AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 640183 10.1155/2013/640183 640183 Research Article Existence and Uniqueness of Solution to Nonlinear Boundary Value Problems with Sign-Changing Green’s Function http://orcid.org/0000-0002-7856-0345 Zhang Peiguo 1 Liu Lishan 2 Wu Yonghong 3 Lai Shaoyong 1 Department of Mathematics Heze University Heze Shandong 274000 China hezeu.edu.cn 2 School of Mathematical Sciences Qufu Normal University Qufu Shandong 273165 China qfnu.edu.cn 3 Department of Mathematics and Statistics Curtin University of Technology Perth WA 6845 Australia curtin.edu.au 2013 21 10 2013 2013 20 07 2013 23 08 2013 2013 Copyright © 2013 Peiguo Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.

1. Introduction

Boundary value problems (BVPs for short) for nonlinear differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory. The study of multipoint BVPs for second-order differential equations was initiated by Bicadze and Samarskiĭ  and later continued by II'in and Moiseev [2, 3] and Gupta . Since then, great efforts have been devoted to nonlinear multipoint BVPs due to their theoretical challenge and great application potential. Many results on the existence of solutions for multipoint BVPs have been obtained; the methods used therein mainly depend on the fixed point theorems, degree theory, upper and lower techniques, and monotone iteration. The existence results are available in the literature  and the references therein.

Recently, by applying the fixed point theorems on cones, the authors of papers  established the existence and multiplicity of positive solutions for the nth-order three-point BVP: (1)u(n)(t)+a(t)f(t,u(t))=0,t(0,1),u(0)=u(0)==u(n-2)(0)=0,u(1)=αu(η), where n2,0<η<1 and 0<αηn-1<1. The nth-order m-point BVP (2)u(n)(t)+a(t)f(t,u(t))=0,t(0,1),u(0)=u(0)==u(n-2)(0)=0,u(1)=i=1m-2αiu(ηi) has been studied in , where n2,   0<η1<η2<ηm-2<1 and αi>0(i=1,2,,m-2) with 0<i=1m-2αiηin-1<1. The existence and multiplicity results of solutions were shown by using various fixed point theorems and fixed point index theory.

By using the cone theory and the Banach contraction mapping principle, the author  established the existence and uniqueness for singular third-order three-point boundary value problems.

The purpose of this paper is to investigate the existence and uniqueness of solution of the following higher-order differential equation boundary value problem: (3)u(n)(t)+f(t,u(t),u(t),,u(n-1)(t))=0,tJ,u(0)=u(0)==u(n-2)(0)=0,u(1)=i=1m-2αiu(ηi), where n2,  fC(J×n,),J=(0,1), i=1m-2αiηin-11,and  0<η1<<ηm-2<1.

Here, we give the unique solution of BVP (3) under the conditions that f is mixed nonmonotone. The methods used in this paper are motivated by , and the arguments are based upon the cone theory and the Banach contraction mapping principle.

2. The Preliminary Lemmas Lemma 1.

For any fL(I), the BVP (4)        u(t)+f(t)=0,tJ,(5)        01(1-t)n-2u(t)dt=i=1m-2αi0ηi(ηi-t)n-2u(t)dt has a unique solution u(t)=01G(t,s)f(s)ds, where (6)G(t,s)={-1+1σ[(1-s)n-1-i=1m-2αi(ηi-s)n-1],dddddddddddddddddd0sη1,st,1σ[(1-s)n-1-i=1m-2αi(ηi-s)n-1],dddddddddddddddddd0tsη1,-1+1σ[(1-s)n-1-i=j+1m-2αi(ηi-s)n-1],ddddddddddddddddddηjsηj+1,st,1σ[(1-s)n-1-i=j+1m-2αi(ηi-s)n-1],ddddddddddddddddddηjsηj+1,ts,-1+(1-s)n-1σ,dddddηm-2st1,(1-s)n-1σ,dddddddddηm-2s1,ts,σ=1-i=1m-2αiηin-1,I=[0,1].

Proof.

First, suppose that uC(I) is a solution to problem (4) and (5). It is easy to see by integration of (4) that (7)        u(t)=u(0)-0tf(s)ds. Substituting (7) into (5), we obtain (8)  01(1-t)n-2[u(0)-0tf(s)ds]dt=i=1m-2αi0ηi(ηi-t)n-2[u(0)-0tf(s)ds]dt, and so (9)u(0)=[01(1-t)n-2dt-i=1m-2αi0ηi(ηi-t)n-2dt]-1×[i=1m-201(1-t)n-20tf(s)dsdt-i=1m-2αi0ηi(ηi-t)n-20tf(s)dsdt]=(1-i=1m-2αiηin-1n-1)-1×[i=1m-201(1-t)n-20tf(s)dsdt-i=1m-2αi0ηi(ηi-t)n-20tf(s)dsdt]=n-1σ[i=1m-201(1-t)n-20tf(s)dsdt-i=1m-2αi0ηi(ηi-t)n-20tf(s)dsdt]=n-1σ[i=1m-201f(s)s1(1-t)n-2dtds-i=1m-2αi0ηif(s)sηi(ηi-t)n-2dtds]=1σ[i=1m-201f(s)(1-s)n-1ds-i=1m-2αi0ηif(s)(ηi-s)n-1ds]. Substituting (9) into (7), we have (10)u(t)=-0tf(s)ds+1σ[i=1m-201f(s)(1-s)n-1dsddddd-i=1m-2αi0ηif(s)(ηi-s)n-1ds]=01G(t,s)f(s)ds. Conversely, suppose that u(t)=01G(t,s)f(s)ds; then it is easy to verify that (4) and (5) are satisfied. The lemma is proved.

For any uC(I), let (11)        (Iiu)(t)=0t(t-s)i-1(i-1)!u(s)ds,i=1,2,,n-1,(Fu)(t)=01G(t,s)f(s,(In-1u)(s),,dddddddddd(I1u)(s),u(s))ds,tI.

Lemma 2.

( i ) If uCn-1(I) is a solution to problem (3), then v(t)=u(n-1)(t)C(I) is a fixed point of F.

( ii ) If vC(I) is a fixed point of F, then u(t)=(In-1v)(t)=0t((t-s)n-2/(n-2)!)v(s)dsCn-1(I) is a solution to problem (3).

By Lemma 1, the proof follows by routine calculations.

Let (12)h1(t)=max{01|G(t,s)|ds,0t01|G(s,x)|dxds},hk(t)=max{01|G(t,s)|hk-1(s)ds,0t01|G(s,x)|hk-1(x)dxds},dddddddddddddddd  k=2,3,,ρ(G)=limk(suptJhk(t))-1/k.

It is easy to see that ρ(G)(suptJhk(t))-1/k(supt,sJ|G(t,s)|)-1>0.

Lemma 3 (see [<xref ref-type="bibr" rid="B27">27</xref>, <xref ref-type="bibr" rid="B28">28</xref>]).

P is a generating cone in Banach space (E,·) if and only if there exists a constant τ>0 such that every element uE can be represented in the form u=v-w, where v,wP and vτu,wτu.

3. Main Results

This section discusses the solution of nonlinear higher-order differential equation BVP (3).

Let P={uC(I)  |  u(t)0,for  all  t[0,1]}. Obviously, P is a normal solid cone of Banach space C(I), by Lemma 2.1.2 in , and we have that P is a generating cone in C(I).

Theorem 4.

Suppose that gC(J×2n,), f(t,x0,x1,,xn-1)=g(t,x0,x0,x1,x1,,xn-1,xn-1), and there exist positive constants K0,M0,K1,M1,,Kn-1,Mn-1 with (13)K0+M0(n-1)!+K1+M1(n-2)!++Kn-3+Mn-32!+Kn-2+Mn-2+Kn-1+Mn-1<ρ(G), such that for any tI,s01,t01,s02, t02,s11,t11,s12,t12,,s1,n-1,t1,n-1,s2,n-1,t2,n-1 with s01t01,s02t02,s11t11,s12t12,,sn-1,1tn-1,1,sn-1,2tn-1,2, one has (14)-K0(t01-s01)-M0(s02-t02)-K1(t11-s11)-M1(s12-t12)--Kn-1(tn-1,1-sn-1,1)-Mn-1(sn-1,2-tn-1,2)g(t,s01,s02,s11,s12,,sn-1,1,sn-1,2)-g(t,t01,t02,t11,t12,,tn-1,1,tn-1,2)-K0(t01-s01)-M0(s02-t02)-K1(t11-s11)-M1(s12-t12)--Kn-1(tn-1,1-sn-1,1)-Mn-1(sn-1,2-tn-1,2), and there exist u0,v0Cn-1(I), such that (15)01g(t,u0(t),v0(t),u0(t),v0(t),,dddddu0(n-1)(t),v0(n-1)(t))dt converges. Then, BVP (3) has a unique solution In-1u* in C(I), and moreover,  for any u0C(I), the iterative sequence (16)um(t)=01G(t,s)f(s,(In-1um-1)(s),,dddddddd(I1um-1)(s),um-1(s))ds,    ddddddddddddddddddddd  m=1,2,, converges to u* in C(I)  (m).

Remark 5.

Recently, in the study of BVP (3), almost all the papers have supposed that Green’s function G(t,s) is nonnegative. However, the scope of αi is not limited to i=1m-2αiηi<1 in Theorem 4, so, we do not need to suppose that G(t,s) is nonnegative.

Remark 6.

The function f in Theorem 4 is not monotone or convex; the conclusions and the proof used in this paper are different from the known papers in essence.

Proof of Theorem <xref ref-type="statement" rid="thm3.1">4</xref>.

It is easy to see that, for any tJ,  G(t,s) can be divided into finite partitioned monotone and bounded function on (0,1), and then, by (15), we have that (17)01G(t,s)g(s,u0(s),v0(s),u0(s),v0(s),,u0(n-1)(s),v0(n-1)(s){s,u0(s),v0(s),u0(s),v0(s)})ds converges. Let p(t)=u0(n-1)(t),q(t)=v0(n-1)(t); then (18)01G(t,s)g(s,(In-1p)(s),(In-1q)(s),,dddddddddd(I1p)(s),(I1q)(s),p(t),q(s))ds converges.

For any u,vC(I), let x(t)=|p(t)|+|u(t)|,y(t)=-|q(t)|-|v(t)| and then xp,yq. By (14), we have (19)-K0(In-1x-In-1p)(t)-M0(In-1q-In-1y)(t)-K1(In-2x-In-2p)(t)-M1(In-2q-In-2y)(t)--Kn-2(I1x-I1p)(t)-Mn-2(I1q-I1y)(t)-Kn-1(x-p)(t)-Mn-1(q-y)(t)g(t,(In-1x)(t),(In-1y)(t),,dddd(I1x)(t),(I1y)(t),x(t),y(t))-g(t,(In-1p)(t),(In-1q)(t),,ddddd(I1p)(t),(I1q)(t),p(t),q(t))K0(In-1x-In-1p)(t)+M0(In-1q-In-1y)(t)+K1(In-2x-In-2p)(t)+M1(In-2q-In-2y)(t)++Kn-2(I1x-I1p)(t)+Mn-2(I1q-I1y)(t)+Kn-1(x-p)(t)+Mn-1(q-y)(t). Hence, (20)  |(I1p)(s),(I1q)(s),p(s),q(s))G(t,s)g(t,(In-1x)(t),(In-1y)(t),,dddddddd(I1x)(t),(I1y)(t),x(t),y(t))-G(t,s)g(t,(In-1p)(t),(In-1q)(t),,ddddddddddddd(I1p)(t),(I1q)(t),p(t),q(t))||G(t,s)|[K0|(In-1x)(t)-(In-1p)(t)|ddddddddddd+M0|(In-1q)(t)-(In-1y)(t)|ddddddddddd+K1|(In-2x)(t)-(In-2p)(t)|ddddddddddd+M1|(In-2q)(t)-(In-2y)(t)|ddddddddddd++Kn-2|(I1x)(t)-(I1p)(t)|ddddddddddd+Mn-2|(I1q)(t)-(I1y)(t)|ddddddddddd+Kn-1|x(t)-p(t)|+Mn-1|q(t)-y(t)|]|G(t,s)|[(K0+K1++Kn-1)x-pddddddddddd+(M0+M1++Mn-1)q-y]. Following the former inequality, we can easily have that (21)01G(t,s)[(I1p)(s),(I1q)(s),p(s),q(s))g(s,(In-1x)(s),(In-1y)(s),,dddddddddd(I1x)(s),(I1y)(s),x(s),y(s))sssssssssssss-g(s,(In-1p)(s),(In-1q)(s),,ddddddddddddd(I1p)(s),(I1q)(s),p(s),q(s))]ds converges, thus, (22)01G(t,s)g(s,(In-1x)(s),(In-1y)(s),,ddddddddd(I1x)(s),(I1y)(s),x(s),y(s))ds=01G(t,s)g(s,(In-1p)(s),(In-1q)(s),,dddddddddddd(I1p)(s),(I1q)(s),p(s),q(s))ds+01G(t,s)[(I1p)(s),(I1q)(s),p(s),q(s))g(s,(In-1x)(s),(In-1y)(s),,dddddddddddddd(I1x)(s),(I1y)(s),x(s),y(s))ddddddddddddd-g(s,(In-1p)(s),(In-1q)(s),,ddddddddddddddddd(I1p)(s),(I1q)(s),p(s),q(s))]ds is converged.

Similarly, by xu,yv, (23)01G(t,s)g(s,(In-1x)(s),(In-1y)(s),,gggggdgggg(I1x)(s),(I1y)(s),x(s),y(s))ds is converged, and we have that (24)01G(t,s)g(s,(In-1u)(s),(In-1v)(s),,ffffdfffffff(I1u)(s),(I1v)(s),u(s),v(s))ds converges.

Define the operator F:C(I)×C(I)C(I) by (25)F(u,v)(t)=01G(t,s)dddd×g(s,(In-1u)(s),(In-1v)(s),,ddddddddd(I1u)(s),(I1v)(s),u(s),v(s))ds,ddddddddddddddddddddddddddddddtI. Let (26)(A0u)(t)=01|G(t,s)|(K0u)(s)ds,(B0v)(t)=01|G(t,s)|(M0v)(s)ds,(Aiu)(t)=01|G(t,s)|(Ki(Iiu))(s)ds,dddddddddddddddddi=1,2,,n-1,(Biv)(t)=01|G(t,s)|(Mi(Iiv))(s)ds,ddddddddddddddddi=1,2,,n-1,(Au)(t)=(A0u+A1u++An-1u)(t),(Bv)(t)=(B0v+B1v++Bn-1v)(t). By (14) and (25), for any u1,u2,v1,v2C(I),u1u2,v1v2, we have (27)-A(u2-u1)-B(v1-v2)F(u1,v1)-F(u2,v2)A(u2-u1)+B(v1-v2),(28)((A+B)u)(t)=01|G(t,s)|[K0u+M0u+K1(I1u)+M1(I1u)+ddddddddddddd+Kn-1(In-1u)+Mn-1(In-1u)](s)ds(K0+M0(n-2)!+K1+M1(n-3)!++Kn-3+Mn-31!ddsdd+Kn-2+Mn-2+Kn-1+Mn-1K0+M0(n-2)!)·uh1(t),((A+B)mu)(t)=01|G(t,s)|(A+B)(A+B)m-1(u)(s)ds(K0+M0(n-2)!+K1+M1(n-3)!++Kn-3+Mn-31!dsddd+Kn-2+Mn-2+Kn-1+Mn-1K0+M0(n-2)!)muhm(t)dddm=2,3,,(A+B)m(K0+M0(n-2)!+K1+M1(n-3)!+ddddddddddddd+Kn-3+Mn-31!+Kn-2+Mn-2ddddddddddddd+Kn-1+Mn-1K0+M0(n-2)!)m·suptJem(t),r(A+B)(K0+M0(n-2)!+K1+M1(n-3)!+ddddddddddd+Kn-3+Mn-31!+Kn-2+Mn-2ddddddddddd+Kn-1+Mn-1K0+M0(n-2)!)ddddddddd×(ρ(G))-1<1. So we can choose β(0,1), which satisfies limk(A+B)k1/k=r(A+B)<β<1, and so there exists a positive integer k0 such that (29)        (A+B)k<βk<1,kk0.

Since P is a generating cone in C(I), from Lemma 3, there exists τ>0 such that every element uC(I) can be represented in (30)        u=v-w,v,wP,vτu,wτu; this implies (31)        -(v+w)uv+w. Let (32)        u0=inf{hhP,-huh}. By (31), we know that u0 is well defined for any uC(I). It is easy to verify that ·0 is a norm in C(I). By (30)–(32), we get (33)        u0v+w2τu,uC(I).

On the other hand, for any hP which satisfies -huh, we have 0u+h2h; thus, uu+h+-h(2N+1)h, where N denotes the normal constant of P. Since h is arbitrary, we have (34)u(2N+1)u0,uC(I). It follows from (33) and (34) that the norms ·0 and · are equivalent. Now, for any u,vC(I) and hP which satisfies -hu-vh, let (35)      u1=12(u+v-h),u2=12(u-v+h),u3=12(-u+v+h), then uu1,vu1,u-u1=u2,v-u1=u3,u2+u3=h.

It follows from (27) that (36)        -Au2F(u,u)-F(u1,u)Au2,(37)-Au3-Bu2F(v,u1)-F(u1,u)Au2+Bu3,(38)-Bu3F(v,u1)-F(v,v)Fu3; subtracting (37) from (36) + (38), we obtain (39)        -(A+B)hF(u,u)-F(v,v)(A+B)h. Let G(u)=F(u,u); then we have (40)        -(A+B)hG(u)-G(v)(A+B)h.

As A and B are both positive linear bounded operators, so A+B is a positive linear bounded operator, and therefore, (A+B)hP. Hence, by mathematical induction, it is easy to know that for natural number k0 in (29), we have (41)-(A+B)k0hGk0(u)-Gk0(v)(A+B)k0h,(A+B)k0hP; since (A+B)k0hP, we see that (42)Gk0(u)-Gk0(v)0(A+B)k0h, which implies by virtue of the arbitrariness of h that (43)  Gk0u-Gk0v0(A+B)k0u-v0βk0u-v0. By 0<β<1, we have 0<βk0<1. Thus, the Banach contraction mapping principle implies that Gk0 has a unique fixed point u* in C(I), and so G has a unique fixed point u* in C(I); by the definition of G,F has a unique fixed point u* in C(I); then, by Lemma 2, In-1u* is the unique solution of (3). And, for any u0C(I), let um=F(um-1,um-1)  (m=1,2,); we have um-u*00  (k). By the equivalence of ·0 and · again, we get um-u*0  (m). This completes the proof.

4. Example

In this paper, the results apply to a very wide range of functions, and we are following only one example to illustrate.

Consider the following nth-order three-point boundary value problem: (44)u(n)(t)+(S0u)(t)+(S1u)(t)+k(t)ln(3+|x(t)|),t(0,1),u(0)=u(0)==u(n-2)(0)=0,u(1)=2u(12), where (Siu(i))(t)=01hi(t,s)u(i)(s)ds,hi,kC(I×I,), i=0,1.

Applying Theorem 4, we can find that (44) has a unique solution Inx*(t)C(n)(I) provided supt,sI|(h0(t,s)/(n-2)!)+(h1(t,s)/(n-3)!)+(k(t)/3(n-2)!)|<1, and moreover, for any u0C(I), the iterative sequence (45)xm(t)=01G(t,s)[S0(In-1xm-1)(s)+S1(In-2xm-1)(s)+k(s)ln(3+|xm-1(s)|)]ds(m=1,2,) converges to x* uniformly for all t in I(m).

To see that, let (46)G1(t,s)={-1+2n-22n-2-1[(1-s)n-1-2(12-s)n-1],dddddddddddddddddddd0s12,st,2n-22n-2-1[(1-s)n-1-2(12-s)n-1],dddddddddddddddddddd0ts12,-1+2n-22n-2-1(1-s)n-1,12st,2n-22n-2-1(1-s)n-1,dddddd12s,ts,e1*(t)=max{01|G1(t,s)|ds,0t01|G1(s,x)|dxds}; then G1(t,s) is Green’s function of (44). It is easy to verify that |G1(t,s)|1, and so ρ(G1)(supt,sIe1*(t))-11.

Let (47)g(t,u(t),v(t),u(t),v(t),,u(n-1)(t),v(n-1)(t))=(S0u)(t)+(S1u)(t)+k(t)ln(3+|v(t)|),(Kiu)(t)=Hi*01u(s)ds,i=0,1,(M0v)(t)=K*30tv(s)ds,(Miu)(t)=0,i=1,,n-1,u0=v0=0, where Hi*=supt,sI|hi(t,s)|(i=0,1),K*=suptI|k(t)|; then it is easy to verify that all conditions in Theorem 4 are satisfied.

Acknowledgments

Peiguo Zhang and Lishan Liu were supported financially by the National Natural Science Foundation of China (11071141, 11371221), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, and the Project of Shandong Province Higher Educational Science and Technology Program (J11LA06, J13LI02). Yonghong Wu was supported financially by the Australian Research Council through an ARC Discovery Project grant.

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