# A Note on the Fixed Point of Generalized Eventual Cyclic Gross Contractive Mappings

Authors: Jin Liang; Sheng-Hua Yan
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Based on the work of Agarwal et al., in press,  we introduce a concept of generalized eventual cyclic gross contractive mapping in metric spaces, which is a generalization of the eventual cyclic gross contractions. Furthermore, we analyze the existence of a fixed point for this type of contractive mapping, and present a new and general fixed point theorem.


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## Body

## 1. Introduction

The following classical Banach contraction principle is well-known.Theorem 1.
IfT is a self-mapping in a complete metric space X satisfying the following condition:
(1)d(Tx,Ty)<λd(x,y),λ∈[0,1),
then T has a unique fixed point.Based on this principle and the idea of its proof, many researchers have presented numerous extensions of the principle and new types of fixed point theorems (cf, e.g., [1–3]).In [3], Karpagam and Agrawal proved the following unique fixed point for cyclic mappings.Theorem 2 (see [3]).
LetA and B be two nonempty closed subsets of a complete metric space (X,d). Suppose ϕ:A∪B→A∪B is a cyclic mapping such that for some x∈A, there exists a kx∈(0,1) such that
(2)d(ϕ2nx,ϕy)≤kxd(ϕ2n-1x,y),n∈N,y∈A.
Then, A∩B is nonempty and ϕ has a unique fixed point in A∩B.Very recently, Agarwal et al. [1] defined a new type of cyclic mappings, “eventual cyclic gross contraction” in complete metric spaces, and established a fixed point theorem for the eventual cyclic gross contraction.Theorem 3 (see [1]).
LetA and B be two nonempty closed subsets of a complete metric space (X,d), and let T:A∪B→A∪B be an eventual cyclic gross contraction. Then A∩B is nonempty and T has a unique fixed point in A∩B.Inspired by the work in [1], in this paper, we introduce a concept of generalized eventual cyclic gross contractive mapping in metric spaces, which is a generalization of the eventual cyclic gross contractions. Furthermore, we obtain a fixed point theorem for this type of contractive mappings, which extends Theorem 3.


## 2. Generalized Eventual Cyclic Gross Contractive Mappings

LetR+ be the set of all nonnegative real numbers and N the set of all natural numbers. Let (X,d) be a metric space, and let A and B be nonempty subsets of X.Now, we introduce the concept of generalized eventual cyclic gross contractive mappings.Definition 4.
A mappingT:A∪B→A∪B is called a generalized eventual cyclic gross contractive mapping if the following are satisfied:(1)
T(A)⊂B,T(B)⊂A,(2)
for somex∈A,
(3)f(d(T2nx,Ty))≤f(λd(T2n-1x,Ty)+(1-λ)d(T2nx,y))-g(d(T2n-1x,Ty),d(T2nx,y)),n≥n0∈N,y∈A,where λ∈[0,1), f:R+→R+ is a monotone increasing and continuous function, g:R+×R+→R+ is a lower semicontinuous mapping such that g(a,b)=0 if and only if a=b=0, and n0 is sufficiently large.Lemma 5.
LetT be a generalized eventual cyclic gross contractive mapping and λ∈(1/2,1]. Then {Tnx} is a Cauchy sequence for every x∈A∪B.Proof.
For everyx∈A∪B, let
(4)n≥n0,y=T2n-2x.
Then, (3) and the monotone increasing property of f imply that
(5)d(T2nx,T2n-1x)≤(1-λ)d(T2nx,T2n-2x)≤(1-λ)d(T2nx,T2n-1x)+(1-λ)d(T2n-1x,T2n-2x),
since
(6)f(d(T2nx,Ty))=f(d(T2nx,T2n-1x))≤f(λd(T2n-1x,T2n-1x)+(1-λ)d(T2nx,T2n-2x))-g(d(T2n-1x,T2n-1x),d(T2nx,T2n-2x))=f((1-λ)d(T2nx,T2n-2x))-g(0,d(T2nx,T2n-2x))≤f((1-λ)d(T2nx,T2n-2x)).
Hence,
(7)d(T2nx,T2n-1x)≤1-λλd(T2n-1x,T2n-2x).
Thus,
(8)d(Tnx,Tn+1x)≤1-λλd(Tn-1x,Tnx).
Since λ∈(1/2,1], we deduce that limn→∞d(Tnx,Tn+1x) exists. Set
(9)limn→∞d(Tnx,Tn+1x)=l≥0.
If l>0, then (5) implies that
(10)l≤(1-λ)limn→∞d(T2nx,T2n-2x)≤2(1-λ)l.
So, we have
(11)1-λ≥12.
This is impossible since λ≤1/2. Therefore,
(12)limn→∞d(Tnx,Tn+1x)=0.
This means that, for any ɛ>0, there exists a natural number N1 such that for any natural number n≥N1,
(13)d(Tnx,Tn+1x)<ɛ.
Moreover, for any natural number m>n>0, we have
(14)d(Tn+N1x,Tm+N1x)≤∑i=nm-1‍d(TN1+ix,TN1+i+1x)≤∑i=nm-1‍(1-λλ)id(TN1x,TN1+1x)<((1-λ)/λ)n1-((1-λ)/λ)ɛ.
This implies that
(15)d(Tn+N1x,Tm+N1x)⟶0
as n→∞, since
(16)1-λλ∈[0,1).
Therefore, {Tnx} is a Cauchy sequence.Lemma 6.
LetT be a generalized eventual cyclic gross contractive mapping and λ∈[0,1/2]. Then {Tnx} is a Cauchy sequence for every x∈A∪B.Proof.
For everyx∈A∪B, let
(17)n≥n0,y=T2nx.
Then,
(18)d(T2nx,T2n+1x)≤λd(T2n-1x,T2n+1x)≤λd(T2n-1x,T2nx)+λd(T2nx,T2n+1x),
since
(19)f(d(T2nx,Ty))=f(d(T2nx,T2n+1x))≤f(λd(T2n-1x,T2n+1x)+(1-λ)d(T2nx,T2nx))-g(d(T2n-1x,T2n+1x),d(T2nx,T2nx))=f(λd(T2n-1x,T2n+1x))-g(d(T2n-1x,T2nx),0)≤f(λd(T2n-1x,T2n+1x)).
So,
(20)d(T2nx,T2n+1x)≤λ1-λd(T2n-1x,T2nx).
Hence,
(21)d(Tnx,Tn+1x)≤λ1-λd(Tn-1x,Tnx).
Since λ∈[0,1/2], we see that for all n≥n0, the nonnegative sequence {d(Tnx,Tn+1x)} is decreasing. Let
(22)limn→∞d(Tnx,Tn+1x)=l≥0.
Then we obtain
(23)l≤λlimn→∞d(T2n-1x,T2n+1x)≤2λl.
This implies that
(24)λ≥12
if l≠0. Clearly, λ>1/2 is impossible. Moreover, λ=1/2 is also impossible by the proof of [1, Theorem 2.4].
Now, we show that{Tnx} is a Cauchy sequence. It follows from
(25)limn→∞d(Tnx,Tn+1x)=0
that for any ɛ>0, there exists a natural number N0 such that for any natural number n≥N0,
(26)d(Tnx,Tn+1x)<ɛ.
For λ∈[0,1/2) and any natural number m>n>0, we have
(27)d(Tn+N0x,Tm+N0x)≤∑i=nm-1‍d(TN0+ix,TN0+i+1x)≤∑i=nm-1‍(λ1-λ)id(TN0x,TN0+1x)<(λ/(1-λ))n1-(λ/(1-λ))ɛ.
So
(28)d(Tn+N0x,Tm+N0x)⟶0
as n→∞, since
(29)λ1-λ∈[0,1).
Therefore, {Tnx} is a Cauchy sequence.
Forλ=1/2, by the proof of [1, Theorem 2.4], we know that {Tnx} also is a Cauchy sequence.The following is our central theorem about cycle mapping in complete metric spaces.Theorem 7.
Assume that(X,d) is a complete metric space, A and B are closed, and T is a generalized eventual cyclic gross contractive mapping. Then, A∩B is nonempty and T has a unique fixed point in A∩B.Proof.
By the virtue of Lemmas5 and 6, we know that for every x∈A∪B, {Tnx} is a Cauchy sequence. Since (X,d) is a complete metric space, A and B are closed, there exists some x0∈A∪B such that
(30)limn→∞Tnx=x0.
Therefore,
(31)x0=limn→∞T2n+1x∈A,x0=limn→∞T2nx∈B.
So, x0∈A∩B; that is, A∩B≠∅.
On the other hand, we obtain(32)f(d(Tx0,x0))≤f(λd(x0,x0)+(1-λ)d(Tx0,x0))-g(d(x0,x0),d(Tx0,x0))=f((1-λ)d(x0,Tx0))-g(0,d(Tx0,x0)),
in view of
(33)f(d(Tx0,T2nx))≤f(λd(x0,T2nx)+(1-λ)d(Tx0,T2n-1x))-g(d(x0,T2nx),d(Tx0,T2n-1x)).
Ifd(Tx0,x0)=0, then x0 is a fixed point of T.
Otherwise, ifd(Tx0,x0)>0, then we have
(34)g(0,d(Tx0,x0))>0.
Equation (32) implies that
(35)f(d(Tx0,x0))<f((1-λ)d(x0,Tx0)).
Hence,
(36)d(Tx0,x0)<(1-λ)(d(x0,Tx0)).
This is impossible since 1-λ∈[0,1].
According to previous discussions, it is concluded that(37)d(Tx0,x0)=0,
and therefore, x0 is a fixed point of T.
If there isy0∈A∩B such that Ty0=y0, then d(x0,y0)=0; that is, the fixed point of T is unique, since
(38)g(d(x0,y0),d(x0,y0))≤0,
in view of
(39)f(d(x0,y0))=f(d(Tx0,Ty0))≤f(λd(x0,Ty0)+(1-λ)d(Tx0,y0))-g(d(x0,Ty0),d(Tx0,y0))≤f(d(x0,y0))-g(d(x0,y0),d(x0,y0)).
The proof is completed.



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*Source: 934758-2013-10-01.xml*