# Iterative Solution for Systems of a Class of Abstract Operator Equations in Banach Spaces and Application

Authors: Hua Su
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

In this paper, by using the partial order method, the existence and uniqueness of a solution for systems of a class of abstract operator equations in Banach spaces are discussed. The result obtained in this paper improves and unifies many recent results. Two applications to the system of nonlinear differential equations and the systems of nonlinear differential equations in Banach spaces are given, and the unique solution and interactive sequences which converge the unique solution and the error estimation are obtained.


---

## Body

## 1. Introduction

Guo and Lakshmikantham [1] introduced the definition of the mixed monotone operator and the coupled fixed point, and there are many good results (see [2–23]). Recently, from paper [6], using the monotone iterative techniques, the iterative unique solution of the following nonlinear mixed monotone Fredholm-type integral equations in Banach spaces E is obtained:(1)ut=∫IHt,s,usds,I=a,b,where I=a,b and H∈CI×I×E,E.In this paper, the following nonlinear abstract operator equations in Banach spacesE are discussed:(2)u=Au,v,v=Bv,u,where A,B:D×D⟶E and D is a partial interval in E which is denoted as the following:(3)D≐u0,v0≡u∈Eu0≤u≤v0.For convenience, the following assumptions are made:H1 There exist positive bounded operators Ti:E⟶Ei=1,2 which satisfy.I+T1−T2x≥θ⟹x∈P, and for any ui,vi∈Di=1,2,u1≤u2,v1≤v2, the following is obtained:(4)Bv2,u1−Bv1,u2≥−T1v2−v1−T2u2−u1,Av2,u1−Av1,u2≥−T1v2−v1−T2u2−u1.H2u0+T2v0−u0≤Au0,v0,Bv0,u0≤v0−T2v0−u0.H3 There exists a positive bounded operator L:E⟶E, and for any u,v∈D,u≤v, the following is obtained:(5)−T1+T2v−u≤Bv,u−Au,v≤Lv−u.H4LT2=T2L,LT1=T1L,T1T2=T2T1 in which the spectral radius satisfies(6)rL+rT1+rT2<infλ:λ∈σI+T1−T2,rT1−T2<1.In this paper, firstly, by using the partial order method, the existence and uniqueness of a solution for systems of a class of abstract operator equations in Banach spaces are discussed. And next, two applications to the system of nonlinear integral equations and the system of nonlinear differential equations in Banach spaces are given, and the unique solution and interactive sequences which converge a unique solution and the error estimation are obtained.


## 2. The Interactive Solution of Abstract Operator Equations

LetP be a cone in E, i.e., a closed convex subset, such that λP⊂P for any λ≥0 and P∩−P=θ. A partial order ≤ in P is defined as x≤y⇔y−x∈P. A cone P is said to be normal if there exists a constant N>0 which satisfies x,y∈E,θ≤x≤y, implying x≤Ny, where θ denotes the zero element of E. And, the smallest number N is called as the normal constant of P and denoted as NP. The cone P is normal iff every ordered interval x,y=z∈E:x≤z≤y is bounded.The following theorem is the main results in this section.Theorem 1.
LetP be a cone in E, u0,v0∈E,u0≤v0. Suppose that A,B:D×D⟶E satisfies conditions H1−H4. Then,(i)
There exists a unique solution of equation (2) u∗,u∗ in D×D, and for any solutions of equation (2) u,u∈D×D, one has u=u∗.(ii)
For any initial valuex0,y0∈D,x0≤y0, the following iterative sequences are constructed:(7)xn=I+T1−T2−1Axn−1,yn−1+T1xn−1−T2yn−1,yn=I+T1−T2−1Byn−1,xn−1+T1yn−1−T2xn−1,which satisfy xn−u∗⟶0,yn−u∗⟶0n⟶∞, and for any δ,(8)rL+rT1+rT2infλ:λ∈σI+T1−t2<δ<1,there exists a natural number n0 which satisfies as n≥n0, the following is obtained:(9)xn−u∗≤2Npδnv0−u0,yn−u∗≤2Npδnv0−u0.Proof.
ByrT1−T2<1, it is known that the operator I+T1−T2 is reversible. And, from condition H1, I+T1−T2−1 is the positive operator. Let(10)Fu,v=I+T1−T2−1Au,v+T1u−T2v,Gv,u=I+T1−T2−1Bv,u+T1v−T2u.
Then, equation (7) can be substituted by the following:(11)xn=Fxn−1,yn−1,yn=Gyn−1,xn−1.
By conditionsH1−H3, it is easy to obtain that operators FandG satisfy the following:(1)
u0≤Fu0,v0≤Gv0,u0≤v0(2)
F,G:D×D⟶E are the mixed monotone operator(3)
θ≤Gv,u−Fu,v≤Hv−u,u0≤u≤v≤v0, where H=L+T1+T2I+T1−T2−1
Lettingun=Fun−1,vn−1andvn=Gvn−1,un−1n=1,2,…, the following two results are obtained by mathematical induction:(12)u0≤u1≤⋯≤un≤⋯≤vn≤⋯≤v1≤v0,(13)un≤xn≤yn≤vn,θ≤vn−un≤Hnv0−u0,n=1,2,….
In fact, from (1) and (3), one has(14)u0≤u1≤v1≤v0,u1≤x1≤y1≤v1,0≤v1−u1≤Hv0−u0.
Suppose that forn=k, one has (12) and (13). Then, as n=k+1, by (2) and (3), the following is obtained:(15)uk+1=Fuk,vk≤xk+1=Fxk,yk≤Gyk,xk=yk+1≤Gvk,uk=vk+1,θ≤vk+1−uk+1=Gvk,uk−Fuk,vk≤Hvk−uk≤Hk+1vk−uk.
Then, it is known that(16)uk≤uk+1≤xk+1≤yk+1≤xk+1≤vk,θ≤vk+1−uk+1≤Hk+1vk−uk.
Then, for any natural numbern, (12) and (13) are obtained by mathematical induction.
Next, it is proved thatxn is Cauchy sequences. From condition H4, it is known that(17)L+T1+T2I+T1−T2−1=I+T1−T2−1L+T1+T2,then by ([14], V 3.9), rH≤rL+rT1+rT2rI+T1−T2−1.
Thus, for anyδ:rL+rT1+rT2/infλ:λ∈σI+T1−t2<δ<1, the following is obtained:(18)limHn1/n=rH≤rL+rT1+rT2rI+T1−T2−1=rL+rT1+rT2infλ:λ∈σI+T1−t2<δ<1.
Then, there exists a natural numbern0 which satisfies(19)Hn≤δn,∀n≥n0.
And, by (12) and (13), it is obtained that(20)θ≤un≤un+p≤xn+p≤yn+p≤vn+p≤vn,θ≤un≤xn≤yn≤vn,n,p=1,2,….
So, by (13), it is known that(21)θ≤xn+p−un≤vn−un≤Hnv0−u0,θ≤xn−un≤Hnv0−u0.
Then, by the normality ofP and (19), it is known that(22)xn+p−un≤NpHnv0−u0≤Npδnv0−u0,(23)xn−un≤NpHnv0−u0≤Npδnv0−u0,n≥n0,p=1,2,….
Thus, the following is obtained:(24)xn+p−xn≤xn+p−un+xn−un≤2Npδnv0−u0,n≥n0,p=1,2,…,i.e., xn is Cauchy sequences. So, there exists u∗∈D (D is bounded), such that limn⟶∞xn=u∗.
And, byθ≤yn−xn≤vn−un≤Hnv0−u0, the normality of P, and (19), one obtains(25)yn−xn≤Npδnv0−u0,therefore(26)limn⟶∞yn=u∗=limn⟶∞xn,xn≤u∗≤yn,n=1,2,….
Thus,xn−un≤Npδnv0−u0, vn−xn≤Npδnv0−u0, and(27)limn⟶∞un=u∗=limn⟶∞vn,(28)un≤u∗≤vn,n=1,2,…,so by (2), (3), and (11), it is also obtained that(29)un=Fun−1,vn−1≤Fu∗,u∗≤Gu∗,u∗≤Gvn−1,un−1=vn.
Lettingn⟶∞ and by (27), Fu∗,u∗=Gu∗,u∗=u∗.
Then, by the definition ofFandG, one obtains u∗=Au∗,u∗,u∗=Bu∗,u∗, i.e., u∗,u∗ is a solution of equation (2).
Lastly, it is proven that the solution is unique. Supposing thatu,u∈D×D also satisfies equation (2), then by (11) and mathematical induction, the following is obtained:(30)un≤u≤vn,n=1,2,….
Thus,u=u∗.
And, lettingp⟶∞ in (24), as n≥n0, the following is obtained:(31)xn−u∗≤2Npδnv0−u0.
Similarly, asn≥n0, the following is obtained:(32)yn−u∗≤2Npδnv0−u0.
The proof is complete.Remark 1.
In Theorem1, it is only supposed that operators AandB satisfy the partial condition, and the unique solution and interactive sequences which converge a unique solution are obtained.


## 3. The Application of Nonlinear Integral Equations

In this section, the following nonlinear integral equations are considered:(33)ut=f1t,ut,vt+∫0tg1s,us,vsds,vt=f2t,vt,ut+∫0tg2s,us,usds,where fi∈I×R+×R+,R+ (here, the continuity of fi is not assumed) and gi∈CI×R+×R+,R+, i=1,2, I=0,+∞, and E is a real Banach space with norm ⋅.In this section, the iterative solution of a nonlinear integral equation (33) is discussed. For convenience, the following assumptions are made:L1 For the nonnegative bounded continuous function at,bt, and nonnegative integrable ct, dt, one has(34)f2t,u,θ≤atu+bt,g2t,u,θ≤ctu+dt.L2 There exists a constant M>0, for any u,v∈E,u≤v, which satisfies(35)fit,v,u−fit,u,v≥−Mv−u,git,v,u−git,u,v≥0,i=1,2.L3 For any u,v∈E,u≤v, the following is satisfied:(36)−Mv−u≤f2t,v,u−f1t,u,v≤ctv−u,0≤g2t,v,u−g1t,u,v≤atv−u.L4maxt∈Iat<1.In this section, the following main theorem is obtained.Theorem 2.
LetP be a normal cone in E. Suppose conditions L1−L4 hold. Then, there exists a unique solution of equation (2) u∗,u∗∈E×E, and there are iterative sequences converging to the unique solution, and corresponding error estimates are given.Proof.
LetE=CI,R. Then, Pc=x∈CI,Rxt≥0,∀t∈I is a cone. Thus, by the normal of P, Pc is also normal.
The following operator is considered:(37)A=F1+G1,B=F2+G2,where for any u,v∈Pc,t∈I,(38)F1u,v=f1t,ut,vt,G1u,v=∫0tg1s,us,vsds,F2v,u=f2t,ut,ut,G1v,u=∫0tg2s,vs,usds.
Then,A,B:Pc×Pc⟶E. It is easy to know that u∗,u∗∈Pc×Pc is a solution of (33) if and only if u∗,u∗ is a solution of the following integral equations:(39)u=Au,v,v=Bv,u.
Next, from conditionsL1−L4, it is obtained that the operators AandB satisfy the whole condition of Theorem 1.
In fact,∀u1,u2,v1,v2∈Pc,u1≤u2,v1≤v2:(i)
Let(40)Lu=atu+∫0tcsusds,h=bt+∫0tdsds,t∈I,L1u=atu,L2u=∫0tcsusds.Then,L1L2=L2L1 and rL1=maxt∈Iat,rL2=0.Thus,(41)rL=rL1+L2≤rL1+rL2=maxt∈Iat<1.Therefore, for the equationI−Lu=h, there exists a unique solution v0=I−L−1h=∑n=0∞Lnh∈P. Then, by L1, for any t∈I, the following is obtained:(42)Bv0,θ=F2v0,θ+G2v0,θ=f2t,v0t,θ+∫0tg2s,v0s,θds≤atv0+∫0tcsv0sds+bt+∫0tdsds=Lv0+h=v0.Obviously,θ≤f1t,θ,v0t+∫0tg1s,θ,v0sd=Aθ,v0.(ii)
ByL2, the following is obtained:(43)Bv2,u1−Bv1,u2=f2t,v2t,u1t−f2t,v1t,u2t+∫0tg2s,v2s,u1s−g2s,v1s,u2sds≥f2t,v2t,u1t−f2t,v1t,u2t≥−Mv2−v1.Similarly,Av2,u1−Av1,u2≥−Mv2−v1.(iii)
FromL3andL4, the following is obtained:(44)Bv,u−Au,v=f2t,vt,ut−f1t,vt,ut+∫0tg2s,vs,us−g1s,vs,usds≥−Mv−u+∫0tg2s,vs,us−g1s,vs,usds≥−Mv−u,Bv,u−Au,v=f2t,vt,ut−f1t,vt,ut+∫0tg2s,vs,us−g1s,vs,usds≤atv−u+∫0tcsv−uds=Lv−u.Then, by (41), it is known that(45)−Mv−u≤Bv,u−Au,v≤Lv−u,rL<1.Therefore, from (i), (ii), and (iii), lettingT1=M1I,T2=0 in Theorem 1, it is easy to know that the condition H4 holds.Finally, for any initial valuex0,y0∈θ,v0,x0≤y0, by constructing the iterative sequences(46)xnt=f1t,xn−1t,yn−1t+∫0tg1s,xn−1s,yn−1sds,ynt=f2t,yn−1u,xn−1t+∫0tg2s,yn−1s,xn−1sds,one hasxn−u∗⟶0,yn−u∗⟶0n⟶∞, and for any α∈0,1, there exists a natural number n0 which satisfies as n≥n0, the following is obtained:(47)xn−u∗≤2Npαnv0−u0,yn−u∗≤2Npαnv0−u0.This completes the proof of Theorem2.


## 4. The Application of Nonlinear Differential Equations

In this section, the following nonlinear initial value problems of the differential equation are considered:(48)u′t=f1t,u,v+∫0Tg1s,u,vds,u0=u0,v′t=f2t,u,v+∫0Tg2s,u,vds,v0=v0,where fi,gi∈CI×R+×R+,R+, i=1,2, I=0,T, and E is a real Banach space with norm ⋅.For convenience, the following assumptions are made:C1 There exists the nonnegative bounded integrable functions at,bt,ct,dt which satisfy(49)f2t,u,θ≤atu+bt,g2t,u,θ≤ctu+dt.C2 There exists constant M>0, for any u,v∈E,u≤v, which satisfies(50)fit,v,u−fit,u,v≥−Mv−u,git,v,u−git,u,v≥0,i=1,2.C3 For any u,v∈E,u≤v, the following is satisfied:(51)−Mv−u≤f2t,v,u−f1t,u,v≤ctv−u,0≤g2t,v,u−g1t,u,v≤atv−u.C4∫0Tatv−udr∫0tKr,sds<eMt,∀t∈I.Then, the following theorem is obtained.Theorem 3.
LetP be a normal cone in E. Suppose that conditions C1−C4 hold. Then, there exists a unique solution of equation (48) u∗,u∗, and there are iterative sequences converging to the unique solution, and corresponding error estimates are given.Proof.
Firstly, differential equation (48) is turned into integral equations. For any fixed η∈C1J,E, the following one-order linear ordinary differential initial value problems in Banach spaces are investigated:(52)u′=f1t,η,η−Mu−η+∫0TKt,sg1s,η,ηds,u0=u0,u′=f2t,η,η−Mu−η+∫0TKt,sg2s,η,ηds,u0=u0.
It is easy to know thatu,u∈C1I,E×C1I,E is a solution of (52) if and only if u,u is a solution of the following integral equations:(53)ut=e−Mtu0+∫0Tg1r,ηr,ηrdr∫0tKs,rds+e−Mt∫0teMsf1s,ηs,ηs+Mηsds,ut=e−Mtu0+∫0Tg2r,ηr,ηrdr∫0tKs,rds+e−Mt∫0teMsf2s,ηs,ηs+Mηsds.
Next, the operatorA,B:C1I,E×C1I,E⟶C1I,E is defined as the following:(54)Aη,η=e−Mtu0+∫0Tg1r,ηr,ηrdr∫0tKs,rds+e−Mt∫0teMsf1s,ηs,ηs+Mηsds,Bη,η=e−Mtu0+∫0Tg2r,ηr,ηrdr∫0tKs,rds+e−Mt∫0teMsf2s,ηs,ηs+Mηsds.
Obviously,η,η is a solution of (48) if and only if(55)η=Aη,η,η=Bη,η.
Next, similar to the proof of Theorem2, it is tested whether the operators AandB satisfy the whole condition of Theorem 1 from conditions C1−C4. Therefore, the result of Theorem 3 is obtained from Theorem 1.



---

*Source: 7820263-2020-07-13.xml*