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.
1. Introduction
The following classical Banach contraction principle is well-known.
Theorem 1.
If T 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]).
Let A 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]).
Let A 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.
Let R+ 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 mapping T:A∪B→A∪B is called a generalized eventual cyclic gross contractive mapping if the following are satisfied:
T(A)⊂B,T(B)⊂A,
for some x∈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.
Let T 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 every x∈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.
Let T 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 every x∈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.
For λ=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 Lemmas 5 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)).
If d(Tx0,x0)=0, then x0 is a fixed point of T.
Otherwise, if d(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 is y0∈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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the NSF of China (11171210).
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