# Existence and Uniqueness of Solution to Nonlinear Boundary Value Problems with Sign-Changing Green’s Function

Authors: Peiguo Zhang; Lishan Liu; Yonghong Wu
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

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.


---

## Body

## 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 [1] and later continued by II'in and Moiseev [2, 3] and Gupta [4]. 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 [5–25] and the references therein.Recently, by applying the fixed point theorems on cones, the authors of papers [5–7] 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 n≥2,0<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 [8–10], where n≥2,   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 [26] 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,t∈J,u(0)=u′(0)=…=u(n-2)(0)=0,u(1)=∑i=1m-2‍αiu(ηi),
where n≥2,  f∈C(J×ℝn,ℝ),J=(0,1), ∑i=1m-2αiηin-1≠1,and0<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 [26], and the arguments are based upon the cone theory and the Banach contraction mapping principle.


## 2. The Preliminary Lemmas

Lemma 1.
For anyf∈L(I), the BVP
(4)u′(t)+f(t)=0,t∈J,(5)∫01‍(1-t)n-2u(t)dt=∑i=1m-2‍αi∫0ηi‍(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],dddddddddddddddddd0≤s≤η1,s≤t,1σ[(1-s)n-1-∑i=1m-2‍αi(ηi-s)n-1],dddddddddddddddddd0≤t≤s≤η1,-1+1σ[(1-s)n-1-∑i=j+1m-2‍αi(ηi-s)n-1],ddddddddddddddddddηj≤s≤ηj+1,s≤t,1σ[(1-s)n-1-∑i=j+1m-2‍αi(ηi-s)n-1],ddddddddddddddddddηj≤s≤ηj+1,t≤s,-1+(1-s)n-1σ,dddddηm-2≤s≤t≤1,(1-s)n-1σ,dddddddddηm-2≤s≤1,t≤s,σ=1-∑i=1m-2‍αiηin-1,I=[0,1].Proof.
First, suppose thatu∈C(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-2‍αi∫0ηi‍(ηi-t)n-2[u(0)-∫0t‍f(s)ds]dt,
and so
(9)u(0)=[∫01‍(1-t)n-2dt-∑i=1m-2‍αi∫0ηi‍(ηi-t)n-2dt]-1×[∑i=1m-2‍∫01‍(1-t)n-2∫0t‍f(s)dsdt-∑i=1m-2‍αi∫0ηi‍(ηi-t)n-2∫0t‍f(s)dsdt]=(1-∑i=1m-2αiηin-1n-1)-1×[∑i=1m-2‍∫01‍(1-t)n-2∫0t‍f(s)dsdt-∑i=1m-2‍αi∫0ηi‍(ηi-t)n-2∫0t‍f(s)dsdt]=n-1σ[∑i=1m-2‍∫01‍(1-t)n-2∫0t‍f(s)dsdt-∑i=1m-2‍αi∫0ηi‍(ηi-t)n-2∫0t‍f(s)dsdt]=n-1σ[∑i=1m-2‍∫01‍f(s)∫s1‍(1-t)n-2dtds-∑i=1m-2‍αi∫0ηi‍f(s)∫sηi‍(ηi-t)n-2dtds]=1σ[∑i=1m-2‍∫01‍f(s)(1-s)n-1ds-∑i=1m-2‍αi∫0ηi‍f(s)(ηi-s)n-1ds].
Substituting (9) into (7), we have
(10)u(t)=-∫0t‍f(s)ds+1σ[∑i=1m-2‍∫01‍f(s)(1-s)n-1dsddddd-∑i=1m-2‍αi∫0ηi‍f(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 anyu∈C(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,t∈I.Lemma 2.
(

i

) If u∈Cn-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 v∈C(I) is a fixed point of F, then u(t)=(In-1v)(t)=∫0t‍((t-s)n-2/(n-2)!)v(s)ds∈Cn-1(I) is a solution to problem (3).By Lemma1, the proof follows by routine calculations.Let(12)h1(t)=max{∫01‍|G(t,s)|ds,∫0t‍∫01‍|G(s,x)|dxds},hk(t)=max{∫01‍|G(t,s)|hk-1(s)ds,∫0t‍∫01‍|G(s,x)|hk-1(x)dxds},ddddddddddddddddk=2,3,…,ρ(G)=limk→∞(supt∈Jhk(t))-1/k.It is easy to see thatρ(G)≥(supt∈Jhk(t))-1/k≥(supt,s∈J|G(t,s)|)-1>0.Lemma 3 (see [27, 28]).
P is a generating cone in Banach space (E,∥·∥) if and only if there exists a constant τ>0 such that every element u∈E can be represented in the form u=v-w, where v,w∈P and ∥v∥≤τ∥u∥,∥w∥≤τ∥u∥.


## 3. Main Results

This section discusses the solution of nonlinear higher-order differential equation BVP (3).LetP={u∈C(I)|u(t)≥0,forallt∈[0,1]}. Obviously, P is a normal solid cone of Banach space C(I), by Lemma 2.1.2 in [29], and we have that P is a generating cone in C(I).Theorem 4.
Suppose thatg∈C(J×ℝ2n,ℝ), 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 t∈I,s01,t01,s02, t02,s11,t11,s12,t12,…,s1,n-1,t1,n-1,s2,n-1,t2,n-1∈ℝ with s01≤t01,s02≥t02,s11≤t11,s12≥t12,…,sn-1,1≤tn-1,1,sn-1,2≥tn-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,v0∈Cn-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 u0∈C(I), the iterative sequence
(16)um(t)=∫01‍G(t,s)f(s,(In-1um-1)(s),…,dddddddd(I1um-1)(s),um-1(s))ds,dddddddddddddddddddddm=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 functionf 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 Theorem4.
It is easy to see that, for anyt∈J,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 anyu,v∈C(I), let x(t)=|p(t)|+|u(t)|,y(t)=-|q(t)|-|v(t)| and then x≥p,y≤q. 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-p∥ddddddddddd+(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, byx≥u,y≤v,
(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 operatorF: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,dddddddddddddddddddddddddddddd∀t∈I.
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,v2∈C(I),u1≤u2,v1≥v2, 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)!)·∥u∥h1(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)!)m∥u∥hm(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·supt∈Jem(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)k∥1/k=r(A+B)<β<1, and so there exists a positive integer k0 such that
(29)∥(A+B)k∥<βk<1,k≥k0.
SinceP is a generating cone in C(I), from Lemma 3, there exists τ>0 such that every element u∈C(I) can be represented in
(30)u=v-w,v,w∈P,∥v∥≤τ∥u∥,∥w∥≤τ∥u∥;
this implies
(31)-(v+w)≤u≤v+w.
Let
(32)∥u∥0=inf{∥h∥∣h∈P,-h≤u≤h}.
By (31), we know that ∥u∥0 is well defined for any u∈C(I). It is easy to verify that ∥·∥0 is a norm in C(I). By (30)–(32), we get
(33)∥u∥0≤∥v+w∥≤2τ∥u∥,∀u∈C(I).
On the other hand, for anyh∈P which satisfies -h≤u≤h, we have 0≤u+h≤2h; thus, ∥u∥≤∥u+h∥+∥-h∥≤(2N+1)∥h∥, where N denotes the normal constant of P. Since h is arbitrary, we have
(34)∥u∥≤(2N+1)∥u∥0,∀u∈C(I).
It follows from (33) and (34) that the norms ∥·∥0 and ∥·∥ are equivalent. Now, for any u,v∈C(I) and h∈P which satisfies -h≤u-v≤h, let
(35)u1=12(u+v-h),u2=12(u-v+h),u3=12(-u+v+h),
then u≥u1,v≥u1,u-u1=u2,v-u1=u3,u2+u3=h.
It follows from (27) that
(36)-Au2≤F(u,u)-F(u1,u)≤Au2,(37)-Au3-Bu2≤F(v,u1)-F(u1,u)≤Au2+Bu3,(38)-Bu3≤F(v,u1)-F(v,v)≤Fu3;
subtracting (37) from (36) + (38), we obtain
(39)-(A+B)h≤F(u,u)-F(v,v)≤(A+B)h.
Let G(u)=F(u,u); then we have
(40)-(A+B)h≤G(u)-G(v)≤(A+B)h.
AsA and B are both positive linear bounded operators, so A+B is a positive linear bounded operator, and therefore, (A+B)h∈P. Hence, by mathematical induction, it is easy to know that for natural number k0 in (29), we have
(41)-(A+B)k0h≤Gk0(u)-Gk0(v)≤(A+B)k0h,(A+B)k0h∈P;
since (A+B)k0h∈P, we see that
(42)∥Gk0(u)-Gk0(v)∥0≤∥(A+B)k0∥∥h∥,
which implies by virtue of the arbitrariness of h that
(43)∥Gk0u-Gk0v∥0≤∥(A+B)k0∥∥u-v∥0≤βk0∥u-v∥0.
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 u0∈C(I), let um=F(um-1,um-1)(m=1,2,…); we have ∥um-u*∥0→0(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 followingnth-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,k∈C(I×I,ℝ), i=0,1.Applying Theorem4, we can find that (44) has a unique solution Inx*(t)∈C(n)(I) provided supt,s∈I|(h0(t,s)/(n-2)!)+(h1(t,s)/(n-3)!)+(k(t)/3(n-2)!)|<1, and moreover, for any u0∈C(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],dddddddddddddddddddd0≤s≤12,s≤t,2n-22n-2-1[(1-s)n-1-2(12-s)n-1],dddddddddddddddddddd0≤t≤s≤12,-1+2n-22n-2-1(1-s)n-1,12≤s≤t,2n-22n-2-1(1-s)n-1,dddddd12≤s,t≤s,e1*(t)=max{∫01‍|G1(t,s)|ds,∫0t‍∫01‍|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,s∈Ie1*(t))-1≥1.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*3∫0t‍v(s)ds,(Miu)(t)=0,i=1,…,n-1,u0=v0=0,
where Hi*=supt,s∈I|hi(t,s)|(i=0,1),K*=supt∈I|k(t)|; then it is easy to verify that all conditions in Theorem 4 are satisfied.



---

*Source: 640183-2013-10-21.xml*