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

Theorem 4.
Suppose that g∈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, ddddddddddddddddddddd m=1,2,…,
converges to u* in C(I) (m→∞).

Proof of Theorem <xref ref-type="statement" rid="thm3.1">4</xref>.
It is easy to see that, for any t∈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 any u,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, by x≥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 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,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)ddd m=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∈J em(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.

Since P 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 any h∈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.

As A 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.