MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2020/78202637820263Research ArticleIterative Solution for Systems of a Class of Abstract Operator Equations in Banach Spaces and Applicationhttps://orcid.org/0000-0002-4578-8714SuHuaChenChuanjunSchool of Mathematics and Quantitative EconomicsShandong University of Finance and EconomicsJinanShandong 250014Chinasdufe.edu.cn202013720202020030620202606202013720202020Copyright © 2020 Hua Su.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.

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.

Project of National Social Science Fund of China18BTY015Shandong Province Higher Educational Science and Technology ProgramJ16LI01
1. Introduction

Guo and Lakshmikantham  introduced the definition of the mixed monotone operator and the coupled fixed point, and there are many good results (see ). Recently, from paper , 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 HCI×I×E,E.

In this paper, the following nonlinear abstract operator equations in Banach spaces E are discussed:(2)u=Au,v,v=Bv,u,where A,B:D×DE and D is a partial interval in E which is denoted as the following:(3)Du0,v0uEu0uv0.

For convenience, the following assumptions are made:

H1 There exist positive bounded operators Ti:EEi=1,2 which satisfy.

I+T1T2xθxP, and for any ui,viDi=1,2,u1u2,v1v2, the following is obtained:

(4)Bv2,u1Bv1,u2T1v2v1T2u2u1,Av2,u1Av1,u2T1v2v1T2u2u1.

H2u0+T2v0u0Au0,v0,Bv0,u0v0T2v0u0.

H3 There exists a positive bounded operator L:EE, and for any u,vD,uv, the following is obtained:

(5)T1+T2vuBv,uAu,vLvu.

H4LT2=T2L,LT1=T1L,T1T2=T2T1 in which the spectral radius satisfies

(6)rL+rT1+rT2<infλ:λσI+T1T2,rT1T2<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

Let P be a cone in E, i.e., a closed convex subset, such that λPP for any λ0 and PP=θ. A partial order in P is defined as xyyxP. A cone P is said to be normal if there exists a constant N>0 which satisfies x,yE,θxy, implying xNy, 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=zE:xzy is bounded.

The following theorem is the main results in this section.

Theorem 1.

Let P be a cone in E, u0,v0E,u0v0. Suppose that A,B:D×DE satisfies conditions H1H4. Then,

There exists a unique solution of equation (2) u,u in D×D, and for any solutions of equation (2) u,uD×D, one has u=u.

For any initial value x0,y0D,x0y0, the following iterative sequences are constructed:

(7)xn=I+T1T21Axn1,yn1+T1xn1T2yn1,yn=I+T1T21Byn1,xn1+T1yn1T2xn1,which satisfy xnu0,ynu0n, and for any δ,(8)rL+rT1+rT2infλ:λσI+T1t2<δ<1,there exists a natural number n0 which satisfies as nn0, the following is obtained:(9)xnu2Npδnv0u0,ynu2Npδnv0u0.

Proof.

By rT1T2<1, it is known that the operator I+T1T2 is reversible. And, from condition H1, I+T1T21 is the positive operator. Let(10)Fu,v=I+T1T21Au,v+T1uT2v,Gv,u=I+T1T21Bv,u+T1vT2u.

Then, equation (7) can be substituted by the following:(11)xn=Fxn1,yn1,yn=Gyn1,xn1.

By conditions H1H3, it is easy to obtain that operators FandG satisfy the following:

u0Fu0,v0Gv0,u0v0

F,G:D×DE are the mixed monotone operator

θGv,uFu,vHvu,u0uvv0, where H=L+T1+T2I+T1T21

Letting un=Fun1,vn1andvn=Gvn1,un1n=1,2,, the following two results are obtained by mathematical induction:(12)u0u1unvnv1v0,(13)unxnynvn,θvnunHnv0u0,n=1,2,.

In fact, from (1) and (3), one has(14)u0u1v1v0,u1x1y1v1,0v1u1Hv0u0.

Suppose that for n=k, one has (12) and (13). Then, as n=k+1, by (2) and (3), the following is obtained:(15)uk+1=Fuk,vkxk+1=Fxk,ykGyk,xk=yk+1Gvk,uk=vk+1,θvk+1uk+1=Gvk,ukFuk,vkHvkukHk+1vkuk.

Then, it is known that(16)ukuk+1xk+1yk+1xk+1vk,θvk+1uk+1Hk+1vkuk.

Then, for any natural number n, (12) and (13) are obtained by mathematical induction.

Next, it is proved that xn is Cauchy sequences. From condition H4, it is known that(17)L+T1+T2I+T1T21=I+T1T21L+T1+T2,then by (, V 3.9), rHrL+rT1+rT2rI+T1T21.

Thus, for any δ:rL+rT1+rT2/infλ:λσI+T1t2<δ<1, the following is obtained:(18)limHn1/n=rHrL+rT1+rT2rI+T1T21=rL+rT1+rT2infλ:λσI+T1t2<δ<1.

Then, there exists a natural number n0 which satisfies(19)Hnδn,nn0.

And, by (12) and (13), it is obtained that(20)θunun+pxn+pyn+pvn+pvn,θunxnynvn,n,p=1,2,.

So, by (13), it is known that(21)θxn+punvnunHnv0u0,θxnunHnv0u0.

Then, by the normality of P and (19), it is known that(22)xn+punNpHnv0u0Npδnv0u0,(23)xnunNpHnv0u0Npδnv0u0,nn0,p=1,2,.

Thus, the following is obtained:(24)xn+pxnxn+pun+xnun2Npδnv0u0,nn0,p=1,2,,i.e., xn is Cauchy sequences. So, there exists uD (D is bounded), such that limnxn=u.

And, by θynxnvnunHnv0u0, the normality of P, and (19), one obtains(25)ynxnNpδnv0u0,therefore(26)limnyn=u=limnxn,xnuyn,n=1,2,.

Thus, xnunNpδnv0u0, vnxnNpδnv0u0, and(27)limnun=u=limnvn,(28)unuvn,n=1,2,,so by (2), (3), and (11), it is also obtained that(29)un=Fun1,vn1Fu,uGu,uGvn1,un1=vn.

Letting n and by (27), Fu,u=Gu,u=u.

Then, by the definition of FandG, 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 that u,uD×D also satisfies equation (2), then by (11) and mathematical induction, the following is obtained:(30)unuvn,n=1,2,.

Thus, u=u.

And, letting p in (24), as nn0, the following is obtained:(31)xnu2Npδnv0u0.

Similarly, as nn0, the following is obtained:(32)ynu2Npδnv0u0.

The proof is complete.

Remark 1.

In Theorem 1, 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 fiI×R+×R+,R+ (here, the continuity of fi is not assumed) and giCI×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,θatu+bt,g2t,u,θctu+dt.

L2 There exists a constant M>0, for any u,vE,uv, which satisfies(35)fit,v,ufit,u,vMvu,git,v,ugit,u,v0,i=1,2.

L3 For any u,vE,uv, the following is satisfied:

(36)Mvuf2t,v,uf1t,u,vctvu,0g2t,v,ug1t,u,vatvu.

L4maxtIat<1.

In this section, the following main theorem is obtained.

Theorem 2.

Let P be a normal cone in E. Suppose conditions L1L4 hold. Then, there exists a unique solution of equation (2) u,uE×E, and there are iterative sequences converging to the unique solution, and corresponding error estimates are given.

Proof.

Let E=CI,R. Then, Pc=xCI,Rxt0,tI 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,vPc,tI,(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×PcE. It is easy to know that u,uPc×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 conditions L1L4, it is obtained that the operators AandB satisfy the whole condition of Theorem 1.

In fact, u1,u2,v1,v2Pc,u1u2,v1v2:

Let(40)Lu=atu+0tcsusds,h=bt+0tdsds,tI,L1u=atu,L2u=0tcsusds.

Then, L1L2=L2L1 and rL1=maxtIat,rL2=0.

Thus,(41)rL=rL1+L2rL1+rL2=maxtIat<1.

Therefore, for the equation ILu=h, there exists a unique solution v0=IL1h=n=0LnhP. Then, by L1, for any tI, the following is obtained:(42)Bv0,θ=F2v0,θ+G2v0,θ=f2t,v0t,θ+0tg2s,v0s,θdsatv0+0tcsv0sds+bt+0tdsds=Lv0+h=v0.

Obviously, θf1t,θ,v0t+0tg1s,θ,v0sd=Aθ,v0.

By L2, the following is obtained:(43)Bv2,u1Bv1,u2=f2t,v2t,u1tf2t,v1t,u2t+0tg2s,v2s,u1sg2s,v1s,u2sdsf2t,v2t,u1tf2t,v1t,u2tMv2v1.

Similarly, Av2,u1Av1,u2Mv2v1.

From L3andL4, the following is obtained:(44)Bv,uAu,v=f2t,vt,utf1t,vt,ut+0tg2s,vs,usg1s,vs,usdsMvu+0tg2s,vs,usg1s,vs,usdsMvu,Bv,uAu,v=f2t,vt,utf1t,vt,ut+0tg2s,vs,usg1s,vs,usdsatvu+0tcsvuds=Lvu.

Then, by (41), it is known that(45)MvuBv,uAu,vLvu,rL<1.

Therefore, from (i), (ii), and (iii), letting T1=M1I,T2=0 in Theorem 1, it is easy to know that the condition H4 holds.

Finally, for any initial value x0,y0θ,v0,x0y0, by constructing the iterative sequences(46)xnt=f1t,xn1t,yn1t+0tg1s,xn1s,yn1sds,ynt=f2t,yn1u,xn1t+0tg2s,yn1s,xn1sds,

one has xnu0,ynu0n, and for any α0,1, there exists a natural number n0 which satisfies as nn0, the following is obtained:(47)xnu2Npαnv0u0,ynu2Npαnv0u0.

This completes the proof of Theorem 2.

4. The Application of Nonlinear Differential Equations

In this section, the following nonlinear initial value problems of the differential equation are considered:(48)ut=f1t,u,v+0Tg1s,u,vds,u0=u0,vt=f2t,u,v+0Tg2s,u,vds,v0=v0,where fi,giCI×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,θatu+bt,g2t,u,θctu+dt.

C2 There exists constant M>0, for any u,vE,uv, which satisfies(50)fit,v,ufit,u,vMvu,git,v,ugit,u,v0,i=1,2.

C3 For any u,vE,uv, the following is satisfied:(51)Mvuf2t,v,uf1t,u,vctvu,0g2t,v,ug1t,u,vatvu.

C40Tatvudr0tKr,sds<eMt,tI.

Then, the following theorem is obtained.

Theorem 3.

Let P be a normal cone in E. Suppose that conditions C1C4 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 ηC1J,E, the following one-order linear ordinary differential initial value problems in Banach spaces are investigated:(52)u=f1t,η,ηMuη+0TKt,sg1s,η,ηds,u0=u0,u=f2t,η,ηMuη+0TKt,sg2s,η,ηds,u0=u0.

It is easy to know that u,uC1I,E×C1I,E is a solution of (52) if and only if u,u is a solution of the following integral equations:(53)ut=eMtu0+0Tg1r,ηr,ηrdr0tKs,rds+eMt0teMsf1s,ηs,ηs+Mηsds,ut=eMtu0+0Tg2r,ηr,ηrdr0tKs,rds+eMt0teMsf2s,ηs,ηs+Mηsds.

Next, the operator A,B:C1I,E×C1I,EC1I,E is defined as the following:(54)Aη,η=eMtu0+0Tg1r,ηr,ηrdr0tKs,rds+eMt0teMsf1s,ηs,ηs+Mηsds,Bη,η=eMtu0+0Tg2r,ηr,ηrdr0tKs,rds+eMt0teMsf2s,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 Theorem 2, it is tested whether the operators AandB satisfy the whole condition of Theorem 1 from conditions C1C4. Therefore, the result of Theorem 3 is obtained from Theorem 1.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

The author read and approved the final manuscript.

Acknowledgments

The author was supported by the Project of National Social Science Fund of China (NSSF) (18BTY015) and the Shandong Province Higher Educational Science and Technology Program (J16LI01).

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