# Fixed Point Theorems of Binary Contraction Comparable Operators and an Application

Authors: Zhan Liu; Chuanxi Zhu
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

The aim of this paper is to present the concept of binary comparable operators in partially ordered Banach spaces and prove several fixed point theorems under some contractive conditions. The results of this paper can be used to investigate a large class of nonlinear problems. As an application, we study the existence of solution of a nonlinear integral equation.


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## Body

## 1. Introduction

The Banach contraction principle [1] as a popular tool for solving problems in nonlinear analysis was invented in 1922. Since then, the study of fixed points of mappings with contractive property has been at the center of various research activities. For example, Liu and Zhu [2] studied the solvability of a binary operator equation satisfying certain contractive conditions; Romaguera [3] obtained several fixed point theorems of mappings satisfying some generalized contractive conditions. For more details on fixed point results of contractive type mappings and applications, we refer to Yan et al. [4], Mukherjea [5], Ran and Reurings [6], Hussain et al. [7], Amini-Harandi [8], Sintunavarat and Kumam [9], Nieto and Rodríguez-López [10, 11], and O’Regan and Saadati [12].One of the common properties of the above results is that the involved operators must satisfy the monotone or mixed monotone conditions. In 2005, Zhang [13] studied an ordinal L-ordering symmetric contraction operator without the mixed monotone property and proved some coupled fixed point theorems. On the other hand, Qiao [14] investigated the fixed point theorems of the ordered contractive operators with the comparable property.In the present paper, we introduce the concept of binary comparable operators, which can be seen as a generalization of the concept of mixed monotone operators and the concept of antimixed monotone operators. Using the iterative techniques ([15, 16]), we obtain several fixed point theorems for such operators under some contractive conditions. The results of this paper generalize several classical results in the literature. As an application, the existence of solution of an integral equation is presented.For the sake of convenience, let us recall the following definitions and lemmas (see [14, 17, 18] for more details and recent results).LetE be a real Banach space. A subset P of E is called a cone whenever the following conditions hold:(i)
Pis closed, nonempty, and P≠0;(ii)
a,b∈R,a,b≥0, and x,y∈P implies ax+by∈P;(iii)
P∩-P=0.Given a coneP⊂E, we define a partial ordering ≤ with respect to P by x≤y if and only if y-x∈P.LetE be a normed Banach space, which is partially ordered by a cone P. The cone P is said to be normal if there exists a constant N>0, such that, for all x,y∈E,  θ≤x≤y implies x≤Ny, where θ denotes the zero element of E.Definition 1 (see [14]).
For somex,y∈E, if x≤y or y≤x holds, x and y are said to be comparable. Moreover, one will write x∨y=y to indicate that x≤y, while x∨y=x stand for y≤x.Lemma 2 (see [14]).
Ifx,y∈E are comparable, then x-y and y-x are comparable, and(1)θ≤x-y∨y-x.Lemma 3 (see [14]).
For somex,y,z∈E, if any two of them are comparable, then (2)x-y∨y-x≤x-z∨z-x+z-y∨y-z.Lemma 4 (see [14]).
If, for each positive integern, x and yn are comparable and limn→∞yn→y0, then x and y0 are comparable.Lemma 5 (see [14]).
If, for each positive integern, xn and yn are comparable and limn→∞xn→x0, limn→∞yn→y0, then x0 and y0 are comparable.Definition 6.
A binary operatorA:E×E→E is said to be comparable if, for all comparable pairs x1,x2∈E and y1,y2∈E,  Ax1,y1 and Ax2,y2 are comparable.Definition 7.
A comparable operatorA:E×E→E is said to be α-contractive, if, for all comparable pairs x1,x2∈E and y1,y2∈E, there exists a constant α∈0,1/2 such that (3)Ax1,y1-Ax2,y2∨Ax2,y2-Ax1,y1≤αx1-x2∨x2-x1+y1-y2∨y2-y1.


## 2. Main Results

Theorem 8.
LetE be a real Banach space, P a normal cone of E with the normal constant N, and ≤ a partial order with respect to P. Let A:E×E→E be demicontinuous. Suppose that the following two conditions are satisfied:(i)
A is α-contractive and comparable, where α∈0,1/2;(ii)
there exists a comparable pairx0,y0 in E, such that x0 and Ax0,y0, y0 and Ay0,x0 are comparable.
ThenA has a fixed point x∗ in E; that is, Ax∗,x∗=x∗. Moreover, the iterative sequences xn=Axn-1,yn-1 and yn=Ayn-1,xn-1 converge to x∗, and(4)x0-x∗≤N21-2αAx0,y0-x0+Ay0,x0-y0,y0-x∗≤N21-2αAx0,y0-x0+Ay0,x0-y0.Proof .
Consider the iterative sequences(5)x1=Ax0,y0,⋮xn=Axn-1,yn-1;y1=Ay0,x0,⋮yn=Ayn-1,xn-1;n=1,2,….Since x0 and y0 are comparable and A:E×E→E is a comparable operator, it is easy to verify that x1=Ax0,y0  and y1=Ay0,x0 are comparable. By inductions, we can prove that xn and yn are comparable for any positive integer n.
SinceA:E×E→E is a α-contractive comparable operator, we have (6)xn-yn∨yn-xn=Axn-1,yn-1-Ayn-1,xn-1∨Ayn-1,xn-1-Axn-1,yn-1≤2αxn-1-yn-1∨yn-1-xn-1.If we continue in the same way, for each n∈N, we have(7)xn-yn∨yn-xn≤2nαnx0-y0∨y0-x0.By the normality of P, we have(8)xn-yn≤N2nαnx0-y0.Since x0,y0 is a comparable pair in E and, moreover, x0 and Ax0,y0, y0 and Ay0,x0 are comparable, we know that Ax0,y0  and AAx0,y0,Ay0,x0, Ay0,x0  and AAy0,x0,Ax0,y0 are comparable, which implies that x1 and x2, y1 and y2 are comparable. By induction, we know that xn,xn+1 and yn,yn+1 are all comparable pairs for each n∈N.
Furthermore, for eachn∈N, we have (9)xn-xn+1∨xn+1-xn=Axn-1,yn-1-Axn,yn∨Axn,yn-Axn-1,yn-1≤αxn-xn-1∨xn-1-xn+yn-yn-1∨yn-1-yn,yn-yn+1∨yn+1-yn=Ayn-1,xn-1-Ayn,xn∨Ayn,xn-Ayn-1,xn-1≤αxn-xn-1∨xn-1-xn+yn-yn-1∨yn-1-yn.By induction we can prove that(10)xn-xn+1∨xn+1-xn≤2n-1αnAx0,y0-x0∨x0-Ax0,y0+Ay0,x0-y0∨y0-Ay0,x0,yn-yn+1∨yn+1-yn≤2n-1αnAx0,y0-x0∨x0-Ax0,y0+Ay0,x0-y0∨y0-Ay0,x0.Since α∈0,1/2, we can prove that xn and yn are Cauchy sequences in E. Then there exist two points x∗,y∗∈E, such that(11)xn⟶x∗,yn⟶y∗n⟶∞.Since A is demicontinuous, xn=Axn-1,yn-1 converges to Ax∗,y∗ weakly and yn=Ayn-1,xn-1 converges to Ay∗,x∗ weakly and thus Ax∗,y∗=x∗,  Ay∗,x∗=y∗; that is, x∗,y∗ is a coupled fixed point of A.
Taking limit in (8) as n→∞, we get(12)x∗=y∗.This means that(13)Ax∗,x∗=x∗.Hence x∗ is a fixed point of A.
Moreover,(14)x0-x∗=limn→∞xn-x0≤limn→∞∑i=1nxi-xi-1≤∑i=1∞N22i-1αi-1Ax0,y0-x0+Ay0,x0-y0=N21-2αAx0,y0-x0+Ay0,x0-y0.By a similar method, we can prove(15)y0-x∗≤N21-2αAx0,y0-x0+Ay0,x0-y0.Then we complete the proof of Theorem 8.Theorem 9.
LetE be a real Banach space, P a normal cone of E with the normal constant N, and ≤ the partial order with respect to P. Suppose that the following two conditions are satisfied:(i)
A:E×E→E is α-contractive comparable one, where α∈0,1/2;(ii)
there exists a comparable pairx0,y0 in E, such that x0 and xn=Axn-1,yn-1, y0 and yn=Ayn-1,xn-1 are comparable for each n∈N.
ThenA has a fixed point x∗ in E; that is, Ax∗,x∗=x∗. Moreover, the iterative sequences xn=Axn-1,yn-1  and yn=Ayn-1,xn-1 converge to x∗, and(16)x0-x∗≤N21-2αAx0,y0-x0+Ay0,x0-y0,y0-x∗≤N21-2αAx0,y0-x0+Ay0,x0-y0.Proof.
By a similar approach as in the proof of Theorem8, we can prove that xn and yn are Cauchy sequences in E, and there exist two points x∗,y∗∈E, such that(17)xn⟶x∗,yn⟶y∗n⟶∞.Furthermore (18)x∗=y∗.We now prove that x∗ is a fixed point of A. For some m,n∈N, assume that n<m; then by condition (ii), we know that x0 and xm-n=Axm-n-1,ym-n-1, y0 and ym-n=Aym-n-1,xm-n-1 are comparable; hence Ax0,y0  and Axm-n,ym-n,  Ay0,x0  and Aym-n,xm-n are comparable. If we continue in the same way, we can prove that xn and xm, yn and ym are comparable. As m→∞, by Lemma 4, we know that, for each n∈N, xn and x∗, yn and y∗ are comparable.
HenceAxn,yn  and Ax∗,y∗, Ayn,xn  and Ay∗,x∗ are comparable; furthermore(19)Axn,yn-Ax∗,y∗∨Ax∗,y∗-Axn,yn≤αxn-x∗∨x∗-xn+yn-y∗∨y∗-ynAyn,xn-Ay∗,x∗∨Ay∗,x∗-Ayn,xn≤αxn-x∗∨x∗-xn+yn-y∗∨y∗-yn.From the normality of P, we have(20)Axn,yn-Ax∗,y∗≤αxn-x∗+yn-y∗Ayn,xn-Ay∗,x∗≤αxn-x∗+yn-y∗.Those imply that(21)xn+1-Ax∗,y∗≤αxn-x∗+yn-y∗yn+1-Ay∗,x∗≤αxn-x∗+yn-y∗.Taking limit in (21) as n→∞, we get Ax∗,y∗=x∗, Ay∗,x∗=y∗; that is, x∗,y∗ is a coupled fixed point of A.
By (18), we can prove that x∗ is a fixed point of A.
Using the same argument as that in the proof of Theorem8, we can obtain(22)x0-x∗≤N21-2αAx0,y0-x0+Ay0,x0-y0,y0-x∗≤N21-2αAx0,y0-x0+Ay0,x0-y0.Then we complete the proof of Theorem 9.Theorem 10.
LetE be a Banach space and P a normal cone in E with the normal constant N, u0,v0∈E with u0≤v0, and u0,v0 an order interval. Suppose that A:u0,v0×u0,v0→u0,v0 is α-contractive and comparable, where α∈0,1/2; then A has a unique fixed point u∗ in E. Moreover, for any initial x0,y0∈u0,v0×u0,v0, the iterative sequences xn=Axn-1,yn-1  and yn=Ayn-1,xn-1 converge to u∗.Proof.
Consider the iterative sequences(23)u1=Au0,v0,⋮un=Aun-1,vn-1;v1=Av0,u0,⋮vn=Avn-1,un-1;n=1,2,….Since u0 and v0 are comparable and A:E×E→E is a comparable operator, it is easy to verify that u1=Au0,v0  and v1=Av0,u0 are comparable. By inductions, we can prove that un and vn are comparable for any positive integer n.
SinceA:E×E→E is a α-contractive comparable operator, we have (24)un-vn∨vn-un=Aun-1,vn-1-Aun-1,vn-1∨Avn-1,un-1-Aun-1,vn-1≤2αun-1-vn-1∨vn-1-un-1.If we continue in the same way, for each n∈N, we have(25)un-vn∨vn-un≤2nαnu0-v0∨v0-u0.By the normality of P, we have(26)un-vn≤N2nαnu0-v0.Since u0≤v0, we know that u0 and u1=Au0,v0, v0 and v1=Av0,u0 are comparable; thus Au0,v0  and Au1,v1, Av0,u0  and Av1,u1 are comparable, which implies that u1 and u2, v1 and v2 are comparable. By induction, we know that un,un+1 and vn,vn+1 are all comparable pairs for each n∈N.
Furthermore, for eachn∈N, we have (27)un-un+1∨un+1-un=Aun-1,vn-1-Aun,vn∨Aun,yn-Aun-1,vn-1≤αun-un-1∨un-1-un+vn-vn-1∨vn-1-vn≤αAun-1,vn-1-Aun-2,vn-2∨Aun-2,yn-2-Aun-1,vn-1+Avn-1,un-1-Avn-2,un-2∨Avn-2,un-2-Avn-1,un-1≤2α2un-1-un-2∨un-2-un-1+vn-1-vn-2∨vn-2-vn-1≤⋯≤2n-1αnu1-u0∨u0-u1+v1-v0∨v0-v1,vn-vn+1∨vn+1-vn≤2n-1αnu1-u0∨u0-u1+v1-v0∨v0-v1.By the normality of P, we have(28)un+1-un≤N2n-1αnu1-u0vn+1-vn≤N2n-1αnv1-v0.For α∈0,1/2, we can prove that un and vn are Cauchy sequences.
Sinceun,vn⊂u0,v0, there exist two points u∗,v∗ in u0,v0, such that(29)un⟶u∗,vn⟶v∗n⟶∞.Taking limit in (26) as n→∞, we get(30)u∗=v∗.We now prove that u∗ is a fixed point of A. For some m,n∈N, assume that n<m; since un,vn⊂u0,v0, we know that u0 and um-n=Aum-n-1,vm-n-1, v0 and vm-n=Avm-n-1,um-n-1 are comparable; hence Au0,v0  and Aum-n,vm-n, Av0,u0  and Avm-n,um-n are comparable. If we continue in the same way, we can prove that un and um, vn and vm are comparable. As m→∞, by Lemma 4, we know that, for each n∈N,  un and u∗, vn and v∗ are comparable.
HenceAun,vn  and Au∗,v∗, Avn,un  and Au∗,v∗ are comparable; furthermore(31)Aun,vn-Au∗,v∗∨Au∗,v∗-Aun,vn≤αun-u∗∨u∗-un+vn-v∗∨v∗-vnAvn,un-Av∗,u∗∨Av∗,u∗-Avn,un≤αun-u∗∨u∗-un+vn-v∗∨v∗-vn.From the normality of P, we have(32)Aun,vn-Au∗,v∗≤Nαun-u∗+vn-v∗Avn,un-Av∗,u∗≤Nαun-u∗+vn-v∗.This implies that(33)un+1-Au∗,v∗≤Nαun-u∗+vn-v∗vn+1-Av∗,u∗≤Nαun-u∗+vn-v∗.Taking limit in (33) as n→∞, we get Au∗,v∗=u∗, Av∗,u∗=v∗; that is, u∗,v∗ is a coupled fixed point of A.
By (30), we can prove that u∗ is a fixed point of A.
For any initialx0,y0∈u0,v0×u0,v0, we construct iterative sequences xn=Axn-1,yn-1  and yn=Ayn-1,xn-1, n=1,2,…. Since u0≤x0,y0≤v0, that is, x0 and u0, y0 and v0 are comparable, we know that Ax0,y0  and Au0,v0, Ay0,x0  and Av0,u0 are comparable, which mean that x1 and u1, y1 and v1 are comparable. By inductions, we know that xn and un, yn and vn are comparable for each n∈N. Moreover,(34)xn-un∨un-xn=Axn,yn-Aun,vn∨Aun,vn-Axn,yn≤αun-xn∨xn-un+vn-yn∨yn-vn≤αAun-1,vn-1-Axn-1,yn-1∨Axn-1,yn-1-Aun-1,vn-1+Avn-1,un-1-Ayn-1,xn-1∨Ayn-1,xn-1-Avn-1-un-1≤2α2un-1-xn-1∨xn-1-un-1+vn-1-yn-1∨yn-1-vn-1≤⋯≤2nαn+1u0-x0∨u0-x0+y0-v0∨v0-y0,yn-vn∨vn-yn≤2nαn+1u0-x0∨u0-x0+y0-v0∨v0-y0.From the normality of P, we have(35)un-xn≤N2nαn+1u0-x0+y0-v0vn-yn≤N2nαn+1u0-x0+y0-v0.This implies that(36)xn⟶u∗,yn⟶v∗u∗n⟶∞.For the uniqueness, we assume that there exists a point ω∗∈u0,v0, such that Aω∗,ω∗=ω∗. Construct the iterative sequence as ω1=Aω∗,ω∗, ω2=Aω1,ω1,…, and ωn=Aωn-1,ωn-1,…, and we know that ωn→u∗n→∞. On the other hand, for ω∗=Aω∗,ω∗, ω∗=ω1=⋯=ωn, and hence ω∗=u∗.
Then we complete the proof of Theorem10.Remark 11.
It is easy to verify that mixed monotone operators and antimixed monotone operators are precisely comparable operators. However, a comparable operator is not necessarily a mixed monotone operator or an antimixed monotone operator. Thus, the fixed point theorems in this work generalize and extend the fixed point theorems of mixed monotone operators and antimixed monotone operators.


## 3. Applications

As an application, we consider the nonlinear Hammerstein integral equation of the following form.LetI be the closed unit interval 0,1 in R. Consider the following integral equation: (37)xs=Axs,ys=∫0sks,ts2-t2·xt+xtyt-11+ytdt,s∈I.Suppose that ks,t:I×I→R+ is continuous about s and bounded measurable about t and, moreover, nonnegative on 0,1×0,1 and α=supt∈I∫01ks,t/s2-t2dt<1/2; then (37) has a unique solution x∗s.Proof.
LetE=C0,1 denote the Banach space of all real valued continuous functions φ on 0,1, with φ=sups∈0,1φs, and let P=φ∈E∣φs≥0,s∈0,1. Clearly, P is a normal cone of E with the normal constant N=1.
We transform integral equation (37) to the form(38)xs=Axs,ys=∫01ks,ts2-t2xt-11+ytdt.Since ks,t/s2-t2 is nonnegative on 0,1×0,1, we get that A is a binary operator from 0,1×0,1 to 0,1, and apparently A is comparable.
For0≤x1s,x2s,y1s,y2s≤1, we get (39)Ax1,y1-Ax2,y2∨Ax2,y-Ax1,y1=Ax1,y1-Ax2,y2=∫01Rs,ts2-t2x1t-x2t+y1t-y2t1+y1tdt≤αx1t-x2t+y1t-y2t.Then we obtain that A is α-contractive. According to Theorem 10, we can prove that the integral equation (37) has a unique solution.Remark 12.
The operatorAx,y defined by (37) is a comparable operator, but A does not satisfy the mixed monotone or antimixed monotone condition. However, by Theorem 10 of this work, we can easily get the conclusion. Thus, from this application, it is shown that some of the results in this work generalize and extend the corresponding results of mixed monotone operators and antimixed monotone operators again.



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