# Fixed Point Theorems on Nonlinear Binary Operator Equations with Applications

Authors: Baomin Qiao
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## Abstract

The existence and uniqueness for solution of systems of some binary nonlinear operator equations are discussed by using cone and partial order theory and monotone iteration theory. Furthermore, error estimates for iterative sequences and some corresponding results are obtained. Finally, the applications of our results are given.


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## Body

## 1. Introduction

In recent years, more and more scholars have studied binary operator equations and have obtained many conclusions; see [1–6]. In this paper, we will discuss solutions for these equations which associated with an ordinal symmetric contraction operator and obtain some results which generalized and improved those of [3–6]. Finally, we apply our conclusions to two-point boundary value problem with two-degree super-linear ordinary differential equations.In the following, letE always be a realBanach space which is partially ordered by a cone P, let P be a normal cone of E, N is normal constant of P, partial order ≤ is determined by P and θ denotes zero element of E. Let u,v∈E,u<v,D=[u,v]={x∈E:u≤x≤v} denote an ordering interval of E.For the concepts of normal cone and partially order, mixed monotone operator, coupled solutions of operator equations, and so forth see [1, 5].Definition 1.
LetA:D×D→E be a binary operator. A is said to be L-ordering symmetric contraction operator if there exists a bounded linear and positive operator L:E→E, where spectral radius r(L)<1 such that A(y,x)-A(x,y)≤L(y-x) for any x,y∈D,  x≤y, where L is called a contraction operator of A.


## 2. Main Results

Theorem 2.
LetA:D×D→E be L-ordering symmetric contraction operator, and there exists a α∈[0,1) such that
(1)A(x2,y2)-A(x1,y1)≥-α(x2-x1),u≤x1≤x2≤v,u≤y2≤y1≤v.
If condition (H1) u≤A(u,v),A(v,u)≤v-α(v-u) or (H2) u+α(v-u)≤A(u,v),A(v,u)≤v holds, then the following statements hold.
(C1) A(x,x)=x has a unique solution x*∈D, and for any coupled solutions x,y∈D,  x=y=x*.
(C2) For any x0,y0∈D, we construct symmetric iterative sequences:
(2)xn=1α+1[A(xn-1,yn-1)+αxn-1],yn=1α+1[A(yn-1,xn-1)+αyn-1],hhhn=1,2,3,….
Then xn→x*,  yn→x*(n→∞), and for any β∈(r(L),1), there exists a natural number m; and if n≥m, we get error estimates for iterative sequences (2):
(3)∥xn(yn)-x*∥≤2N(α+βα+1)n∥u-v∥.Proof.
SetB(x,y)=(1/(α+1))[A(x,y)+αx], and if condition (H1) or (H2) holds, then it is obvious that
(4)u≤B(u,v),B(v,u)≤v.
By (1), we easily prove that B:D×D→E is mixed monotone operator, and for any x,y∈D,  u≤x≤y≤v,
(5)θ≤B(y,x)-B(x,y)≤H(y-x),
where H=(1/(α+1))(L+αI) is a bounded linear and positive operator and  I, is identical operator.
By the mathematical induction, we easily prove that(6)θ≤Bn(y,x)-Bn(x,y)≤Hn(y-x),u≤x≤y≤v,
where Bn(x,y)=B(Bn-1(x,y),Bn-1(y,x)),  x,y∈D,  n≥2.
By the character of normal coneP, it is shown that
(7)∥Bn(y,x)-Bn(x,y)∥≤N∥Hn∥∥y-x∥,u≤x≤y≤v.
For anyβ∈(r(L),1), since lim⁡n→∞||Hn||1/n=r(H)≤(α+r(L))/(α+1)<(α+β)/(α+1)<1, there exists a natural number m, and if n≥m, we have ||Hn||<((α+β)/(α+1))n, and N||Hm||<1. Considering mixed monotone operator Bm and constant N||Hm||, Bm(x,x)=x has a unique solution x* and for any coupled solution x,y∈D, such that x=y=x* by Theorem 3 in [3].
FromBm(B(x*,x*),B(x*,x*))=B(Bm(x*,x*),Bm(x*,x*))=B(x*,x*), and the uniqueness of solution with Bm(x,x)=x, then we have B(x*,x*)=x* and A(x*,x*)=x*.
We take note of thatA(x,x)=x and B(x,x)=x have the same coupled solution; therefore, a coupled solution for B(x,x)=x must be a coupled solution for Bm(x,x)=x; consequently, (C1) has been proved.
Considering iterative sequence (2), we construct iterative sequences:
(8)un=B(un-1,vn-1),vn=B(vn-1,un-1),
where u0=u, v0=v, it is obvious that
(9)xn=B(xn-1,yn-1),yn=B(yn-1,xn-1),θ≤vn-un≤Hn(v-u),
by the mathematical induction and characterization of mixed monotone of B; then
(10)un≤x*≤vn,un≤xn≤vn,un≤yn≤vn.
Hence,
(11)∥xn(yn)-un∥≤N∥vn-un∥,∥x*-un∥≤N∥vn-un∥,n=1,2,3,….
Moreover, if n≥m, we get
(12)∥xn(yn)-x*∥≤2N∥vn-un∥≤2N∥Hn∥∥v-u∥≤2N(α+βα+1)n∥u-v∥.
Consequently, xn→x*,yn→x*(n→∞).Remark 3.
Whenα=0, Theorem 1 in [4] is a special case of this paper Theorem 2 under condition (H1) or (H2).Corollary 4.
LetA:D×D→E be L-ordering symmetric contraction operator; if there exists a α∈[0,1) such that A satisfies condition of Theorem 2, the following statement holds.
(C3) For any β∈(r(L),1) and α+β<1, we make iterative sequences:
(13)un=A(un-1,vn-1),vn=A(vn-1,un-1)+α(vn-1-un-1),hhhn=1,2,3,…,
or
(14)un=A(un-1,vn-1)-α(vn-1-un-1),vn=A(vn-1,un-1),hhhn=1,2,3,…,
where u0=u,  v0=v.Thus,un→x*,vn→x*(n→∞), and there exists a natural number m, and if n≥m, we have error estimates for iterative sequences (13) or (14):
(15)∥un(vn)-x*∥≤N(α+β)n∥u-v∥.Proof.
By the character of mixed monotone ofA, then (1) and (C1), (C2) [in (1), (C2) where α=0] hold.
In the following, we will prove (C3).
Consider iterative sequence (13); since u≤x*≤v, we get
(16)u1=A(u,v)≤A(x*,x*)=x*≤A(v,u)=v1-α(v-u)≤v1.
By the mathematical induction, we easily prove un≤x*≤vn, n≥1, hence
(17)θ≤x*-un≤vn-un,θ≤vn-x*≤vn-un.
It is clear that(18)θ≤vn-un≤(L+αI)(vn-1-un-1)=(L+αI)n(v-u),n≥1.
For anyβ∈(r(L),1), α+β<1, since
(19)lim⁡n→∞∥(L+αI)n∥1/n=r(L+αI)≤r(L)+α<α+β<1,
there exists a natural number m, if n≥m, such that
(20)∥(L+αI)n∥<(α+β)n.
Moreover,(21)∥un(vn)-x*∥≤N∥(L+αI)n∥∥u-v∥≤N(α+β)n∥u-v∥,(n≥m).
Consequently, un→x*, vn→x*, (n→∞).
Similarly, we can prove (14).Theorem 5.
LetA:D×D→E be a L-ordering symmetric contraction operator; if there exists a α∈[0,1) such that (1-α)u≤A(u,v),A(v,u)≤(1-α)v, then the following statements hold.
(C4) Operator equation A(x,x)=(1-α)x has a unique solution x*∈D, and for any coupled solutions x,y∈D, x=y=x*.
(C5) For any x0,y0,w0,z0∈D, we make symmetric iterative sequences
(22)xn=11-αA(xn-1,yn-1),yn=11-αA(yn-1,xn-1),hhn=1,2,3,…,(23)wn=A(wn-1,zn-1)+αwn-1,zn=A(zn-1,wn-1)+αzn-1,hhln=1,2,3,….
Then xn→x*, yn→x*, wn→x*, zn→x*(n→∞), and for any β∈(r(L),1),α+β<1, there exists a natural number m, and if n≥m, then we have error estimates for iterative sequences (22) and (23), respectively,
(24)∥xn(yn)-x*∥≤2N(β1-α)n∥u-v∥,∥wn(zn)-x*∥≤2N(α+β)n∥u-v∥.Proof.
SetB(x,y)=(1/(1-α))A(x,y) or C(x,y)=A(x,y)+αx; we can prove that this theorem imitates proof of Theorem 2.Similarly, we can prove the following theorems.Theorem 6.
LetA:D×D→E be L-ordering symmetric contraction operator; if there exists a α∈[0,1) such that u+αv≤A(u,v),  A(v,u)≤v+αu, then the following statements hold.
(C6) Equation A(x,x)=(1+α)x has a unique solution x*∈D, and for any coupled solutions x,y∈Dx=y=x*.
(C7) For any x0,y0∈D, we make symmetric iterative sequence:
(25)xn=11+αA(xn-1,yn-1),yn=11+αA(yn-1,xn-1),hhn=1,2,3,….
Then xn→x*, yn→x*(n→∞); moreover, β∈(r(L),1), and there exists natural number m, and if n≥m, then we have error estimates for iterative sequence (25):
(26)∥xn(yn)-x*∥≤2N(βα+1)n∥u-v∥,
(C8) For any β∈(r(L),1)(α+β<1),w0,z0∈D, we make symmetry iterative sequence wn=A(wn-1,zn-1)-αzn-1,zn=A(zn-1,wn-1)-αwn-1,  n≥1; then wn→x*,  zn→x*(n→∞), and there exists a natural number m, and if n≥m, we have error estimates for iterative sequence (24).Remark 7.
Whenα=0, Corollary 2 in [4] is a special case of this paper Theorems 2–6.Remark 8.
The contraction constant of operator in [5] is expand into the contraction operator of this paper.Remark 9.
OperatorA of this paper does not need character of mixed monotone as operator in [6].


## 3. Application

We consider that two-point boundary value problem for two-degree super linear ordinary differential equations:(27)x′′+a(t)xm+11+b(t)x=0,t∈[0,1],(m≥2)x(0)=x′(1)=0.Letk(t,s) beGreenfunction with boundary value problem (23); that is,
(28)k(t,s)=min⁡{t,s}={t,t≤ss,s<t.Then the solution with boundary value problem (23) and solution for nonlinear integral equation with type ofHammerstein(29)x(t)=∫01k(t,s){a(s)[x(s)]m+11+b(s)x(s)}ds
are equivalent, where max⁡t∈[0,1]∫01k(t,s)ds=1/2.Theorem 10.
Leta(t), b(t) be nonnegative continuous function in [0,1],  p=max⁡t∈[0,1]a(t), q=max⁡t∈[0,1]b(t). If p<1,mp+q<2, then boundary value problem (23) has a unique solution x*(t) such that 0≤x*(t)≤1(t∈[0,1]). Moreover, for any initial function x0(t),y0(t), such that
(30)0≤x0(t)≤1,0≤y0(t)≤1(t∈[0,1]),
we make iterative sequence:
(31)xn(t)=∫01k(t,s){a(s)[xn-1(s)]m+11+b(s)yn-1(s)}ds,yn(t)=∫01k(t,s){a(s)[yn-1(s)]m+11+b(s)xn-1(s)}ds,hhhhhhhhhlln=1,2,3,….
Then xn(t) and yn(t) are all uniformly converge to x*(t) on [0,1], and we have error estimates:
(32)|xn(t)(yn(t))-x*(t)|≤2(mp+q2)n,hlt∈[0,1],n=1,2,3,….Proof.
LetE=C[0,1],  P={x∈E∣x(t)≥0,t∈[0,1]},  ∥x∥=max⁡t∈[0,1]|x(t)| denote norm of; then E has become Banach space, P is normal cone of E, and its normal constant N=1. It is obvious that integral Equation (24) transforms to operator equation A(x,x)=x, where
(33)A(x,y)(t)=∫01k(t,s){a(s)[x(s)]m+11+b(s)y(s)}ds,t∈[0,1].
Setu=u(t)≡0,  v=v(t)≡1; then D=[0,1] denote ordering interval of E, A:D×D→E is mixed monotone operator, and 0≤A(0,1),A(1,0)≤(1+p)/2<1.
Set(34)Lx(t)=∫01k(s,t)[ma(s)+b(s)]x(s)ds,t∈[0,1].
ThenL:E→E is bounded linear operator, its spectral radius r(L)≤(mp+q)/2<1, and for any x,y∈E,0≤x(t)≤y(t)≤1 such that 0≤A(y,x)(t)-A(x,y)(t)≤L(y-x)(t), A is L-ordering symmetric contraction operator, by Theorem 2 (where α=0); then Theorem 10 has been proved.



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