λ1=Λ/(8−B3),Λ2=Λ/(8−B1),B1=(m∑r=1σr)(m∑d=1κd)(m∑n=1γn),B2=(m∑p=1ζp)(m∑l=1βl)(m∑q=1ρq),B3=(m∑k=1ξk)(m∑j=1αj)(m∑e=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(t−a)+c2(t−a)22+∫ta(t−s)22f1(s)ds, | (2.13) |
v(t)=c3+c4(t−a)+c5(t−a)22+∫ta(t−s)22g1(s)ds, | (2.14) |
w(t)=c6+c7(t−a)+c8(t−a)22+∫ta(t−s)22h1(s)ds, | (2.15) |
where ci∈R,i=1,…,8 are arbitrary real constants. Using the boundary conditions (1.2) in (2.13), (2.14) and (2.15), we obtain
2c0+(b−a)c1+(b−a)22c2−(m∑j=1αj)c3−(m∑j=1αj(ηj−a))c4−(m∑j=1αj(ηj−a)22)c5=−∫ba(b−s)22f1(s)ds+m∑j=1αj∫ηja(ηj−s)22g1(s)ds, | (2.16) |
2c1+(b−a)c2−(m∑l=1βl)c4−(m∑l=1βl(ηl−a))c5=−∫ba(b−s)f1(s)ds+m∑l=1βl∫ηla(ηl−s)g1(s)ds, | (2.17) |
2c2−(m∑n=1γn)c5=−∫baf1(s)ds+m∑n=1γn∫ηnag1(s)ds, | (2.18) |
2c3+(b−a)c4+(b−a)22c5−(m∑e=1δe)c6−(m∑e=1δe(ηe−a))c7−(m∑e=1δe(ηe−a)22)c8=−∫ba(b−s)22g1(s)ds+m∑e=1δe∫ηea(ηe−s)22h1(s)ds, | (2.19) |
2c4+(b−a)c5−(m∑q=1ρq)c7−(m∑q=1ρq(ηq−a))c8=−∫ba(b−s)g1(s)ds+m∑q=1ρq∫ηqa(ηq−s)h1(s)ds, | (2.20) |
2c5−(m∑r=1σr)c8=−∫bag1(s)ds+m∑r=1σr∫ηrah1(s)ds, | (2.21) |
−(m∑k=1ξk)c0−(m∑k=1ξk(ηk−a))c1−(m∑k=1ξk(ηk−a)22)c2+2c6+(b−a)c7+(b−a)22c8=−∫ba(b−s)22h1(s)ds+m∑k=1ξk∫ηka(ηk−s)22f1(s)ds, | (2.22) |
−(m∑p=1ζp)c1−(m∑p=1ζp(ηp−a))c2+2c7+(b−a)c8=−∫ba(b−s)h1(s)ds+m∑p=1ζp∫ηpa(ηp−s)f1(s)ds, | (2.23) |
−(m∑d=1κd)c2+2c8=−∫bah1(s)ds+m∑d=1κd∫ηdaf1(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=18−B1{−4∫baf1(s)ds−2(m∑n=1γn)∫bag1(s)ds−S1∫bah1(s)ds+S1(m∑d=1κd∫ηdaf1(s)ds)+4(m∑n=1γn∫ηnag1(s)ds)+2(m∑n=1γn)(m∑r=1σr∫ηrah1(s)ds)},c5=18−B1{−S3∫baf1(s)ds−4∫bag1(s)ds−2(m∑r=1σr)∫bah1(s)ds+2(m∑r=1σr)(m∑d=1κd∫ηdaf1(s)ds)+S3(m∑n=1γn∫ηnag1(s)ds)+4(m∑r=1σr∫ηrah1(s)ds)},c8=18−B1{−2(m∑d=1κd)∫baf1(s)ds−S2∫bag1(s)ds−4∫bah1(s)ds+4(m∑d=1κd∫ηdaf1(s)ds)+2(m∑d=1κd)(m∑n=1γn∫ηnag1(s)ds)+S2(m∑r=1σr∫ηrah1(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−(m∑l=1βl)c4=18−B1{−∫ba[(b−s)(8−B1)+S3E3−4(b−a)]f1(s)ds−∫ba[4E3−2(b−a)(m∑n=1γn)]g1(s)ds−∫ba[2E3(m∑r=1σr)−S1(b−a)]h1(s)ds+m∑d=1κd∫ηda[2E3m∑r=1σr−S1(b−a)]f1(s)ds+m∑n=1γn∫ηna[S3E3−4(b−a)]g1(s)ds+m∑r=1σr∫ηra[4E3−2(b−a)(m∑n=1γn)]h1(s)ds}+m∑l=1βl∫ηla(ηl−s)g1(s)ds, | (2.25) |
2c4−(m∑q=1ρq)c7=18−B1{−∫ba[2E6(m∑d=1κd)−S3(b−a)]f1(s)ds−∫ba[(b−s)(8−B1)+S2E6−4(b−a)]g1(s)ds−∫ba[4E6−2(b−a)(m∑r=1σr)]h1(s)ds+m∑d=1κd∫ηda[4E6−2(b−a)(m∑r=1σr)]f1(s)ds+m∑n=1γn∫ηna[2E6(m∑d=1κd)−S3(b−a)]g1(s)ds+m∑r=1σr∫ηra[S2E6−4(b−a)]h1(s)ds}+m∑q=1ρq∫ηqa(ηq−s)h1(s)ds, | (2.26) |
−(m∑p=1ζp)c1+2c7=18−B1{−∫ba[4E9−2(m∑d=1κd)(b−a)]f1(s)ds−∫ba[2E9(m∑n=1γn)−S2(b−a)]g1(s)ds−∫ba[(b−s)(8−B1)+S1E9−4(b−a)]h1(s)ds+m∑d=1κd∫ηda[S1E9−4(b−a)]f1(s)ds+m∑n=1γn∫ηna[4E9−2(m∑d=1κd)(b−a)]g1(s)ds+m∑r=1σr∫ηra[2E9(m∑n=1γn)−S2(b−a)]h1(s)ds}+m∑p=1ζp∫ηpa(ηp−s)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(8−B1)(b−s)+A1]f1(s)ds−∫ba[2(8−B1)(b−s)(m∑l=1βl)+A2]g1(s)ds−∫ba[S6(8−B1)(b−s)+A3]h1(s)ds+A3m∑d=1κd∫ηdaf1(s)ds+A1m∑n=1γn∫ηnag1(s)ds+A2m∑r=1σr∫ηrah1(s)ds+S6(8−B1)(m∑p=1ζp∫ηpa(ηp−s)f1(s)ds)+4(8−B1)(m∑l=1βl∫ηla(ηl−s)g1(s)ds)+2(8−B1)(m∑l=1βl)(m∑q=1ρq∫ηqa(ηq−s)h1(s)ds)},c4=1Λ1{−∫ba[S8(8−B1)(b−s)+A7]f1(s)ds−∫ba[4(8−B1)(b−s)+A8]g1(s)ds−∫ba[2(m∑q=1ρq)(8−B1)(b−s)+A9]h1(s)ds+A9m∑d=1κd∫ηdaf1(s)ds+A7m∑n=1γn∫ηnag1(s)ds+A8m∑r=1σr∫ηrah1(s)ds+2(8−B1)(m∑q=1ρq)(m∑p=1ζp∫ηpa(ηp−s)f1(s)ds)+S8(8−B1)(m∑l=1βl∫ηla(ηl−s)g1(s)ds)+4(8−B1)(m∑q=1ρq∫ηqa(ηq−s)h1(s)ds)},c7=1Λ1{−∫ba[2(m∑p=1ζp)(8−B1)(b−s)+A4]f1(s)ds−∫ba[S7(8−B1)(b−s)+A5]g1(s)ds−∫ba[4(8−B1)(b−s)+A6]h1(s)ds+A6m∑d=1κd∫ηdaf1(s)ds+A4m∑n=1γn∫ηnag1(s)ds+A5m∑r=1σr∫ηrah1(s)ds+4(8−B1)(m∑p=1ζp∫ηpa(ηp−s)f1(s)ds)+2(8−B1)(m∑p=1ζp)(m∑l=1βl∫ηla(ηl−s)g1(s)ds)+S7(8−B1)(m∑q=1ρq∫ηqa(ηq−s)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−(m∑j=1αj)c3=1Λ1{−∫ba[Λ1(b−s)22+((8−B1)(b−s))(S8E1−4(b−a))+J1]f1(s)ds−∫ba[((8−B1)(b−s))(4E1−2m∑l=1βl(b−a))+J2]g1(s)ds−∫ba[((8−B1)(b−s))(2E1m∑q=1ρq−S6(b−a))+J3]h1(s)ds+J3m∑d=1κd∫ηdaf1(s)ds+J1m∑n=1γn∫ηnag1(s)ds+J2m∑r=1σr∫ηrah1(s)ds+(8−B1)(m∑p=1ζp∫ηpa(ηp−s)[2E1m∑q=1ρq−S6(b−a)]f1(s)ds)+(8−B1)(m∑l=1βl∫ηla(ηl−s)[S8E1−4(b−a)]g1(s)ds)+(8−B1)(m∑q=1ρq∫ηqa(ηq−s)[4E1−2m∑l=1βl(b−a)]h1(s)ds)+Λ1(m∑j=1αj∫ηja(ηj−s)22g1(s)ds)}, | (2.28) |
2c3−(m∑e=1δe)c6=1Λ1{−∫ba[((8−B1)(b−s))(2E4m∑p=1ζp−S8(b−a))+J4]f1(s)ds−∫ba[Λ1(b−s)22+((8−B1)(b−s))(S6E4−4(b−a))+J5]g1(s)ds−∫ba[((8−B1)(b−s))(4E4−2m∑q=1ρq(b−a))+J6]h1(s)ds+J6m∑d=1κd∫ηdaf1(s)ds+J4m∑n=1γn∫ηnag1(s)ds+J5m∑r=1σr∫ηrah1(s)ds+(8−B1)(m∑p=1ζp∫ηpa(ηp−s)[4E4−2m∑q=1ρq(b−a)]f1(s)ds)+(8−B1)(m∑l=1βl∫ηla(ηl−s)[2E4m∑p=1ζp−S8(b−a)]g1(s)ds)+(8−B1)(m∑q=1ρq∫ηqa(ηq−s)[S6E4−4(b−a)]h1(s)ds)+Λ1(m∑e=1δe∫ηea(ηe−s)22h1(s)ds)}, | (2.29) |
−(m∑k=1ξk)c0+2c6=1Λ1{−∫ba[((8−B1)(b−s))(4E7−2m∑p=1ζp(b−a))+J7]f1(s)ds−∫ba[((8−B1)(b−s))(2E7m∑l=1βl−S6(b−a))+J8]g1(s)ds−∫ba[Λ1(b−s)22+((8−B1)(b−s))(S6E7−4(b−a))+J9]h1(s)ds+J9m∑d=1κd∫ηdaf1(s)ds+J7m∑n=1γn∫ηnag1(s)ds+J8m∑r=1σr∫ηrah1(s)ds+(8−B1)(m∑p=1ζp∫ηpa(ηp−s)[S6E7−4(b−a)]f1(s)ds)+(8−B1)(m∑l=1βl∫ηla(ηl−s)[4E7−2m∑p=1ζp(b−a)]g1(s)ds)+(8−B1)(m∑q=1ρq∫ηqa(ηq−s)[2E7m∑l=1βl−S6(b−a)]h1(s)ds)+Λ1(m∑k=1ξk∫ηka(ηk−s)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(b−s)2+μ1(8−B1)(b−s)+L1]f1(s)ds−∫ba[Λ1(m∑j=1αj)(b−s)2+μ2(8−B1)(b−s)+L2]g1(s)ds−∫ba[Λ1S11(b−s)22+μ3(8−B1)(b−s)+L3]h1(s)ds+L3m∑d=1κd∫ηdaf1(s)ds+L1m∑n=1γn∫ηnag1(s)ds+L2m∑r=1σr∫ηrah1(s)ds+μ3(8−B1)(m∑p=1ζp∫ηpa(ηp−s)f1(s)ds)+μ1(8−B1)(m∑l=1βl∫ηla(ηl−s)g1(s)ds)+μ2(8−B1)(m∑q=1ρq∫ηqa(ηq−s)h1(s)ds)+Λ1S11(m∑k=1ξk∫ηka(ηk−s)22f1(s)ds)+2Λ1(m∑j=1αj∫ηja(ηj−s)2g1(s)ds)+Λ1(m∑j=1αj)(m∑e=1δe∫ηea(ηe−s)2h1(s)ds)},c3=1Λ{−∫ba[Λ1S12(b−s)22+μ4(8−B1)(b−s)+L4]f1(s)ds−∫ba[2Λ1(b−s)2+μ5(8−B1)(b−s)+L5]g1(s)ds−∫ba[Λ1(m∑e=1δe)(b−s)2+μ6(8−B1)(b−s)+L6]h1(s)ds+L6m∑d=1κd∫ηdaf1(s)ds+L4m∑n=1γn∫ηnag1(s)ds+L5m∑r=1σr∫ηrah1(s)ds+μ6(8−B1)(m∑p=1ζp∫ηpa(ηp−s)f1(s)ds)+μ4(8−B1)(m∑l=1βl∫ηla(ηl−s)g1(s)ds)+μ5(8−B1)(m∑q=1ρq∫ηqa(ηq−s)h1(s)ds)+Λ1(m∑e=1δe)(m∑k=1ξk∫ηka(ηk−s)2f1(s)ds)+Λ1S12(m∑j=1αj∫ηja(ηj−s)22g1(s)ds)+2Λ1(m∑e=1δe∫ηea(ηe−s)2h1(s)ds)},c6=1Λ{−∫ba[Λ1(m∑k=1ξk)(b−s)2+μ7(8−B1)(b−s)+L7]f1(s)ds−∫ba[Λ1S13(b−s)22+μ8(8−B1)(b−s)+L8]g1(s)ds−∫ba[2Λ1(b−s)2+μ9(8−B1)(b−s)+L9]h1(s)ds+L9m∑d=1κd∫ηdaf1(s)ds+L7m∑n=1γn∫ηnag1(s)ds+L8m∑r=1σr∫ηrah1(s)ds+μ9(8−B1)(m∑p=1ζp∫ηpa(ηp−s)f1(s)ds)+μ7(8−B1)(m∑l=1βl∫ηla(ηl−s)g1(s)ds)+μ8(8−B1)(m∑q=1ρq∫ηqa(ηq−s)h1(s)ds)+2Λ1(m∑k=1ξk∫ηka(ηk−s)22f1(s)ds)+Λ1(m∑k=1ξk)(m∑j=1αj∫ηja(ηj−s)22g1(s)ds)+Λ1S13(m∑e=1δe∫ηea(ηe−s)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:X3→X3 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(t−s)22ˆf(s)ds+1Λ{−∫ba[2Λ1(b−s)2+G1(t)(b−s)+P1(t)]ˆf(s)ds−∫ba[Λ1m∑j=1αj(b−s)2+G2(t)(b−s)+P2(t)]ˆg(s)ds−∫ba[Λ1S11(b−s)22+G3(t)(b−s)+P3(t)]ˆh(s)ds+P3(t)(m∑d=1κd∫ηdaˆf(s)ds)+P1(t)(m∑n=1γn∫ηnaˆg(s)ds)+P2(t)(m∑r=1σr∫ηraˆh(s)ds)+G3(t)(m∑p=1ζp∫ηpa(ηp−s)ˆf(s)ds)+G1(t)(m∑l=1βl∫ηla(ηl−s)ˆg(s)ds)+G2(t)(m∑q=1ρq∫ηqa(ηq−s)ˆh(s)ds)+Λ1S11(m∑k=1ξk∫ηka(ηk−s)22ˆf(s)ds)+2Λ1(m∑j=1αj∫ηja(ηj−s)2ˆg(s)ds)+Λ1m∑j=1αj(m∑e=1δe∫ηea(ηe−s)2ˆh(s)ds)}, | (3.3) |
H2(u,v,w)(t)=∫ta(t−s)22ˆg(s)ds+1Λ{−∫ba[Λ1S12(b−s)22+G4(t)(b−s)+P4(t)]ˆf(s)ds−∫ba[2Λ1(b−s)2+G5(t)(b−s)+P5(t)]ˆg(s)ds−∫ba[Λ1m∑e=1δe(b−s)2+G6(t)(b−s)+P6(t)]ˆh(s)ds+P6(t)(m∑d=1κd∫ηdaˆf(s)ds)+P4(t)(m∑n=1γn∫ηnaˆg(s)ds)+P5(t)(m∑r=1σr∫ηraˆh(s)ds)+G6(t)(m∑p=1ζp∫ηpa(ηp−s)ˆf(s)ds)+G4(t)(m∑l=1βl∫ηla(ηl−s)ˆg(s)ds)+G5(t)(m∑q=1ρq∫ηqa(ηq−s)ˆh(s)ds)+Λ1m∑e=1δe(m∑k=1ξk∫ηka(ηk−s)2ˆf(s)ds)+Λ1S12(m∑j=1αj∫ηja(ηj−s)22ˆg(s)ds)+2Λ1(m∑e=1δe∫ηea(ηe−s)2ˆh(s)ds)}, | (3.4) |
H3(u,v,w)(t)=∫ta(t−s)22ˆh(s)ds+1Λ{−∫ba[Λ1m∑k=1ξk(b−s)22+G7(t)(b−s)+P7(t)]ˆf(s)ds−∫ba[Λ1S13(b−s)22+G8(t)(b−s)+P8(t)]ˆg(s)ds−∫ba[2Λ1(b−s)2+G9(t)(b−s)+P9(t)]ˆh(s)ds+P9(t)(m∑d=1κd∫ηdaˆf(s)ds)+P7(t)(m∑n=1γn∫ηnaˆg(s)ds)+P8(t)(m∑r=1σr∫ηraˆh(s)ds)+G9(t)(m∑p=1ζp∫ηpa(ηp−s)ˆf(s)ds)+G7(t)(m∑l=1βl∫ηla(ηl−s)ˆg(s)ds)+G8(t)(m∑q=1ρq∫ηqa(ηq−s)ˆh(s)ds)+2Λ1(m∑k=1ξk∫ηka(ηk−s)2ˆf(s)ds)+Λ1m∑k=1ξk(m∑j=1αj∫ηja(ηj−s)2ˆg(s)ds)+Λ1S13(m∑e=1δe∫ηea(ηe−s)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,ˆmi≥0,(i=1,2,3) and m0>0,ˉm0>0,ˆm0>0 such that ∀u,v,w∈R, 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,w∈R, 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,wi∈R,i=1,2 there exist ℓi>0(i=1,2,3) such that
|f(t,u1,v1,w1)−f(t,u2,v2,w2)|≤ℓ1(|u1−u2|+|v1−v2|+|w1−w2|), |
|g(t,u1,v1,w1)−g(t,u2,v2,w2)|≤ℓ2(|u1−u2|+|v1−v2|+|w1−w2|), |
|h(t,u1,v1,w1)−h(t,u2,v2,w2)|≤ℓ3(|u1−u2|+|v1−v2|+|w1−w2|). |
For the sake of computational convenience, we set
Θ1=Δ1+Δ4+Δ7,Θ2=Δ2+Δ5+Δ8,Θ3=Δ3+Δ6+Δ9, | (3.6) |
where
Δ1=(b−a)36+13|8−B3|[2(b−a)3+S11(m∑k=1ξk(ηk−a)32)]+1|Λ|[Q1(b−a)22+Υ1(b−a)+Υ3(m∑d=1κd(ηd−a))+Q3(m∑p=1ζp(ηp−a)22)], | (3.7) |
Δ2=m∑j=1αj3|8−B3|[(b−a)3+2(ηj−a)3]+1|Λ|[Q2(b−a)22+Υ2(b−a)+Υ1(m∑n=1γn(ηn−a))+Q1(m∑l=1βl(ηl−a)22)], | (3.8) |
Δ3=13|8−B3|[S11(b−a)32+(m∑j=1αj)(m∑e=1δe(ηe−a)3)]+1|Λ|[Q3(b−a)22+Υ3(b−a)+Υ2(m∑r=1σr(ηr−a))+Q2(m∑q=1ρq(ηq−a)22)], | (3.9) |
Δ4=13|8−B3|[S12(b−a)32+(m∑e=1δe)(m∑k=1ξk(ηk−a)3)]+1|Λ|[Q4(b−a)22+Υ4(b−a)+Υ6(m∑d=1κd(ηd−a))+Q6(m∑p=1ζp(ηp−a)22)], | (3.10) |
Δ5=(b−a)36+13|8−B3|[2(b−a)3+S12(m∑j=1αj(ηj−a)32)]+1|Λ|[Q5(b−a)22+Υ5(b−a)+Υ4(m∑n=1γn(ηn−a))+Q4(m∑l=1βl(ηl−a)22)], | (3.11) |
Δ6=m∑e=1δe3|8−B3|[(b−a)3+2(ηe−a)3)]+1|Λ|[Q6(b−a)22+Υ6(b−a)+Υ5(m∑r=1σr(ηr−a))+Q5(m∑q=1ρq(ηq−a)22)], | (3.12) |
Δ7=m∑k=1ξk3|8−B3|[(b−a)32+2(ηk−a)3)]+1|Λ|[Q7(b−a)22+Υ7(b−a)+Υ9(m∑d=1κd(ηd−a))+Q9(m∑p=1ζp(ηp−a)22)], | (3.13) |
Δ8=13|8−B3|[S13(b−a)32+(m∑k=1ξk)(m∑j=1αj(ηj−a)3)]+1|Λ|[Q8(b−a)22+Υ8(b−a)+Υ7(m∑n=1γn(ηn−a))+Q7(m∑l=1βl(ηl−a)22)], | (3.14) |
Δ9=(b−a)36+13|8−B3|[2(b−a)3+S13(m∑e=1δe(ηe−a)32)]+1|Λ|[Q9(b−a)22+Υ9(b−a)+Υ8(m∑r=1σr(ηr−a))+Q8(m∑q=1ρq(ηq−a)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:Y→Y be a completely continuous operator (i.e., a map restricted to any bounded set in Y is compact). Let Ξ(T)={x∈Y: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:X3→X3 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 Φ⊂X3 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(t−s)22ˆf(s)ds+1Λ{−∫ba[2Λ1(b−s)2+G1(t)(b−s)+P1(t)]ˆf(s)ds−∫ba[Λ1m∑j=1αj(b−s)2+G2(t)(b−s)+P2(t)]ˆg(s)ds−∫ba[Λ1S11(b−s)22+G3(t)(b−s)+P3(t)]ˆh(s)ds+P3(t)(m∑d=1κd∫ηdaˆf(s)ds)+P1(t)(m∑n=1γn∫ηnaˆg(s)ds)+P2(t)(m∑r=1σr∫ηraˆh(s)ds)+G3(t)(m∑p=1ζp∫ηpa(ηp−s)ˆf(s)ds)+G1(t)(m∑l=1βl∫ηla(ηl−s)ˆg(s)ds)+G2(t)(m∑q=1ρq∫ηqa(ηq−s)ˆh(s)ds)+Λ1S11(m∑k=1ξk∫ηka(ηk−s)22ˆf(s)ds)+2Λ1(m∑j=1αj∫ηja(ηj−s)2ˆg(s)ds)+Λ1m∑j=1αj(m∑e=1δe∫ηea(ηe−s)2ˆh(s)ds)}|≤ϱf{(b−a)36+13|8−B3|[2(b−a)3+S11(m∑k=1ξk(ηk−a)32)]+1|Λ|[Q1(b−a)22+Υ1(b−a)+Υ3(m∑d=1κd(ηd−a))+Q3(m∑p=1ζp(ηp−a)22)]}+ϱg{m∑j=1αj3|8−B3|[(b−a)3+2(ηj−a)3]+1|Λ|[Q2(b−a)22+Υ2(b−a)+Υ1(m∑n=1γn(ηn−a))+Q1(m∑l=1βl(ηl−a)22)]}+ϱh{13|8−B3|[S11(b−a)32+(m∑j=1αj)(m∑e=1δe(ηe−a)3)]+1|Λ|[Q3(b−a)22+Υ3(b−a)+Υ2(m∑r=1σr(ηr−a))+Q2(m∑q=1ρq(ηq−a)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[(τ−s)22−(t−s)22]ˆf(s)ds+∫τt(τ−s)22ˆf(s)ds−∫ba[(τ−t)(4(b−s)(8−B2)+A1Λ1)+2(8−B1)(τ2−t2)]ˆf(s)ds−∫ba[(τ−t)(2m∑l=1βl(8−B2)(b−s)+A2Λ1)+m∑n=1γn(8−B1)(τ2−t2)]ˆg(s)ds−∫ba[(τ−t)(S6(8−B2)(b−s)+A3Λ1)+S12(8−B1)(τ2−t2)]ˆh(s)ds+m∑d=1κd∫ηda[A3Λ1(τ−t)+S12(8−B1)(τ2−t2)]ˆf(s)ds+m∑n=1γn∫ηna[A1Λ1(τ−t)+2(8−B1)(τ2−t2)]ˆg(s)ds+m∑r=1σr∫ηra[A2Λ1(τ−t)+m∑n=1γn(8−B1)(τ2−t2)]ˆh(s)ds+S6(8−B2)(τ−t)(m∑p=1ζp∫ηpa(ηp−s)ˆf(s)ds)+4(8−B2)(τ−t)(m∑l=1βl∫ηla(ηl−s)ˆg(s)ds)+2∑ml=1βl(8−B2)(τ−t)(m∑q=1ρq∫ηqa(ηq−s)ˆh(s)ds)|≤ϱf[(τ−t)33+13!|(τ−a)3−(t−a)3|]+(τ−t)|8−B2|[(b−a)2(2ϱf+ϱgm∑l=1βl+12ϱhS6)+ϱfS6(m∑p=1ζp(ηp−a)22)+2ϱg(m∑l=1βl(ηl−a)2)+ϱh(m∑l=1βl)(m∑q=1ρq(ηq−a)2)]+(τ−t)|Λ1|[(b−a)(ϱf|A1|+ϱg|A2|+ϱh|A3|)+ϱf|A3|(m∑d=1κd(ηd−a))+ϱg|A1|(m∑n=1γn(ηn−a))+ϱh|A2|(m∑r=1σr(ηr−a))]+(τ2−t2)|8−B1|[(b−a)(2ϱf+ϱgm∑n=1γn+12ϱhS1)+12ϱfS1(m∑d=1κd(ηd−a))+2ϱg(m∑n=1γn(ηn−a))+ϱh(m∑n=1γn)(m∑r=1σr(ηr−a))]→0independent of(u,v,w)∈Φasτ−t→0. |
Similarly, it can be established that
|H2(u,v,w)(τ)−H2(u,v,w)(t)|≤ϱg[(τ−t)33+13!|(τ−a)3−(t−a)3|]+(τ−t)|8−B2|[(b−a)2(12ϱfS8+2ϱg+ϱhm∑q=1ρq)+ϱf(m∑q=1ρq)(m∑p=1ζp(ηp−a)2)+ϱgS8(m∑l=1βl(ηl−a)22)+2ϱh(m∑q=1ρq(ηq−a)2)]+(τ−t)|Λ1|[(b−a)(ϱf|A7|+ϱg|A8|+ϱh|A9|)+ϱf|A9|(m∑d=1κd(ηd−a))+ϱg|A7|(m∑n=1γn(ηn−a))+ϱh|A8|(m∑r=1σr(ηr−a))]+(τ2−t2)|8−B1|[(b−a)(12ϱfS3+2ϱg+ϱhm∑r=1σr)+ϱf(m∑r=1σr)(m∑d=1κd(ηd−a))+12ϱgS3(m∑n=1γn(ηn−a))+2ϱh(m∑r=1σr(ηr−a))]→0independent of(u,v,w)∈Φasτ−t→0, |
and
|H3(u,v,w)(τ)−H3(u,v,w)(t)|≤ϱh[(τ−t)33+13!|(τ−a)3−(t−a)3|]+(τ−t)|8−B2|[(b−a)2(ϱfm∑p=1ζp+12ϱgS7+2ϱh)+2ϱf(m∑p=1ζp(ηp−a)2)+ϱg(m∑p=1ζp)(m∑l=1βl(ηl−a)2)+ϱhS7(m∑q=1ρq(ηq−a)22)]+(τ−t)|Λ1|[(b−a)(ϱf|A4|+ϱg|A5|+ϱh|A6|)+ϱf|A6|(m∑d=1κd(ηd−a))+ϱg|A4|(m∑n=1γn(ηn−a))+ϱh|A5|(m∑r=1σr(ηr−a))]+(τ2−t2)|8−B1|[(b−a)(ϱfm∑d=1κd+12ϱgS2+2ϱh)+2ϱf(m∑d=1κd(ηd−a))+ϱg(m∑d=1κd)(m∑n=1γn(ηn−a))+12ϱhS2(m∑r=1σr(ηr−a))]→0independent of(u,v,w)∈Φasτ−t→0. |
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+m1‖u‖+m2‖v‖+m3‖w‖)+Δ2(ˉm0+ˉm1‖u‖+ˉm2‖v‖+ˉm3‖w‖)+Δ3(ˆm0+ˆm1‖u‖+ˆm2‖v‖+ˆm3‖w‖)≤Δ1m0+Δ2ˉm0+Δ3ˆm0+(Δ1m1+Δ2ˉm1+Δ3ˆm1)‖u‖+(Δ1m2+Δ2ˉm2+Δ3ˆm2)‖v‖+(Δ1m3+Δ2ˉm3+Δ3ˆm3)‖w‖, |
|v(t)|≤Δ4(m0+m1‖u‖+m2‖v‖+m3‖w‖)+Δ5(ˉm0+ˉm1‖u‖+ˉm2‖v‖+ˉm3‖w‖)+Δ6(ˆm0+ˆm1‖u‖+ˆm2‖v‖+ˆm3‖w‖)≤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‖+‖w‖≤Θ1m0+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
x⩾max{12Θ1‖ϕ‖,12Θ2‖ψ‖,12Θ3‖χ‖,(12Θ1ϵ1)11−λ1,(12Θ1ϵ2)11−λ2,(12Θ1ϵ3)11−λ3,(12Θ2ϵ4)11−λ4,(12Θ2ϵ5)11−λ5,(12Θ2ϵ6)11−λ6,(12Θ3ϵ7)11−λ7,(12Θ3ϵ8)11−λ8,(12Θ3ϵ9)11−λ9} |
Firstly, we prove that H:Γ→Γ. For that, we consider
|H1(u,v,w)(t)|=|∫ta(t−s)22ˆf(s)ds+1Λ{−∫ba[2Λ1(b−s)2+G1(t)(b−s)+P1(t)]ˆf(s)ds−∫ba[Λ1m∑j=1αj(b−s)2+G2(t)(b−s)+P2(t)]ˆg(s)ds−∫ba[Λ1S11(b−s)22+G3(t)(b−s)+P3(t)]ˆh(s)ds+P3(t)(m∑d=1κd∫ηdaˆf(s)ds)+P1(t)(m∑n=1γn∫ηnaˆg(s)ds)+P2(t)(m∑r=1σr∫ηraˆh(s)ds)+G3(t)(m∑p=1ζp∫ηpa(ηp−s)ˆf(s)ds)+G1(t)(m∑l=1βl∫ηla(ηl−s)ˆg(s)ds)+G2(t)(m∑q=1ρq∫ηqa(ηq−s)ˆh(s)ds)+Λ1S11(m∑k=1ξk∫ηka(ηk−s)22ˆf(s)ds)+2Λ1(m∑j=1αj∫ηja(ηj−s)2ˆg(s)ds)+Λ1m∑j=1αj(m∑e=1δe∫ηea(ηe−s)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)Θ3≤x, |
where Θ1,Θ2 and Θ3 are given by (3.6). Therefore, we conclude that H:Γ→Γ, 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
Θ1ℓ1+Θ2ℓ2+Θ3ℓ3<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 HBς⊂Bς, where Bς={(u,v,w)∈X3:‖(u,v,w)‖≤ς} with
ς≥Θ1M1+Θ2M2+Θ3M31−(Θ1ℓ1+Θ2ℓ2+Θ3ℓ3). |
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)|≤ℓ1(‖u‖+‖v‖+‖w‖)+M1≤ℓ1‖(u,v,w)‖+M1≤ℓ1ς+M1. |
In a similar manner, we have
|g(s,u(s),v(s),w(s))|≤ℓ2ς+M2,|h(s,u(s),v(s),w(s)|≤ℓ3ς+M3. |
Then, for (u,v,w)∈Bς, we obtain
|H1(u,v,w)(t)|=|∫ta(t−s)22ˆf(s)ds+1Λ{−∫ba[2Λ1(b−s)2+G1(t)(b−s)+P1(t)]ˆf(s)ds−∫ba[Λ1m∑j=1αj(b−s)2+G2(t)(b−s)+P2(t)]ˆg(s)ds−∫ba[Λ1S11(b−s)22+G3(t)(b−s)+P3(t)]ˆh(s)ds+P3(t)(m∑d=1κd∫ηdaˆf(s)ds)+P1(t)(m∑n=1γn∫ηnaˆg(s)ds)+P2(t)(m∑r=1σr∫ηraˆh(s)ds)+G3(t)(m∑p=1ζp∫ηpa(ηp−s)ˆf(s)ds)+G1(t)(m∑l=1βl∫ηla(ηl−s)ˆg(s)ds)+G2(t)(m∑q=1ρq∫ηqa(ηq−s)ˆh(s)ds)+Λ1S11(m∑k=1ξk∫ηka(ηk−s)22ˆf(s)ds)+2Λ1(m∑j=1αj∫ηja(ηj−s)2ˆg(s)ds)+Λ1m∑j=1αj(m∑e=1δe∫ηea(ηe−s)2ˆh(s)ds)}|≤(ℓ1ς+M1){(b−a)36+13|8−B3|[2(b−a)3+S11(m∑k=1ξk(ηk−a)32)]+1|Λ|[Q1(b−a)22+Υ1(b−a)+Υ3(m∑d=1κd(ηd−a))+Q3(m∑p=1ζp(ηp−a)22)]}+(ℓ2ς+M2){∑mj=1αj3|8−B3|[(b−a)3+2(ηj−a)3]+1|Λ|[Q2(b−a)22+Υ2(b−a)+Υ1(m∑n=1γn(ηn−a))+Q1(m∑l=1βl(ηl−a)22)]}+(ℓ3ς+M3){13|8−B3|[S11(b−a)32+(m∑j=1αj)(m∑e=1δe(ηe−a)3)]+1|Λ|[Q3(b−a)22+Υ3(b−a)+Υ2(m∑r=1σr(ηr−a))+Q2(m∑q=1ρq(ηq−a)22)]}≤(ℓ1ς+M1)Δ1+(ℓ2ς+M2)Δ2+(ℓ3ς+M3)Δ3, |
which, on taking the norm for t∈[a,b], yields
‖H1(u,v,w)‖≤(ℓ1ς+M1)Δ1+(ℓ2ς+M2)Δ2+(ℓ3ς+M3)Δ3. |
Similarly, we can find that
‖H2(u,v,w)‖≤(ℓ1ς+M1)Δ4+(ℓ2ς+M2)Δ5+(ℓ3ς+M3)Δ6, |
and
‖H3(u,v,w)‖≤(ℓ1ς+M1)Δ7+(ℓ2ς+M2)Δ8+(ℓ3ς+M3)Δ9, |
where Δi(i=1,…,9) are defined in (3.7)−(3.15). In consequence, it follows that
‖H(u,v,w)‖≤(ℓ1ς+M1)Θ1+(ℓ2ς+M2)Θ2+(ℓ3ς+M3)Θ3≤ς. |
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(t−s)22|f(s,u1(s),v1(s),w1(s))−f(s,u2(s),v2(s),w2(s))|ds+1|Λ|{∫ba[2|Λ1|(b−s)2+|G1(t)|(b−s)+|P1(t)|]×|f(s,u1(s),v1(s),w1(s))−f(s,u2(s),v2(s),w2(s))|ds+∫ba[|Λ1|m∑j=1αj(b−s)2+|G2(t)|(b−s)+|P2(t)|]×|g(s,u1(s),v1(s),w1(s))−g(s,u2(s),v2(s),w2(s))|ds+∫ba[|Λ1|S11(b−s)22+|G3(t)|(b−s)+|P3(t)|]×|h(s,u1(s),v1(s),w1(s))−h(s,u2(s),v2(s),w2(s))|ds+|P3(t)|(m∑d=1κd∫ηda|f(s,u1(s),v1(s),w1(s))−f(s,u2(s),v2(s),w2(s))|ds)+|P1(t)|(m∑n=1γn∫ηna|g(s,u1(s),v1(s),w1(s))−g(s,u2(s),v2(s),w2(s))|ds)+|P2(t)|(m∑r=1σr∫ηra|h(s,u1(s),v1(s),w1(s))−h(s,u2(s),v2(s),w2(s))|ds)+|G3(t)|(m∑p=1ζp∫ηpa(ηp−s)|f(s,u1(s),v1(s),w1(s))−f(s,u2(s),v2(s),w2(s))|ds)+|G1(t)|(m∑l=1βl∫ηla(ηl−s)|g(s,u1(s),v1(s),w1(s))−g(s,u2(s),v2(s),w2(s))|ds)+|G2(t)|(m∑q=1ρq∫ηqa(ηq−s)|h(s,u1(s),v1(s),w1(s))−h(s,u2(s),v2(s),w2(s))|ds)+|Λ1|S11(m∑k=1ξk∫ηka(ηk−s)22|f(s,u1(s),v1(s),w1(s))−f(s,u2(s),v2(s),w2(s))|ds)+2|Λ1|(m∑j=1αj∫ηja(ηj−s)2|g(s,u1(s),v1(s),w1(s))−g(s,u2(s),v2(s),w2(s))|ds)+|Λ1|m∑j=1αj(m∑e=1δe∫ηea(ηe−s)2|h(s,u1(s),v1(s),w1(s))−h(s,u2(s),v2(s),w2(s))|ds)}≤ℓ1(|u1−u2|+|v1−v2|+|w1−w2|){(b−a)36+13|8−B3|[2(b−a)3+S11(m∑k=1ξk(ηk−a)32)]+1|Λ|[Q1(b−a)22+Υ1(b−a)+Υ3(m∑d=1κd(ηd−a))+Q3(m∑p=1ζp(ηp−a)22)]}+ℓ2(|u1−u2|+|v1−v2|+|w1−w2|){∑mj=1αj3|8−B3|[(b−a)3+2(ηj−a)3]+1|Λ|[Q2(b−a)22+Υ2(b−a)+Υ1(m∑n=1γn(ηn−a))+Q1(m∑l=1βl(ηl−a)22)]}+ℓ3(|u1−u2|+|v1−v2|+|w1−w2|){13|8−B3|[S11(b−a)32+(m∑j=1αj)(m∑e=1δe(ηe−a)3)]+1|Λ|[Q3(b−a)22+Υ3(b−a)+Υ2(m∑r=1σr(ηr−a))+Q2(m∑q=1ρq(ηq−a)22)]}≤(ℓ1Δ1+ℓ2Δ2+ℓ3Δ3)(|u1−u2|+|v1−v2|+|w1−w2|), |
which implies that
‖H1(u1,v1,w1)−H1(u2,v2,w2)‖≤(ℓ1Δ1+ℓ2Δ2+ℓ3Δ3)(|u1−u2|+|v1−v2|+|w1−w2|), |
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)‖≤(ℓ1Δ4+ℓ2Δ5+ℓ3Δ6)(|u1−u2|+|v1−v2|+|w1−w2|), |
and
‖H3(u1,v1,w1)−H3(u2,v2,w2)‖≤(ℓ1Δ7+ℓ2Δ8+ℓ3Δ9)(|u1−u2|+|v1−v2|+|w1−w2|), |
where Δi,(i=4,…,9) are given by (3.10)−(3.15). Thus we have
‖H(u1,v1,w1)−H(u2,v2,w2)‖≤(Θ1ℓ1+Θ2ℓ2+Θ3ℓ3)(‖u1−u2‖+‖v1−v2‖+‖w1−w2‖), | (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)=531√t3+24+|u(t)|2204(1+|u(t)|)+3342sinv(t)+1t2+97w(t),t∈[1,3],v‴(t)=e−(t−1)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+26√4356tu(t)+w(t)|v(t)|810(1+|v(t)|),t∈[1,3], | (3.20) |
supplemented to the following boundary conditions
u(1)+u(3)=4∑j=1αjv(ηj),u′(1)+u′(3)=4∑l=1βlv′(ηl),u″(1)+u″(3)=4∑n=1γnv″(ηn),v(1)+v(3)=4∑e=1δew(ηe),v′(1)+v′(3)=4∑q=1ρqw′(ηq),v″(1)+v″(3)=4∑r=1σrw″(ηr),w(1)+w(3)=4∑k=1ξku(ηk),w′(1)+w′(3)=4∑p=1ζpu′(ηp),w″(1)+w″(3)=4∑d=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 B1≈2.444444≠8,B2≈6.875556≠8,B3≈4.545452≠8, and Λ≈21.580256 (Λ is given by (2.11)). Also, Δ1≈21.294227,Δ2≈22.603176,Δ3≈11.800813,Δ4≈7.983258,Δ5≈12.996835,Δ6≈8.497948,Δ7≈10.977544,Δ8≈14.165941 and Δ9≈12.745457 (Δi(i=1,…,9) are defined in (3.7)−(3.15)). Furthermore we obtain Θ1≈40.255029,Θ2≈49.765952 and Θ3≈33.044218(Θ1,Θ2 and Θ3 are given by (3.6)). Evidently,
|f(t,u,v,w)|≤131+1204‖u‖+1114‖v‖+198‖w‖,|g(t,u,v,w)|≤1192+1114‖u‖+196‖v‖+1128‖w‖,|h(t,u,v,w)|≤150+1198‖u‖+1810‖w‖. |
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+Θ3ˆm1≈0.800762<1,Θ1m2+Θ2ˉm2+Θ3ˆm2≈0.871509<1 and Θ1m3+Θ2ˉm3+Θ3ˆm3≈0.840357<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)(tan−1(u(t))+v(t)+|w|1+|w|)+e−(t−1),t∈[1,3],v‴(t)=1610πsin(2πu)+42t+1218sin(v(t))+73+1305w(t),t∈[1,3],w‴(t)=322√999+90t(u(t)+|v(t)|1+|v(t)|+tan−1(w(t)))+cos(t−1),t∈[1,3], | (3.22) |
subject to the coupled boundary conditions (3.21). It is easy to see that ℓ1=1/219,ℓ2=1/305 and ℓ3=1/242 as
|f(t,u1,v1,w1)−f(t,u2,v2,w2)|≤1219(|u1−u2|+|v1−v2|+|w1−w2|),|g(t,u1,v1,w1)−g(t,u2,v2,w2)|≤1305(|u1−u2|+|v1−v2|+|w1−w2|),|h(t,u1,v1,w1)−h(t,u2,v2,w2)|≤1242(|u1−u2|+|v1−v2|+|w1−w2|). |
Using the values obtained in Example 3.1, we find that Θ1ℓ1+Θ2ℓ2+Θ3ℓ3≈0.483526<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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